A TR.IEIATISE' N T R I GO N O A E T R Y PLANE AND SPHERIICAL, W I T I T' I' A ILIC AT (N T Ol NAVIGATION AN) SIJlVV!'Y{LN'G, NAUTIGA [ AND PRIACTICAL ASTRONO)MY AN]) -EOD SY, W I T II LO3ARITHMIIO, TRIGONOMETRICAL, AND NAUTIOAL T AB B L E S TF;R'eifE USE OF SCHOOLS AND COLLTEJO. NEW EDITION, WITH EXTENSIVE ADDITIONS ANTD IM/PROVEMENTS. B TY r 11t > REV. CH[ARLES W. IHACKLEY, S.T.D., Prof of Mathematicas ain Aastroriomy in Columbia College, FOU1l If ERi 11ION. NEW YORK: GEOjRGE P. PUTNAM & CO., 10 PARK PLACE. 1853. E.NTERED, according to Act of Congress, in the year 1850, by CHIAIRLES W. IIACKLEY, in the Clerk's Office of the District Court of the Soulttern District of New York. R. Cti.AlI(IEAD, Printer and Seretypwr, 63 VeJey Strst. PART OF PREFACE TO THE FIRST EDITION. ANALYTTICAL TRIGONOMETRY has always been to the majority of students a dry and difficult study. A conviction that it might be rendered easy and interesting to all who have a tolerable acquaintance with Algebra and Geometry, has led to the production of the present work. The faults of former treatises on this subject, which have detracted from their usefulness as books of instruction, appear to be these: 1. A too sudden transition from Geometry to Trigonometry, in consequence of which, the first efforts of the learner are in the dark as to the object of his pursuit. 2. A tedious succession of general formulas at the commencement, the use and application of which are so long delayed as to produce weariness and discouragement before there is any apparent fruit to reward labor. 3. Too much abridgment in the demonstration, and particularly in the derivation of the algebraic results. The author is aware of the importance attached to the exercise of intellect required to discover the connexion between propositions whose mutual dependence is shown by intermediate links which the mind must supply unaided, but it will be admitted on the other hand, that the ordinary term of study is too limited, and the field of knowledge in this department too extensive, to afford the loss of time which such a mode occasions. Besides, there will be abundant scope for this kind of exercise, in a more matured familiarity with mathematical reasoning, for which the shortening of labor here, will leave additional room. PREFACE TO THE NEW EDITION. THE old edition has been entirely remodelled, and vast additions of valuable matter which. have been some years in collecting, or are the results of recent improvemcnts in science, have been made. The present work begins with some constructions of triangles according to the rules given in geometry, followed by others in which scales of equal parts and protractors are employed, showing at once and distinctly, what is to be understood by the solution of a triangle, and the value of trigonometry in the measurement of inaccessible heights and distances. The evident inaccuracy in the use of instruments leads the learner to perceive the necessity of a more exact and certain method, and prepares him to enter with satisfaction upon the study of Analytical Trigonometry. The explanation of the.Trigonometrical Lines has been prepared with great care, and it is believed that considerable improvement in the method of exhibiting their changes will be observed. Their application to the solution of triangles is imme-~ diately shown in a few cases, with the help of a table of natural sines and cosines at the end. Then follows a full exposition of the theory and use of logarithms, with every variety of example, including an explanation of the Tables at the end. The use is also taught of the tables of Callet, the tables in highest repute, an American edition of which is known as Hassler's tables.* Part I. concludes with the application of logarithms and logarithmic sines, tangents, &c., to a number of practical examples in heights and distances, involving every case in the solution of plane triangles. After this a few pages of miscellaneous exercises occur, which, with those in fine print scattered through the 1st Part, will serve to give greater skill to the better class of students. Appendix I., which follows next in order, contains a vast variety of general formulas, succinct methods of solution, and methods advantageous in particular cases, methods of treating small arcs, resolutions of Algebraic equations by the aid of Trigonometry, various expressions for the area of a triangle in terms of its angles and sides, effects of errors of observation on results; in short, everything necessary for a comprehensive knowledge of Trigonometry.t * The German tables most in use are those of Vega and those of K6hler, t Navigation and Surveying, if to be studied, should be taken up immediately after Plane Trigonometry. PREFACE, V Part II. contains Spherical Trigonometry. Particular care has been taken to render the demonstrations here, plain and easy, and to avoid all unnecessary repetition and complication. ft was found that the introduction of a few celestial circles, such for the most part as the study of geography may be supposed to have already rendered familiar, would afford an opportunity for making all the examples of Spherical Trigonometry Astronomical. The use of the hour angle and of different kinds of time has led also to the introduction of a full description of the transit instrument and its various adjustments, the theory of which depends on Spherical Trigonometry, and is given in all its details. The practical character of the problems is a peculiar feature in the plan of the present work. The* consideration which led to it was that since Trigonometry had grown out of the actual wants of men in these very particulars, if they were sufficiently interesting to stimulate discovery, they would also incite to the study of what is already known. The analytic method, though not always practicable before the mind is somewhat furnished, is doubtless by far the best method of training. Besides this general reason for introducing Astronomical problems here, it was deemed useful thus to prepare the way for the study of Astronomny whilst the formulas and rules of Trigonometry were fresh in the memory, and to prevent that neglect of the Trigonometrical Solutions of Astronomy, which is apt to result from the trouble of recalling what has been long laid aside. It was thought, too, that this foretaste of Astronomy might excite a relish for that study. The examination questions will be found convenient for students preparing for examination on Trigonometry, or for those studying without the aid of a teacher. Appendix II., which follows Spherical Trigonometry, is of a character analogous to Appendix I. Part III. exhibits a pleasing and useful application of Plane Trigonometry to the principles of navigation. This will be found a very complete treatise on the subject in small compass. The appendix to this part, App. III., includes great circle sailing, a method not usually treated in works on navigation, nor much used at present at sea; but as it serves to shorten voyages, and has no practical inconveniences in the case of steamers, which class of vessels is becoming numerous on the ocean, it cannot longer with propriety be omitted. Sumner's method is also here introduced. Part IV. is a very complete treatise of surveying, which, by reducing the subject rigorously to its essential elements, is brought within a small space. Besides what is contained in ordinary treatises, including a full description of all the surveying instruments, will be found the methods of surveying railroads and canals, the principles of Topography, and a new method of Hydrographic Surveying. Part V., which treats of Nautical and Practical Astronomy, contains a complete description of all the Astronomical instruments used at sea and in observatories, a thorough investigation of the theory of their adjustments, and of the corrections to be applied to the observations for errors of adjustment, the use of nearly every part VI PREFACE. of the nautical almanac and tables of corrections for determining the co-ordinates of the true places of the heavenly bodies, and the solutions in Spherical Trigonometry necessary for converting one set of co-ordinates into another, with all the best methods of determining latitude and longitude, either on land or at sea. App. V. contains the description of the reflecting circle and mural circle, the determination of latitude by circummeridian altitudes, by the method of Littrow, and by an altitude of the pole star out of the meridian. Part VI. contains the necessary instruction for conducting a geodetic survey on a scale of sufficient magnitude to require not merely the spherical figure, but also the spheroidal figure of the earth to be taken into consideration. When the formulas in this part involve the theory of conic sections, they are given and the use taught, but the demonstrations are reserved for the last appendix, in which the calculus is freely introduced when necessary. The subject commences with the modes of measuring bases, with an account of the beautiful improvements in the base apparatus recently made in this country, and the formulas of reduction to the level of the neighboring seas. Then follows a description of the great theodolite, and the methods of conducting the observations of the great or primary triangulation, the modes of verifying and correcting the observed spherical angles, and of computing the elements of the spherical triangles. Then the methods of determining geodetically the differences of latitude, longitude, and azimuth of the stations at the vertices of the triangles, with the construction of maps and the explanation of the necessary tables. Then the best methods of conducting the Astronomical observations for latitude, longitude, and azimuths. The description of the instruments and modes of conducting the magnetic observations, and the use of the formulas for determining the elements of terrestrial magnetism. App. VI. describes the equatorial, the altitude and azimuth instrument, the prime vertical transit, and gives theorems for determining the size and figure of the earth, &c. The methods given in this geodetic treatise are those employed upon the coast survey of the United States.* The tables include a table of logarithms, of numbers, of logarithmic sines, tangents, cosines, cotangents, secants and cosecants;t a table of natural sines and cosines, a table of difference of latitude and departure for every point and quarter point of the quadrant, a table of Rhumbs, a table of meridional parts, Workman's table for the correction of the middle latitude, a table of refractions, with corrections for the states of the barometer and thermometer, a table for dip or depression of the horizon, a table of the sun's parallax in altitude, of the contraction of the: These are in some respects superior to the latest and best European methods. The author has to acknowledge the politeness of the accomplished superintendent of the coast survey in furnishing every facility for obtaining information. t The last two are not usually found in the best tables. The method of taking out the difference for the seconds in these tables is new and expeditious. PREFACE. V11 sun's or moon's vertical semi-diameter from refraction, ot the augmentation of the moon's semi-diameter vith its altitude, a table of proportional logarithms, a table of the reductions of the moon's equatorial parallax for the spheroidal figure of the earth, and finally a table of natural versed sines for reducing observations to the meridian. Besides these, other small tables and specimens of tables are scattered throughout the w,rk. Most of the tables are printed from the beautiful and accurate stereotype plates of the tables accompanying Bowditch's Navigator, by permission of the proprietor, Mr. G. W. Blunt. The author has to acknowledge the kindness of Prof. CHAUVENET, of the U. S. Naval Academy, in permitting the use of his valuable paper on Unlimited Spherical Triangles, first introduced by Gauss. It will be found in Appendix II., as contained n the Astronomical Journal, with some slight modifications and explanatory notes. N. B. The fine print in the following pages may be omitted without breaking the continuity of the treatise. O N TE N T S. PART 1. PLANE TRIGONOMETRY. Pdf Definition of Trigonometry,..... Number of data for determination of a triangle,. ib. Geometric solutions of triangles,.... 2 Scales of equal parts,..... ib. Sectoral scale,.. e 3 Diagonal scale,.......... 4 Measurement of angles,......... ib. Centesimal division,...... 6 Conversion of the two modes of division,,.. ib. Engineer's method of expressing angles,....... 7 Circular protractor,...... ib, Semicircular protractor,....... 8 Rude instrument for observing angles,..... 9 Construction of triangles observed on ground,.. 1 Construction in the measurement of heights,... 11 Nature of Analytical Trigonometry,.,.. Nature of trigonometric lines,..... 12 Definition of the sine,.... 13 Definition of a supplement,.. ib. Exercise,.....,..... 14 Variation of the sine for values of the arc from zero to 3600,.ib. Method from an arc > 900 to find one < 900 having the same sine,. 15 Sines of negative arcs,..... 16 Definition of tangent,...... ib. Variations in the value of the tangent for arcs from zero up to 3600. ib. Definition of the secant,... 18 Variation of the secant for variation of the arc from zero to 3600,.. ib Definition of a complement and exercises,...... 19 Definitions of the cosine, cotangent, and cosecant,. ib. The cosine,............ 20 Variations in the value of the cosine,..... ib. Trigonometric lines of negative arcs...... 21 The cotangent's and cosecant's variations of value,. 22 X CONTENTS. Pay Trigonometric lines have the same signs in pairs, 22 Algebraic notation for trigonometric lines,...... ib Expressions for the tangent, cotangent, secant, and cosecant, in terms of the sine and cosine,.......... 23 Relations of tangent and cotangent,... 25 Exercises in the trigonometric functions,..... 26 Derivation of formulas for the solution of right angled plane triangles, ib. Use of the tables of natural sines,...... 28 Application of the formulas to practical examples,..29 THEORY OF LOGARITHMS,...... 33 Definitions, log. of the base 1 and 0,....... ib. Characteristic in the common system,...... 34 EXPLANATION OF THE TABLE,.......35 Method of finding the logarithm of a number between 100 and 10,000,. ib. Examples,..... 36 Multiplication by logarithms,...... ib Division,............ 37 Mode of determining the characteristic,. 38 Method of finding the logarithm of a number beyond the limit of the tables,....... 39 Of finding the number corresponding to any given logarithm,. 42 Mode of finding the logarithms of a number by the tables of Callet,. 46 Examples in multiplication and division by logarithms,. 44 Raising of powers and extraction of roots by logarithms,. 45 Description of table of logarithms, sines, tangents, &c.,.... 47 Rules for finding the log. sin. tan., &c. of any given arc, expressed in degrees and minutes,..... 48 Of an arc expressed in degrees, minutes, and seconds,.. 49 Rules for obtaining log. see from log. cosine, and by log. cosec for log. sin ib. Logarithmic relations of tangents with cotangents,....ib. Description and mode of using the trigonometric tables of Callet,.. 51 Method of finding the degrees, minutes, and seconds corresponding to any given log. sine, tangent, &c.,... 52 SOLUTION OF RIGHT ANGLED TRIANGLES BY THE AID OF LOGARITHMS,. 54 Definitions of trigonometric lines in terms of the ratios of sides of right angled triangles and consequent rules of solution by two logarithms, ib. Rule for adding or rejecting 10 in the introduction of radius for homogeneity, 57 Examples in the solution of right-angled triangles,... ib. Exercises in ditto,..... 59 Practical examples,... 60 Practical exercises,...... ib Use of the arithmetical complement,.... 61 SOLUTION OF OBLIQUE ANGLED PLANE TRIANGLES,... 62 Practical problems,...... 63 Deduction of the formula for the cosine of an angle in terms of the three sides of a triangle....... 56 CONTENTS. XI Demonstration of formulas for the sine and cosine of the sum and difference of two arcs,..... 68 Of formulas for the sine and cosine of twice an are,.. o 71 Of half an are,... o e., Derivation of formula for the sum of half an angle, in terms of the three sides of a triangle,.. 73 Example under the above formula,.... 74 Blank form for the same,...,.. 75 Formulas for the sine of an angle, the cosine of half an angle, and the tangent of half an angle in terms of the three sides,.. o 76 Examples,......... ib, Derivation of formulas for the sum and difference of the sines of two arcs and their ratio,.. o. 77 Demonstration of formulas for the solution of a triangle when two sides and the included angle are given,..... 78 Example of their application,.. o, o 79 Blank form for this case, o..... s Demonstration of formula to be used in the last case, when only the unknown side is required,. o.... 81 Example of its use,. o.. o. 82 Practical exercise,.. o. ib. Miscellaneous exercises in Plane Trigonometry,. 83 APPENDIX I. Demonstration of formulas for the tangent of the sum and difference of two arcs,....... 87 Exercise,......... ib. Of the sine and cosine of 450. Of the sine, &c., of 300 and 600,.. 88 Extension of the demonstration for sine and cosine of (a + b) to all the quadrants,............ 89 Demonstration of a formula for sin (n + 1) a,.. 90 CC S.... cos (n +1) (,... 91 Application of the formulas for computing a table of sines, &c.,.. lb. Formulas of verification in the computation of the tables,. o. 92 Series for calculating sines, cosines, &c.,.... ib, Forms for finding the trigonometric lines and their logarithms for very small arcs,....... 94 Derivation of series for calculating the logarithmic sines, cosines, &c., of arcs,...... 96 Forms for the sum and difference of the sines and cosines of two arcs,. 98 Table of forms for the trigonometric lines in the different quadrants,. 99?orms for trigonometric lines of arcs greater than 360,... 100 Table of formulas expressing the most useful general relations of arcs,. ib. Tables of formulas of less frequent occurrence,.... 102 Table'f analytical values of sin, cos, and tan, 104 .o XiA CONTENTSo Patr Derivation of formulas similar to cos a 1= (s -+ ) 106 Of formula for (cos a - sin a V/- l)n... ib. Of formulas for sinn a and coss a,..... ib. Area of a plane triangle in terms of any three parts,... 107 An angle and the logarithms of the sides containing it being given, to solve the triangle,........... 108 Various problems admitting particular solutions,.... 109 Three point problem,..... 110 Ditto for a height,.....112 Demonstration of formula for solution of a triangle when two sides are nearly equal,..... 114 Use of subsidiary angles,......... ib. Evasion of tabular errors for very large or very small angles, e. 113 Resolution of quadratic equations trigonometrically,... 115 Determination of the increment of the sine, tangent, &c., from a small increment of the angle,.... i.b. Exercises in Plane Trigonometry and its applications,... 116 PART II. SPHERICAL TRIGONOMETRY AND PRACTICAL ASTRONOMY. Definitions,.. 121 Demonstration of sine proportion for right angled triangles,. 122 Definitions of circles in the celestial sphere,.... 123 Applications of right angled trigonometry to finding the sun's longitude when his declination is given,.... 125 Sine proportion in oblique angled spherical triangles,.. 126 Astronomical example of its application,...127 Demonstration of the formula for the cosine of an angle in terms of the three sides of a spherical triangle,.... 128 Of a formula for the cosine of half an angle in terms of the three sides,. 130 Formulas for the sine and tangent of half an angle in terms of the three sides of a spherical triangle,.... 131 Blank form for the application of one of these formulas,.... 132 Derivation of the formulas for sine and cosine of a a side in terms of the three angles of a spherical triangle,.. 133 Gauss equations,......... 135 New mode of deriving Napier's analogies from the Gauss equations,.. 136 Geographic example of the application of Napier's Analogies,.. 137 Astronomic example of the same, viz. the determination of the latitude and longitude of a heavenly body from its right ascension and declination,............ 139 Mode of solution when two sides and the included angle are given, and the unknown side only is required,.... ib CONTENTS. Xlii Paye Mode of solution when two angles and the included side, the unknown angle only being required,... 140 Napier's rules for the solution of right angled triangles,.. 141 Astronomical example, from the sun's right ascension and declination, ta find his longitude,.. 1 43 Exercises in right angled spherical trigonometry,.... 144 Astronomical Problem. Given the sun's declination, to find the time of his rising or setting,.... e 145 Exposition of the different kinds of time,... ib. DESCRIPTION/OF THE TRANSIT INSTRUMENT,... 147 Method of collimating the instrument,.... 148 " " levelling ".....149 " " adjusting to the meridian, o... 150 " " observing a meridian transit,... ib. Rules for determining the meridian altitude of a star,.. ib. Mode of applying electro-magnetism to record the observation,.. 152 Reduction of time of transit over any wire to time over imaginary middle wire,............ ib. Example of this reduction,..... 154 Demonstration of formula for determining the inclination of the axis, ib. Mode of computing the effect of this inclination on the time of meridiar transit,... 156 Demonstration of a formula for azimuth error,.. 157 Mode of computing its effect on the time of meridian transit,. 160 Mode of computing the effect of error of collimation,. ib. Example illustrating the application of all the corrections for instrumenta; error,.. 162 Conversion of siderial into mean solar time, and vice versa,.. 164 Astronomical Problem. Given the latitude of a station and the declinatior of a heavenly body, to find the time of its being on the six o'clock hour circle,...... 165 Solution of an oblique angled spherical triangle when two of the three give parts are a side and its opposite angle,..166 Example of solution,.... 168 EXAMINATION QUESTIONS ON TRIGONOMETRY,.... 169 Questions on the general relations of trigonometric lines,.. ib. On the resolution of right angled plane triangles,... 171 On the resolution of plane triangles in general,. 172..I' " right angled spherical triangles.... ib. cs " " spherical triangles in general,.... ib. Questions on logarithms,..... 173 " the celestial sphere,......174.C 6. the transit instrument,...... ib. inv^ CONTENTSo APPENDIX II. ON UNLIMITED SPHERICAL TRIANGLES AND THEIR SOLUTION. Pang The various triangles formed with the same three points on the sphere,. 177 Demonstration of spherical formulas of unlimited application,.. 178 Proof that there may be 128 triangles, each the first term of an infinite series formed by three great circles on the surface of the sphere,. 179 Ambiguity in the solution of the general spherical triangle,.. 181 Formulas required for the solution of the general spherical triangle. Gauss Theorem,..... 82 Mode of using the Gauss equations,... 184 Proof that the parts of the triangle may be interchanged in the Gauss equations,........... ib Auxiliary angles,....... 186 Solution of the several cases of the general spherical triangle, with the use of checks,.. 187 SOLUTION OF QUADRANTAL TRIANGLES,... 193 DEMONSTRATION OF NAPIER'S RULES,... 194 Formulas to be used in place of Napier's rules, when the required part expressed by its sine is very small, or expressed by its cosine is very large,.......... ib Rules for deciding upon the ambiguity of the result in the use of the sine proportion,...... 196 Modes of determining the effect of small errors of observation in the data upon the queesita, and under what circumstances these will be minima,........ 197 PART III. NAVIGATION. Definitions,............ 203 Description and use of the mariner's compass,.. 204 ". " the log,.....205 THEORY OF PLANE SAILING,... 206 Examples and exercises,...208 Theory of traverse sailing, and use of the traverse table, 209 Example,. 210 Construction of the traverse.....211 Exercises in traverse sailing,....... 214 THEORY OF PARALLEL SAILING,.. o. ib. Examples and exercises,... 216 THEORY OF MIDDLE LATITUDE SAILING,... 217 Use of Workman's Tables,......... 219 e'xamples and exercises,...... ib. CONTENTS. XV Page THEORY OF MERCATOR'S SAILING,...... 221 Construction and use of the table of meridional parts,... 222 Examples and exercises,..... 224 APPENDIX III. THEORY OF GREAT CIRCLE SAILING,. o.. 227 Example and exercises,.... ib. Great circle courses from New York to Liverpool or Havre,.. 228 From the Chesapeake to Bordeaux,..... 229 PART IV. SURVEYING. General principles,.... 233 Description and use of the plane table,...... 234.". ".surveyor's compass,. 237 Surveying by offsets,.238 Description and use of the theodolite and its adjustments,.. 239 Theory of the vernier,....... 243 I'riangulation of a country,....... 244 Survey of a large estate,... ib. Map and field book, 246 and 247 LEVELLI NG,........ 247 Description and use of the level,.... ib..c it'. levelling staves,. 249 Difference between true and apparent level,... 250 Section of ground,...... ib. Contour of ground,......... 251 SURVEY OF ROADS, RAILWAYS, AND CANALS,...... 253 Reconnaissance by the courses of streams on maps,... ib. Personal reconnaissance of the country,.... 254 Surveys with compass and chain and level,.. ib. Survey of cross sections and computation of excavation and embankment, 255 Computation of the contents of fields,...... 256 By reduction of irregular to regular boundaries,..... 257 Without plotting by traverse table,........ 258 Examples of the above method,... 259 HYDROGRAFHIC SURVEYING,....... 262 Norris's method by signals,.... ib. Determination of remote points at sea, e... 263 ivl CONTENTS, PART V. NAUTICAL ASTRONOMY. Pd Definitions,........... 267 Mode of determining the apparent right ascension and declination of a star from the British Catalogue,..... 269 Of the corrections to be applied to the observed altitudes of heavenly bodies,....... 270 Theory of the dip or depression of the horizon,., 271 Correction for semi-diameter,..... ib. Augmentation of the moon's semi-diameter,. 272 Correction for refraction,.. o.. 273 Use of Table,....... o ib. Correction for parallax,.... 274 Formula for computing parallax in altitude, f o 275 Contents of the Nautical Almanac,.. 276 Examples of corrections,..... 277 Example of corrections for the moon's altitude,.. ib. DETERMINATION OF THE LATITUDE AT SEA BY MERIDIAN ALTITUDE, 279 Example and exercises,..... 281 Determination of the time of the moon's meridian passage,. 282 Determination of the latitude by two altitudes,.. o 283 Example,........ 285 ON FINDING TIHE LONGITUDE,....... 287 Method of finding the local time by an altitude of the sun or other heavenly body,....... ib. Exercises in finding the time,. 289:Example of finding the longitude by difference of time,. 290 Description and use of the sextant-Note,... ib.:longitude by lunar distances,. o ib. Example,.. 294 Variation of the compass,.....295 Examples of finding the variation,.... 296 APPENDIX TO PART V. THE REFLECTING CIRCIE,...... 299 Description of Troughton's Circle and mode of observing with it,. ib. Dolland's Circle,....... ib. Mode of using the repeating process,...... ib. Example of the repeating process,.. e 300 CONTENTS' XVUi VARIOUS MODES OF DETERMINING LATITUDE,..... 301 Latitude by a single altitude,.... lb. Method when the latitude is approximately known,... ib. Latitude by circum-meridian altitudes,.... 302 Use of one of Bowditcths tables, and tables of versed sines,.. ib. Example of circumn-meridian altitudes with repeating circle,. 303 Method of Littrow by three altitudes near the meridian,. 304 Latitude by an altitude of the pole star out of the meridian,,. 305 THE MURAL CIRCLE,......306 Description and use of the ghost apparatus for adjustment,.. 307. " " the reading microscopes,... 308 Method of taking the readings,..... 309 Reduction to the meridian of an observation with the mural, ib. Meridian circle,...... 31 I PART VI. GEODESY. Definitions,............ 315 General account of triangulation,. ib. MEASUREMENT OF BASES,......... ib. Description of Base Apparatus,.. o.ib. Optical contact of rods,.. 316 Lever of contact,....... ib Compensating arrangement,.... ib, Level sector for determining inclination of rod,... 31 Reduction of base measures to the horizon,.. ib. Reduction to the level of the neighboring seas,.... ib. Mode of making the horizontal alignment,. 318 Method of testing the rods by Saxon's Pyrometer,.. ib. " equalizing capacity of heating in the brass and iron bar,. ib. Reduction of a broken base to a straight line.... 319 THE GREAT THEODOLITE,........ ib. Description,.... ib. Method of observing,....... ib. SELECTION OF STATIONS AND SIGNALS,...... 320 Day Signals,..... ib. Night Signals,.......... Ib. Formula for reduction of phase in tin cones, o... ib. The heliotrope,...... o.. 321 REDUCTION TO CENTRE OF STATION,;.... ib. Formula of reduction,...... 322 VERIFICATION OF OBSERVED ANGLES, a,. e. lb. Legendre's theorem,. o.. o o ib, XV1ii CONTENTS. Mode of conmputing the spherical excess,.. 323 Ditto between the latitude of 450 and 250,.... 432 Form and example of the above used on the U. S. Coast Survey, 325 DEATERMINATTON 01 LATITUDES, LONGITUDES, AND AZIMUTIHS OF TH.E SATIONS, ib. Demonstration of' formulas for the spherical figare of the earth,.. b. Necessary modifications for the spheroidal figure,.. 327 Specimens of table from which co-efficients of the formula lo; difib-rcee of' latitude are taken,........ Example of geodetic determination of latitude,. 3:' Indication of eases in whic th the third and fourth tlrnms jo the i;rnl 111a may be omitted,... Number of places to which the logs. of the dififrernt terms imu -ti e carrjcil, 33 Demonstration of the forlmula for difibrenee of longitude,. i. b. F'orm and rationale of subsidiary table for its correctioln whe tho? odl ti line exceeds 16- miles,........ 33 Demonstration of the formula for difierenee of azimuthl,. 3 Rule for the direction in which azimuths are to be estimated, 333 Form used on the U. S. Coast Survey and example of gceodetic deko- il in,tion of latitude, longitude, and azimuth,... b. PROJECTION OF IIArs,..o 334 Use of blank maps by plane table parties,.. ib. Method of projection by tangent cone,... i. Rule for drawing the ap,....... 335 Modification of the method of a partly inscribed cone,. 334 Further modification of Delisle,....... i PROJECTION OF FLA-ISTEED....... 33 Spherical projection,...... ib introduction of She earth's oblatenes,.... MEITIOD OF Trii. FREnNCI DErOT DE L. GUERE,..... if. Demonstration of the formula,...... LATITUDE Er As'RONOMIC OBSERVA1aTON.A..s 3 40 By observations with zenith sector,......... i; telescopes,... 341 Correction for refraction in zenith telescopes,.... 3 ". in-lclination of vertical axis,..3. 3 Reduction to meridian,........ 44 General formula, including all the corrections,.. Example and form used in the U1. S. Coast Survey,..... 34 longitude by celestial observations,..... 3-8 by moon culminations,.... Example,. 3.. By difference in time of culmination of the moon and a star,... ib. Example,..... 350 Rule when the meridian is distant from that of Greenwich,... ih. Formula for interpolation,..... Longitud. by eclip:es of Jupiter's satellites,.... ib. CONTENTS, XIX Astronomical determination of azimut,... 352 Azimuth of the sun or a star,.........Azimuth by Polaris at its elongation,... 3. 3'Trigonometric le velling........ 355, MAGNETIC OBSERVATIONS,.... ib. Description of the declinometer,..... 3 Absolute horizontal intensity,...... 357 The inclination,.. 358 A P P E N DIX TO PART VI. INSTRUl'MENTS I OR EXTRA MErTIIAN OBSIERVAlTIONS,.... 361 The equatorial instrument,...... ib. The position micrometer,....362 Angles of position and distance of double stars,.. 363.Altitude and azimluth instrument,.... ib. Form for recording- its observations,....... 364 Conversion of astronomic and geocentric latitude,. 365 Demonstration of the formulm for the radius of curvature in terms of the latitude,. 366 Demonstration of formula for determinilng the figure and dimensions of the earth by the lengths of two measured degrees in distant latitudes, ib. Prime vertical transit,.. 36S Folrmulas for latitude, altitude, and time of observation lused in prime verlical transits...... 3 Description of the Pulkova and Washington instruments... ib. Struve's formula,....... i Examiple from Struve's observations,... 370 Concluding note,..... 372 PART I. PLANE TRIGONOMETRY. 1. THE term TRIGONOMETRY is compounded of two Greek words, rPgywvo;, a triangle, and p.,gov, measure, signifying literally the measurement of triangles. It has for its object to determine the unknown parts of a triangle when a sufficient number of the parts is known. By parts or elements of a triangle are understood commonly the sides and angles, though trigonometry properly includes the measurement of the surface also. There will accordingly be six elements of every triangle, namely the three sides and the three angles. 2. It has been proved (Plane Geom., Theorems 1, 2, and 5), that when two triangles have three elements, one of which is a side, in the one, equal respectively to the corresponding elements in the other, the triangles are identical One element must be a side, because if the three angles only were equal respectively in the two triangles they would be but similar (Plane Geom., Theorem 63); that is, alike in shape but not necessarily in size. Since all triangles which have three elements equal, are by consequence equal, it is said that three given elements determine a triangle; that is, with these three given elements, but one triangle can be formed. There is one exception to this principle, pointed out in Prob. 8, Plane Geom., where two sides and the angle opposite one of them are given, in which case two triangles can be constructed with the given elements. 3. Three elements of a plane triangle being given then (except they be the three angles), it ought to be possible to find the other three, since these are fixed by their dependence upon the three given. ~This may be accomplished with sufficient accuracy for many purposes, by means of constructions such as are exhibited at Problems 5 and 8 of Plane Geometry.! 2 PLANE TRIGONOMETRY, We shall repeat one of these constructions, enunciating the problem somewhat differently. The two sides and included angle of a triangle being given, let it be required to find the remaining side and the other two angles. A B Let A and B be the two given sides, and c the given included angle. Draw two lines DH and DG of indefinite length, making with c / each other an angle equal to the given angle c. Lay off on the P first of these the given line A from D to E, and on the second the. given line B from D to F. Join D E rE. The only possible triangle DEF will thus be formed with the three given elements, in which EF will be the required side, and E and F the required angles. The finding the unknown elements of a triangle by means of those which are given is called its solution. 4 The method of solution just exhibited is rendered more practically useful by the employment of scales of equal parts and protractors. The most simple form of the scale of equal parts is shown in the annexed figure. 9 8 7 6 5 4 3 2 1 0 I I, I I. I,'I I... I __ It is a straight rule divided into any number of equal parts; in this example ten, and one of these again into ten, so that the smallest division is one hundredth of the whole length of the rule. The following is the manner of using it. Suppose that it is required to draw upon paper a line equal in length to 56. Place one foot of a pair of dividers at the line of division marked 5, and extend them till the other foot reaches exactly to the sixth smaller division mark on the right of 0 the feet of the dividers will then be at a distance of 56 apart. A To draw now the required line upon paper, let A be the point from which it is to be drawn. Placing one foot of the dividers at A, extended the THEORY OF THE SCALES. 3 d.,tance 06 obtained from the scale, describe with the other an are of a circle on the side towards which the line is to be drawn; then from A draw the line in the proper direction, terminating it at the are before described, and it will be the line required. Another line of 42 being measured from the scale and laid down upon the paper, the two lines will be in the ratio of 56 to 42. If they are lines upon a map, and the first corresponds to a line of 56 feet upon the ground, the second will correspond to a line of 42 feet. If the first represent 6;6 yards, or chains, or miles, the second will represent 42 yards, or chains, or miles. And in general lines upon the same drawing which are measured in parts of the same scale must be understood to be expressed in units of the same kind. The sectoral scale of equal parts consists of a ruler of two arms moving on a hinge, each arm being divided into a number (usually 100) of equal parts. To set this scale to any size, say 40 parts to the inch, the arms =: must be separated by turning them round the hinge, till a pair of dividers opened to the distance of an inch will extend exactly from the division marked 40 on one arm, to that marked 40 on the other. If now any other distance be required upon a scale of 40 parts to the inch, as for instance the distance 65, the dividers must be opened till they will extend from the 65th division on the one arm, to the 65th division on the other. This kind of scale is constructed on the principle that lines drawn parallel to each other between the sides of an'angle are proportional to the parts into which they divide the sides. (See Plane Geom., Theorems The3, 65d.) The diagonal scale of equal parts is constructed as seen min the diagram, 4 PLANE TRIGONOMETRY. by parallel lines drawn along a ruler in sufficient number to embrace 10 spaces between them. Transverse lines are drawn perpendicularly to 2 I _ I........ these at intervals usually but not necessarily equal to the whole breadth of the 10 spaces. The last one of these intervals is divided into ten spaces, and the first division at top is joined to the second at bottom, the second at top to the third at bottom, and so on by diagonal lines. To measure any distance on this scale, as 456, place the dividers on the 6th horizontal line from the top with one foot upon the 4th of the larger divisions, from the 1st on the right, and extend the'other foot of the dividers, till it reaches to the 5th smaller division in the right hand square. 5. Before describing the protractor, which is an instrument for laying off angles, it will be necessary to explain the method of estimating the magnitude of angles. In Geometry, it is shown that angles are proportional to the arcs included between their sides, the arcs being described with equal radii, and it is also there stated that hence such arcs are properly the measures of angles. So that if an are included between two sides of one angle be double, or triple, or sextuple, an are described with the same radius included between the sides of another angle, the first angle is double, triple, or sextuple the second. The relative magnitudes of angles may therefore be correctly expressed by means of the relative magnitudes of the arcs which measure them. The relative magnitudes of quantities are commonly given by referring the quantities to be compared to some known standard of measure, which must be always of the same kind with the quantities themselves. This standard is called a unit. Thus a foot, a yard, &c., are units of length, and the idea of the relative lengths of two lines is obtained by its being said that one is seven feet or yards, and the other nine. Or the just conception of the length of a single line is had by being told how many feet, yards, or miles it contains. The mind compares it with one of these well known units, which in imagination it repeats along its length. MEASUREMENT OF ANGLES. 5 Now the unit of measure, which is employed in a similar manner for giving the conception of the magnitude of an are, is called a degree. A. degree is the -'-0 part of the circumference of a circle. The relation which any given arc bears to the whole circumference may be conveniently expressed by stating the number of degrees which the arc contains. Thus an arc of 90 degrees will be one fourth the whole circumference. An arc of 45 degrees will be one eighth. An arc of 30 degrees will be somewhat less. And it is plain that the length-of the arc, as compared with the whole circumference, may be readily conceived, as soon as the number of degrees which it contains is mentioned. So also the magnitude of D,,d so so / the angles subtended by these E,.o' - ares will, after a little fami- liarity, be rendered easily sen- sible to the mind. To speak of an angle of 10 degrees for instance (A C B in the T' annexed diagram), will suggest the image of a very acute angle, one of 60 degrees (A C D) a muchl larger acute angle, one of 140 degrees (A C E) an obtuse angle. A degree being always the -i — part of a circumference, a single degree will be larger in a larger circle than in a smaller, and this, so far from being inconvenient, is particularly advantageous in the measurement of angles; for since arcs described about the vertex of an angle as a centre with different radii, and included between the sides of the angle, bear the same relation to each other as the radii, and since the entire circumferences are also proportional to their radii, it follows that two concentric* arcs included between the sides of the same angle, and having the vertex of that angle for a centre, are the same aliquot parts of their respective circumferences. Consequently, two such ares will contain the same number of degrees. Hence, to find the number of degrees contained in a given angle, the arc described for the purpose about the vertex, and extending fiom side to side of the angle, may be with any radius at pleasure.* Having the same centre. 06 P~PLANE TRIGONOMETRY. This may be distinctly seen in the following diagram. ACB is the angle; the larger arc AB included between its / sides contains 50 degrees of the ~ / whole circumference; the arc ab /.' with the lesser radius also con- | | /. o tains 50 degrees, and so would c an arc included between the sides of the given angle described with any other radius whatever. Where the size of an angle is such that it does not embrace an exact even number of degrees of the circumference, smaller divisions called minutes, 60 of which make a degree, are employed. The angle is then said to contain as many degrees and minutes as there are degrees and parts of a degree, each 6t over, between its sides. If the second side of the angle does not pass exactly through one of these smaller divisions, a still smaller kind termed seconds, 60 of which form a minute, or 360 a degree, must be introduced. 900= 5400t-324000", 180~=10800'= 648000", 2700=1 6200'-=922000", 360o=21600t=1296000". When it becomes necessary to use more minute divisions, the same system is continued. The next denomination is thirds, 60 of which make a second; the next fourths, and so on.* The notation for these denominations is as follows. Degrees are written thus ~; minutes thus'; seconds thus "; thirds thus "', &c.; 30~ 20' 10" is read thirty degrees, twenty minutes, and ten seconds. 6. It is evident that the numbers used in the system of division, for the circumference of the circle, are entirely arbitrary. Others might be employed with equal propriety, provided the same principles were observed. In fact the attempt has been made, and probably will be successful in France, to subvert the old system of division, and to adopt a decimal system in this as well as in every other sort of measurement. Thus a right angle, which is the unit of angles, is made to contain 1000 instead of 90; and the circumference will then contain 400~ instead of 360. 100' instead of 60=1~, 100"=1'. Degrees in the centesimal division of the circumference are called grades; and the notation in this division is g I ". Grades are converted into degrees by multiplying by — % or'9. It will be found more convenient to subtract T of the given * Instead of thirds, fourths, &c., the almost universal practice now is to use decimals of a second, viz. tenths, hundredths, and thousandths. MEASUREMENT OF ANGLES. 7 number of grades from the given number itself. The two methods above described are called the sexagesimal and the centesimal divisions. EXERCISES. 1. Convert 42s 34' 56" or 42g3456 into degrees, &c. Ans. 380.11104 or 380 6' 39"'74. 2. Convert 24~ 511 45" into grades, &c. Ans. 27g 62' 50". 3. Prove 45~ 15' 20"-=50g 28' 39'*50. 4. Also, 10~ 15' 46" —11g 40' 3"'09. 5. Also, 18~ 10' 48t-=20g 20\'. The semi-circumference of a circle whose radius is 1 is 3'14159265=180~.. 1~= 40.017453293:. 1=-0'0002908882.'. 1"=0~000004848137. 180~ Again, -3141 5965-7~2957795=3437~'74677=206264'"806, is the length of t>14159265 the radius of any circle expressed in degrees or minutes or seconds as units of length. 7. Another method of expressing the magnitudes of angles is as follows. A distance at pleasure is laid off from the vertex of the angle upon one of the sides, and a perpendicular there drawn to this side till it meets the other side of the angle. The ratio of this perpendicular to the distance from its foot to the vertex, serves to indicate the size of the angle. For example, if the line BCDE be perpendicular to the line AB, and BC E be one fourth AB, the angle BAC is said to be an angle of ~. If BD be one half AB, the angle BAD is said to be an angle of -. If BE be equal to AB, BAE is said to be an angle of 1; and so on for other magnitudes. An angle of 1 is plainly half a right / / angle, or 450. This kind of measurement is much A B used by engineers, to express the degree of slope in excavations and embankments. 8. The protractor which we are now prepared to describe is an instrument for drawing upon paper an angle of any given number of degrees. This instrument is made in a variety of forms; sometimes with a full circle divided into degrees, sometimes comprising only a semicircle, sometimes upon a rectangular rule having not the circumference but the radii drawn, as they would be through the divisions of the circumference if it were actually described. The first kind is made usually of brass. * It is quite unnecessary to use the symbols' and ". 148g'5926 would sometimes be written lq'485926, where the symbol q denotes a quadrant or right angle 8 PLANE TRIGONOMETRY. It has a metallic radius movable about the centre of the circle, and extending beyond the circumference. This prolonged radius serves to point out the number of degrees, and is armed with a sharp pin under the outer extremity for the purpose of pricking the paper, so that when the instrument is removed a line may be drawn with pencil through this point, and that upon which the centre was placed. THE SEMI-CIRCULAR PROTRACTOR, A B which is the one most commonly seen, is a semicircle of brass (or other metal), having the greater part of the interior cut out to render the instrument less heavy. The semi-circumference is divided into degrees by marks made in the metal, and these are numbered from 0~ to 180 (the number in a semicircumference) both ways, in order that the counting may commence with convenience at either end. The degrees are also sometimes divided into half degrees, and lines of liffelent length are employed to mark more distinctly every five and every ten degrees.* The centre is marked by a notch in the straight side of the instrument, which side is a diameter of the semicircle.t 9. In order to explain the use of the instrument here described, suppose it be required to draw at the point A in the line AB a line making with AB an angle of 220. * Such a division of instruments is termed graduation. t This instrument may be made out (d paper, and a large one so made is very accurate. SEMI-CIRCULAR PROTRACTOR. 9 Place the protractor so that its c centre shall be upon the point A, and its straight edge or diameter upon the line AB. Then mark the paper at the point c against the 22d A B division of the protractor, and a line joining c and A will form with AB the angle required. 10. We are now prepared to construct triangles when three of their six elements are given, the angles in degrees and the sides in feet, yards, or other linear units. In order to show. the practical utility of trigonometry at the same time that we explain the solution of a triangle, let us take the following problem in the calculation of distances to inaccessible objects. Suppose a fort situated upon an island, and a light-house upon the main shore, and let the distance from the light-house to the nearest salient of the fort be required. Measure a line along the shore of any length at pleasure, say 500 yards, beginning at the light-house. Then if two lines be imagined to be drawn from the extremities of the line jiust measured, to the salient of ^ the fort, a large triangle will be formed having its two longest sides resting upon the / 1 sea. If now the angles which these two a sides form with the first side, which we will call the base, could be determined by observation upon the shore, there would be known in this triangle a side and the two adjacent angles, which would be sufficient data to construct the triangle on a small scale, and to obtain the length of the required side extending from the light-house to the salient of the fort. A somewhat rude instrument for the purpose of observing such angles as those alluded to above, might easily be made. Let there be a circle, or flat circular ring of wood, divided into degrees, and having a tin tube movable upon a pivot at the centre of the circle; the tube being closed at one end except a very small orifice, and having two threads crossing at right angles in the centre of the other end, so that in looking through the tube with the eye at the small orifice, the line of sight may coincide with the axis. Let this apparatus be mounted upon a three-legged stand called a tripod, so that the plane of the circle shall be horizontal; then, 10 PLANE TRIGONOMETRY. by placing the instrument thus formed at the light-house, in the example above, and: sighting with the tube, first to a staff at the other extremity of the base, and then to the salient of the fort, keeping the circle stationary, the number of degrees passed over upon its circumference by the tube will indicate the angle of the triangle at the light-house. This angle we shall suppose to be 105. The angle at the other extremity of the base might be found in the same manner, and suppose it 47~.* To construct the triangle with these data, draw on paper a line AB, and make it equal in length to five hundred divisions of some scale of equal parts.t Then draw an indefinite line AC making with AB an angle of 105. Also lay off in a similar manner at the point n an angle of 47~; \ the two lines AC and Bn will meet at c. Take the line AC in the dividers and apply them to the scale. The number of equal parts upon the scale between the feet of the dividers, will show the number of yards from the light-house to the fort. The number is 791. If the angle at c were required, it might be measured by applying to it the protractor; or it is equal to 180 —(A+B)= 271 o. The side B c, if among the sought parts, might also be measured from the scale. 11. The instrument described above may be rendered suitable for application to the determination of heights. If a round bar be made to project horizontally from the top of the tripod, so that the graduated circular fiame can be suspended by the socket at its centre in a vertical position, it will then serve to Measure angles in a vertical plane.t * The instrument here described is of course very rude. It was deemed advisable to postpone a description of more accurate instruments to a subsequent part of the work. t This may be done conveniently by taking 50 divisions, and considering each division as equal to ten. t A vertical plane is one perpendicular to the surface of the earti. SOLUTIONS BY CONSTRUCTION. I. To show the use of the instrument thus prepared take the following problem. Required the height of a tower which stands upon horizontal ground, and the base of which is accessible. Measure back a distance from the [I I base of the tower, say 200 feet; call this distance the base line; at the _ 0 extremity of the o feet. base line place the instrument arranged for taking vertical angles: suspend a plumb line from the centre of the circle, and the point 900 distant from that in which the plumb line cuts the circumference will be thei point through which a horizontal radius would pass. Then sight with the tube to the top of the tower: the number of degrees between the tube and the horizontal radius just mentioned, will be the measure of the angle included between a line drawn to the top of the tower and the base line; let this number be 30~. Constructing a right angled triangle upon paper, having its base 200 and angle at the base 30~, the perpendicular of this triangle, measured by a scale of equal parts, will be the height of the tower. The height of the instrument must be added to the result found. N. B. The sides found will always be expressed in units of the same kind as the base. 12. It is evident that when any three parts of a triangle, one of whicl is a side, are given, the other three may be discovered by a process similar to those just exhibited. This kind of solution is said to be by construction. The accuracy of the results must depend upon the niceness of the instruments, and the care with which the construction is made. A degree of accuracy so uncertain and so variable is quite inadequate for many purposes to which Trigonometry is applied. A method of calculating the required from the given parts of a triangle, which should produce always the same results from the same data, and be either perfectly, or so nearly exact, as to leave an error of no importance, however great the dimensions employed, would be evidently a desideratum. Such a method we have, and it is that which it will be the object of the residue of the present treatise to unfold. To give the student a general view of what is before him, it will be 12 PLANE TRIGONOMETRY. well to state that a number of equations will be found, each containing four quantities, which quantities will be general expressions for the measures of elements of a triangle. The equation will express the true relation between these elements. By making one of these elements the unknown quantity and resolving the equation with respect to it, its value will be expressed in terms of the other three. If now these three were given, the value of the fourth would be known the moment the values of the three given were substituted for their general representatives. It is plain that as many such general equations will be required, as there can be formed essentially different combinations of four out of the six elements of a triangle. Equations like those here alluded to are called formulas, because each is a general form, under which a multitude of particular examples are included. As these general formulas require of necessity the use of algebraic symbols and processes, and as algebra, from its power and application to decompose combinations of quantity so as to extricate their elements, is often called analysis, the subject upon which we are now about to enter is called ANALYTICAL TRIGONOMETRY. 13. The sides and angles of a triangle are not quantities of a similar kind, and therefore do not admit of direct comparison. Since angles are expressed in degrees, and sides in units of length, one of the first principles which governs the formation of equations, namely, that the members and terms should express quantities of the same kind, would be violated by the introduction of angles and sides together, without some modification of one or both. The expedient which has been invented to accommodate these heterogeneous quantities to each other, is that of employing straight' lines, so related to the arcs which measure the angles of a triangle, as to depend upon these arcs for their length, in such a manner that when the arcs are known, these straight lines may be known also; and vice versa. The chords of arcs are plainly lines of this description, aad chords were at one time used for the purpose of which we here speak; but there is a more convenient kind of lines, of which there are three principal sorts, termed sines, tangents, and secants, of an arc or angle, called, when spoken of collectively, trigonometrical lines, the nature and use of which we shall presently explain. These lines being straight and expressed, as they will TRIGONOMETRICAL LINES. 13 be found to be, in linear dimensions, like the sides of a triangle, they may be employed with the latter in equations or formulae; and when, by the resolution of an equation of this description, one of these trigonometrical lines is found in terms of one or more sides of the triangle, the angle to which the trigonometrical line belongs may also be supposed to be known. How the former is known from the latter will be hereafter explained. Let it be taken for granted here that the knowledge of a trigonometrical line is equivalent to the knowledge of its arc or angle, and vice versa. The trigonometrical lines are sometimes called trigonometrical functions' of an arc or angle. Of these trigonometrical lines, we now proceed to explain the nature and properties. THE SINE. 14. The sine of an arc is a perpendicular let fall from one extremity of the arc upon the diameter drawn through the other extremity. Thus the line MP is the sine of the arc AM. The same line MP is likewise the sine of the arc BM, because it is a perpendicular let fall from one extremity M of the arc upon the diameter drawn through the other ex- tremity B. 15. Two arcs, which together make a semi-circumference, have, it thus appears, the same sine. Two such arcs are called supplements of each other. A semicircle contains 1800. The supplement of an arc is therefore what is left after taking the arc from 1800 or 200g. Thus 80~ is the supplement of 1000. 70~ is the supplement of 1100. 85' is the supplement of 115g; in general 90C-a, or 200g-a is the supplenent of the arc a. * One quantity is said to be a function of another, when the former depends in ahy way upon the latter for its value. It is said to be an increasing function when it increases as the quantity upon which it depends increases; and a decreasing function when it diminishes as the other increases. The latter is called the argument, 14 PLANE TRIGONOMETRY, EXERCISES. 1. The supplement of 560 20 1230 40'. 2. " 1860 12t — (6o 12'). 3. " 370 41 3 = 1420 55t 57". 4. " 115o 13' 24I. 66 640 46t 351"34. 5. c 2260 14' 17".- (460 14' 17"). 6. " 23g 25" 176g 75\. 7, " 110g 40' 50\" - 89 59' 50\. Two arcs, then, which are supplements of each other, have the same sine, or, as it is sometimes expressed, the sine of an are is equal to the sine of its supplement. If a represent an arc of any number of degrees, the notation employed to express the sine of that arc is sin. a. The proposition* above, stated algebraically, will stand thus, sin a =sin (180 0 a.) The sine of an arc is also the sine of the angle measured by that arc. 16. When the arc is very small, it is plain that its sine will be very small also, and that when the are is 0, the sine will be 0. As the arc increases the sine increases till the arc is 900, whiclh, being a quarter of the circumference, is called a quadrant, the sine of which is R. (R signifying radius; which line this letter, whenever employed hereafter, will be understood to represent.) As the are increases beyond 90, the sine diminishes, i. e. becomes a decreasing function of the arc, till the arc reaches 180~, when the sine is 0 again. Beyond this value of the arc the sine again increases till the are reaches 270~, or three quadrants, when the sine is again equal in length to R. From 2'70 to 360~ the sine decreases, till at the latter value it is a third time 0. Beyond 360~ we pursue the same round again, and no new variations are developed. 17. The leastvalue ofthe sineis 0. It has this value at 00, at 180~, and at 360~. The greatest value of the sine is R. It has this value at 90~ and at 270~. It has all possible values between 0 and R, but it has no different ~ The word proposition is here used in the enlarged sense of anything propounded'wtrre. TRIGONOMETRICAL LINES. 15 values, as the arc increases to two, three, and four quadrants, from those which it had in the first. So that when the sine of an arc greater than 90~ is required, an arc, having an equal sine, may be found in the first quadrant. To find this arc we have the following rule, the correctness of which the annexed diagram will show. Observe how many degrees distant the termination of the given arc is from 180~ or 360~, according to which of these two is nearest, and that number of degrees and fractions of a degree, will be the arc in the first quadrant, having the same sine as the given arc. For example, let the given arc be 200~' This is nearest 180~, and differs 20~. The sine of 200 is equal in length to the sine of 200~. Or M P, which is the sine of A B M, is also the sine of B M.'-, B.- A Again, let the given aicr be 300~. This is nearest 360~, and differs 60~. The sine of 600 is equal in length to the sine of 300o. If the given arc exceeds 3600, subtract 360, and then apply the rule just given. If the arc contains a number of circumferences, divide by 360, and apply the rule to the remainder. 18. It is customary, for the purpose of being able to bring the trigonometrical lines as they appear in the figure, the more readily before the mind when the figure is not present, to begin all arcs at the same point; and the point commonly chosen is the extreme right of the circumference, determined by the intersection of the horizontal diameter of the circle with the circumference. This is the point A, in the last figure. An arc of 900 will then reach to the top of the circle, or the upper extremity of a vertical diameter. An arc of 180~ will terminate at the left of the circle, or of the horizontal diameter. An arc of 270~, at the lowest point of the circle, or lower extremity of the vertical diameter.' An arc of 3600, at the right of the circle, or point of beginning. One advantage of this plan will readily appear. Since the arc always commences at the same point, namely, the right of the circle, the horizontal diameter will be the diameter which passes through one extremity of the arc, and wherever the arc may terminate, the perpendicular from the other extremity of it, which is the definition of the sine, will be a per 16 PLANE TRIGONOMETRY. pendicular to the horizontal diameter; so that the sines of all arcs, in a diagram so constructed, will be perpendiculars to the horizontal diameter. The sines of arcs between 00 and 1800 will be drawn downwards; and those of arcs between, 180~ and 360~ will be drawn upwards. According to the general principle of analysis, that quantities estimated in a contrary sense are distinguished by contrary signs, if the sines of arcs between 00 and 1800 be considered as positive, those of arcs between 180~ and 3600 must be regarded as negative.* THE TANGENT. 19. The tangent of an arc is a perpendicular drawn to the radius at one extremity of the arc, and terminated by the radius produced, which passes through the other extremity.. In the annexed diagram A T is the tangent of the arc A M. It is also the tangent of the angle A C M, measured by the arc. The shorter the arc is, the shorter will be the tangent. When the are is 0, the tangent will evidently be 0. As the arc increases, the tangent increases, and very rapidly as the arc approaches 900. In order to trace the tangent through its various changes, we shall suppose the arc to commence at the point on the extreme right of the circle, and the degrees to be counted upwards, towards the left, as in a fomer case —the tangent of every arc will then be drawn at the extremity of the horizontal radius on the right of the centre, and be terminated by the radius produced, passing through the other extremity of the arc, which extremity wvill vary its position as the arc varies its magnitude. * See Algebra, page 182. TRIGONOMETRICAL LINES. I7 When the arc is 90~, the per- T pendicular to the radius at one extremity is parallel to the radius through the other extremity. These lines will never meet, and the tangent will have no termination. It is in this case said to be infinite. The sign employed to A. express infinity oo is also called the sign of impossibility. The value of the tangent of 90~ is expressed algebraically thus, tan. 90 = o~. The tangent of an arc, terminating in the second quadrant, will be. cut off below the origin* of the arc. Thus A T is the tangent of A M; and according to the principle adopted when treating of the sine, l this tangent, being in the opposite direction to that of the tangent of an arc in the first quadrant, is negative. When the arc is 1800, the' negative tangent, which became shorter and shorter as the second extremity of the arc approached this point, again reduces to 0. Beyond 1800, or in the third quadrant, the tangent is cut off above the origin again. Thus A T in the annexed diagram, is the B tangent of the arc A B M. The tangent of an arc in the third quadrant is, therefore, positive. When the arc is 2700 or 3 quadrants, the tangent becomes parallel to the radius which produced * A term applied to the point a, where the are commences. 2 18 PLANE TRIGONOMETRY. ought to terminate it, and the tangent is again oo.* The tangent of an arc in the fburth quadrant is negative, as may be seen from the annexed diagram. 20. The least value of the tangent is 0. The greatest value is o. So that the tan_gent has all possible values. But these it has, if we do' not regard the sign, in the first quadrant; and the same rule applies to finding the length of the tangent belonging to any given arc, from that of an arc in the first quadrant, as was given for the sine. The tangent changes its sign in every quadrant, that is four times in going round the circle. It is positive in the first and third, two diagonal quadrants, and negative in the second and fourth, the other two diagonal quadrants. The tangent is oo at the top and bottom of the circle, and 0 on the right and left. THE SECANT. 21. The secant of an arc is a line drawn from the centre of the circle to the extremity of the tangent. In the preceding diagrams, CT is the secant of the arc AM. It is also,the secant of the angle measured by the arc. As the arc with its tangent diminishes, the secant diminishes; and when the arc and tangent are 0, the secant is equal to R. The secant can never be less than radius, because the tangent cannot pass within the circumference, and consequently the line from the centre to the extremity of the tangent, must extend at least to the circumference. When the arc is 90~ the secant is or. When the arc is 180~ the secant is R. And when the arc is 2700 or three quadrants, the secant is again o. All which will appear from an inspection of the last diagrams. * The infinity here has the doubtful or double sign + oo. Zero may have always the double sign + 0. Infinity only when it is the transition from a + to a — value or vice veri. (See Art. 36.) TRiGONOMETRICAL. LINES. 9'The tangent and secant have their greatest values, namely co, together; that is, at the top and bottom of the circle. They have also their least values, that of the tangent being 0, and that of the secant R, together, to wit, at the right and left points of the circle. 22. In the first quadrant the secant is estimated from the centre towards. the second extremity of the arc. In the second and third quadrants it is estimated in the opposite direction. According to the principle which it is necessary to observe, and of which we have before spoken, the secant must in these quadrants be considered as negative. In the fourth qua.drant the secant is again estimated towards the second extremity of the are. and is therefore positive. The vertical diameter separates the positive from the negative secanlt, the positive being in the quadrants on the right of this diameter, and the negative being on the left. 23. We have now exhibited three of the trigonometrical lines. There are three others closely connected with these in character, called the cosine, the cotangent, and the cosecant; the reason for which names will presently appear. The difference between an are or angle and a right angle or 90 -= 100, is called the complement of the are or angle. Thus 400 is the complement of 500; 600 is the complement of 30; 75g is the complement of 259 and in general 90 —-a, or 100 —a, is the complement of the are a. EXERCISES., The complement of 24~ 3~ -= 650 28'. S., 1100 151'-(200 15'). 3. " 170 36' 43"- =720 23I 17". 4. " 290 27' 6"'32 600 32' 53i.68,. 5. - 2160 451 =- 126o 45/. C!. 65g 34 -2 34g 65' 73". 7. l 107s 44' 20\ = -71 44' 20". The cosine, cotangent, ald cosecant, are the sine, tangent, and secant of the complement. Thus the cosine of 50~ is the sine of 40~; the cotangent of 300 is the tangent of 600; and in general the cosine, cotangent or cosecant, of the arc a, is the sine, tangent, or secant of 90~-a. 20 PLANE TRIGONOMETRYo THE COSINE. 24. In the annexed diagram DM is the complement of the arc AM; and MQ being a perpendicular from one extremity M of the are DM upon the diameter which passes through the other ex- tremity D, is the sine of the a P arc DM. Therefore by the \ definition it is the cosine of the are AM. But MQ c.P Hence cp is also the cosine of the arc AM. We have then another definition for the cosine of an arc, viz. the distance from the foot of the sine of the arc to the centre of the circle. 25. If the arc terminate on the right of the vertical diameter, i. e. in the first or fourth quadrant, the foot of the sine will fall on the right of- the centre; but if the arc terminate on the left of the vertical diameter, i. e. in the 2d or 3d quadrant, the foot of the sine will fall on the left of the centre. The cosine being estimated in opposite directions in these two cases, must have opposite signs. It is therefore positive in the 1st and 4th quadrants, and negative in the 2d and 3d. It will be recollected that the positive were separated from the negative secants, as the positive are here seen to be from the negative cosines, by the vertical diameter. The secant and cosine have therefore always the same algebraic sign. It was shown (Art. 15), that sin (180~-a)= sin a; so also cos. (1800-a) is equal in length to cos. a, since they are both the distance from the foot of the same sine (MP in the diagram of Art. 14) to the centre, i. e. if we suppose one of the arcs to originate at A, the other at n, and both to be extended towards M in opposite directions.* But if a < 90~, it follows that 1800-a terminates in the second quadrant, hence its cosine is negative; if a > 90~ then cos. a is negative, and 180 —a being in the first quadrant, its cosine is positive; therefore, the cosine of an arc and the cosine of its supplement are equal with contrary signs. a Both arcs a and 1800-a are now supposed to originate at the same point A, and to be estimated in the same direction. TRIGONOMETRICAL LINES. 2I 26. The cosine of 00 (being equal to the sine of the complement of 0o which is 900) is R. The cosine of 900 is equal to the sine of 0o; which is 0. The cosine of 180~, being the distance from the foot of the sine to the centre, and being also on the left of the vertical diameter, is R- n, as may be seen from the preceding diagram. The cosine of 2700, being the distance from the foot of the sine to the centre, since the sine falls on the centre, is 0. The least value of the cosine is 0; the greatest value is R. When the sine has its least value, the cosine has its greatest; and vice versa. The versed sine of an arc, which is seldom employed in Trigonometry, but often in Mechanics, is the distance from the foot of the sine to the origin of the are, thus PA in the last diagram is the versed sine of the arc AM. 27. Before noticing the cotangent and cosecant, let us consider the manner of treating negative arcs. Such arcs commencing at the point A in the diagram ought evidently, on the general principle already repeatedly mentioned, to be laid off upon the circumference in A. the opposite direction from the positive arcs, i. e. downwards. Let us for simplicity suppose the arc in question to be less than a quadrant; being laid off downwards, such an arc will terminate in the fourth quadrant. Hence we see that the trigonometrical lines of a negative are must be affected with the same signs as those of an are in the fourth quadrant. Thus the sine of a negative arc will be -, the cosine +, the tangent -, the secant +. Secondly, suppose the given negative arc to be greater than a quadrant; were it positive, some of its trigonometrical lines would be negative. The rule given above, which determines the signs of its trigonometrical lines, by those of an arc in the 4th quadrant, will apply with this modification, that when the trigonometrical line is + in the fourth quadrant, the corresponding trigonometrical line of the negative arc has the same sign as that of a positive arc of the same magnitude, and when the trigonometrical line is - in the fourth quadrant, a contrary sign. The truth of this assertion may be seen, by trying negative arcs of ~2 ~:s PLANE TR:GONOMETRY. various magnitudes upon the diagram, laying them off downwards from the right point of the circle, and observing in which quadrant their extremities fall. They will be found in every case to give results agreeable to the rule just stated. THE COTANGENT AND COSECANT. 28. The cotangent of 00 is equal to the tangent of 900 (Art. 23) and is therefore oo. The cotangent of 900 is equal to the tangent of 00 and i 0o The cotangent of 1800 is equal to the tangent of 90~ - 1800 = the tangent of-90~= oo, since-90~ is a negative arc, and terminates at the bottom of the circle, or the 270~ point. The cotangent of 2'70= the tangent of 90~ - 2'70~the tangent of- 1800 = 0. When the tangent has its least value, which is 0, the cotangent has its greatest which is o, and vice versa. 29. The cosecant of 00~-the secant of 900 = o. The cosecant of 90~= the secant of 0~ = R. The cosecant of 1800 = the secant of 90~ 1S80 = oo. The cosecant of 2700 = the secant of- 1800 - - R. Then the secant has its least value, which is R, the cosecant has its greatest, which is o, and vice versa. The cotangent and cosecant have their greatest values together and their least values together, viz. that of the one 0, of the other R, at the top and bottom of the circle, and both o at the right and left points. 30. With regard to the signs of the cotangent and cosecant in the different quadrants, they will be most conveniently discovered from the analytical expressions for these lines which we shall presently have. We add here, however, which so far as the cotangent and cosecant are concerned must be for a moment taken for granted, that the six trigonometrical lines may be arranged in three pairs, each pair having always the sa,mse algebraic sign. We have seen that the secant and cosine go together in this way; so do also the cosecant and sine; and so do the tangent and cotangent. The positive sines and cosecants are separated from the negative by the horizontal diameter; the positive cosines and secants from the negative, by the vertical diameter; and the tangent and cotangent are together -- and - alternately in the successive quadrants. 31. The following algebraic notation is employed for the six trigonometrical lines. Let a be the algebraic expression for the number of degrees in any arc, then the trigonometrical lines of the arc a will be expressed thus; sin a, tal a, sec a, cos a, cot a. cosec a. TRIGONOMETRICAL LINES. 23 Cot a tan a == R is read, the cotangent of the arc a multiplied by the tangent of the same arc is equal to the square of the radius of the circle in which these trigonometrical lines are supposed to be drawn. Cot a and tan a are expressions for straight lines, and the equation above expresses that the rectangle formed by the tangent and cotangent of an arc is equivalent to the square formed upon the radius. The two members of the above equation contain the same number of dimensions, and are therefore homogeneous. This ought to be the case in all trigonometrical equations; because a line cannot be equal to the rectangle of two lines or a surface, nor either of these to a solid. Sometimes in analytical investigations R is supposed to be equal to 1I Rn and X' would also be equal to 1. -Whether this 1 is a unit of length, of surface, or of solidity, must be determined by what is required to, preserve the homogeneity of the equation. 32. The tangent, secant, cotangent, and cosecant may be expressed in terms of the sine and cosine. The values of the four former in terms of the two latter are derived geometrically as follows: D E Call the arc AM in / the diagram, a; then DM = 900- a = com- plement of a, DE = cot a and CE =cosec a. In the similar triangles CPM and CAT, since homologous sides are proportional, we have P PM:: CA: AT or cos a: sin a:: R: tan a whence multiplying the means and dividing by the first term, we obtain the last R X sin a tan a - cos a that is, the tangent of any arc is equal to radius multiplied by the sine divided by the cosine of the same arc. If R be made equal to 1, then 24 PLANE TRIGONOMETRY. sin tan = - cos 33. In the same similar triangles we have CP: CM: CA: CT or, cos a: R: a: sec a hence, Ra sec a = - cos a when R 1 1 sec =cos 34. In the triangles cMP and CED, which have their sides respectively parallel, and are therefore similar, we have the proportion,* MP: CD: CP: DE or, in a: R:: cos a cot a whence, R COS a cot a = sin a when R = 1 Cos cot = sin 35. The same triangles give also the proportion MP: D:: CM CE or, sin a: R:: R: cosec a whence, cosec a =. sin a i being I 1 cosec = sin * The homologous sides are those which are parallel. TRIGONOMETRICAL LINES. 25 36. In the expressions for the tangent and cotangent which we have here derived, it will be observed that we have the quotient of the sine and cosine, and that therefore when the sine and cosine have contrary signs, the tangent and cotangent will be negative. This occurs in the second and fourth quadrants. It appears hence, that the cotangent changes its sign always with the tangent. Also that both the tangent and cotangent of an arc are equal to those of its supplement with contrary signs. From the expressions for the secant and cosecant, it appears that the former must always have the same sign as the cosine, and the latter the same as the sine. The formulas derived in the last four articles should be committed to memory. Quantities in changing their signs pass through zero or infinity. (See Alg. Note 3d, p. 176.) Thus the sine changes from + to - or vice versa, twice in going round the circle; viz. in passing through 0 at 00 and 180; the cosine twice in passing through 0 at 90~ and 270~; the tangent four times, in passing through 0 at 00 and 1800, and oo at 900 and 2700; the cotangent four times, in passing through 0 at 900 and 2700, and through infinity at 00 and 1800; the secant twice, in passing through co at 900 and 2700; the cosecant twice, in passing through oo at 00 and 1800. 37. Multiplying the expression for the tangent given in Art. 32 by that of the cotangent in Art. 34, we have tan a cot a = R whence, Pa2 tan = — cot and, cot -- tan when R = 1 the above expressions become tan= - cot cot = tan i. e. the tangent and cotangent are reciprocals of each other. 26 PLANE TRIGONOMETRY. EXERCISES. 1. Express each of the six trigonometrical lines in terms of each one of the other five by itself. 2. The sine of an angle being 0*856, find the other trigonometrical functions. 3. The tangent being 2-34, find the others. 4. The cotangent being 1'203. 5. State the equivalents of the following functions of angles greater than 900, or obtuse angles in equivalent functions of angles less than 900 or acute angles, sin 1700, cos 1470, tan 980 31', cot 1710 14t, sec 1710, cosec 1550, see 215g, cos 318s 10', tan 271g 81', cot 204g 18' 94\. 6. Find the versed sine of the angle, the cosine of which is'9o358. 7. Prove the vers. = 1 - cos to be always positive. 8. Prove the greatsst value of the versed sine to be 2R. 38. We are now prepared to find formulas for the solution of right angled plane triangles, in all cases, and plane triangles in general in a few particular ones. The remaining cases of triangles in general will require further preliminary matter. DERIVATION OF FORMULAS FOR THE SOLUTION OF RIGHT ANGLED PLANE TRIANGLES. M.....'i, Let ABC be any right angled triangle. With c / [ I as a centre describe, with?y/'\ \ any radius at pleasure, the I \ arc MN terminating at the /?^~ \ \] \ [ sides of the angle. This' I \ arc will be the measure C.................... J P N P N of the angle c. Draw MP perpendicular to cN. MP will be the sine of the arc MN because it is drawn from one extremity M of the arc perpendicular to the diameter which passes through the other extremity N. MP is also the sine of the angle c. CM is the radius of the circle to which the are MN belongs. The two triangles CMP and CBA are equiangular and similar, and give the proportion CM: MP:: CB: BA or R: sin:: Cn: BA. Had an arc been described with n as a centre in a similar manner we RIGHT ANGLED TRIANGLES. 27 should have had R: sin B':: BC: CA, from which it appeals that the radius of any circle whatever* bears the same proportion to the sine in that circle of the arc which measures one of the acute angles of a right angled triangle, that the hypothenuse of the triangle does to the side opposite the acute angle. It is customary, for conciseness, to represent the sides opposite the angles of a triangle by small letters of the same name with the large letters which are placed at the angles; which large letters are also employed as the algebraic representatives of the angles. Thus in the triangle above, A being the right angle, the hypothenuse opposite is expressed by a; the side AC opposite B is represented by b, and so the other. The above proportions would, according to this method, be written thus n: sinB:: a: b(1) R: sin c:a: c Both these proportions are expressed in the single rule printed in italics above. When R = 1, multiplying the second and third terms, and dividing by the first, in the preceding proportions we have b - a sin B and (2) c a sin c That is either perp. side =the hypoth. X the sine of the angle opposite. The two acute angles of a right angled triangle are together equal to a right angle or 900 (Plane Geom. Theorem 15, Cor. 5), therefore they are complements of each other; hence sin c = cos B; and the second of the above proportions (1) may be changed into R: cos:: a: a (3) which may be translated into ordinary language thus; radius: the cosine of one of the acute angles of a right angled triangle:: the hypothenuse: the side adjacent the acute angle. When any three terms of a proportion are given, the remaining telm can be found. If the unknown term be one of the extremes, multiply * It is important to observe that the same trigonometrical lines of angles or arcs containing the same number of degrees in two different circles bear the seme relation to each other. Thus in the diagram above, CM: MP: CM: Mr, or, (R: sin) of the smaller cire:: (R: sin) of the larger; also, CMt: Cr:: ca: CP or (R: cos) of the one:: (R: cos) of the other. 28 PLANE TRIGONOMETRY. the two means and divide by the other extreme; if the required term be a mean, multiply the two extremes, and divide by the other mean. When R = 1 we have from proportion (3) above c = a cos B (4) i. e. either perpendicular side of a right angled triangle, equal to the hypoth. X cos of the adjacent angle. The above formulas contain each of them two of the sides of a triangle, the sine or cosine of an angle, and radius. If the lengths of the sides be given in numbers, these numbers may be put in place of the small letters which represent the sides in the proportion, and the general form becomes so far adapted to a particular case in the solution of rightangled triangles; but if the angle be given in degrees, how are we to know its sine or cosine, for that is the quantity which enters into the formula; and how are we to know the numerical value of R? For the present the student must be satisfied with the reply, that he can find the numerical value of any trigonometrical line corresponding to an angle of any given number of degrees, in-a table at the end of the work. This is TABLE XXIV.* of Natural Sines. The degrees for angles or arcs of every magnitude within the quadrant will be found at the top -of the columns of the table, and the minutes in the column marked M on the left, if the given angle or arc be less than 45~; but if it be greater than 450, the degrees will be found at the bottom of the page, and the minutes on the right; the length of the sine or cosine will be found in the column under or over the degrees, as the case may be, and on the same horizontal line with the minutes. The title of the column must be looked for at top if the arc be less than 450, and at bottom if the arc be greater.f The other trigonometrical lines may be easily calculated from the sines and cosines, as will be seen in the examples. The trigonometrical lines of this table are computed, by a rule which will be hereafter demonstrated, for a circle whose radius is 1. So far as the principles for the solution of triangles are concerned, the length of the radius is entirely immaterial, as it will be recollected that the arc in the last diagram was described with any radius at pleasure. When, in cases of the solution of right angled triangles, the hypothenuse and one of the acute angles are either given or required by the problem, one of the above formulas is always employed. * The tables are selected and printed from the stereotype plates of a very large collection. t A decimal point must be understood at the left of all the numbers in the columns of the table entitled N. sine and N. cos. RIGHIT ANGLED TRIANGLES. 29 39. Let us take an example by which to illustrate their application, and as upon a former occasion, one which shall at the same time exhibit the practical utility of Trigonometry. A roof is to have a height of 15 feet in the interior at the centre, and an inclination of 350. Required the length of the inner line of the rafters. A right angled triangle will be formed in which the angle at the ~ base will be 350, and the side opposite 15 feet, and of which the hypothenuse is required. Formula (1) of the last article applied to this case givesn I: sin 350: a 15 Multiplying the extremes and dividing by the first mean the value of the other mean which is a, the hypothenuse required, will be obtained I X 15 a = sin 350 Or formula (2) by a simple transformation gives the same thing. 15 a = sin 35~ Looking out the sine of 350 in the tables and performing the operations indicated in the last equation, the value of a will be known, which will be the length of the rafters required. The answer will be in feet. Sin 350 is found from the tables to be *57358. thus 15 a = 78 = 26.1 feet'57358 If (to vary the problem) half the interior breadth of the roof had been given, say 20 feet, and the angle of inclination, instead of 350 as in the last example, had been 15~, then to find the length of the rafters, it would * Since in the demonstration of the formulas, the sides and angles of the triangle were supposed to have no particular values, it follows that any numbers, or any other letters compatible with the properties of a triangle, may be put in the place of those employed, and the formulas will still be true. This must be borne in mind throughout the work. 30 PLANE TRlIGONOMI ETiRY, be necessary to find first the angle opposite the given side 20 feet; which is done by subtracting the given angle 15~ from 900, since the two acute angles of a right angled triangle are complements of each other. The remainder is 753. Applying the same formula as before, there results the proportion 1:sin 75"::a:20 or the equation 20-==a sin 750 whence, 20 20 a - - 2007 sin 75"~ 96600 The same result might be obtained by using the given angle 15, and employing formula (3) above, which contains the cosine of one of the acute angles. The proportion would stand thus I: cos 15: a: 20 or its equivalent (4) 20-a cos 15 whence, 20 a _~ ~ —207 cos 150 the same as before. In fact cos 15===sin 75~. (See Art. 23.) 40. Had the height and half the breadth of the interior of the roof been given, the length of the rafters might have been obtained, by employing the property of the right angled triangle demonstrated at Theorem 26, of Plane Geometry, that the square on the hypothenuse is equivalent to the sum of the squares upon the other two sides. Let the height of the roof be 12 feet, and the semi-breadth 16 feet, then a =- 122 + 162= 400 whence, a =20 If the length of the rafters had been given equal to 20 feet, and the height of the roof equal to 12 feet, then the semi-breadth would have been expressed thus b = 2o - 12 = (20+12) (20-12)* = 32X8 = 206 whence, b = 16 * See Algebra, p. 16, ex. 2, and note. RIGHT ANGLED TRIANGLES. 31 41. Had the semi-breadth or base of the triangle and the inclination of the roof been given, and the height of the roof or perpendicular of the triangle been required, the hypothenuse not entering into the problem, neither of the above formulas, all of which contain the hypothenuse, would serve to find the side required in a direct manner. It might, however, be found indirectly by first finding the hypothenuse, using one of the above proportions, and then by means of the hypothenuse, using the same proportion, the required side might be obtained. It is, however, objectionable to find one of the required parts in terms of the part which has itself been calculated from the given parts; because in the use of the tables which give the trigonometrical lines of the different angles not with perfect accuracy, but truly for as many decimal places as the table employs, a small error arises from the decimals neglected beyond the last place, and this, though so small as to be unimportant, becomes magnified by repetition, as in the case where one part of a triangle itself not perfectly accurate, is employed to calculate another. It is therefore desirable to find each of the required parts, in terms of the given parts; and this may always be done in right angled triangles. We proceed, therefore, to demonstrate a formula for the direct solution of the last case supposed above. Let ABC be a right angled triangle. With any radius at pleasure describe an arc ML!. which shall be the measure of M the angle c. At the point L draw a perpendicular LT to the line CL, terminating at the line CT. LT is evidently the tan- gent of the arc ML, since it is a perpendicular to the radius at one extremity of the arc, and is terminated by the radius which passes through the other extremity, according to definition of Art. 19. It is also the tangent of the angle c. The equiangular and similar triangles CLT and CAB give the proportion CL: LT: CA: AB or, n: tan c: b c (1) If - 1 this proportion gives c =b tan a (2) 32 PLANE TRIGONOMETRY. or, C tan c -= b (8) Let the angle of the roof in the above problem be 20~ and the semf breadth 25 feet, then: tan 20: 25:: c whence, tan 200 X25 1 Had the angle B been used instead of c, the resulting proportion would have been R: tan B::c: b (4) Both proportions may be expressed together in common language thus: Radius: the tangent of one of the acute angles of a right angled triangle:: the side adjacent that angle: the side opposite. This last rule applied to the problem at Art. 11, gives I: tan 300:: 200: whence,* c=tan 30 X200 —'5735 X200 = 115*'4000 c is the height of the tower. The same rule will evidently serve to determine either of the acute angles of a triangle when the two perpendicular sides are given. If the side c were given and the angle B, the side b might be found in the same manner, using the proportion (4) which contains the angle B. 42. We have now exhibited all the cases which can possibly occur in the solution of right angled triangles, with some specimens of their application. The right angle of the triangle is fixed; and any two of the five remaining parts being given, the other three may be found. Let the student select at pleasure any two of the five parts, the two selected to be considered as given, and he will find the case for solution with which he will then be presented, solvable by some one of the formulas above. The operations in the cases already exhibited, though of the most simple kind, nevertheless involve multiplications and divisions, which, from the number of places of figures, are somewhat tedious. Tn more complicated cases this evil would be much increased. * The tangent is found from Table XXIV. by dividing the sine by the cosine. (Art. 32.) Should the cotangent be required, divide the cosine by the sine. (Art. 34.) To find the secant divide I by the cosine. (Art. 33.) For the cosecant divide one by the sine. (Art. 35.) THEORY OF LOGARITHMS, 33 On tons account it is customary to employ in trigonometrical calculations, that ingenious invention of Lord Napier's for facilitating numerical calculations, the table of logarithms;' before explaining the use of which we shall give some exposition of the THEORY OF LOGARITHMS. 43. The logarithm of any given number is the exponent of the power to which it is necessary to raise some particular number in order to produce the given number. Thus, let 10 be the number raised to the power; then 2 is the logarithm of 100, because 102 = 100 and 3 is the logarithm of 1000, because 10' = 1000. Every given number will have a corresponding logarithm or exponent of the power to which it is necessary to raise 10 in order to produce the given number. The number 10, which is the only number that does not change in the above equalities, is called a constant. Should the constant number which has been employed be changed for another, the logarithms of numbers would be different from those derived by the use of the first constant. Logarithms derived from different constants are said to belong to different systems of logarithms, and the constant number belonging to each system is called the base of that system. The system most in use has the number 10 for a base, and is called the common system. The relation which this number sustains to the decimal system of notation will readily shggest some reasons for its selection; it will be found, as we proceed, to have many advantages. 44. If 6 be the base of a system, n a number, and I its logarithm, then by the definition If we put b in the place of n, this equation becomes I= b Here I is evidently equal to 1. Hence the logarithm of the base of every system is 1. * Table XXVI. at the end. The tables most in repute are the French tables of Callet. N. B. The tables in this volume having been printed from the accurate stereotype plates of Bowditch's tables, by permission of the proprietor, are numbered as in Bowditch's edition, and as but a part of his tables are necessary to the present work, the Nos. of the table must not be expected to occur in regular order. 3 34 PLANE TRIGONOMETRY. If in the equation b= n be made equal to 1, we have bl- 1 Here I is evidently equal to zero. (Algebra, Art. 17.) Hence in every system, log. of 1 = 0. 45. Suppose now the system be the common system; b will be equal to 10. If we substitute for n all possible numbers successively, we shall have a series of equations like the following, 10t= 2 101 = 3 &c. In the first I is the common logarithm of 1, in the second of 2, in the third of 3, &c. If I be made the unknown quantity, and these equations be successively resolved, we shall have the common logarithms of all numbers.* If now a table be formed having the series of natural numbers, 1, 2, 3, 4, &c., in one column, and their logarithms calculated as above placed in a second column against them, this would be a table of logarithms. The tables in actual use do not differ from such an one in principle, though some arrangements are adopted in them to avoid unnecessary repetitions.f In the common system the logarithm of 10 is 1, the logarithm of 100 is 2; and the logarithms of all numbers between 10 and 100 are between I and 2, that is, they are 1 and a fraction. The logarithm of 1000 is 3, and the logarithms of all numbers between 100 and 1000 are between 2 and 3, that is, they are 2 and a fraction. In the same manner it may be shown that the logarithms of all numbers between 1000 and 10,000 are 3 and a fiaction; of all numbers between 10,000 and 100,000, 4 and a fraction, and so on. The logarithms of most numbers, therefore, are mixed numbers. The fractional part is written in the tables; the whole number part, which is called the characteristic, is not written, nor is it necessary that it should be; for numbers between 10 and 100, or those composed of two figures, have I for a characteristic, as has just been X The method of resolving them is given in Art. 325, Alg. t For a more full exposition of the theory of logarithms, see Algebra, Art. 210, page 258, et seq. THEORY OF LOGARITI'HMS. seen,; numbers between 100 and 1000, or those containing three fgures, have 2 for a characteristic; numbers containing four figures have 3 for a characteristic, and so on. Whence it appears that the characteri tic is always I less than the number of digits in the number to which the logarithm belongs. So that if against any given number, the decimal part of its logarithm be found in the tables, the entire part or characteristic may be supplied by counting the figures in the given' number, ano making the characteristic, one less. In proceeding to explain the tables, we will premise that the logarithns of several consecutive numbers, if the numbers be somewhat large, will differ so little as to have several of their first figures the same. Hence, by a proper arrangement of the tables, the first figures of the logarithm may be written but once for several numbers, provided all be designated to which they refer, and thus much repetition be avoided. The manner in which this is accomplished will be shown S the EXPLANATION OF THE TABLES, PROBLEM I. 46. To find jfrom the tables the logaarithm'of any given number. CASE I.-When the nunmber is between 100 and 10,000, if it be composed of three figures, find it in Table XXVI. at the end of the volume, and in the column at the left entitled No.; in the next column marked 0 at top and pn the same horizontal line you will find the decimal part of the logarithm required. This contains five places.* If the given number contain fdiour figures, find the first three of it in the column No. as before, and the fourth in one of the columns marked 0, 1, 2, 3, &c., at top; under the latter, and on the same horizontal line with the first three, you will find the decimal partj of the logarithm sought. N. B. The characteristic is always one less than the number of figures in the given number. * In the tables of Callet seven places, the first three of which being the same fol several numbers are not repeated, but must be understood before those which follow the number that has them expressed until you come again to seven places. t In the tables of Callet substitute the word five for four in the above rule, and the word four for three. t In the tables of Callet you find only the last four places, the first three to be prefixed to them must be taken from the numbers projecting to the left in the' column marked 0 at top. $6 PLANE 1RIGONOMETRY. EXAMPLES. 1. Required the logarithm of 217o In the column entitled No. on page 171 of Table XXVI. I find 217; in the next column marked 0 at top, and on the same horizontal line, I find 33646 for the decimal part of the logarithm required. The characteristic is 2, since 217 contains three figures, and the whole logarithm 2*33646. 2. Required the logarithm of 1122 On page 170 and in the column No. I find 112; in the column having the last figure 2 of the given number at top, and on the same horizontal line with the 112 before found, I find 04999 or with the proper characteistic 3*04999.* 3. Required the logarithm of 2188. Aim 3*34005. 47. We proceed now to show the use of logarithms in numerical calculations. MULTIPLICATION. Let b be the base of the system of logarithms, n any number, and I its logarithm. Then by the definition Let n' be another number, and 1' its logarithm, we have also b' =n' Multiplying these two equations, member by member, and observing the rule for exponents in multiplication, which is to add them together, we have From this last expression, it appears that 1+1' is the exponent of the power to which it is necessary to raise the base a, in order to produce the number nn'. - But nn' is the product of n and n'. Hence the logarithm of the product is equal to the sum of the logarithms of the multiplier and multiplicand. * At the tops of the pages in the table will be found catch numbers for the eye, in taming over the leaves, to show what numbers and logarithms are contained on the page. TABLES OF LOGARXIT'HMS 37 EXAMPLE. Multiply 2421 by 1613. The logarithm of 2421 is 3383909 The logarithm of 1613 is 3S20763 The logarithm of 2421x 1613 or 3905073 is* 6'59162, or the sum of the logarithms. 48. If in addition to the numbers n and nzabove, wesuppose a third number n" of which the logarithm is I" we shall have in a similar manner -` " = - frn,"l and so on. Or, in general, the logarithm of a product of several factors is equal to the sum of the logarithms of those factors separately. DIVISION, 49. Dividing the equation. b = n by the equation.b' __ m' we have, observing the rule of division, to subtract the exponent of the divisor from that of the dividend in order to obtain that of the quotient. Since — I' is the exponent of the power to which it is necessary to raise b the base, in order to produce n it follows that I-' is the logarithm of n i. e. the logarithm of the quotient is equal to the difference between the logarithms of the divisor and dividend. EXAMPLE. Divide 3905073 by 2421 The logarithm of 3905073 is 6'59162 r " " 2421 is 3'38399 3905073 - The logarithm of 242 1 1613 is 3'20763 diff. of logs. 2421 * As the number 3905073 is too large to be found in the tables, the method of finding its logarithm from the tables must be postponed till the explanation of such cases, further in advance. 38 PLANE TRIGONOIMETRYo Before explaining other operatioes by means of logarithms, we shall exhibit some principles derived from those just demonstrated. 50. The base of the common system being 10, the common logarithm of 10 is 1. (Art. 44.) Hence if any number be multiplied or divided by any number of 4imes 10, the logarithm of the result will be equal to the logarithm of the given number increased or diminished by the same number of times 1. This 1 being an entire number, the decimal part of the logarithm of the given number will not be altered by this addition or diminution, but only the characteristic. Thus 39794, which is the decimal part of the logarithm of 2500, is also of 25000, and of 250000, or of 250, or of 25. The claracteristics belonging to these different numbers are different. That of the log. of 2500 is 3; that of the log. of 25000 is 4; that of the log of 25 is 1. (See-Art. 47.) Any number is divided by a multiple of 10, by pointing off from the right as many places for decimals, as the divisor is times 10. Thus 2348 divided by 10, by 10 twice, by 10 three times, becomes successively 234*8, 23*48, 2*348. The decimal part of the logarithms of these last three numbers, will be the same, viz. 37070, the characteristic being one less each time that we divide by 10 or remove the decimal point one place to the left. Because to divide by 10 it is necessary (see the last Art.) to subtract 1, which is the log. of 10, from the log. of the dividend. The characteristic of the first, 234'8, which is between 100 = 102 and 1000 = 103, is 2. The characteristic of the second is 1; and the characteristic of the last is 0, since 2'348 is less than 10, or 101. The decimal part of the logarithm of a number consisting of significant figures, either followed or preceded by ciphers, will be the same as if the ciphers were absent. Thus the decimal part of the logarithm of 482000 or of *00482 is the same as the decimal part of the logarithm of 482. The following table illustrates the theory of the characteristic. The characteristic of the log. of 482000 is 5 of 482is2 of 4*82 is 0 of *482 is- 1 of *0482 is- 2 of' 00482 is- 3 From the above, it appears that the characteristic of the logarithm of a decimal fraction is negative; the decimal part of the same logarithm is, however, positive. The actual value of the whole logarithm will be therefare a negative quantity somewhat less than the characteristic. That the THEORY OF LOGARITHMS. 39 logarithms of proper fractions ought to be negative, appears from the fact, that since a fraction expresses the quotient of the numerator divided by the denominator, applying the rule for division bv logarithms, the greater logarithm would have to be subtracted from the lesser, and the remainder would of course be negative. From the above principles are derived the following rules: 1. To find the logarithm of a number consisting of significant figures with any number of ciphers annexed, find the logarithm of the significant figures, and make the characteristic one less than the number of figures in the given number including the ciphers. 2. To find the logarithm of a decimal or mixed number, consider the number as entire; find the decimal part of its logarithm, and make the characteristic one less than the number of figures in the entire part of the given number. 3. To find the logarithm of a decimal number having ciphers at the left; look for the logarithm of the significant figures, and make the characteristic negative* and one more than the number of ciphers at the Ift of the given decimal. EXAMPLES. The logarithm of 3266000 is 6*51402 of 114*1 is 2-05729 of'001684 is 3'22634 51. CASE II.-We proceed now to the method of determining the logarithm of a number beyond the limits of the table. This method is by a simple calculation fiom the logarithms of numbers which the table contains, and depends upon the fact that the difference of any two numbers bears the same proportion to the difference of their logarithms, that the difference of two other numbers does to the difference of their logarithms, which is nearly true. (See Algebra, p. 284, note.) Take two numbers in the table differing fiom each other by 100, as the numbers 843700 and 843800, and a third number 843742 differing from the first of these by 42. The logarithm of the first number 843700, being the same as that of 8437, is given by the tables, and is 5.92619 The logarithm of the second number 843800 is 5'92624 Their difference is 5 * It is customary to write the negative sign over the characteristic, thus, 2.1756348. It affects the characteristic alone and not the decimal part of the logarithm, which must be considered as -. 40 PLANE TRIGONOMETRY. which may be found by subtraction, but to save this trouble the subtraction is performed, and the difference is written in the margin (in the tables of Callet in the right hand column marked dif.) Then on the principle that the difference of numbers is proportional to the diffe:ence of their logarithms, we have diff. of numbers. diff. of logs. diff. of num. diff. of logs 100: *0000:: 42 x ]lence, -*00005 X 42 X- - = 0000210 100 adding this to the logarithm of 843700 which is 5*92619 ~0000210 the sum, rejecting the two last places 10 which go beyond the usual number is 5'92621 which is the logarithm of 843742. HEad the first two numbers differed by 1000 instead of 100 the divisor in the value of would have been 1000, and the quotient would have extended three places beyond the usual. Had they differed by 10, the quotient would have extended one place beyond. The inaccuracy of this method increases with the number of additional ligures beyond four, in the number the logarithm of which is to be found.* From the above process may be observed the following rule: To find the logarithm of a number beyond the limits of the table. Enter the table with the first four figures of the given number, and find the corresponding logarithm. From the right hand margin take out the difference between this logarithm and the next in order in the table, and multiply it by the remaining figures of the proposed number, reject from the product as many figures to the right as there are in the multiplier, and add the rest of the product to the logarithm already found. EXAMPLE. 1. Required the logarithm of 739245. The decimal part of the log. of 7392 is 86876. * The same process applies to most other tables as well as tables of logarithms. In all tables the column corresponding to the column of numbers here is called the column of arguments. The other columns contain functions depending on these arguments. TABLES OF LOGARITHMS. 41 The number in the margin is 6 Multiplying this by the remaining figures of the given number 45 Product, 270 From this product reject as many figures to the right as are contained in the multiplier, that is two in this case, and add the rest to the logarithm before found, namely, 86876 The sum is 86879* which is the decimal part of the log of 739245 required. Prefixing the proper characteristic, we have 5*86879. EXAMPLE II. Required the log. of 8193217 log. of 8193 = 91344 217 1 dif. 5 log. of 8193217 == 6.91345 1'085 To facilitate the above calculations several smaller tables will be found in the right hand margin of the page, the use of which may be explained from an example. Suppose it be required to find the logarithm of 276738. After cutting off 38 on the right, I find the logarithm of the first four figures, 2767, to be 44201. The difference between this logarithm and the next is 16. In the left hand column of the little table in the margin headed 16, I find the first of the two figures cut off, 3, and against it the number 5, which I place under the logarithm already found, as seen in the scheme below. Again I find in the same way the second of the two figures cut off, 8, and against it 13, which I place as in the scheme below one place to the right. Adding these numbers taken from the little table to the first logarithm found, the sum 44207 is the logarithm sought. 44201 5 13 44207 or with the characteristic. 544207 The right hand figure 3 of the 13, which would carry the decimal places beyond five, is rejected. If it were any figure greater than 5, we should add 1 to the last figure of the result, 7. The following example is worked with the tables of Callet in the same manner. * We add 1 for the I rejected which is more than a. 4t2 PLANE TRIGONOMETRY. Find the logarithm of 8193217. log. of 81932 = 9134536 5 37 6'91.34545 The 7 neglected in adding up the above numbers being more than 5, or - of the preceding place, 1 is added to that place. For the theory of the above see Algebra, at the end of Art. 214. PROBLEM II. To find the number corresponding to any given logarithm. 52. By referring to the proportion of Art. 51, and putting the value of x for the fourth term, we have diff. of num. diff. of logs. diff. of num. diff. of logs. 100: 00005:: 42: 000021 Instead of the 42 being given and the 000021 required as before, the 000021 is now given and the 42 required. The first term of the proportion is 100 or 1000, &c., and the second term is diff. in the margin, to find the third term multiply the extremes and divide by the second term 000021 X 100 42 -= 00005 Hence the following IULE.-To find the number corresponding to any given logarithm. Seek for the decimal part of the given logarithm. If we find a logarithm exactly agreeing with that given, then the number, which the table shows us to belong to the logarithm found, will be the required number. If, however, as is most likely, we do not find the proposed logarithm exactly, then we are to take out the number corresponding to the next less logarithm; this number will of course fall short of that required, but the deficiency may be supplied as follows. Subtract the tabular logarithm from the given one, annex ciphers to the remainder at pleasure, and divide it by the diff. number in the margin, and annex the quotient to the number already taken from the table. N. B. Should there be a quotient figure without annexing a cipher to the dividend, this quotient figure must be added to the last figure of the lumber taken from the table. Should it be necessary to annex two ciphers before obtaining a quotient figure, a cipher must be placed in the OPERATIONS OF LOGARITHMS. 43 quotient, and annexed with the figures that come after, to the number taken from the table. The logarithm next greater than that given may be taken from the tables, and the latter subtracted from the former, in which case you would subtract the quotient obtained by dividing the difference as above, instead of adding it. EXAMPLES. 1. Find the number the log. of which is 5s86879 The decimal part of the next less log. is that of 7392 = *86876 Their difference is 3 Annex ciphers to this diff. and divide by the diff. number in the margin, which is 6. 61300 6) 6 Annex the quotient 50 to the number 7392 before found, and you have the number required corresponding to the given logarithm, namely, 739250. This number contains six figures, one more than the characteristic of the given logarithm. In every case a sufficient number of ciphers must be annexed to obtain quotient figures enough, when appended, to make the whole of the number which thus results contain one more figure at least than is expressed by the characteristic of the given logarithm. If more quotient figures still be obtained, they will occupy the place of decimals. 2. Find the number of which the log. is 2'91345 Next less log. that of 8193 91344 Number required is 819*32 5)10 2 53. To find the number corresponding to any given logarithm by the tables of Callet, seek the nearest logarithm in the tables, and subtract it from the given as directed above, then seek the remainder in the right hand column of the little table nearest, and if it be found, or a number not differing more than unity from it, the figure on the left of this number will express the sixth figure of the number required. EXAMPLE. 2-5386717 The nearest log. is *5386617 its number is 34567 Subtract and 100 is the remainder. The nearest number in the right hand column of the little table adjoining is 101, against which on the left is the figure 8, and the number sought is 345.678 44 PLANE TRIGONOMETRY. If the diff. (100 above) is not found exactly in the little table, take the nearest number to it, and take the difference between these again, and annex to it a cipher, seek this result again in the right hand column of the little table, the figures on the left of the two numbers taken out of the right hand column of the little table will be the sixth and seventh figures of the number required A third remainder might be found in the same manner, and an eighth figure of the required number be found. EXAMPLE. 0-4971499 The nearest log. is 4971371 Diff. 128 nearest to which in the little table is 124 number on its left is 9. diff. with cipher annexed 40 nearest to this in the little table is 41 number on its left is 3. The last two figures of the number sought are 93, and the number itself is 3.141593. The same method is applicable to our tables, though not with the same degree of accuracy. EXAMPLE. 4*90835 Nearest log. 90832 corresponding number 8097 Difference 3 on its left in the marginal table is 5. The number required is 80975. EXAMPLE. 0~06180 Nearest log. 06145 corresponding No. 1152. Diff. 35 The nearest No. to which in the little table is 34 which subtract; on its left is 9. The rem is. 10 on left of which in the marginal table is 3. The number required is therefore 1-15293. EXAMPLES IN MULTIPLICATION AND DIVISION BY LOGARITHMS. 54. 1. Required the product of 26784 and 7'865. log. of 26784* is 4-42787 log. of 7'865 is 0'89570 Their sum is 5S32357 5'323574 is the log. of 210656, which last number is, therefore, the product required. In looking for the log. of this number, look first for that of 2678, multiply the tab. diff. by 4, the last figure of the given number, and cut off one figure from the product. OPERATIONS OF LOGARiTIMS. 4 2. Required the product of 3'586, 2*1046,'8372, and'0294. log. of 3'586 is 0'55461 of 2*1046 is 0*32317 of *8372 is 1'92283 of *0294 is 2*46835 Product ~18576 1*26896 Instead of using negative characteristics, a method is sometiniea employed of taking the difference between the negative characteristic and 10, which is really adding 10 to the negative characteristic, and writing this difference as a positive characteristic; thus, in the above example, 3'586 logo 0*55461 2*1046 log. 0*32317 *8372 log. 9*92283 *0294 log. 8'46835'185764 log. 9'26896 twice 10 must be rejected from the sum. That is a 10 for each positive characteristic employed in the place of a negative. The result thus obtained 1*26896 is written 9*26896, 10 being added again to avoid the negative characteristic. 3. Divide 28*654 by 127'34. log. of 28*654 is 1*45718 of 127*34 is 2*10496 difference 1*35222 1*35222 is, therefore, the log. of the quotient which from the tables, observing the converse rule for pointing off decimals according to the characteristic (3 Art. 50), is'225020. Divide'06314 by'007241. log. of'06314 is 2*80030 of.007241 is 3*85980 Quotient 8'719'7 0'94050 55. We shall now demonstrate rules for raising numbers to powers, and for extracting the roots of numbers, by means of logarithms. Resume the equation, raising both members to the m power, we have, observing the rule of Algebra, which is to multiply the exponent by the degree of the power, bn -- =a ni From this last equation, it appears that 1m is the power to which it 46 PLANE TRIGONOMETRY. is necessary to raise the base b in order to produce n'; hence the following RULE.-To raise a number to any power, by means of logarithms, multiply the logarithm of the given number by the exponent, of the power, and the product will be the logarithm of the power. EXAMPLES; 1. Required the 4th power of *09163 log. of *09163 is 2'96204 Multiply by 4 Product 5'84816 5'84816 is the log of'000070494, which last number is the power required. 2. Required the tenth power of *64. log. of*64 1*80618 10 Power 0115293 2*06180 In multiplying the first decimal place by 10, the product is 80, then 10 times I is 10, and 8 to carry is 2. The same example, with positive characteristics, according to the method pointed out on p. 45 would stand thus 9*80618 10 8*06180 The 10 added to the 1 is repeated 10 times and therefore 100 must be rejected from the product, which leaves 2 for the characteristic to be written 8. The rule for placing the decimal point in the number corresponding to the logarithm will be to place one less cipher after the decimal point than is expressed by the difference between the characteristic and 10. 56. To find a rule for extracting the root of a number by means of logarithms, assume again the equation b= n Take the mlt root of both members, applying in the first member the rule to divide the exponent by the number expressing the degree of the root, and there results b = ^/n lainly the logarithm of no LOGARITHMIC SINES, &C. 4C RULE.-To extract the root of a number by means of logarithms, divide the logarithm of the given number, by the index of the root, and the quotient will be the logarithm of the root. EXAMPLES. 1. Required the 4th root of'434296. log. of *434296 1'63779 ~ of this logarithm is obtained by observing that the index, which alone is negative, must be divided separately, as we should divide a minus term, followed by a plus term in Algebra; the 1 can be rendered divisible by borrowing 3, and afterwards carrying + 3 before the 6, rendering it 36; that is, the proposed logarithm is viewed under the form 4 - 3'63779. The quotient is 1.90945* which is the logarithm of'8118, the fourth root required. 2. Required the 10th root of 2. log. of 2 0'30103 Divide this by 10 0'03010 quotient. which is the log. of 1'07177, the root required. 3. Required the cube root of'00048. log. of *00048 4*68124 A of it 2*89375 = log. of'078297, the root. TABLES OF LOGARITHMIC SINES, TANGENTS, A&C. 57. This is Table XXVII. It contains the logarithm of the sine, tangent, cosine and cotangent, secant and cosecant, corresponding to every degree and minute in the quadrant. These logarithms are those of the trigonometrical lines in a circle, the. radius of which is 10000000000, or the tenth power of 10, the common logarithm of which is 10. As the sine is never greater than radius, its logarithm will always be less than 10, except for the arc 90~, the logarithmic sine of which is equal to 10. * The last quotient figure is nearer 5 than 4. t Without this table we should have been obliged to employ the other two tables which have been already described, as follows. First we must have found the natural sine, tangent, &c., of the given arc or angle in Table XXIV. then with this have entered Table XXVI. and found its logarithm. The tables of Callet do not contain the columns of secants and cosecants. ~48 PLANE TRIGONOMETRY, PROBLEM. 58. To find from the table the logarithm of the sine, tangent, or cosine of the number expressing any arc. CASE I. If the given number be composed of degrees and minutes, seek first for the number of degrees among those which are written at the top or bottom of the pages; at the top and on the left of the page if it l)e less than 450; at the bottom and on the right of the page if it be greater. Run the eye down the first column which goes on increasing from top to bottom, if the number of degrees is found at the top of the page; or up the last column, which goes on increasing from the bottom upwards, if the number of degrees is found at the bottom; run the eye, I say, through one or the other of these columns in the direction in which it increases until you have found the number of minutes given; upon the same horizontal line with the minutes thus found you will find the logarithm of the sine, cosine, tangent, or cotangent which you seek. In order not to mistake the column, it is necessary to consult the title at the head of the column, if the number of degrees given is at the top of the page, but if it is at the bottom, the inferior title must be consulted.* EXAMPLES. 1. Required the logarithmic sine, tangent, secant, cosine, cotangent, and cosecant of 190 55'. I find 190 at the top of page 204, I descend the first column at the left marked M, which goes on increasing downwards till I find 55'; upon the same horizontal line, and in the column entitled sine at top, I find 9*53231, in the column entitled cosine 9'97322, in the column of tangents 9*55910, and in that of cotangents 10*44090, that of secants 10'02678, that of cosecants 10'46769; and these numbers are therefore the numbers required. 2. Required the logarithmic sine and tangent of 70~ 10'. I find 700 at the bottom of p. 204t; I ascend the last column marked M at bottom which goes on increasing upwards; I find 10' in that column; upon the same horizontal line I find in the column marked sine at bottom 9.97344, and in the column marked tangent at bottom 10944288, which are the logarithms sought. * The columns marked Hour A. M. and P. M. are connected with Nautical and Practical Astronomy, and will oe explained under the proper head. t For the reason mentioned in note p. 33, the paging of the tables will be found to exhibit gaps. TABLES OF SINsES, &C. 49 3. Required the logarithmic sine of 159~ 20'. This will be the same with that of its supplement 200 40'. For convenience the supplements of arcs in the 1st quadrant are placed on the right of the page in the tables at top, and on the left at bottom. By looking therefore for 159~ at top on the right, and immediately under for 20' in the column M, on the range with this in the column entitled sine at top I find 9'54769, the same that would have been found for 200 40'. 59. CASE II. If the given number be composed of degrees, minutes, and seconds, find the logarithm of the degrees and minutes as above, and then to know how much this should be increased for the given number of seconds, in case of the sine or tangent, or diminished in case of the cosine or cotangent, observe that the number in the column marked Diff. is the increase of the logarithm for the number of seconds in the column headed M, against which it stands, and will be the quantity to add to the logarithmic sine or tangent before found, or to subtract from the logarithmic cosine or cotangent. The number in the column Diff. is calculated by subtracting one of two consecutive logarithms in the table, which differ by 1', from the other,* and dividing the remainder by 60, the number of seconds in a minute; the quotient is the difference of logarithms corresponding to a difference of 1" in the numbers to which they belong, or is the increase of the logarithm for 1" increase of arc. This quotient, multiplied by any number of seconds, will give the increase of the logarithm for that number of seconds. This calculation depends upon the principle mentioned at Art. 51, that the differences of logarithms are proportional to the differences of their corresponding numbers. See also Art. 23 App. I. 60. The logarithmic sines and cosecants, cosines and secants, tangents and cotangents, have each pair but one column of differences between them, the reason of which will appear from the following demonstration. The secant and cosecant may be easily computed from the cosines and sines. Thus (Art. 33): sec. cos hence, log. sec - 20- log. cos br log. of the quotient - difference of logs. (Art. 49); and log. of the * This diff. is given in the first five pages of the table so that the seconds must be calculated as here described when the given are is 40 or less, and any number of minutes. 4 50 PLANE TRIGONOMETRY. square of a number equal to twice the log. of the number (Art. 56), and log of R = 10. To obtain the log. secant, therefore, we have this RULE. —Subtract log. cosine from 200 Also (Art. 35), R2 cose — = sin hence, log. cosec = 20 log. sine whence this RULE.-To obtain log. cosec, subtract log. sine from 20. From the above it appears that as the log. cos decreases with the increase of arc the log. sec increases by the same amount, and as the log. sin increases the log. cosec decreases by the same amount. Again, by Art. 37, we have tan X cot = R2 applying logarithms to this equation, since the log. of a product = the sum of the logs. of the factors, and the log. of a power = log. of the number raised to the power multiplied by the index of the power, we have log. tan + log. cot = 2 log R = 20'log. it being 10. Therefore having two arcs a and b, since log. tan + log. cot in both is 20 we have log. tan a + log. cot a = log. tan b + log. cot b, or transposing, log. tan a log. tan b = log. cot a log. cot b, that is, the difference of the logarithmic tangents of two arcs is equal to the diference of their logarithmic cotangents. EXAMPLES. 1. Required the logarithmic sine of 40~ 26' 28". I find the log. sine of 40~ 26' to be 9'81195; in the column M, at the left, I find the given number of seconds 28, and on the same horizontal line in the column of diff. I find 7, which added to the log. before found 9'81195 7 gives 9.81202* The logarithmic tangent of any given number of degrees, minutes, and seconds, is found in a similar manner from the column entitled tangent. * A similar arrangement for finding the difference corresponding to any number of seconds will be found in the table of natural sines and cosines, for the former on he left and for the latter on the right of the page. TABLES OF SBNES, &C. 5 2. Required the logarithmic cosine of 8~ 40' 40". I find the cosine of 8~ 40' to be 9*99501; the tabular difference in the adjoining column against the seconds 40" is 1; subtracting* ttis result from 9*99501, the remainder is 9'99500, the logarithmic cosine sought. The difference for the seconds may be calculated as follows. 3. Required the log. sine of 320 10' 23". 320 10' log. sin 9~72622 32~ 11' " " 9e72643 DiA 21 23 63 42 60)483 Diff, for 23"' 8 32~ 10' log. sin 9'72622 320 101 23" log. sin 9'72630 61. In the tables of Callet are found the logarithmic sines, tangente, cosines, and cotangents for every 10" in the quadrant; and the columns of differences contain the differences of the consecutive logarithms, or the:ncrease of the logarithm for 10" at that part of the quadrant. To take out therefore a logarithmic sine, &c., from these tables, take out for the degrees, minutes, and tens of seconds, and take out also the number from the column of diff.; cut off one figure on the right of the latter, which is equivalent to dividing by 10, and multiply it by the number of seconds by which the given arc differs from an exact number of tens of seconds. T.he product will be the number by which to increase or diminish the logarithm already taken out, according as the trigonometrical line to which it corresponds is an increasing or decreasing function of the arc. EXAMPLES OF TIE APPLICATION OF THE TABL:ES OF CALLET. 1. To find the log. tan of 49~ 12' 25"'8 Of 490 12' 20" log, tan is 1000639854f Tab. diff. 425 X 5'8 246*5 10*0640100 log. required. * It will be recollected that as the arc increases in the first quadrant the cosine diminishes. t The characteristic in the column of tangents and cotangents when 10, is printed 0. and when 11, is printed 1.,in the tables of Callet PLANE TRIGONOMETRY. 2. To find the log. cot of 101~ 25' 43" = log. cot of 780 34' 17"' Of 78~ 34' 10" log. cot is 9'3057605 Diff. for 10" or 1084 X7 758*8 9*3056846 log. required. In the tables of Callet are to be found the logarithms of trigonometrical lines of arcs given in grades, &c., of the centesimal division of the circle. The following is an example of the use of the table. To find the log. sine of 929 75' 84" of 92g 75' the log. sin is 9'9971776 tab. diff. 78 X 84 = 6552 log. required is 909971842'The first 4 figures of the log. 9'997 are taken from the top or bottom of the column, the former when the logarithm is above a black horizontal line drawn across the column, the latter when below. As the diff. 78 is the diff. of two consecutive logarithms corresponding to 100" two figures must be pointed off to the right after multiplying by the 84." PROBLEM. 62. To find the degrees, minutes, and seconds answering to any given logarithmic sine, cosine, tangent, or cotangent. The method is, of course, exactly the reverse of that just given. Look for the given logarithm in the proper column, which you will know from its title, either at the top or bottom, and if you find it exactly, the degrees will be found at the top of the page, and the minutes on the same horizontal line with your logarithm, in the first column at the left, if the title of the column be at top, but the degrees will be found at the bottom of the page, and the minutes in the column at the right, if the title of the column which contains your logarithm be at the bottom. If the given logarithm cannot be found, take the next less logarithm contained in the tables, subtract it from the given, and seek the remainder in the column marked diff.; the number on the same horizontal line in the column M is seconds, which add to the degrees and minutes belonging to the logarithm found in the tables, if your given logarithm be that of a sine or tangent, but which subtract from the degrees and minutes, if a ine or cotangent. Or more accurately, to find the seconds multiply the remainder above TABLE'OF SINES, &C. 5 mentioned, by 60, and divide the product by the difference between two consecutive logarithms in the table.* EXAMPLESo 1e Required the number of degrees, minutes, and seconds, of which the logarithmic sine is 9'88005. I find the next less logarithm in the column marked sine at bottom, to be 9'87996, which subtracted from the given logarithm, leaves 9; this found in the column diff. adjoining, against it in the column M is 50, which is seconds. Taking the degrees from the bottom of the page, and the minutes from the column at the right, and in the same horizontal line with the logarithm 98'7996, I have 490 20' 50" for the number required. Or more accurately, since the diff. 9 corresponds to any number of seconds from 48" to 52"' calculate as follows: Given log. 988005 98'7996 Nearest log. 98'7996 consec log. 9888007 Diff. 9 Diff. 11 60 11)540 49 2. Required the number of degrees, minutes, and seconds, of which the log. cotangent is 10*00869. I find the next less logarithm in the table to be 10*00859, that of 440 26', which subtracted from the given logarithm, leaves 10, corresponding to which in the column M is either 23" or 24"; more accurately 10 60 25)600(24" 50 100 100 amd the required number is 440 26'-24" or 44~ 25' 36'" This last rule is on the principle that the difference of the logarithmic functip.s is proportional to the difference of their arguments, the difference of the arguments in this case being 60". s^ P wPLANE TRIGONOMETRY 3, Required the log. sec of, 480 35' 27". BY TABLE XXVII. 480 35' log. sec 10'17945 Diffi for 27" 6 48~ 35' 27" log. sec 10'17951 The tables of Callet not containing the logarithmic secants and cosecants, the calculation of this example from his tables would be as follows, by- the rule at Art. 60. log. cos 48~ 35' 27" = 9082049 log. sec 48~ 35' 27" = (20 — 9*82049) = 10.17951 4. Required log. cosec 35~ 27' 24". Ans. by Tab. XXVII., 10'23651 or log. sin 350 27' 24" = 9'76349 log. cosec 350 27' 24" = 20-9'76349 = 10*23651 A method of finding with greater accuracy the sine and tangent of a very small arc, or the cosine and cotangent of one near 900, is pointed out at Art. 8 App. I. To find the trigonometrical lines of arcs greater than 900, observe the nire at Art. 17. SOLUTION OF RIGHT ANGLED TRIANGLES, WITH THE AID OF LOGARITHMS. EXAMPLE, 63. Referring to the example of Art. 39, where the hypothenuse 1010 X 15 sin 350 employing 100 as a, instead of 1, because the tables which we are about to use are constructed with that value of R, we have, by the rules for multiplication and division of logarithms log. of 10O = 10'00000 add log. 15 = 1'17609.11 17609 Subtract log. sin 350 = 975859 Remainder 1'41750 log. of 26*15 = the hyp. RIGHT ANGLED TRIANGLES. 55 EXAMPLE II. 5200 t1et 1referring to Art. 41 where the height of the tower c is c = tan 300 X 200* which becomes, when the radius 1 in the first term of the proportion i changed into 101 tan 300 X 200 1010 we have, by applying logarithms log. tan 30~ = 9*76144 add log. 200 = 2*30103 12206247 Subtract log. 10~ = 10 206247 = log. 11554 64. I. Another definition besides that given at Art. 14 for the sine of an angle, is the ratio of the hypothenuse to the side opposite the angle in any right angled triangle of which the angle forms one of the elements. For from formulas (1) Art. 38 may be obtained. b c sin B =- and sin c = (1) aC a II. A corresponding definition for the cosine of an angle is the ratio of the hypothenuse to the side adjacent the angle in any right angled triangle of which it forms an element. For,form. (3) Art. 38 cos B = and cos c - (2) a a * It is evident that radius must be understood in the second member of this expression, because a line c cannot be equal to the rectangle of two lines. (Art. 31.) t The hypothenuse, which is the greater side, is evidently the denominator of the ratio, as the sine and cosine to radius unity are always fractions. 5PLANE TRIGONOMETRY. III. In a similar manner the tangent of an angle is the ratio of the side adjacent to the side opposite the angle in any right angled triangle of which the angle forms an element. For Art. 41, (3) and (4) tan B =- and tan c - (3) c b From these definitions the following consequences flow. IV. The hypothenuse multiplied by the sine of one of the acute angles of a right angled triangle will give the side opposite to the angle. For from (1) above clearing of fractions we have a sin B = b and a sin c = c (4) V. The hypothenuse multiplied by the cosine of the angle will give the side adjacent. For from (2) above a cos B = Cn and a cos c =b (5) VI. The side adjacent multiplied by the tangent will give the side opposite. For from (3) above ctan B = b and b tan c = c (6) VII. To find the hypothenuse when a side and angle are given, we must still use the sine and cosine of the angle, but as a divisor; the sine when the side opposite is given and serves for a dividend, the cosine when the side adjacent. For from formulas (4) and (5) above may be obtained c c I b a,, = a=,and a= — (7) sin n sin c cos B cos c VIII. When one of the perpendicular sides is given with an angle to find the other, the tangent of the angle is the multiplier of the side adjacent the angle to find the side opposite, and is the divisor of the side opposite, to find the side adjacent. See (6) above from which also may be obtained b c c == - and b - (8) tan B tan c * When the hypothenuse is given with an acute angle it will be necessary always to multiply the hypothenuse by either the sine or cosine of the acute angle to obtain the other side; by the sine when it is the side opposite which is required, and by the cosine when it is the side adjacent. The memory will be aided by observing that for radius unity the sine and cosine are always fractions, and therefore the hypothenuse, which is the larger side, must evidently be multiplied by these in order to obtain the other sides. RIGHT ANGLED TRIANGLES. 57 IX. Radius unity has no effect either as a multiplier or divisor, nevertheless when using the tables in which radius is 10"~ or 10000000Q00, it is necessary to know how the radius enters into the products or quotients formed by the above rules. And the following consideration will always show, viz. that two equal quantities must be homogeneous. A line cannot be equal to a surface or the rectangle of two lines. Therefore all the products formed by the above rules must be understood to be divided by radius. Neither can a line be equal to the ratio or quotient of two lines, for this is an abstract number. Therefore all the quotients formed by the above rules must be understood to be multiplied by radius. In the use of logarithms, therefore, for the solution of right angled plane triangles, in every case two logarithms only will have to be employed, and their sum or difference taken according as the rule which applies to the case requires multiplication or division. When the sum of the two logarithms is taken, 10 must be rejected from the characteristic, which is in effect dividing by radius, and when the difference is taken, 10 must be added to the characteristic of the minuend, which is in effect multiplying by radius. All this will be easily comprehended from the following examples. EXAMPLE 1. In the right angled triangle ABC right angled at A, given the hypothenuse a = 493*7, and the angle c = 65~ 40' to find the side c. By Rule IV. above, observing that addition of logarithms corresponds to multiplication of the corresponding numbers, and rejecting C - JA 10 from the sum, according to IX., the calculation will be as follows: a 493'7 log. 2'69346 c 650 40' log. sin 9'95960 c 449'8 log. 2'65306 The sum of the logarithms 2'69346 and 9*95960 whichis 1265306 after rejecting 10 from index will be the logarithm of 449*8 as may be seen by inspecting the tables, and this number is the value of the required side c. ~~58 ~ PLANE TRIGONOMETRY. EXAMPLE II. The same thing being given to find the side b. By Rule V. above a 49387 log. 269346 c 65~ 40' log. cos 9*61494 b 203*4 log. 2*30840 Both the above operations may be conveniently connected together by preparing first a blank form thus: a log. log. c log. sin log. cos c log. b log. in which the arguments or elements of the triangle occupy the first column, the trigonometrical functions of these arguments necessary for the calculation of the side c the second, and those necessary for the calculation of the side b the third. This form is then to be filled up as follows a 493*7 log. 2'69346 log. 2*69346 c 65~ 40' log. sin 9-95960 log. cos 9161494 c 449*8 log. 2'65306 - b 203*4 log. 2*30840 The log. of 493 when found in the tables is written in two columns, and the log. sin and log. cos of 65~ 40' can be taken out at one opening of the tables. EXAMPLE III. Let 6b 7394, c = 57~ 20', to find a by VII., subtracting the logarithms for division, adding 10 to the index of the minuend (see IX.) before subtracting, the calculation will be as follows: b 73'94 log. 1'86888 c 57~ 20' log. cos 9'73219 a 136*9 log. 2'13669 EXAMPLE IV. Let c - 115'3, c - 570 20', to find a c 115.3 log. 2*06183 c 570 20' log. sin 9'92522 a 136*9 log. 2*13661 RIGHT ANGLED TRIANGLES. 5 EXAMPLE V. Given b = 6784, c = 41~ 4' 36", to find c and a. By VII. VIIL uld IX. b 67'84 log. 1'8314858* log. 1'8314858 c 410 4' 36" log. cos 9'8772740 log. tan 9'9403365 a 89-993 1*9542118 e: 9*132 1*7718223 EXAMPLE VI. Given a = 84e9, b = 534, to find B, c, and c. For the rule applicable see I. a 84*9 log. 1'92891 b 53*4 logo 172754 B 38~ 58' 28' log. sin 9'79863 a+b 138'3 log. 2'14082 ab 31*5 log. 1*49831 c0-= b21 ~ 2log. 3'63913 + 2 c 6600035 log. 1'81957 c =900- B =51 01' 32' EXERCISES IN THE SOLUTION OF RIGHT ANGLED PLANE TRIANGLES, 1. Given a =- 49'63, b 25'42, to find c, B, and c..As. c - 42*625, c-=59~ 11' 25", n= 300 48' 35", 2 "S a='723'1, c = 95'4, to find b, c, and n. Ans. b -= 71675, B =- 820 25' 2", c 7 34' 58" 3. " a — 853. B == 490 31' 22", to find the remaining parts. Ans. b = 648*8463, c 553'725. 4. " a=-940, c =300 20' 10", to find the other elements.Ans. c=474*769, b 811'32. 5. " b=25, c = 7, to resolve the triangle. Ans. c 34~ 12' 58" a 30*226, B 550 47' 02". * The logarithms in this example are taken from the tables of Callet, and extend to seven places of decimals. t The product of a - b and a - b which is obtained by adding this logarithm, is equal to a2 --. (See Alg. Art. 13.) 60 PLANE TRIGONOMETRY. 6. Given — 42173, c-=40 20'. Ans. a= 55324, c 358'08, B 490 40'. 7. " b= 328, B =740 25' 18"7. Ans. a 340'5025, c 91-436. 8, " c- 8876213, == 890. Ans. a 8'7637 b 15294, B 1~. 9. " c - 82'94, B = 40~ 50' 20"'6. Ans. a 109'612, b 71'684, c 490 9' 39/'4. 10. " b.- 81'5, c=-92'19. Ans. B 41~ 28' 42", c 480 31' 18" a 123333. 65. We shall finish the subject of.right angled triangles by presenting a case of their practical application which is likely often to occur. 77f030 At the top of a mountain whose height was known by the barometer or otherwise to be 1000 feet above the level of the sea, a ship was observed through the tube of the instrument described at Art. 10, and the number of degrees between the tube and a plumb line from the centre of the circle was found to be'77 30'; required the distance of the ship. A right angled triangle is here formed, in which are given the perpendicular and angle at the vertex, and the base is required. Referring to Rule VI. tan 77 30' X 1000 = dist. required. 1000 log. 3*00000 770 30' log. tan 10'65424 3*65424 -log. of 4510*71 feet. Therefore the distance of the ship is 4510'71 feet, or a little over f of a mile. EXERCISES, An upright post being 90 feet high, and its horizontal shadow 117 feet, to find the altitude of the sun. Ans. 370 34' 5.. The alt. of the sun being 480 10' and the length of a horizontal shadow = 201, to find te height of the tower which casts it. Ans. 224-54. RIGHT ANGLED TRIANGLES. 61 To find the horizontal parallax of the moon (angle M), its distance MC being 240,000 miles, and the rad. of the earth BC 3956. Ans. 56' 40". The lat. of a place D being given, DCA = 38~ 14' and the radius DC of the earth 3956 miles, to find the radius of the parallel DE. Ans. 3107'43. Ajb -^' )~ ~ To find the side of a regular inscribed figure of 13 sides, when the radius of the circle is 6,7. Ans. 3-206. The side of an inscribed regular heptagon being 5*73, find the rad. of the circumscribing circle. Ans. 6-6032. 66. It is customary where the subtraction of logarithms, corresponding to the division of numbers, is to be performed, to change this operation into addition by means of what is called the arithmetical complement of each subtractive logarithm. The arithmetical complement of a logarithm is the remainder after taking the logarithm from 10; thus the arithmetical complement of the logarithm 2*32245, is 7-67755. 10o00000 2'32245 7.67755 It may be formed most conveniently, instead of beginning at the right and subtracting each figure from 10 and carrying one each time throughout, by beginning at the left and subtracting each figure from 9 till you come to the last figure, which subtract from 10. When we have a logarithm to subtract, we shall obtain the same result by adding its arithmetical complement, and afterwards subtracting 10. Which may be proved as follows: By the definition arith. comp. log. b- 10 - logo b. Now add this arith. compo to some other log. as log a, the result will be log. a+ 10 I-log. b stubtract 10 and there remains log. a - log. b The same result as would have been obtained by subtracting log.o from log. a. Hence to perform operations containing a number of multiplications and divisions, by means of logarithms, we have the following 62 PLANE TRIGONOMETRY. RULE.-Write the logarithms of all the multipliers, and the arith. comlplements of those of the divisors in a column. Add up the whole, and reject as many times 10 from the characteristic of the sum as there have been atith. complements employed. "'This use of the arithmetical complement is not expedient in the solution of right angled triangles, but inl the solution of oblique angled triangles it always saves one line of numerical work." SOLUTION OF PLANE TRIANGLES IN GENERAL. 67. Let AB3 be any triangle. A From the vertex of one of the angles A, let fall upon the side opposite, the perpendicular AD; the given triangle will be divided into two right angled triangles ABD and ACD. 3) ) In the first of these (Art. 64, IV.) we have A D = A B X sin B Again, in the right angled triangle A c D, we have A D = A c X sin c The first members of the two equations above being the same, thie second members are equal, hence sin B X A B =sin c X A c Turning this equation into a proportion, by making the first product the extremes, and the second the means, we have sin B: sin c A c A B That is, the sines of the angles of any plane triangle, bear the same proportion to each other as the opposite sides.* From the nature of the above demonstration the selection of the vertex from which to let fall the perpendicular being entirely unrestricted, it is plain that this rule applies to all the angles and sides alike. * The same demonstration will apply when the perpendicular from the vertex of one of the angles falls upon the side opposite produced. OBLIQUE ANGLED TRIANGLES. 63 When therefore two of the three given elements of a triangle, are a side and its opposite angle, the element opposite to the third given element may be found by the proportion which has just been established. We shall according to our custom suppose a practical problem which introduces the case of solution in question. Let it be required to ascertain accurately the - distance from the town in the figure, across an impas-,, sable marsh, for the purpose of es- \o timating the expense of a causeway. Plant two staves with small flags, the one at the border of the marsh opposite the town where the causeway is to terminate, and the other at some convenient point from which the first and the town may,both be seen. Measure the distance between the staves, and let it be 1000 yards; observe also the angle at the second staff by turning the tin tube of the instrument for taking angles first to the staff on the border of the marsh and then to the town; let this angle be 102; observe also the angle at the town subtended by the line of the two staves, and let this last be 30~. A triangle will be formed in which are known an angle 30~, the side opposite 1000 yards, and another angle 1020, the side opposite to which is required. Then applying the above proportion sin 300 sin 102:: 1000: length of causeway required. Arith. comp. log. sin. 300 = 0'30103 (Art. 15) log. sin 1020 = log. sin'78~ - 999040 log. 1000 -- 3*00000 3*29143 -log. of 1956 The length of the causeway must be 1956 yards. EXERCISE. In an oblique angled triangle given A 300 20t 10", B- 400 10' 30", a - 9754, to find c, b, and e. Ans. c = 1090 29t 20.", b 12458*28, c = 18205,71. 67. This proportion is also applicable to the case where two angles and the interjacent side of a triangle are given. Such is the case in the problem at Art. 10. 6 PLANE TRIGONOMETRY. One of the angles of that triangle being given equal to 47~, and the other equal to 1050 30', it will be easy to find the third angle, by recollecting that the sum of the three angles of a triangle is equal to two 9 right angles (Geom. Theorem 15), or 1800; / therefore subtracting the sum of the two __ o | given, 47~ + 1050 30' 1520 30' from 1800, the remainder 270 30' is the third angle of the triangle, and is opposite the given side 500 yards. Thus by the proportion sin 27~ 30': sin 47o: 500: dist. from It. house to fort. aritho comp. log. sin 270 30' = 0'33559 log. sin 470 = 9*86413 log. 500 2*69897 2*89869 = log. of 791*9 The distance from the light-house to the fort is 791*9 yards. The importance, where any considerable degree of accuracy is required, of the method of solution by calculation, instead of that by construction, will appear from the fact that with tolerably accurate instruments, and some care in the construction, we made the required side, which we here find to be 791X9 yards, to be 800 yards upon the scale; thus committing an error in the construction of about 8 yards. EXERCISE. Given in an oblique angled triangle, A 750 30' 18".5, B = 450 16', c = 1145~3, to find the other parts of the triangle. Ans. c = 590 13' 41"o5, a = 1290-55, b = 946-949. We add another practical EXAMPLE Involving the same case of solution combined with the solution of a right angled triangle. An observer upon a plain desires to find the height of a neighboring hill above the level of the plain. Near the foot of the hill let him take the angle of elevation to the top, and suppose it to be 550 54'; then let him measure back a distance, say OBLIQUE ANGLED TRIANGLES. 100 yards, and again take the angle of elevation, which let be 330 20'. Then in the triangle of which 100 yards is the base, and 33 20' is one of the angles at fhe base, we may _ go /__ have the angle at the vertex, and: "Oyda opposite to the given base, by observing that the exterior angle of a triangle being equal to the two interior and opposite (Geom. Theorem 13), one of the interior is equal to the exterior, minus the other interior, and therefore the angle at the vertex here is equal to 550 54'-330 20' - 220 34'; then say, as in the last example, sin 22' 34' sin 330 20': 100: side opp. to 330 20'; but in the right angled triangle of which 550 54' is the angle at the base, and the height of the hill one of the perpendicular sides, we have the proportion 10lo:sin 550 54':: hypoth.: height required. from which, multiplying means and dividing by the first term, sin 55~ 54' X hypoth height req. = -. 1010 but from the preceding proportion sin 338 20' X 100 hypoth. or side op. 330 20'sin 33 20' X 1 sin 22~ 34' substituting this value in the last equation, we have sin 550 54' X sin 330 20' X 100 height required l. ~~-n - 10"' X sin 220 34' arith. comp log. 01" = 000000 arith. comp. log. sin 22C 34' = 041594 log. sin 55~ 54' = 9*91806 log. sin 330 20'- 9'73997 log. 100 2 sum rejecting twice 10 =2 2'0739'7 =log. of 118 6,. 118*6 yards, or 355*8 feet is the height of the hill. * To this result the height of the eye or of the instrument should be added. ~ 66 PLANE TRIGONOMETRY. EXAMPLE III. 68. Let there be a street A 216.geee- G in which the front of a tri- /2 angular block is 216 feet, /. and another street making an angle of 22~ 37' with the first; under what angle B. must a third street be laid out from the extremity A of the first, so that the front of a complete row of buildings upon it shall be 117 feet in length? SOLUTION. 17: sin 220 37': 216:sin of the angle B 117 arith comp. of log. 7.93181 220 37' log. sin 9'58497 216 log. = 233445 45~ 13' 55" log. sin -- 9.85123 B=-45~ 13' 55" and 1800 —B-0c=1120 9' 5"=angle A required. It must be observed that 9'85123 is also the log. sine of the supplement of 450 13' 55", because the sine of an arc is equal to the sine of its supplement. (Art. 15.) Hence 134~ 46' 5" is also a value of the angle B, and there are two solutions to the problem, which are both exhibited in the diagram. This case corresponds to Problem 8, Geom. If the given angle were right or obtuse, there could be but one solution, and the required angle must be acute. The same is the case if the given side opposite the given angle be greater than the other given side, because in every plane triangle the greater angle is opposite the greater side. 69. We shall next derive a formula for the solution of a triangle when the three sides are given, and one or all of the angles required. Let ABC be any triangle; from A the. vertex of one of the angles A, let fall a perpendicular A D upon the,side opposite. This perpendicular may fall either within or without >the triangle. First suppose that it &Ills within; then (Geom. Th. 29.) B. C OBLaQUE ANGLEDB TRANGLES. 67 AC - A B-+:B C~- 2 B c X B D A B2+ B C2 — A C whence B D == + _2 B c Also in the right angled triangle A B Dr we have:cos:: A B: B D (Ar't 38) whence multiplying the extremes and dividing by the third term R X BD COS B = A B substituting in this expression for B D its value obtained above, we have A 1B2 + B C2- A CA COS B ==- R X 2ABXBO an expression for the cosine of an angle in terms of the three sides of a triangle. Suppose now that the perpeidicular falls without the triangle. A Then (Geom. Theorem 28) AC = AB2-~+B-2+2 B DX BC hence A B +- B C2- A 0 Again, in the right angled triangle A B D R: COS A B D -A 1B B D hence, R X B D COS A B D AB But A B B is the supplement of the angle B of the triale A B c, hence COS A B D - = COS B substituting cos B for cos A B D above, and chnging the signs we: have -— R X B cOS B -A A B c~b68 ]PLANE TRIGONOMETRT'.uabstituting for -B D in the second member of this equation its value found above, we have as before A B -+ B C2- A O2 Cos B -= R X 2 AB ) B C or employing the small letters to represent the sides opposite the angles which are expressed by the large letters of the same name as + c2- b COS B == a - 2 ac That is to say, the cosine of either angle of a triangle is equal to the sum of the squares of the two sides which contain it, minus the square of the side opposite, divided by twice the rectangle of the containing sides. Let us apply this formula to an EXAMPLE. Suppose the three sides of a triangular plat of ground are to be 50, 60, and 70 yards, under what angle must the first two be laid out? 2500 -+ 3600 - 4900 cos required angle -== X -0 _ 2 X 50 X 60 if we make R=- I; hence I or *20000 is the nat. cosine of the angle required. This angle will be found from the table of sines and cosines, to be 780 27' 47".* This case might also be solved with the table of log. sines, &c., by subtracting the log. of the whole denominator from that of the whole numerator, and adding 10 the log. of R, or adding at once the arith. comp. of the denominator, the logs. of the numerator and R, rejecting 10 from the sum; in either case the result would be log. cosine of the angle required. The solution is left as an exercise for the student. 70.'We now proceed to demonstrate some formulas which express relations between the different trigonometrical lines of the same arc, and The seconds are found as follows: Take the difference between the two cosines next greater and next less than yours, and also the difference between yours and the next greater, multiply the latter difference by 60 and divide the product by the former fference; the quotient will be the seconds sought. The reason appears from the lowing proportion. diff, of cosines. diff. of numbers. diff. of cosines. diff. of numbers. in the tab. 60":: yours and the tab.: seconds reqred. GENERAL FORMULAS. 69 between the trigonometrical lines of two different arcs. They are introduced here because necessary for the solution of the few cases of plane triangles which remain. We shall first derive formulas by means of which, when the sines and cosines of two arcs are known, the sine ald cosine of their sum or difference may be found. Thus if the sine and cosine of 30~, and also those of 20~ be given, those of 50~ -- 30o + 200 or of 100 == 300 - 20 may be found. Let A ==a in the diagram B be bne of the given arcs, and BM- b, be the other. Then M P==sin a, B I==sin b, since it is the perpendicular let fall from one extremity of the arc b, upon the radius which passesP through the other extremity. c - = cos a and c I the distance fiom the foot of the sine to the centre ==cos b. A B = A M +a B === a - b, B E =sin (a + b) and E c -- cos (a + b). In the triangle c K i we have (Art. 64, IV.) IK sin C X c I or, E L==- sin acos (1) and in the triangle B I L (Art. 64, V.) B L -= B I COS B or since B = c their sides being respectively perpendicular B L = sin b cos a (2) Adding equations (1) and (2) and observing that E Lr B L —B - =ssin A B we have sin (a + b) -si a cos b + sin b coss a (3) The second member must be understood to be divided by R= 1, for a lkie cannot be equal to the sum of two surfaces (See Art. 64, IX.). Formula (3) is read thus: the sine of the sum of any two arcs is equal to the sine of the first into the cosine of the second plus the sine of the secend into the cosine of the first, divided by radius4 Again, C K -= os c X c i= cos a cos b (4) E R -=I L 1 sin B X B I sin a sin b (5) T7O PLANE TRIGONOMETRY. Subtracting (5) from (4), and observing that K — E K r= c =s e A B (Art. 24), we have cos (a + b) = cos a cos b- sin a sin b (6) The second member of (6) must be understood to be divided by R or the first member to be multiplied by R to produce homogeneity. Formula (6) is read thus: the cosine of the sum of any two arcs is equal to the rectangle of their cosines minus the rectangle of their sines divided by radius. In formula (3) let a = 600 and b - 20~ then by the first sin 600 cos 200 + sin 20~ cos 60~ sin 80~ I or 10" as the case may be Performing the operations by the aid of logarithms log. sin 600 9- 93753 log. cos 20~ = 9.97299 log. sin 20~ = 9.53405 log. cos 60~ 9'69897 9'91052* log. of 881380 9925302 log. of'17101 sin 800 = - 98481 By formula (6) cos 60~ cos 20 -sin 60~ sin 20 cos 800~ = - 1 or 101~ or whatever R may be t W(1e shall derive expressions for sin (a - b) and cos (a -b) or the sine and cosine of the difference of two arcs in terms of the arcs themselves, by making, in the formulas just derived, for sin (a +b) and cos (a +b); 5 -- b, observing that cos ( b) = cos b and sin ( b) = - sin b (Art. 27). By this substitution there results * From each of the logs. 9*91052 and 9.25302, 10 must be rejected in order to pass from the table of logarithmic sines, &c., in which the radius is 10000000000, the logarithm of which is 10, to the table of natural sines, &c., in which the radius i 1. The characteristics of these logarithms would thus become 1, but they may be mead so as they stand according to Art. 54, Ex. 2: t The student may develope this as an exercise, and compare the result with cos 800 as given by the table of natural sines and cosines. GENERAL FORMULAS. 71 sin (a -b) = sin a cos b- sin b cos a (7) and cos (a-b) = cos a cos b + sin a sin b (8) EXERCISES. Find the sine of 570 from sin of 150 -= 25882, and cos = *96593, and sin 420 -= 66913, and cosine 7=== 4314. Find the cosine of 90 from sine and cosine of 240 =.40674 and.91355 and sine and cosine 150 =.25882 and -96593. These four formulas for the sine and cosine of the sum and difference of two arcs should be committed to memory, as they are constantly recurring in trigonometry, and in the higher analysis. The four may be expressed in two by the use of the double sign, thus sin (a ib) - sin a cos b + sin b cos a cos (a +b) = cos a cos b T sin a sin b 71. From formula (3) sin (a+b) = &c., we derive one much used in the higher analysis for expressing twice an arc in terms of the are itself, by simply making b = a the result is sin 2a = 2 sin a cos a (1) the two terms of the second member becoming the same. We also get an analogous expression for the cosine of twice an arc by making b = a in formula (6) of the last article cos (a + b) = &c. This expression is cos 2 a = cos2 a- sins a (2) Thus knowing the sine and cosine of 200, these last two formulas would give us the sine and cosine of 40~. These two formulas may be modified so as to express the sine and cosine of an arc in terms of half the arc, under which last form they are much used. This is accomplished by making a = A a, which is legiti. mate, since a is supposed to have no particular value; then 2 a becomes a and we have from (1) sin a = 2 sin - a cos A a (3) and from (2) cos a = cos a-sin' 2 a (4) 72 PLANE TRIGONOMETRY. EXERCISES. Find the sine of 320 from the sine and cosine of 160, the former being,27564 and the latter.96126. Find also the cosine of 320 from the same data. NOTE.-This last is best done by observing that the product of the sum and difference is equal to the difference of the squares. 72.i By means of this last, and a very simple formula depending upon the well-known property of the right angled triangle, that the square of the hypothenuse is equal to the sum of the squares of the other two sides, a formula expressing the value of the sine of half an arc in terms of the are itself may be obtained. The formula depending upon the property of the right angled triangle, will be found by referring to the last diagram, in which the triangle c P M is right angled at P, whence (Geom. Th. 26). c P2 + P Ma — M2 or calling A M - a cos2 -I a + sin" a a = it (5) Introducing R into equation (4) according to the rule for homogeneity at Art. 31, and changing the order of the members it becomes cos2 a- sin ~ a = R cos a (6) Subtracting (6) from (5) we have 2 sina a = R2 - R cos a Dividing by 2 and taking the square root of both members of this last equation, we have the formula required. aine a= — = _ i n R cos a (7) making R =1 sin - a -- cos a (8) 73. We resume the solution of triangles, having now a formula, by means of which we shall be able to derive an expression for one of the angles of a triangle in terms of the three sides; an expression which will be found much more convenient for the application of logarithms than that contained in Art. 69. By Art. 69, making R =l1 as B c -- b1) cos = 2ac 2 a'c OBLIQUE ANGLED TRIANGLE. 73 putting B in the place of a in the formula for sin -. a (formula (8) of the last article) we have sin If B = / _ - cos B substituting for cos B in this, its value in (1), we have / a2+ c'2 -b2 sin I B r= / i - 2 ~ ^2 2ac reducing the terms under the radical to a common denominator, there results /2a c-a2 —c2 +bV sin 1 B = / 4 a c but (Algo Art. 13, Note 2) 2 ac -a -ca - (a -c)' hence sin B = 4 a c but the difference of the squares of two quantities is equal to the product of their sum and difference (Alg. Art. 13, Note 2), hence b- (a~-c) = (b + a - c) (b- a + c) substituting the second member of this in place of the first in the preceding equation, and separating the 4 of the denominator into two factors 2 X 2, we have ( (b+a-c) (b+c-a) B- 2 2 a c but b+-a-c b-+a-+c b+- c — b+c+ a _~ - — c and - a 2 2 2 2 representing b + a -+ c the sum of the three sides of the triangle by s, the second members of the two last equalities become - s-c and i s-a 74 PLANE TRIGONOMETRY. substituting these for their equals in the preceding equation it becomes sin -? B ls- - ( s-)a)( a c the formula sought.* As the angles have each the same relations to the corresponding sides of a triangle, the same formula by a proper modification will furnish the values of the angles A and c. It may be expressed in ordinary language thus, the sine of half either angle of a triangle is equal to radius into the square root of half the sum of the three sides minus one of the adjacent sides, into half the sum rninsus the other adjaceOt side, divided by the rectangle of the adjacent sides. To apply this to an EXAMPLE. Let there be three places at distances from each other respectively of 50, 60, and 70 miles. Required the angle under which two roads must depart from that which is 60 and 70 distant from the other two, in the direction of these last. 60 and 70 will be the sides of a triangle adjacent the required angle, and 50 the side opposite; then sin t the angle - R /(- 70) ( s- 60) 60 X 70 180 i..80.90, - s - 0 o=20 2 and i s 60- 30 log. of 20- 1*30103 log. of 30 - 1 47712 ar. comp. of log. of 60 — 822185 ar. comp. of log. of 70 == 815490 sum rejecting twice 10 = 1,15490 Divide this sum by 2 for V, quot. -= 05 + o07745 Add 10 to multiply by R sum -- 9.57745 = log. sin of - the required angle. * Radius must be understood as a factor of the second member for the sake of homogeneity, since the quantity under the radical is the ratio of a surface to a surface, and therefore an abstract number. OBLIQUS. ANGLED TRIANGLE. 75 From the tables we find i the angle to be 220 12' 28". The whole angle required will be double this or 440 24' 56" The other two angles may be found in a similar manner; the one is 67~ 07' 18", the other 780 27' 40". If R in the above formula should be made to pass (by squaring it) under the radical sign, it would be necessary to add twice 10 in order to effect. the multiplication by this factor R2 before taking the square root. But as on the other hand twice 10 must be rejected for the arithmetical complements used, these two operations exactly counterbalance each other, and neither of them need be performed. By adding together the four logarithms, therefore, and dividing by 2, the same result will be obtained. The operation in the above example would be as follows: log. of 20 1'30103 log. of 30 = 1-47712 ar. comp. of log. of 60 = 8'22185 ar. comp. of log. of 70 = 815490 2)19'15490 log. sin. of 220 12' 28" = 9'57745 The best mode of proceeding in the solution of a plane triangle when three sides are given, is to prepare a blank form similar to that on p. 58, by ruling four columns, the first for the arguments, and each of the other.three for the trigonometrical functions of those arguments necessary to be employed in the calculation of one of the three angles. Thus, a, b, c, denoting the given sides, s their sum, and A, B, c the required angles, Column of Arguments. Column for calculation of A. Column for calculation of B. Column for calculation of c. ar. co. log. ar. co log. 5 ar. co. log. ar. co. log. c ar. co. log. ar. co. log. 8 is s-a log. og. log 8 s-b log. log. 4 s — log. log. 2) 4 A log. sin 2) ~ B log. sin 2) c log. sin ~ —--------------.- _. —-------.-. —-- _-.._..............__ 76 PLANE TRIGONOMETRY. To show how this form is filled up take the following example. Given a == 33, b5- 42*6, c == 536, required A, B, Co Column of Arguments. Column for calculation of A. Column for calculation of B. Column for calculation of c. a 33 ar. co. log. 8*4814861 ar. co. log. 8-4814861 b 42-6 ar. co. log. 8*3705904 ar. co. log. 8.3705904 c 53.6 ar. co. log. 8*2708352 ar. co. log. 8-2708352 s 129.2 is 64.6 a s- a 31-6 log. 1'4996871 log. 1'4996871 As —b 22 log. 1'3424227 log. 1'3424227 s-c 11 log. 1'0413927 log. 1'0413927 2)19'0252410 A 180 59' 56"I5 log. sin 9'5126205 2)19'2934011 B 260 18 5311.3 log. sin. 9-6467005 2)19*6941863 A c 440 41' 10" log. sin 9'8470931 A - 370 59f 53", B - 520 37' 46"6, c -- 890 22' 20". There are other forms which have some advantages over the above, and which may be derived in an analogous manner. They are as follows. 2R sin A =- s ( s-a) ( s- b) (is~ -) (8) be c3 / S (A s -a)* C03oAs K / ^-be (4) ( s- b) ( c s-)t tan I A = / sA b A) ( 5a) The student may write the blank forms for these formulas as an exercise, EXAMPLES. 1. Given in a plane triangle the three sides 120, 112'65, and 112, to find the three angles. S 570 27' Ans. 570 58' 39" ( 640 34' 21" * Derived from (5) and (6) of Art. 72, and (1) of Art 73. A convenient form. t By dividing the formula sin A = — / cos A by the formula cos A A _ -. ____._ ~~___ sin A V + -- ncos A is found tan A A =1+ - os A from which (4) above may be derived; or at once dividing (3) by (4). In a similar manner may be found cot a= _ sin sec ~^+" n oe =f / see / 2 sec a 1-~sa see u / see and cosec A a / - -cos a seca+1 V sec a -1 GENERAL FORMULAS. 77 2. Given a a 6876, b = 4231, c - 8913'24, to find A, B, and c. (A 48~ 24' 36" Ans. n B = 27 24 c= 104 11 24 74. Before treating of the only remaining case in the solution of triangles, it will be convenient to demonstrate some additional general fornulas which shall present certain important relations of the trigonometrical lines of two different arcs; which formulas are of frequent use in the higher analysis, are employed in the subsequent parts of the work, and will be immediately of service in deriving a formula for the last case of plane trigonometry which we have to consider. Add together equations (3) and (7) of (Art. 70) which express the values of sin (a + b) and sin (a- b) and the resulting equation, cancelling the second terms of the second members which. are similar with contrary signs, is sin (a+ b) + sin (a- b) 2 sin a cos (1) make a + b =p and a -b q add these last two equations; there results 2a -=p + q whence a -- (p + q) subtracting the same equation, the second from the first, 2b p -q whence b - (p - q) substituting in equation (1) the values of a + b, a b, a and b in terms of p and q, that equation becomes sin p si q 2 sin s (p + q) cos (p q) (2) Which may be translated into ordinary language thus: the sum of the sines of two arcs is equal to twice the sine of half the sum into the cosine of half the difference of those arcs. By subtracting the latter of the same equations (3) and (7) of (Art. 69) from the former, and reducing similar terms, there results sin (a + b) - sin. (a -- b)==2 sin b cos a (3) P R must be understood either as a divisor of the second member or multiplier of the first, because the sum of two lines cannot be equal to a rectangle, 78 PLANE TRIGONOMETRY. making the same substitutions as above in equation (1) this last equation becomes sin p- sin q == 2 sin I (p - q) cos i (p + q) (4) or, the difference of the sines of two arcs is equal to twice the sine of ha.' their difference into the cosine of half their sum.* Divide' equation (2) by equation (4) sin p sin q sin - (p q) cos-(p — q) sin p — sin q cos (p +q) sin (p-q) but:in~ ~ (p + q) cosj (p+- ) tangl (p+ q) (Art. 32) and s~i n(Pq)tang3 (P-) cos (p -- q) ox inverting this last cos (p —q)_ l sin P (p- q) tan (p- q) substituting in (5) the values of sin (P + q) and cos ~ ( - q) cos (p + q) sin ~ ( —q) tle equation becomes sin p4- sin q tan (p + q) sin p-sin q tan j (p- q) which may be expressed in a proportion thus: the sum of the sines of any two arcs is to the difference of their sines as the tangent of half their sum is to the tangent of half their difference. A 75. Let A B c be any triangle; then (Art. 6 7) a: b: sin A: sin B or by composition, (Alg. Art. 133, IX., Geom. Theorem 47), B a+ b:a b:: sin A+ sinl n:s in A-sin * The same remark applies to this form as to (2)o GENERAL FORMULE, 79 b it by equation (6) Art. 74. sin A + sin B: sin A-sin B:3 tan { (A + B): tan I- (A B) hence, a+ 6: a -b: tan i (A + B): tan -j (A -B) That is to say, the sum of two of the sides of a plane triangle is to their diference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. 76This proportion is employed when two sides and the included angle of a triangle are given to find the other parts. Since the three angles of every triangle are together equal to two right angles or 1800, subtracting the given included angle from 180~, the remainder is the sum of the two angles opposite the given sides; then substituting for a and b in the above proportion the two given sides, three terms of it are known and the fourth may be found. After which, having half the sum and half the difference of the unknown angles, these angles themselves can be found by adding half the sum to half the difference for the greater, and subtracting half the difference from half the sum fiom the lesser: when all the parts of the triangle will be known, except one side, which may be found by the proportion the sines of the angles are as the opposite sides. (Art. 67.) EXAMPLE. An observer wishing to know the length of a small lake, measured two lines from the same point to the two extremities of the lake, which he found to be respectively 153 and 137 / yards; he also observed with an instrument for,/ taking angles the angle subtended from this point by the lake to be 400 33' 12". // I'4;* 80 PLANE TRIGONOMETRY, SOLUTION. Io. To find the twio other angles. As sum of the given sides 290 3 arithl comp. of log. —~* 7'53760: diff. of the sides 16 =- log.-' 120412: tan of - sum op. angs. - ( (180 400 33' 12") -690 43' 24" log.-' 10*43245 tan of I diff. of op. angs. log. of which is sum rejec. 10. 9117417- log. tan 8~ 29' 37" Add and subt. with I sum of angs. 69~ 43' 24" sum = greater angle 780 13' 1' diff. -lesser angle 601 13' 47" II. iTo find the remaining side.f As sin 78~ 13' 1" — arith comp. log.-' 0*00925 e opp. side 153T= log.-' 2*18469: sin 400 33' 12" - logr.-' 9'81302:side opp. 400 30' 12" log.-' sum rejec. 10 = 2*00696 log of 101'616L The length of the lake is 101*616 yards. The blank form for this case would be as follows: a' alog. b a + b ar. co log. a -b log. c log. sin k (A + B) log. tan (A ^ - B) log. tan A ar. co. log. sin c ___________ _____________________ log. " This notation " log.-1' signifies the number whose log. is. t A more direct mode of finding this side when it is the only part required, is given at Art. 77. t 153 is known to be the side opposite 78~ 13't 1 because the greater angle of a triangle is always opposite the greater side. (Geom. Theorem 9.) GENERAL FORMVULJ. 81 The same form filled up with the given example above is given below. a 153 log. 2.18469 b 137 a + b 290 ar. co. log. 7.53760 a-b 16 log. 120412 c 400 33' 12" log. sin 9.81302 A+B 1390 26 48" % (A + B) 690 43' 24" log. tan 10.43245* (A-B) 80 29' 37" log tan 9-17417 A 780 13' 1" ar. co. log. sin 0*00925 - 610 131 47"' c 101*616 log. 2*00696 77. Given two sides and the included angle of a plane triangle to determine the third side, without finding the remaining angles. The general expression for the side c, in terms of the two sides a, b, and the included angle c, is (Art. 69) on the supposition of R 1, c a2 + b2 -- 2 ab cos C =(a - hb)3 2 ab (1 cosc) = (a - b)2 + 2 ab *2 sin2. c (a-b)2 1 4ab sin2 i c (1) =(a.)2 (-) Asume the second term within the brackets equal to tan 20, then, since 1 + tan's = Q 0e _= _ we have cos 2' c (a-b) rad (2) cos 0 Hence c is determined by these two formulas, viz., log. tan 0 - log. 2 +- log. a -+- log. b + log. sin c - log (a — /b) log. c = log. (a - b) + 10- log. cos 0.t * Log. cot c c might be used instead of log. tan -( (A -+ ) since 1800 - c A - B and.-. 900 ~- c - (a + B) and tan (900- c) = cot 3 c. But nothing would be gained, since - (A +-3) must be employed to add and subtract with i (A -B). t When b is nearly equal to a, the following formulas will give c with greater exactness. For demonstration see App. I. Art. 23. 2 ab sia = -- cos I e (a) a +b s=(a+-b) cos# (4) 5 82 PLANE TRIGONOMETTRY EXAMPLE. Given a 562, b = 320, and c = 1280 4', to find c. log. 2 0*30103 l log. 562 1-37487 l log. 320 1*25257 log. sin 640 2' 9-95378 ar. comp. log. 242 7-61618 log. 242 +10 12-38382 log. tan 0 10.49843.'. log. cos 0 - 9.48072 log. c 800'01 2'90310 EXERCISES. Given a 5891, b 4562-34, c 300 20' 10"'3 to find the other parts of the triangle, Ans. A 990 57' 5"'25, B 490 42' 44-45, c 3020-823. Given A 400 55' 31", b 83-25, c 100, to find a. Ans. a 65'9574. This case of the solution of a triangle combined with that exhibited at Art. 67, serves to determine the horizontal distance between two inaccessible objects. Let the distance between two towns which are in sight be required. Measure a line upon the ground (which is called the base line) of 2 miles. Take the angles at each extremity formed by this base line and a line to each of the towns. Two triangles will be formed, in each of which a side, viz. the base line, and two adjacent angles will be given. Let the angles in the triangle of which the \ upper town is the vertex be \ 1590 and 140; and those in that of which the lower town is the vertex be 250 and 1490. Calculate the distance from one extremity of the base line, say the upper extremity, to each of the towns, as in Art. 67. Then you will know two sides of a triangle the third side of which is the dis. tance between the towns required. MISCELLANEOUS EXAMPLES. 83 The included angle between these two sides is 159 ~- 25~ 1834. Having then two sides and the included angle, the remainder of the solution is the same as in the last case. We leave it as an exercise for the learner. Ans. 12'932 miles. MISCELLANEOUS EXAMPLES. (I.) From the equation sin a +- 5 cos2 a = 3 to find the value of sin a. Ans. sin a = ~(2.) If sin" a = m cos a~ n to determine cos a. Ans. cos a — ~ m: m n m2+ 1. (3.) Given sin a =. m sin b, tan a = n tan b to find sin a and cos b. q/.m -- n 11 — m'Ans. sin a =, e- cos b = -_ - -1- m1s 1-1 2 (4.) Prove sin (a +b — c) sin a cos b cos c+sin b cos a cos c+sin e cos a cos b-sin a sin b sin c. (5.) Given tan" a + 4 sin a = 6 to find a. Ans. a = 600. (6.) Sin a - sin 2a to find sin a. Ans. sin a = = / (7.) Given the base of a triangle 87'75, the vertical angle 730 20', and the difference of the angles at the base 130 4', to find the other parts. Ans. S 460 48', 590 52'. 79-2194, 66.7721. (8.) Given the base 117o3, the vertical angle 190 18', and the ratio of the other two sides 8: 11 to solve the triangle. Aus. 1230 13' 23'"7, 370 28' 36'-3, 215'94, 296-89. (9.) Find into what two parts the perpendicular from the vertex of the angle c, divides the side c of a triangle when A - 330, c = 750, and a = 2134. Ans. 3125.2 and 659*44. (10.) The side of a regular polygon of 41 sides is 0-736. What is the ratio of the radii of the inscribed and circumscribed circles? Ans. -99708. (11.) Find the extent of the circle of vision, viz. the arc TO, from the M top of a mountain M., whose height is 5460 feet, supposing the radius T of the earth to be 3956,1 miles. Ans. 91 miles, 155888 feet. (12.) When the sun's altitude is 190 16', the peak of a mountain casts its shadow at a certain point, and at another point when the sun's altitude is 200 42', distant from the former point 937 ft. Now supposing the sun to be vertical on the top of the mountain at noon, what is its height? Ans. 4369.1 ft. 84 PLANE TRIGONOMETRY. (13.) Two forces, one of 410, the other of 320 pounds, act under an angle of 51~ 37', required the direction and intensity of their resultant.* The resultant makes an angle of 290 13' 46"*7, with Ans. the less force, and 220 23' 13"-3 with the greater. Intensity 702*39838 pounds. (14.) From the edge of a ditch, the width of which was 36 feet, the angle of elevatiortthe top of an opposite wall was 620 40'; to find the height of the wall and the length of a ladder which would reach obliquely across the ditch to the top of the wall Ans. Height of wall, 69*64. Ladder, 78'4 feet. (15.) To find the length of a shoar, which, projecting 11 feet from the perpendicular face of a building, will support a jamb 23 feet 10 inches above the ground? Ans. 26 feet 3 inches. (16.) Suppose that a ladder, 40 feet long, will reach a window 33 feet from the ground on one side of a street, and on being turned over, without moving the foot, it will reach a window 21 feet high on the other side; to find the breadth of the street? Ans. 56.649 feet. (17.) A liberty-pole whose top was broken off strikes the ground at 15 feet distance from the foot of the pole; to find the height of the whole if the broken piece measures 39 feet in length? Ans. 75 feet. (18.) At 170 feet distance from the bottom of a tower, suppose the angle of elevation to be 520 30'; to find the altitude of the tower? Ans. 221 feet. (19.) From the top of a tower by the sea-shore, 143 feet high, the angle of depression of a ship was observed to be 350; to find the distance of the ship from the bottom of the tower? Ans. 204*22 feet. (20.) To find the height of a hill, the angle of elevation at the bottom being 460, and 200 yards distant from the bottom 310? Ans. 286.28 yards. (21.) To find the height and distance of an inaccessible tower, on a horizontal plane, the angle of elevation being 580, and at a point 300 feet more distant, the angle being only 320? Ans. Height, 307*53. I Distance, 192*15. (22.) To find the height of a tower on the top of an inaccessible hill, the angle of elevation to the top of the hill being 400, the top of the tower 510, and 200 feet further back the angle to the top of the tower being 330 45'? Ans. 93*33148 feet. (23.) From a window on a level with the bottom of a steeple, the angle of elevation of the top of the steeple being 400; and from another window, 18 feet directly above the former, the angle was 370 30'; to find the height and distance of the steeple? Ans. Height, 210.44. Distance, 250*79. * The resultant of two forces is a single force equivalent to them, and is the diagonal of a parallelogram of which the two forces given are sides. MISCELLANEOUS EXAMPLES. 85 (24.) A balloon being directly over one of two towns whose distance apart was 8 miles, the angle of depression of the second was observed to be 10o. Required the height of the balloon? Ans. 1P41 of a mile. (25.) The horizontal angles were observed from each of two stations, 3000 feet apart, by first sighting to the other station, and then to a balloon, and the angle of elevation at one, as follows: 1st Station. Hor. angle, 750 2 640 30 Angle of elev. 180 2 Required the height and horizontal distance of the balloon from the first station. Ans. Distance, 4205 feet. H eight, 1366 feet. (26.) Two vessels of war anxious to cannonade a fort, are so remote from it that their guns cannot reach it with effect. In order to find the distance they move a quarter of a mile apart, then each vessel observes and measures the angles which the other and the fort subtend.; the angles being 830 45', and 850 15', required the distance between each vessel and the fort? i2292'26 Ans. 22 yards. 2298*05 (27.) Wishing to know the distance to an object on the other side of a river, I measured a base line of 400 feet in a right line by the side of the river, and found that the two angles, one at each end of this line,subtended by the other end and the object, were 680 2' and 730 15'. Required the distance between each station and the object? As. 593-08 feet. 612'38 (28.) Wanting to know the breadth of a river, I measured a base line of 500 feet in a right line close by its bank; the angles subtended by lines connecting each extremity of this line and an object on the opposite bank, were 530 and 790 12'. Required the perpendicular breadth of the river' Ans. 529*48 feet. (29.) Suppose it be required to'find the distance between two headlands, measure from each of them to any point inland, and supposing the distances respectively to be 735 feet and 840 feet, also the horizontal angle subtended between these two lines to be 550 40', what was the required distance? Ans. 741*2 feet. (30.) Wishing to know the distance between a church and tower, situated at a distance on the other side of a river, I measured a base line along the side where I was, of 600 feet, and at each end of it took the angles subtended by the other end and the church and tower; at one end the angles were 580 20' and 950 20', and at the other end the angles were 530 30', and 980 45'. Required the distance? Ans. 959*5866 feet. (31.) To determine the intensity and direction of a force which, combined with another force expressed by 128, shall produce a resultant of 200, which shall make an angle with the direction of the given force of 180 24'. Ans. Intensity, 88*32714. Angle, 270 13' 16".6. 86 PLANE TRIGO NOMETRY. (32.) To determine the force with which a body weighing 516 pounds moves down a plane inclined to the horizon under an angle of 140 10'.* Ans. 126'288 lbs. (33.) The angle of incidence of a ray of light falling upon a surface being 46% and the angle of refraction being 350 11', to find the index of refraction. NOTE.-The index of refraction is the ratio of the sine of the angle of incidence to the sine of the angle of refraction. Ans.'8.' The force down an inclined plane is to the force of gravity as the height of tha plane is to its length. APPENDIX I. 1. We have postponed to this place the investigation of a few formulas requisite for the study of Analytical Geometry, with other matters of interest. By resuming the expression for the tangent (Art. 32), and putting a + b for a, we have sin (a + b) tan (a ~ b)= — cos (a b) But by Art 70 sin (a 4 b) = sin a cos b cos a sin b and cos (a i b) = cos a cos b: sin a sin b substituting these values the first equation becomes (sin a cos b - sin b cos a) tan (a b ~ cos a cos b:] sin a sin b dividing both numerator and denominator of the second member of this equation ev cos a cos b we have cos a cos b tan (a ~ b) = ~ sin a sin b cos a cos b substituting for n a and s_ their values (Art. 32) tan a and tan b, the last exprescos a cos b sion becomes tan (a::b)-= (tan a +- tan b) 1 f tan a tanb ( i. e., the tangent of the sum or difference of two arcs is equal to rad. square into the sum or difference of their tangents divided by rad. square minus or plus the rectangle of their tangents.* a The mode in which Rt enters is derived from the principle of homogeneity. 88 6APPENDIX Lo If a represent the tangent of a and a' the tangent of a', the tan (a- a') = 1 + aa' Using the upper sines and making b a in equation (1), we have' tan a tan 2a - taa (2) 1- tan2 a Tan 3a, tan 4a, &c., may be found by making b successively equal to 2a, 3a, &c. EXERCISE. tan a - tan b +- tan c - tan a tan b tan c Prove tan (a + b +c). an-nbn ro-ve -tal a tan b tan a ta n c tan b ta n c 2. The sine and cosine of 450 are equal, since'the complement of 450 is 45o. These two lines form two sides of a right-angled triangle of which radius is the hypothenuse....sn 450 = cos 450 R =_ RV2 3. The sine of ~ an arc is equal to j the chord of the arc. For let MN be the arc; draw the diameter st perpendicular to the chord MN of tiis arc; this perpendicular bisects the chord, and also the arc subtended by it (Geom. Theorem 34), but MP half the chord is the sine of MA half the arc, since iP is a perpendicular from one extremity,a to the diameter which passes through the other extremity A. Corollary.-The chord of 600, or I of the circumference, which is the side of the xegular hexagon, is equal to R (Geomo Prob. 31), hence the sine of 300 is equal to i a. Again, cos 300/1 - -sin s 300 = /J —t t3 i tax 300 sin 300 cos 300 1 V3 cot 300 tan 300 _ V3 ~ The form employed in Analytical Geometry. GENERAL FORMULAS. 89 4. Referring to Art. 33, it will be observed that sec = - COS cos 60 sin 300~ = bence, sec 600 =- R _ 2 R - the diameter of the circle. R To find the numerical value of the sine, cosine, &c. of 600. sin 600 = cos (900 — 600) cos 300 3, by last art. 2 Again, cos 600 - sin (900 - 600) sin 300 1 Also, tan 600 = V3 cot 600_ V3 5. Making a - 450 in equation (1) of Art. 1, App. 1., we have 1 E tan b tan (45~ b) =- 1: tan b 6. Our demonstration for the sine and cosine of the sum of two arcs at Art. 70, rmight seem to want generality, since the arcs a and b are there supposed to be less than 900. That these arcs may extend to the other quadrants, can be shown as follows: Let a = 900 + m, then will the formula sin (a + b) = sin a cos b - sin b cos a (1) till be true, for substituting 900 -+ m for a, we have sin (900 +- m + b) in the place of the first member, which is equal to cos (m + b)t; for the second member by the same substitution we have sin (900 + m) cos b +- sin b cos (900 + m) * Tan 450 = cot 450 = = 1. t By referring to either of the diagrams in which a sine is drawn, it will bo evident that sin (900 + a), a being any arc less than a quadrant, is equal in length to ia (900 — a) = cos a. Also that cos (900 + a) - - cos (90~ -a) -- sin a. 90 APPENDIX I. but sin (900 +-m)= cos m and cos (900 +-m) — sin m, hence equation (1) becomes cos (m - b) = cos m cos b-sin m sin b which, since m and b are less than 900, we know to be true, by Art. 70; hence (1), from which it is derived, is true also. Assuming (1) to be true with a > 900, which we have just proved, make b = 900 - m, and in a similar manner the truth of the formula may be established on the supposition of both a and b > 900. Afterwards make a = 1800 + m and observe that sin (1800 +- m + b) =sin am + b and cos (1800 +- m) -=- cos m, and you will show that the formula extend& to the third quadrant, and so on. 7. Not having been sufficiently advanced in the theory of trigonometrical lines to explain the construction of the tables of sines, tangents, &c., at Art. 38, the explanation is here given. The diameter of a circle be;ig multiplied by -ir= 31415926 the product is the length of its circumference; this divided by 360 gives us the length of one degree, and this by 60 the length of one minute of the circumference. So small an arc as I' may be considered as equal to its sine, without sensible error. Having thus found the sine of 1' the sines of other arcs may be found by a formula which will now be deduced. To determine the sine of (n +- 1) a, in terms of n a, (n - 1) a and a By formulas (3) and (7) of Art. 70. sin (b + a) = sin b cos a-+ sin a cos b (a) sin (b-a) = sin b cos a-sin a cosb (b) Adding these two equations there results sin (b -- a) +- sin (b- a) = 2 sin b cos a () Subtracting sin (b - a) from each member there remains sin (b-+a) = 2 sin b cos a-sin (b -a) Let b -n a, then the above becomes sin (n + 1) a = 2 sin n a cos a -sin (n - 1) a ( In the last expression let n = 1; then n - 1 = 2, n — - 0. sin2 a = 2 sin a cos a —sin 0 (m) = 2 sin a cos a the same result as in (g), if a = b. Let n 2; then n 1 =3, n-1 1; and sin 3 a = 2 sin 2 a cos a-sin a = X 2 sin a cos a Xcos a-sin a = 4 sin a cos2 a-sin a = 4 sin a (1-sin2 a)-sin a = 3 sin a -4 sin3 a (n) GENERAL FORMULAS, 91 Let -=3; then n- 1 4, n-1- =2; And by formula (m): sin4a 2 X sin 3 a X cos a-sin 2 a 2 (3 sin a - 4 sinS a) cos a-2 sin a cos a (8 cos3 a —4 cos a) sin a. Jontinuing the same process, we find successively sin 5 a, sin 6 a, &c. To determine the cosine of (n + 1) a, in terms of n a, (n — 1) a, and a. By formulas (6) and (8) of Art. 70. cos (b -+ a) = cos b cos a-sin b sin a (c) cos(b —a) = cos b cos a + sin b sin a (d) Adding these two equations, we have cos (b - a) + cos (b-a) = 2 cos b cos a Subtracting cos (b - a) from each member there results cos (b - a) = 2 cos b cos a - cos (b —a) If b'= n a, this becomes cos (n- 1) = - 2cos n a cos a- cos (n-1) a () a the formula (o) let z = 1;.. n +-1 =, n —I 0 Then, cos 2 a = 2 cos a cos a —cos 0 2 cos2 a 1 et n 2, then n-+ l =-3, n- l=; and cos 3 a = 2 cos 2 a cos a —cos a = 2 (2 cos'2 a- 1) cos a- cos a = 4cos3a-3 cosa () Let n=3; thenn+-l -4, n-l 2; and cos4a = 2cos 3 a cos a cos2 a - 2 (4 - cos: a 3 cos a) cos a-(2 a 1 8= 8C04 os4a —8 32 a+l Continuing the same process, we may find, successively, cos 5 a, cos 6 a, &c, Making a equal to 1t (m) becomes sin 2' 2 sin 1t cos I' observing that cos 1' =- v/T-sin2 1' (Art. 72); we have thus the value of the san 2'. From (n) sin 3'- =3 sin 1'-4 sins 13 and so on. The cosines are calculated in the same manner from (o) or from tha sines by the formula cos = 1- sin2 92 APPENDIX 1. She tangents by the formula (Art. 32), a sin tan = Cos the cotangents by (Alt. 34), R COS cot = - sin It is manifest that by continuing the above processes the numerical values of the sines and cosines, &c., of all angles from 1' up to 900 will be obtained. The above arithmetical operations are laborious. It is evident that an error in the sine or cosine of the first arcs found will involve errors in the sines, cosines, &c. of all succeeding arcs. Hence the necessity of some check on the computation. For this purpose formulas are employed, called formulas of verification'o Sin2 a -- cos' a - 1 And 2 sin a cos a = sin 2 a Hence, sin a-: sVl — sin 2 a: Vi-s in 2 a cos a = Vi +- sin 2 a F: -- sin 2 a Now if we suppose a = 120 30' sin 120 30' ^V1 + sin 250 -j V 1 —sin 250 cos 120 30' -,/1 +sini l- sin 25 If the values of the sine and cosine of 120 30', and of the sine of 250 obtained by the method previously explained, when substituted in these equations, render the two members identical, the results are supposed correct. The values of the sine and cosine of 300, 450, 600, &c., may be used as formulas of verification. The iollowing is known as Euler's formula of verification, which we give without demonstration. sin a - sin (360 - a) + sin (720 - a) - sin (360 +-a) + sin (720 - a) The following is Legendre's formula. cos a = sin (540 + a) + sin (540 - a) -sin (180 - a)- sin (180 -a) To exemplify the latter, make a = 130, then cos 130 =sin 670o - sin 410 —sin 31o-sin 50 Substituting for these their values from tab. nat. sines 97436 = 92050 + 65606 -51504 - 08716 8, The iaes, cosines, &c., may be calculated by series. GENERAL FORMUTLAS 93 T develop sin x and cos x in a series containing the ascending powers of x, by MeA method of undetermined co-efficients. The series for sin x must vanish when x = 0, therefore it can contain no term independent of x, nor can the even powers of x enter into the series; for suppose, sin x = A + t-B x2 + c 3 + D x' — E X3 mbstitute (- x) for x and the above becomes sin ( —) = -A x +.B X — c - D X x s 4 -x - but sin (- x) = - in r -A -B X -cx C s x -c - E x -.'. B — B, D = - D, &c., absurd unless B = 0, D = 0, &e.'. sin x = A x - c x3 + E x + G x - (1) Again, the series for cos x must = I when x = 0, its first term must be I; and it ean contain no uneven powers of x, for suppose cosx - 1- A x+ B 2 -+ c x31- D t then cos (-~ x) = -A xZ B x_ -c x3 + D X41 - but cos (~ x) cos x = + 4A X ~- B 2 — 2 C X -+ D X4 -.'A a A C C A,,. A 0, C = 0'. cOS =1 + - B 2 + Dx4 + (.) Adding and subtracting (1) and (2) cos x +- sin x = -- A X -{+ B 2 - c X3 -+ D -X4 E x5 - (3) Cos -sin X = 1 - A X- B x- C X3 + D 4 -E X +( (4) In equation (3) substituting x -- h for x, it becomes cos (x - h) -- sin (+-{ h) — A (x h)+B (x-h)2+ (x+h)- (5) but cos (x + h) -+- sin (x -- A) = cos x co -sin sin sin x cos h - cos sin h = cos h (cos x + - sin x) + sin h (cos x- sin x) = (1+ B h2 + D 4+) ( + A X - B 2 -+ 3 +) + (A + -c h +E hS +- ) ( AX + B X2~ C X +) + A X + B X2 +C X3 + + A h -- A2h + A B X2 h - - B h2 + AB Ah2 + ( +c h3 + +J braparing (5) and (6) we have 1-AX+B X2+CC3X + 1-AX- B2 4-cX3 + \ + A h + 2B xh 3c xhA - - A - - A A A xh + A B X2h - + B A2 + 3C xh2 + B 2 +AB h2+ + ch3 + -+c h - I J $ - +J and equating the coefficients of the terms containing the same powers of x ho ~~~g~94 ~APPENDIX I. AA A2 2B=- AA;.-. B=' 2 1-2 AB A3 3C. AP C = A 3 1.~23 A C A4 4D= AC D =- - + — 4 1-2*3.4 A D A"5 5E A D E -= 5 1.2'3.4'5 A3 A5 A7 hence sin x x 3 123.4 x 1.2.3.45 627 3 - A2 A4 A6 cos 1 - 13 X2 + 123.4 1- 234^ 6 + It remains now to determine the value of A. For this purpose we have A3 A5 sinl - = A- - +~x 23x3'45 x A2 A4 =A X - 1 x2.3 q12'3'4'5 x4 If z be very small, the first member sin x - x and the terms in the parenthesis: after the first vanish. Hence x - A x.'. A 1, and xS sx x" sin x - x-'23 345 1 2 3'4* 6 (1) Xz X4 X6 Cos X 1 -12 + 12.3.4 -234-5- 1. + (2) Now, the length of an are of one degree is so small that if x be equal to this, the third term of the above series will contain no significant figure in the first ten places of decimals. Retaining therefore only the first two terms of (1) we have, when x is small, X3 X2 2 X4 sin x =x — x- (1- 23 )= 1 -- 23.4 nearly that is, the quantity within the brackets being - cos x, by (2) sinx=xcos x; therefore introducing radius to render the expression homogeneous,we have log. sin x =log. x- (10-log. cos x) (3) Suppose the are x to contain n seconds, then n' 180 X 60 X60 " Dividing (1) by (2) we have tan xx' x + x. x SMALL ARCS,. etroducing radius and applying logarithms log. x = log. n + log. 3*14159, &c. 10 — log. 180 X 6b0 = log. n + 4*68558; substituting this value of log. x in (3) it becomes log. sin x =log. n +- 468558 -- ar. comp. log. cos x. (4) Hence this rule. To the logarithm of the are reduced into seconds add the constant 4'68558, and from the sum subtract one-third of the arithmetical complement of the log. cosine; the remainder will be the logarithmic sine of the given are. 9. To find the logarithmic tangent of a very small are. By the last article sin x sin x = x cos x. -- tan x =-. - - COS x cos a; Introducing the radius and applying logs. log. tan x = log. x +- - (10- log. cos x) The second member of this equation may be formed from the second member of (3) in the last article, by adding the arithmetical complement of the log. cos x; therefore from (4) log. tan x = log. n +- 4*68558 + - arith. comp. log. cos x (5) hence this rule. Add the logarithm of the arc reduced to seconds, the constant 4-68558, and two-thirds of the arithmetical complement of the log. cosine, the sum is the log. tangent required. 10. To find a small arc from its log. sine. From (4) Art. 8, Appendix I., log. n = log. sin x — 4"68558 + ] arith. comp. log. cos x = log. sin x + 5'31442 + -t arith coimp. log. cos x- 10 Hence the rule is this. Add the log. sine of the arc, the constant 5*31442, and ~ of the arithmetical complement of the log. cosine; subtract 10 from the index of the sum, and the remainder will be the logarithm of the number of seconds in the arc. To find a small arc from its log. tangent. From (5) last art. log. nP log. tan x -468558 - i arith. comp. log. cos x = log. tan x + 5*31442 - ~ arith. comp. log. cos x - 10; RULE.-Add the log. tangent of the are, the constant 5131442, and subtract ~ of ile arithmetical complement of the log. cosine, reject 10 from the index, and th result will be the logarithm of the arc in seconds. If we substitute for cos x in the last fraction its value -- we may VI - tan2 xasily deduce x = tan z - 1 tan3 x, &c. 89{6~~ AIrrPENDIX 11. The values of a, which satisfy tho equation sin a == 0 are 0, + r, I 2rf, fi: 3.,.. ffin ir Consequently the series which is the development of sin a must be divisible by a, a --, a - 7, a — 2, a + 2.... (Alg. Art. 238, Prop. II.) If therefbre k be a constant whose value is afterwards to be determined, we have -: k a i (1 a ) (1 + ) 2, (1 +.)... if f 21? 2ir I22 2, k - a2,2 r'32,2..(1 — 2) (1-=-/ —-)... -.. k a,.22 7.32 rr-.. (1a_ ) (.-. ). T ~2 22 2 fa=0, sin a 2 If a 0,, -- 1.'. L =: k,922 7r2*32 f2c... a.'si a=. a - a) (1- ) (... (1),2a2 a3ga' 22 f 322 In a similar manner may be derived 22a2 22 a2 2202) cos=( 1 — 2 ) (1- ) (1 -. ). ( ) 2 32if2 52n2 If a = s r, (1) becomes 1 = ~ (1 i —) (1~42) (1~ ) (1 ~)... 2) ~;)... 2r ~-1 42- 1 62 1 82 1 2 22 42 6 82 (f (2-1) (2+1) (4-1) (4+1) (61) (6+1).=.'.r - - S~)'~ D.' 2 22 42 22 42 6" 82 * 1'3 * 3 5 5.7 7 9 Applying logarithms to (1) we have, making a =m n 2 log. sin 2 =log. ( lo) ltog. ( -log ^ n 2 n 2 22n2 42 n' - log. (1-i ) + * i being 180o, or the semi-circumference whose radius is 1 t This is the theorem of Wallis for determining o. Its successive factors approxi. mate each a more and more to 1. SMNALL ARCS. 97 Developin gog. ( 1 — 4 log. m- 6' &c. Ag. Art. 224 rn m' 22 n m log s - = log., -t-!og. (- ) ma m:* m^ ^^n^ 44n4in 4'/ 8mn4 m6 -— M {''"_. " s~1 C, + f, 7 — + ~ 4-' ) But, log. (- -) = log. mn 4- log. 7 log. n -log. o And log. -- - — log (2n+ m n ) (2n- t)- log. 23 n = log. (2n -4- n) - log. (2n - m) - 2 (log. 2 +- log. n) m -.o. log. sin -- log. nm + log. (2n +- m) +- log. (2n - m) + log a -3 (log. n + log. 2.) o-g c )8. -... ) ~ ( 44 (6 8' ) 71 I+ (+i-+ )a r Y L 46 ^c-, +&c. The e aboe form erves to compute a logarithmic sine at once, without first com2 puting the natural sine. f From (2) in a sirnilar manner may be derived the following formula for calculating logarithmic cosines. lo- expg.resses what fracti )g. n of) log. quadrant th) loar. is. 7 (32 + n +t ( 4 ^ * ) r | + ( +a, 7c T' ) l + &C. ^ M la tle modulus of the systern, equal for the common to "43429448, f - expresses what fraction of a quadrant the are is. n 7 U9 APPENDIX 1. 12. Adding and subtracting (3) and (7) of Art. 70, and adding and subtracting (6) and (8)' of the same article, the following formulas are obtained. sin (a - )-+ sin (a — b) 2 sin a cos b sin (a - b) -sin (a - b) 2 sin b cos a cos (a — b) + cos (a-b) 2 cos a cos b cos (a b) - cos (a -b) -- sin a sin 6 The following two are identical equations a a+ b a b a+b a-b 2 2.~.sina=sin +2 ~ a -b aa —b a-b a b sin - cos 2 -~ sin ~ cos ( 2 2 2 sin b sin a+2 a-~t a. b a- b a - b a+ b a'4~b a-b a-b a~-~ sin coss os sin s a+b a-b. a-b 3 2 2 2 2 2.Co^os a C a=b a -~ - b C B~P h COS 1 2 2 a+. b a -b a+ b a-b - cos 2 cos- +sin —- sin 2 (4) ~osb~=oso2 I 2 2 22 a -b a —b a 4-b - a — b = COS cos + sin sin 3 Adding together (1) and (2) there results a+b a-b ( sin a + sin b = 2 sin a cosa- 2 2 Subtracting (2) from (1),.a-b a ba (6 sin a - ssin b =2 sin a2 - cb ost 2 2 Add together (3) and (4), a +.b. a -b os a + cos b 2 s c os cos - GvNEKIAL FORMULAS. 99 Subtract (4) from (3), cos a - cos b -= 2 sin a b sin < (8 2f 2 These formulas might have been immediately deduced from the group (q)~ by a+ b a-b changing a - b into a, a - b into b, a into 2-b into 13. The following are various forms of expressing the principles deduced from Art. 14 to Art. 31 inclusive. sin 0 0- sin (1800 - a) -sin a cos 0 I cos (1800 q- a) -cos a tan 0 =0 tan (1800 + a) =tan a cot 0 m o cot (1800 + a) cot a sec 0 = sec (1800 a) — sec a cosec 0 = O cosec (1800- a) -= cosec a sin (900 a) = cos a sin (2700 -a) - cos a cos (900 — a) sin a cos (2700 - a) =-sin a tan (900 ~a) cot a tan (2700 -a) cot a cot (90 -a) = tan a cot (2700-a) = tan a see (900 - a) - cosec a sec (2700 -a) = -cosec a cosec (900-a) = sec a cosec (2700-a) =- see a sin 900 =1 sin 2700 1 os 900 = 0 cos 2700 0 tan 900 =a o tall 2700 — oo cot 900 0 cot 2700 - 0 sec 900 o sec 2700 - oo cosec 900 - 1 cosec 2700 = — 1 sin (900 + a) = cos a sin (2700 + a) -cos a cos (900 + a) - -sin a cos (2700 o- a) sin a tan (900 + a) = -cot - tan (2700 - a) = — cot a cot (900 + a) = — tan a cot (2700 +a) = - tan a sec (900 + a) -cosec a sec (2700- - a) = cosec a cosec (900 + a) see a cosec (2700-Ft) - see a sin (1800- a)' sin a sin (3600 -a) — sin a cos (1800 -- a) - co a cos (3600 - ) - cos a tan (1800 - a) -= tan a tan (3600 -- a) = -tan a cot (1800 - a) - cot a cot (3600 a) - cot a sec (1800 - a). sec a sec (3600 -a) - sec a cosec (1800 —a) = cosec a cosec (3600-a) - cosec a sin 1800 = 0 sin 3600 0 cos 1800 =-1 cos 3:600 tan 1800 = 0 tan 3600 = 0 cot 1800 =- oo cot 3600 - oc sec 1800 = —1 sec 3600'- 1 cosec 180 = 0 cosee 3600 -- oa * All the infinities in the above table should perhaps strictly have the doubtful sign since they are the transition value. 100 APPENDIX 1. mn a = (sin n 3600 + a) - sin (2n 1800 — + sia sin (2n + 1) 1800-~a I in a = — in (2n + ) 1800 + a Wi a - sin (2n*1800 —a) ooa = cos (2n.1800 + a)or-cos (2n + 1)1800-a or — cos (2n + 1) 1800 +a or cos (2n1800-a) an a ^ tan (2n.180 + a) or- tan I (2n + 1) 1800 —a or tan 02n + 1) 1800 } a or - tan (2n.1800 - a) a.. see (2n.1800 + a) or - sec (2n + 1) 1800-a - or- sec (2n + 1) 1800 + a } or see (2n1800 - a) 14. The following are the most useful general relations of arcs or angles deduced m, the preceding pages or deducible from the formulas which they contain, TABLE I1. (1.) sim (a _ b) sin a cos b i sin b cos-a (2.) cos(a - b) cosa cos b sin a sin b tan a + tan b (3.) tan (a:: b) 3.) tao( ) = -1 + tan a tan b 2 tana 2a ec a (4.) 2 a - 2 sin a cos a also _ _ 2 ec I I + tana a seea a 2 cot a 2Vcosec a-. /cosec2 a 1 + cot2 a cosec2 a (5.) c a. cos2 a — c sin2 a =2 cos2 a-1 - sin2 a 1-tan2 a 2 sec2 a cot'2 a - I + tan2 a sec2 a cot2 a +- I cosece a - 2 == 1 - 2 (2 vers a -- vers2 a) cosec2 a 2 tan a ) ta a 1 -tan2 a (7.) sin ao,' a /1o +cos a * 1 Cos a -sinC a s 1 +si a ~' 2 — 1~H1,i+cosa sina 1+cosa GENERAL FORMULAS. 101 a a (10.) il a S 2 sin cos - 2 2 (11.) sin2 a -- (cos2a- 1) (See Art. 18, App. 1.) 1 (12.) sin3 a - (sin 3a - 3 sin a) 1 4-3 (13.) si14 a - -~ (cos 4a-4 cos 2a + - 1 2) 1 5*4 (14.) sins a - 4 (sin 5a- 5 sin 3a + f- sin a) 1 6-5 6.5,4 (15.) sin6 a - (cos 6a -6 cos 4a 6- co s 2a- 123-) 1 (16.) cosa a 2 (cos 2a + 1) (17.) cos3 a - 24 (cos 3a +- 3 cos a) 1 4*3 (18.) cos- a ^ - (cos 4a + 4 cos 2a +- -) 1 5*4 (19.) cos5 a 24 (cos 5a + 5 cos 3a 1- cos a) 1 6,5 C-5,4 (20.) cose a - 2 (cos a + 6 cos 4a + 1-c2 os + 2a.-1 ) (21.) sin (n 1) a = 2 sin na cos a —sin (n-1) a (22.) cos (n + 1) a = 2 cos na cos a - cos (n- ) a a-b a-b (23.) sin a +inb = si n 2 cos 2 a-b a-b (24.) sin a-sin b 2 sin 2 cos 2 a + b a-b (25.) cos a-4- cos - 2 cos 2- os 2 a+b a-b (26.) cos a cosb -- 2 - s i a + b - + b b tal sin a + sin b 2 (7.) sin asin b a-b tan 2 (28.) sin a sin b tan (a b) cos a - cos b 2 si9 a: in b cot - (a: b) cos b - cos a (30.) sin (a + b) + sin (a-b) = 2 sin a cos b (31.) sin (a- b) - sin (a- b) = 2 sin b cos a (32.) cos (a + b) + cos (a -b) = 2 cos a cos b (33.) cos(a+b) cos(a —b) = — 2 sin a sin b (34.) 2 cos 2 -= 2 - &c7. to + 1 rica3. 10Q2 APPENDIX L, (3o) eos 360 = sin 540 = 1 (I + 5) 4 (36.) sin 450 = cos 450 ~-2 (7.) tan 450 = cot 450 1 (38) sin 300 = cos 600 = /3 2 ($9.) os830~ = sin 600 2/3 00.) tan 300 cot 600 - (4a.) cot 300- = tan 600 =_ 3 T1he formulas of Trigonometry may be varied to almost any extent, and the same quantity expressed in many different ways. 15. The following, of less frequent occurrence, may be readily deduced from the ~aWove. (2), $ sin (450 k a) _ cos a i sin a cos (450~:L a / - (3.) tan (450 a) 1 tan a a 1 sin a (44.) tan2 (450 1 f si a a I - sin a cos a (45.) tan (450 2 co) s sin sin (a + b) tan a - tan b cot b - cot a 6.) sin (a - b) tan a-tan b cot b-cota (cos a + b) cot b- tan a 1 -tan a tan b cos (a b) - cot b 4- tan a 1 + -tan a tan aV —1 -.V2 a ^ -Sag-i (48,) sin a =2 - aV-1 —.a T (49,) cos a = (50.) Ea -1 Cos ao+ / —lsin a (51.) -a/-1 cos a- v'~-1 in a cos a os a b a -b (5.) ~ ~~- =-= cot cot cos a cos b 2 2 sin (a - b) (53.) tan a + tan bs a os b ~^~''cos a cos b sin (a + b) (54.) cot a + cot b - sin (a b~^lgD~'50& * -~- G~t sin a sin b a s is the base of the Naperian system of logarithms. For the demonstration of te first two of these four formulas see next article. GENERAL FOrMULAS. 103 sin (a-b) (55.) tan a — tan b cos a cos b sin (a- b) (56.) cot a-cotb sin a s sm a smin b (57.) cos a sina l1 +-sin 2 a (58.) cos a-sin a 1-sin 2 a If a be less than 450 then (59.) cos a = ~(59.) 5C03 =- {t1 + sin2a + /1-sin 2 at (60.) sin. 1+ sin 2 /l 2a - sin 2 a} tan a - tan b sin (a- +b) ) tan a -tan b sin (a-b) cos a - sin a (62.) -- sec 2 a - tan 2 a cos a - sin a (63.) sin (a +- b) sin ( - b) = sin2 a -sins b cos2 b -cos a (64.) cos (a + b) cos (a - b) = cosa a-sin2 b = cos2 b - sin a (65.) sin 3a = 3 sin a —4 sin3 a (66.) cos 3a = 4 cos3 a -3 cos a (67.) cos 29 a 2t 1 + tan 2 a tan a sin (a - b) sin (a-b) (68.) tan2 a - tan2 b = -s2 a cos2 a cos3 b sin (a +- b) sin (a- b) (69.) cot2 a -cot b= -s 2 - sin' a sin2 b 1 cos a + -- cos a sin a (71.) 1- -cos = tan a 1- cos a 1 + cos a (72.) sin =cot sin a sinl a (73.) - -- = cot a6 I- cos a 1-cos a (74.) -si -a tan ~ a ^' sin a a3 a5 a? (75.) sin a'-a- + i*21345 1234567 + &c. a2 (t4 a6 (76.) cos a -- + 1-4 - 3- 6 &c. - 12, + -25-1.2.3.4' 5.6 1q23 4&5. 1 2 17 (77.) tan a= — - a3+ - a5-+ 5a7 + &. -11 3 (78.) z= are (sin =y) or sin / Y 6 y 3 + 40 y5 + &e. (79.) z - cos x (I-x)+ ( - X3) + -0 (I &C. 6 40 -1 1 1 (80.) z =-tan t = t - 3 t3 + 5 t &c. 15. The following will be found a useful table of reference. n o sin y are read the ar whose si y * Arc (sin = y) or sin y are read the are whose sin is y. 104 APPENDIX I. TABLE OF THE MOST USEFUL ANALYTICAL VALUES OF SIN a, COS a, TAN a. YALUES OF SIN a. VALUES OF COS a. VALUES OF TAN a. 1. cos a tan a sin a sin a 16. — 31. eos a tan a cos a.32. 3. V-~.coS, a 18. 1 - sin a ot a i4! 19. 33 / - 1 ==::= ~ ~ ~-3. -c —os a' / I+cotV I -1- tan2 (T cos a.1+ 1+cat2 a ltana. tan a 20. co - 34. a — 5',/lfTan2^ 1 -+cotia V 1 -sin. a VI v/l+tan2 a, 62 sin - cos 2 2 35. — cs a 2 2 3 cos a 7. V1-cos2a 22. 1-2sin 2 2 tan a 2 _ 36.! t a 23. 2 cosa 1 1-tan a 2 2 2 I8. I + tant /1 -cos2 a i'"',~ 99 37. _ cora a_1 9.c a 1tan2a 2 2 9 2 2 5. 2 10 a^ts^~o^-j-F) —sln^ ) 1 an+ a 38. a a -ll.~OfZ2 cot tan /3 2 2 11. 2 sin's (450 q a)~_I cot tan a 39. cot a-2cot 2 a 26. -- a cot a tan a1 cos a 121. 1 f sin2 (450 ) 2 2 40. --'1~~~~ 2^~~~ ~ ~sin 2 a t 1 tan2 (45~ --- )1 i _t an (4.50~) 27. sin 2 a 13. I- 1 + tan a tan a 41. 1a 21 - cos 2 a 1 1 -tanr (4-50 —-) ) tan (450-+ ) — tan (450o -- ) 8+c' /(5+- I cos 2 a'~2', tan (450+ )+ecot (450+') I 14.. ita1 (450 + )an (450-2) 43.tan (450-+ -) tan (450-) i 2)29. 2cos(450o+)cos(450-) 22 2 2 2 ~ ~"15. s(600+a) —in(600-a) 30. coF (60~+a)+cos (60~-a).. w. _. u _ _ _ _.... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ GENERAL FORMULAS. 105 16. To develop A* in a series of terms arranged according to the ascending vowers of x. a~ 1 + -(a 1) by the binomial theorem x- 1 (x- 1) (x-2) I +x (a-1)+x 2 (a —1)2+ - X - (a- )3+.,. Representing the coefficient of x, which is (a - 1)- - (a - 1)2 3. (a 1)3. by A, those of x, x3... by B, c, &c. the above becomes a" - 1 -+ x -+ BX2 + CX3 +.. (1).'. az = 1 + AZ + Bz2 + cz3 +.. + 2.-. a - (1 + AX -+...) (1 +AZ + B2...) by ( 1) I + A (X + ) + B (X + Z)2 +.. Equating the coefficients of xz, x2z, x3z, &c. 2B = A A.'. B - ~ A3 3c = n B C 1-2-3 &c. But by Alg. Art. 222, A Nap. logarithm of a. Denoting a Napierian log. by i (1) becomes (la3 (1()3 l + la x + (1- X2 + c-.23 X3. * * Making a = the base of the Napierian system this becomes X22 2X3 + = + _ 2- +1 12.3 + (3) From the above may be deduced remarkable expressions for the sine, cosine, and tangent of an arc. From (3) ~a /- - l a2 a3 a4 1*2 1 2-3 122-3'4'V-1 ~, 2,23 I23a V4 a +- - a +v-13 a2 aa + 6 = 2 (1 — a- ) 2 cos a 1-2 1-2-3-4 (App. I. Art. 8.) a i~ -a ~ -_ -— 3 1.2*3 106 APPENDIX I. /a / V- 1 (4).'. Cos a = ~ (s +- ) 1 a i - Va-ia T sin a ( - s ) (5) Dividing (5) and (4) and multiplying numerator and denominator by g we have 2aV-1 1 s -1 tan a V-1I 2aV/-1 +1 17. Squaring the expression cos a + sin a /1 (1) Lhere results co2 a-sin a+ 2cos asin a - 1 =cos2a +sin 2a V/2 Art. 71, (1) (2) Multiplying by (1) we get, applying the forms for sin and cos of (a + b) (cos a- sin a - )3 = cos 3 a - sin 3 a /Finally, proceeding in the same way would be obtained (cos a + sin a /-1 )n = cos na + sin na /-1 (2) Making a negative (2) becomes (cos a-sin a V/- l)= cos na- sin na /~- (V) 18. To find the n' power of the sine and cosine. Assume cos a + sin a /- 1 =z' cosa-sin a - I =u ( By addition, 2 cos a = z + u e, 2" cosa ar Xn nz uz u z 3 + ~.2. n + - 1 2 -43+ - - uu n nl(n U2 2 n 6za a U+ But z u —.'. z 2 u- 1 &c... these factors will disappear. Also by the preceding article, form. (2) z" = cos na q- sin na V - 1 zn -- - cos (n -2) a + sin (n - 2) a / GENERAL FORMULAS. 107 Making these substitutions n (n- 1) n (n-1) (n —2) cos a = cos nc o - n cos (n-2) a +. os (n- 4) a (~ 2 3 cos (n —6) a +.. (2) n (n- 1) Tn(n —1) (n-=2) + sin na nn (n a sin (n — 4) a +,n 2n 3 -~ 2- sinn —) 3 sin (nm-6) a +-.. o 1 ~.1 When n is a positive whole number, the second part of the series vanishes, and the first part extends only as far as a term containing the factor (n -n). The reason why the second part vanishes will be seen by considering that the arcs in the terms preceding that which with the following terms contains n- n will be negatives of the first, second, &c. terms; their sines will be equal therefore with contrary signs, and the coefficients are also evidently equal at equal distances from the centre, and if n be even, the arc in the middle term, and consequently its sine, will be zero. Again, take the difference of equations (1) and multiply it by - -- 1 2 sin a= (z- u) (- -- 1) and the binominal formula proceeding as above gives cos na- n cos (n - 2) a - ( con ( -X4) a.. 2o sin" a = + sinna-n sin (n-2) a+ n (n -) sin (n~-4)a v 1 1*2.1 (- V/- )" (3) If n be even the second part of the series vanishes. If also n be of the form 4n, (-V/1)n = 1. But if n = 4m + 2, then ( —/-1) = - 1, and our formula becomes 2 sin a=- - CO S con s1?os(n- 2) a+n ( 1 cos (n- 4) a-. The upper sign applying when n = 4m, the lower when n - 4m + 2 If n be an odd number, the first part of (3) vanishes, and the formula becomes, /=-l outside and inside the parenthesis uniting, 2" sin^ a -= sinna -nsin (n-2) a+ n(n sin (n-4) a... the upper sign of which is to be used when n = 4m + 1 and the lower sign when n=4m + 3. For examples of the applications of these forms see page 101o PROBLEM I. 19. To determine the area of a plane triangle when any three parts except the three angles are given. 1. Let two sides, a, c, and the included angle B be given. (See fig. Art. 67.) 108 APPENDIX 1. The area of the triangle is expressed by j BC.AD; but AD = AB sin B; hene e't expression for the area, in terms of the given quantities, is, area a c sin n (1) 2. Let two angles, B, A, and the interjacent side c, be given Then, since sin c n n A: c: a we have sin A sin A sin a —. c.'. ac sin B e sin c sin c hence the expression for the area is by (1) sin A sin B area. c2 sin c 3. Let the three sides be given. By Art. 73, (2) sin ( s-a) ( s- c) V?/ ac Also, by (4) of the same article cos B s (i s-b) ac 2.____.______________ o. 2 sin i B cos A B, or (Art. 71) sin B- a Va s (As.- b) (A s —a) (A s-c) Consequently, by substituting this value of sin B in (1) we have area - =V s (Is- a) ( - sb) (-s- c) which formula furnishes the well known rule, given in all books on mensuration, foa the area of a triangle when the three sides are given. These expressions for the area of a plane triangle are all adapted to logarithmic computation. 20. In the solution of certain astronomical problems involving the case of soltion of triangles at Art. 76, the logarithms of a and b are given, but not the A -a ides themselves; we can very easily calculate - — W- without knowing the sides A-B a-b c tan - cot 2 a + b 2 1-_ a c -— = -- cot -+ b -a Assume tan a a A-B 1-tan 0 c ta _T 1 1 tan 2 PRACTICAL PROBLEMSo!. tan (450 - ) cot o. iog. tan log. tan (450 - + log. cot c _ log. a. 2 "2 The angle 0 is known from the equation tan a Whence log. tan $ = log. R + log. b - log. a The angle A —_ thus becomes known from the logs. of a and b, without calca iating a and b. In the same way we may have A-B C cot __ = tan (450 - t) tan c 2,oD oo n _ d log. C..Ad.g. co. log. tan. (450 - +) + log. tan log C 21. A person on one side of a river observes a building CD on the opposite side, and takes the angle of elevation B = 550 54P, at the place wlhere ho stands, then going back the distance BA = 100 feet, he again takes the angle /~f~: /: ~':of elevation A, and finds it to be A.....ny___ /_ ~ ~ 330 20' The height of the building is ~,''':' I I. required. The problem may be solved as follows. Taking CD for radius, Dn will be the tangent of the angle DCB, and DA, the tangent of acA, therefore, AB is the difference of those tangents. By the table of natural sines *ad cosines,* nat. tan 560 40' = 1.520426 nat. tan 340 6' = -677091 difference ='843335.'843335: 1: 100: 118'57 Ans, PROBLEM II. From the top of a mountain three miles high, the angle of depression of a line tangent to the earth's surface is taken, and found to be 20 13' 27"; it is required thence to determine the diameter of the earth, supposing it to be a perfect sphere. * A table of natural tangents which some collections of tables contain is often eonvenient. 110 APPENDIX I. Let o be the centre of the earth, BA the mountain, A AC the visual ray or line touching the earth's sur-..........- E face at c. Draw the tangent BD, and join oD, oc; then the angle of depression EAc being given, we C have also the angle BAD, the complement of it, equal to 870 46' 33". Also since the tangents BD, CD, are equal (Geom. p. 83), we have the angle BOD = DOC = ~ comp. A = 10 6' 43k", and, there- O fore, BDO = 880 53' 16l". Now in the right-angled triangle ABD we have BD AB tan A; and in the right-angled triangle oBD, OB = DB tan BDO; hence, by substitution, OB R AB tan tan BADO the computation is, therefore, as follows: A B 3 log. o047712 A 870 46' 33" log. tan 11'41074 BDO 880 53' 16-}" log. tan 11'71193 OB 3979.15 log. 3*59979 hence the diameter is 7958'3 miles. PROBLEM 117. Given the distances between three objects, A, B, c, and the angles subtended by these distances at a point D in the same plane with them; to determine the distance of D from each object. Let a circle be described about the triangle ADB, and join AE, EB, then will the angles ABE, BAE, be respectively equal to the given angles ADE, BDE (Geom. p. 44), thus all the angles of the triangle AEB are known, as also the side AB; we B, may find, therefore, the remaining sides AE, EB. A Again, the sides of the triangle ABC being known, \ we may find the angle BAC; hence the angle CAE becomes known, so that in the triangle CAE we shall have the two sides AE, AC, and the \\ included angle given, from which we may find the angle AEc in fig. 1, or the angle ACE in fig. 2, PRACTICAL PROBLEMS. Il and thence its supplement AED or ACD; this with. R^^~~> ^^^ ~ the given side AE and angle ADE, in the first figure, or with the given side AC, and angle ADC in the second, will enable us to find AD, one of the required lines, and thence DC and DB the other two. Or the solution may be conducted more analytically as follows: Put x for the angle DAc, and x' for the angle Hor^ _^^ DBC; also call the given angles ADC, BDC a and a' then a, b, c, representing as usual the side, opposite to A, B, c, we have sin a b sin a a = ~ (1) sin X DC sin x' DC sin a sin x' b (, =.'.i. a sin a sin xI = b sin a' sin x ( sin a' sin x a This is one equation between the unknown quantities x, x'. Another is easily obtained; for since the four angles of the quadrilateral ABCD make up four right angles or 3600, we have x -- + xI a + a' - ACD + BCD = 3600; the sum of the two latter angles may become known, since in the triangle ABC the angle c is determinable from the three given sides; therefore all the terms in the first member of this equation are known except x and x'. Call the sum of the known quantities 0, and we shall thus have x'= - Ax, and, consequently by substitution, equation (2) becomes a sin a sin (f - x) - b sin a' sin x a sin a (sin A cos x — cos sin x) or dividing by sin x b sin a' = a sin a (sin, cot x- cos,') b sin a' cos f.. cot x =-. + __~ a sin a sin 3 sin 0 b sin a' = - cot f a sin a sin / The first term of this second member may be easily calculated by logarithms, and this added to the natural cotangent of 0 gives the nat. cot. of x, and thence x' is known from the equation x' = B- x, and CD from either of the equations (1). This problem has a useful application in the survey of harbors. Let the angles be taken with a sextant, from a boat, at a point where a sounding is made, to three stations on the shore. After having drawn upon a map the triangle, of which these three stations are the vertices, the following simple and elegant construction will determine the point where the sounding was made. Upon the line joining two of the stations, on the map, make a segment, capable of containing the angle observed from the place of sounding, and subtended by this line (Plane Geom., Prob. 21); upon a line joining one of these two stations and tlh 1 12 APPENDIX Lo third, make another segment that will contain the angle observed to be subtended by this last line, and the intersection of the arcs of these two segments \will determine the point on the map, corresponding to that at which the sounding was made. PROBLEM IV. Given the angles of elevation of an object taken at three places on the same horizontal straight line, together with the distances between the stations; to find the height of the object and its distance from either station. Let AB be the object, and c, c', c", the three stations, then the triangles CA, B cIA, sCA A, will all be right angled at A; and, therefore, to radius 1A, AC, AC', AC", will be the tangents of the angles at B, or the cotangents of the angles of elevation; hence, putting a, at, a't, for the angles of elevation, x for the height of the object, and a, b, for the distances cc', c'c", we shall have C AC — Ct a, cot a', AC" == x cot a" Now, if a perpendicular aLr be drawn from A to cc", we shall have (Geom., p. 34,) from the triangle AC= AC.c2+ C- c2C 2 cc. c'r; azld from the triangle AC'c" AC"2 = AC't + Cfc'2 + 2 Ct2"c' * CP; that is, we shall have the two equations x" cot2 a -x- cot2 a' -+ a"-2 a Ct' x2 cota a"l - x cot2 a' + b2 - 26b c' in order to eliminate c'p, multiply the first by b, the second by a, and add, and ws shall have x2 (b cot6 a+ a cot" a") (a + ) 4x cot a' + ab (a + b) ab (a + b) b cot2 a + a cot' a" - (a q- b) cot' a' If the three stations are equidistant, then a = b, and the expression become. a V/- cot2s a + g cot'2 a -_ cot2 a The height A B being thus determined, the distances of the stations from the objeet are found by multiplying this height by the cotangents of the angles of elevation. EVASION OF TABULAR ERRORS,;:11 EXAMPLE. 22 Given the hypothenuse a = 6512'4 yards, b= 6510*6, to find c By Art. 64 log. cos c = log. +- log. b- log. d Now log. R= 10 log. b= 3-8136210 13.8136210 log.a- 3,8137411 to.cos c- 9.9998799 c o10 20'50"t Upon inspecting the tables that are calculated to seven places of decimals only, it will be seen that, when the angles become very small, the cosines differ very little from each other. The same remark applies, of course, to the sines of angles nearly 900. In cases, therefore, where great accuracy is required, we may commit an important error by calculating a small angle from its cosine, or a large one from its sine. We must consequently endeavor to avoid this, by transforming the expression employed. In the example before us, c is a small angle which has been calculated from its cosine; we must, therefore, if possible, calculate this angle by means of its sine, or some other trigonometrical function. Now, by formula (8), Art. 72, we have generally /1. cos c In the prcsent case, cos c =- substituting this in the above equation, sin g c = - log. sin i c =. log. (b - a) - - log. 2a- + log. L From which we find * 40' 24" And o c 10 20' 48" Instead of 10 20' 50", as obtained by the former process. Or c might first be calculated from a and b, and then c by means of its sine. No angle which is nearly 900 ought to be calculated from its tangent, for the tangents of large angles increase with so much rapidity, that the results, derived from the column of proportional parts found in the tables, cannot be depended on as accurate. 8 114 APPENDIX L. 23. The following is the demonstration of formulas (3) and (4) of Art. 77. By Art. 69 " =' - b~-Y ab cos c = a-{- b 2- 2ab (2 cos c-1) (a -- - 4.ai) 4ab coa i c ( c + 5)4 I-(_ ( cos c Making M-ft __ COsAC()..nrt. _.cos ~ c (1) a+b we have c? (a4- +b (- ) ( sn ip) (a + b3) cos? c = (a -+ b) cos.: (2),'iTse, forms (1) and (2) are more suitable than (1) and (2) of Art. 77, if b be a~b is very large, or 0 is near 900, unless c is very small, and when such is the case the increase of the tangent is not proportioned to the increase of the arc, so that the ordinary mode of calculating logarithms not exactly found in the tables would bh inaccurate, US& OF SUBSIDIARY ANGLES. 24. Formulas not adapted to logarithmic computation may often be rendered so by tle use of subsidiary angles. Specimens have been given in the last Art. and Art. 77. The following is another example. To adapt sin a cos I cos d cos a -4- sin d sin I to logarithmic computation. It may be written cos d cos a sin a sin d (sin -4 cos --- ) sin d cos d cos a Put the fractional part -t- d =- tan < (0) / Our form thus becomes sin, sin a i snl d (sin -p cos I ) cos' ~- ~ oY +(sin I cos + cos ai ) sin d -s. sin (1 +,) cos i sin a cos, QUADARATIC EQUATIONS. 115 may be computed from (1) by logarithms, and then' -- froom (2), and fronr and I + >, I becomes known. 25. To resolve a quadratic equation by the aid of Trigonometry. T'he general form of such an equation is xa + pr q and the values of x (see A.lg. Art. 183) are X=-2 i 4 q V \ /4q' Put r' 2 1an ) (1).'. x = — /q Ta _P~ ta'-n~ ~ q tan 9 _ a'qsee Cos tan S sin rp But cos 0 + 1 2 cos 0 c o cot sin 0 sin 0 c cos ~ 9 and cos ~-1 -2 sin2 t. ~~in ~= tan k sin 0 2 sin 0 9 cos - 9 I'he two values of x therefore will be x an (2), x2 V/q tran + ( 3) Logarithms may be applied to the formulas (1), (2), and (3). If p and q be negative the following forms should be used, which may easily be 4q deduced in a similar manner. Put ~ sin2 f, then P2. Xi = p (1-1- COS ^) ssp^ CeOS ^ r X, - P (1 - cos p) -p o0s, - 9 26x o sof the spine. ir Foren the coeffsolution of a cubic ation by e large, t trigonometry, mode l sog. A 378. 26. To find the inc'rement of the s-ine, tangent, bc. co'responding to a emal increment of the angle. Let a represent the angle, i its increment, and $ sin a the corresponding inaremeni of the sine. Then w When the coefficients of a quadratic are large, the trigonometric mode of: solutionl is convenienLt ^.$ 'l16 APPENDIX. s 5in a in (a + i) — sin a sin a cos i + cos a sin i- sin- a a cos a sin i- sin a (1- cos i) 2 sin' ~ i cos a sin i ( 1-tan a. -: (Art. 72.) (2 sin" 3 i) = cos a sin i (1 -tan a 2 s i c i)' 2 sin I i cos A v - cos a sin i (1 — tan a tan 2 i) The increment i being supposed very small, tan g i will be very small also, and unles tan a be large, the second term in the parenthesis may be omitted. Then since sin i is equal to i very nearly, i being very small, it follows that the ratio of 5 sin a to i is eos a. In other words, the difference of the sines of two angles is proportional to the difference of the angles when the difference is small. When tan a is large this principle fails, which is the case with ares near 900. The reasoning for a cos a is very similar. For the tangent it is as follows: sin (a - i) sin a J tan a tan (a + i) tan a s= ~ cos (a -+) cos a sin (a - i) cos a - cos (.a + i) sin a cos a (cos a cos i- sin a sin i) The numerator is equal to sin (a + i)- a sin i sin i 1 6. ta. na c a o - - -- sea tan i -t - - cos a cos i ( - tan atanii) 1 tan a tan which if i be very small, and a not near 900, reduces to $ tan a = see2 a tan L EXERCISES. (1). Prove vers (1800 - a) = 2 vers, (1800 + a) vers 2 (1800 - a) (2). Find the numerical values of sin 150, vers 150, sin 90, cos 120, (3). Prove tan 500 + cot 500 = 2 sec 10~. 4). If a + b + c = 900, prove tan a tan b - tan a tan c - tan b tan c = 1 d cot a cot b + cot b = cot a cot b cot c and tan ca +- tan b -+ tan c atan b tan c + ea see se see cos u —-e / (5). If cos v a - c- prove tan v tan uv (6). Prove the radii of the inscribed and circumscribed circles of a regular poly1800 gon of any given number (n) of sides to be for the former r == a cot, a being the' 1800 Rongth of one side of the polygon, and for the latter R = 4 a cosec l This is the formula used in the solution of Kepler's problem in Astronomy. EXERCISESo 17 (.) Prove thn area of a regular circumscribed polygon to be 1800 n r2 tan n That of a regular inscribed polygon to be n 360o - r2 sin 2 n (8). Prove the area of a regular polygon of n sides, one of which a to b 1800 n a2 cot (9). Prove the radii of the inscribed and circumscribed circles of a triangle to be -. /( 8- a) (4 s-b) (js-e) aR - ^' 4 It has already been mentioned that the diedral angles of the trihedral angle correspond in the same manner to the angles of the spherical triangle; and that these diedral angles are measured by the angle of two lines, drawn one in each plane, perpendicular to the common intersection of the two planes at the same point. In order to draw these lines so as to be used most conveniently in the following demonstration, take o M = the radius of the tables; draw M P perpendicular to o A, it will be perpendicular to the plane A o n (Geom. of Planes, Prop. 19), since the two planes A o B and A o c are perpendicular to each other, A being by hypothesis a right angle; from P draw P D perpendicular to o B, a line of the plane A B; join D M; M D will be perpendicular to o B (Geom. of Planes, prop. 8); M D and D P being both perpendicular to o B at the same point D, the angle M D P is the diedral angle of the planes A o B and B o c; or M D P = the angle B of the spherical triangle; o M being equal to radius, M D is the sine of the plane angle a, and M P is the sine of the plane angle b; in the triangle M D P, right angled at r, we have the proportion (Art. 38) R: sin:: M D: M P CELESTIAL CIRCLES. 123 substituting for D its equal B, for M D, its value sin a, and for M i,, its value sin b, we have I: sill B: sill a: sin b That is, the radius is to the sine of either of the oblique angles of a right angled spherical triangle as the sine of the hypothenuse is to the sine of the side opposite that angle. 79. The solution of astronomical problems forms one of the most useful and agreeable applications of the theory of spherical trigonometry, which branch of mathematics has grown out of the wants of Astronomy. To illustrate therefore the above and subsequent formulas of spherical trigonometry we shall introduce a few great circles of the celestial sphere. They are so well known that to define them is perhaps superfluous.. The equator is that great circle the plane of which is perpendicular to the axis of the earth. The axis being the line about which the earth performs its diurnal rotation. This produced to the celestial sphere becomes the axis of the heavens about which all the stars appear to revolve daily. The ecliptic is a great circle which makes an angle of about 230 28' with the equator. It is the path which the sun appears to describe among the stars once a year. The points in which the two great circles above defined intersect are called equinoctial points. The one at which the sun crosses the equator in the spring about the 21st of March, is called the vernal equinox. The other, which is where the sun crosses in the autumn, viz. about the 23d of September, is called the autumnal equinox. Declination circles are great circles, the planes of which pass through the axis and the circumferences of which all intersect in the poles or points where the axis meets the sutface of the celestial sphere. They are alsc called hour circles. The sun appears to move about the earth once in 24 3600 hours; 24 15 is the number of degrees through which the sun moves in an hour. That declination circle, the plane of which passes through any place on the surface of the earth and the earth's centre, is called the meridian of the place. The angle contained between the meridian of a place and that declination circle which passes through the sun at any given moment, is called the hour angle of the sun, and converted into hours, 15~ to the hour, will k124 SPHERICAL TRIGONOMETRY. show the time of day, if we reckon from noon instead of midnight as astronomers do. This time may be either A. M. or p. M. It is what is called apparent time, which varies a little fiom mean time, the time given by the clocks, in consequence of the slightly unequal motion of the sun in its annual revolution. The hour angle of a star is similar to that of the sun. The horizon of any place is a great circle whose plane touches the surface of the earth at that place, and extends to the celestial sphere. This is called the sensible horizon; the real horizon is a plane parallel to this through the centre of the earth. When any of the fixed stars are in question, the distances of which from the earth are so great that its radius is as nothing comparatively, these two horizons may be regarded as coincident. The zenith is the pole of the horizon directly overhead. The nadir is the opposite pole. Great circles passing through the zenith and nadir are called vertical circles. They are secondaries to the horizon. The position of a heavenly body is fixed on the celestial sphere, like that of a place on the globe, by its latitude and longitude, only it must be observed that on the former these are measured from and upon the ecliptic instead of the equator. Similar measurements from and upon the celestial equator are called the declination and the right ascension, the former corresponding to the latitude, and the latter to the longitude.* Longitude upon the earth is reckoned from some fixed meridian, as that of Greenwich. Longitude upon the celestial sphere is reckoned from the vernal equinox which is called the first of Aries; right ascension also from the same point; the former upon the ecliptic, the latter upon the equator. The azimuth of a celestial object is an arc of the horizon, comprehended between the meridian of the observer and the vertical circle which passes through the object. Or it is the angle which these two vertical circles make with each other having its vertex at the zenith. 80. We are now prepared with materials for a practical application of the formulas of spherical trigonometry, and we commence with that already demonstrated. The symbol for right ascension is AR or R. A.; for declination D, or Dec. SUN 8 LONGITUDE. 125 S Let E ill the annexed diagram be the equinoxial point, EQ a portion of the equator, ES a portion of the ecliptic, a the place of the sun, and SQ a portion of a dec. circle through the sun; then SQ will be the o's declination, which denote by S, EQ his right ascension, which denote by a, and ES his longitude, which denote by 1. Given the o's declination* equal to 200, required his longitude. In the right angled triangle EQS right angled at Q we,know E = 230 28' the opposite side SQ = 20~ required the hypothenuse rs. Hence the proportion n: sin 230 28' sin I: sin 20~ R X sin 200 sin I =. sin 23~ 28' 1 200 log. sin 9'53405 iE 23 28' log. sin 9e60012 1 590 11' 26" log. sin 9'93393 Hence =s 590 11' 26" the longitude of the sun required. * The declination of the sun may be found rudely by taking its meridian altitude with the same instrument and in the same manner as was described at Art. 11. More accurate instruments and methods will be described hereafter. This observation should be made about noon repeatedly, and the greatest observed altitude will be the meridian altitude. A piece of colored glass will be required for the purpose. Let p be a place on \ the earth; pq its distance from the equator will be the latitude; this contains the same number \ of degrees as the are zQ between the zenith and celestial equator. - 1 )- Q Let s be the place of the sun, then SQ will be his declination. Let HO be the horizon, then so is equal o's meridian altitude, sz = complement of his altitude, and 0 is called the zenith distance, or coaltitude: SQ = zq - sz or declination = latitude - zenith distance. N. B. The altitude of the uppermost point of the circumference' of the bsu should be first taken, then of the lowermost point, and half their difference added to the latter,or simply half their sum will give the altitude of:the e's centre. 126 SPHERICAL TRIGONOMETRY. Let the student try the following modification of the problem as an exercise. Given o - 90~ to find his declination,* 81. By means of the proportion for right angled triangles, and of which an application has just been given, one may be derived for triangles in general. Let ABC be any spherical triangle; A let fall fron A the arc AD perpendicular to the side BC, the given triangle will be divided into two right angled triangles ABD and ACD. In the right angled triangle ABD we C have te proportion (Art. 78) D: sin: sin AB: sin AD snd in the right angled triangle ACD, the proportion sinc:: sinAC sin AD Multiplying the extremes and means of each of these proportions, we have the equations R X sin AD = sin B X sin and R X Sill AD = sin c X sil AC The first members of these equations being the same the second numbers are equal, hence sin B X sin A B = sin c X sin ac substituting for the sides AB and AC the small letters of the same name with the angles opposite to them the last equation may be written sin B sin c = sin c sin b sin B sin c sin b gin c * This symbol ~ signifies longitude of the sun. :TRIANGLES IN GENERAL. -127 sin nB sin b:: sin c: sin c* that is, the sines of the angles of a spherical trianyle are as the sines of the opposite sides. EXAMPLE. Let z be the zenith, p the pole of the equator, and s the place of a star; zs will be the zenith distance of the star, zrP its hour angle, Ps its co-declination or polar distance, and szP its azimuth. Let the azimuth, zenith distance, and hour angleJ be given, to find the polar distance, which is the complement of the declination; z, zs and a are given, and PS required. SOLUTION. sin: sin z: sin p sin z Let P = 32~ 26' 6", z = 490 54' 38", and zs or p = 440 13' 45" P 32~ 26' 6" ar. comp. log. sin 0'27056 z 490 54' 38" log. sin 9'88369 p 440 13' 45" log. sin 9'84357 z 840 16' log. sin 9'99782 or declination of the star = 5 44' Since the sine of an arc is equal to the sine of its supplement (Art. 15), the required side may be also the supplement of 84~ 16', or 950 44' The dec. would then be 50 44' south of the equator. To illustrate this double solution by the dialgram, let the student make or conceive to be made the following construction. Draw an arc from s making with rz an angle equal to z, meeting pz in a point which we will call z'. sz' will then be equal to sz; prolong Pz and Ps till they meet in the opposite pole, which we will call p'; a triangle will be formed Z'P's * The student will recollect that a proportion is an equality of ratios, and that ratio, As commonly understood, is the quotient of two quantities. The above is familiarly called the sine proportion. t The zenith dist. and azimuth may be observed with a theodolite or altitude and azimuth instrument, to be described hereafter; the hour angle by a sidereal clock. ib28 sSPHERICAL TRIGONOMETRY. in which the angles z' and p', and the side sz' will be equal to those given in the above example, but in which the side P's is the supplement of Ps.* The polar distance of the fixed stars will be found to be always the same, hence they describe circles about the poles in their apparent daily motion. EXERCISE. Given a 420 32' 19", A 480 12', b 550 7' 32" to find B. Ans. B 640 46' 10OB 82. We shall next demonstrate a formula which will express one of the angles of a spherical triangle in terms of the three sides. Let ABC be any spherical tri-. angle, o the centre of the sphere; join OA,, oc; a trihedral angle is formed" having its vertex at o. The plane angles of this trihedral may be called by the same letters as the sides of the spherical triangle for the reason given in Art. 77. By referring to the note of Art. 38, - / o it will be seen that we may choose at pleasure the length of a radius, and the trigonometrical lines will lave the Same relation as those corresponding to the radius of the tables or any other radius. Let us take OA as radius, and draw the perpendiculars AD and ATE / at its extremity and in the planes Aoc and AOB; produce these perpendiculars till they meet the lines o oc and on in D and E. AD Will be the tangent and OD the secant of tle side b of the spherical triangle, and aE the tangent, and oE the secant of the side c. This being premised, let us take the value of DE, in terms of the other sides and one angle, of each of the two plane triangles DAE and DOE, to both of which it belongs. This may be done by means of the formula * A rule for determining when there are two solutions, and when but one in suet ases,:is given at p. 196. TRIANGLES IN GENERAL. 129 os A = bR 2 (Art. 69), from which, taking the value of the square of the side opposite the angle in the formula, we have 2 b c cos A in the triangle EAD this formula becomes 2 9 a 2 ADX A C0S A ED AD2- +- AE2 R and in the triangle EOD t ao 2 DO X EO COS a ED = + DO 2+ E-O ~ X csubtracting the former from the latter of these two equations and obseriang that in the right angled triangles OAD and OAE DO2 -AD =2- OA2 and EO2- AE2 OA there results 2 D X E o cos 2ADXAE COS A 0 = 2 o AO~a -2~ R:R Finding the value of cos A from this equation, we have D 0 X E o COS a- X 0 A' COS A = ~AD XA E substituting for o A its value R for D o its value sec b — = ~ (Art 33), cos b a2 I -sin b for E o, sec C - for A D its value tan b (Art. 32), and for A its value tan c = thR e above expresion becomes COS c - X - COS a -R cR COS a -R Cos bcos e cos b COS c COS A =..,.. R sin b Rsinc R sinbsin cos b cos c or striking out Ra R COS R osa s b cos c COS A = - sin b sin C 9 130 SPHERICAL TRIGONOMETRY. r if R = I cos a - cos b cos c COS A - - sin b sin c But the angle A of the plane triangle D A E is the same as the angle A of the spherical triangle (Spher. Geom., Prop. 4) hence, translating the above formula, The cosine of either* angle of a spherical triangle is equal to radius square into the cosine of the side opposite, minus radius into the rectangle of the cosines of the adjacent sides, divided by the rectangle of the sines of the adjacent sides. The above formula will serve to calculate one of the angles of a spherical triangle when the three sides are given, if we employ the table of natural sines and cosines; but is unsuitable for the application of logarithms, in consequence of the sign — in the numerator requiring a subtraction to be performed, which operation is impracticable by means of logarithms. We shall therefore derive from this another formula, involving only multiplications, divisions, &c., of the trigonometrical lines contained in it, to which operations logarithms apply. 83. At Art. 74 were derived formulas (2) and (4) for the sum of the sines and difference of the sines of two arcs. In a similar manner two others may be derived for the sum and difference of the cosines. See (7) and (8) of Art. 12, App. I. These four forms would be as follows: sin m +{- sin n = 2 sin - (m + n) cos ~ (m -n) (1) sin m -sin n = 2 cos ~ (m - n) sin {m (m n) (2) cos m +- cos n - 2 cos i ( m+ n) cos ( n- n) (3) cos n- cos m = 2 sin ~ (m + n) sin i- (m n) (4) The last is read, the difference of the cosines of any two arcs is equal to twice tae sine of half their sum into the sine of half their difference. 84. A formula for the cosine of half an angle of a spherical triangle, in terms of the three sides, may now be derived. Resume from Art. 82, cos a - cos b cos c COS A sin b sin c Add I to both members it becomes by (6) of Art. 70, cos a - cos (b +c) 1 + cos A= - sin b sin c But 1 + cos A 2 cos2 ~ At ~ We say either angle, because in the above demonstration no particular agh was selected: t Deduced similarly to (8) of Art. 72, by adding (5) and (6) of that article. 'TRIANGLES IN GENERAL. S31 (4) Art. -83, A sin (b - +c + ) sin ~ (b - c - a) cos2 s A ) sin sin cos a x - /sin a s sin A (s —a) (1) i. e. The cosine of half an angle of a spherical triangle is equal to the squ P132) 30 CD is- log. sin 2) l, s o r -c l oog. sin 2) " l - log. sin si |'R EXAMPLE I.-Given a = 1200 17, b = 750 3' c =48o 56to - C2,. a 1200 17' ar. co. log. sine 0.06372 ar. co. log. sine 0'06372 o o' I ~ b 750 3' ar. co. log. sine 0.01495 ar. co. log. sine 0'01495 o 1 e 480 56' ^ ar. co. log. sine 0.12266 ar. co. log, sine 0.12266 11 c 8 244016' 16 o (a s 1220 8' I; 3 | 36 sa 10 51' log. sine 8-50897 log sin 850897 o E 8o - b 470 5' log. sine 9'86472 lo sine 98642 IS II 181 ~~ 3 s - ~c 730 12' log. sine 9*98106 log. sine 9*98106 g. g CRo- r 2)19*98339 0oo o 0 _, | A0 = 780 49' 40f log. sine 4 9-99169 2)18.67641 _6 Q 1 B = 12~ 35' 2".1 log. sine 933820 2)1845236 o I o r > c = 9041' 28'*4 log. sine 9*22618 0J e 9o 3428 dAns. A = 1570 39' 20", B = 250 10' 4"2, c = 190 22' 56'*8 ~ ~~~ TRIANGLES IN GENERAL. 133 This formula, for the solution of a spherical triangle when the three sides are given, is very convenient for calculation by logarithms. Applied to each of the angles separately, it will serve to determine them all. An interesting astronomical application of this case of solution is to finding the time of day and error of a watch by an altitude of the sun or other heavenly body. See Nautical Astronomy, p. 287. 85., It is proved (Spher. Geom., Prop. 13), that the three angles of a triangle being given, the triangle is determined. A formula for calculating either of the sides when the three angles are given, may be easily derived from that of the last article, by means of the polar triangles. It is necessary first to premise that the polar triangles of the wh6ole' range of triangles will include all possible triangles: for as each side of a triangle passes through all values from 1806 to 0o, the opposite angle of the polar triangle will pass through all values from 00 to 1800. Wherefore whatever can be proved of the polar triangles of all possible triangles may be considered as proved for all triangles. Resume the equation before (1) of the preceding article. os- 1;= / sin / l (a + b+c) sin (b + c-a) sin b sin c For the parts of the triangle in this formula, substitute their equivalents in the polar triangle. It will be remembered (Spher. Geom., Prop. 6), that each angle of a spherical triangle-is the supplement of the side opposite in the polar triangle, and vice versa; hence if a', b', and c' represent the sides, and A', Bn, c0 the angles of the polar triangle, we have A - 1800 - a' a 1800 -', b -= 180 —n' and c= 1800-~' putting these values of the letters A, a, b, and c in their places in the formnila above, it becomes )os I (1800 —-a') /sin (5400 -(A +B' + C')) sin (1800~- (B' + I'-A)) sin (180 --') sin (180 - c') But i (180 - a') = (900 -- a') and cos (90 —- a) = sin A a' (Art. 23); also } (5400 -(A'+B' + c') = 270 — ~ ((A' + B'+') and sin (2~70 — (A'+ B'+ c'))= -cos - (A' + B'+ c'); also (180~(Bn'+'-A') =900~- (B' +'-A) and sin (900 - (B' +- c -A')) = cos I (B' + o' A'); also sin (1800 B') = sin B' (Art. 15) and sin (1 80~ - 0') = sin c\. 1 4 SPHERICAL TRIGONOMETRY. Makin, these substitutions, the formula become3..,/ COS (A'-+ I Bi-') CO - (B'+' — ) (A sin B'sin C a formula for the sine of half a side in terms of the three angles of a tangle. We may leave out the accents over the letters, which we have employed only to distinguish the polar from the triangle to which it corresponds, and which are superfluous in a general formula. This formula will undergo a similar modification to that made in the form.ula preceding (1) of the last article. Represent A + B + c by s, and the formula becomes sin-i a - a ii..(_l ). (2) sin B sin c or the sine of half either side of any spherical triangle is equal to radius* into the square root of minus the cosine of half the sum of the three angles into the cosine of half the sum minus the opposite angle, divided by the rectangle of the sines of the adjacent angles. A form for the cosine of half a side may be derived in the same manner from the form from which (2) of Art. 84 is derived. eosa i /- cos.(A -B- -C) cos 0 (A+c — (B) 3) C-oS 4 is L v X (3) sin B sin c kIonm which cos s e / (s-1 c) sina B sin c 86. We shall next derive two sets of proportions applicable to the solution of a spherical triangle, the first set when two sides and the included angle are given, and the second when two angles and the included side. It will be found convenient in the longer analytical processes to represent the angles of a spherical triangle opposite the sides a, b, and c, respectively by s, 3, y. Formula (1) of Art. 84, applied to each of the three angles of a triangle will give * For homogeneity. In the above form - cos a s is always positive, because s is always more than 180P. (See Spher. Geonm, Prop. 14.) TRIANGLES IN GENERAL. 135 sin (b + c + a) sin - (b + c-a) cos* c ~ (1) sill b sin c sin. (a + c + b) sin -I (a + c- b) (2 A COS- ff (2 sin a sin c os sin (a + b + c) sin I (a + b-c) sin a sin b and the formula from which (2) of the same article is derived. 1i - s sin (a + c- b) sin I (a+ b — c) sina: 6 (4) I 2 sin b sin c i3.in2 sinl ( + c-a) sin { (a+ b - c) B^ sin, ~ ~~ (5) sin a sin c Sin ~ (b+ -- a) sin - (a+cb)6) sinL - sin a sin b By multiplying two of these formulas, and dividing by a third will be obtained fo(2) X (3) ct f os i cos sin (a + b +c) from the formula, ~ () 1 4 sin I a sin a i ~ () X (6) sin i 3sin sin (b +c -a) () f (4) sin ac sin a, (2) x (6) cos sin 7 sin - (a + c —b) ] (1) cos a sin ae,(3j ) X (51) sin /cos 7 sin (a+ b —c),, (3 ) -- (10) cos a sin a Subtracting and adding (7) and (8), and also (9) and (10), applying to the first members of the results of these operations the formulas deduced in Art. 70, and to the second members formulas (5), (6), (7), (8), of Art. 12, App. I., and afterwards to these same second members formula (3) Art. 71, we obtain the following forms. os. (3 + y) cos (b + c) sin c a cos I a Tcos - ( -^~) sin - (b + c) sin si ( + + ) _ s c (b —c) cos j a cos a 136 SPHERICAL TRIGONOMETRY. n V. ( —7) sin si (b -c) cos - a sin i a These four forms (and four others derived similarly from repeating for each side formula (1) and (3) of Art. 85), are known as the Theorem of Gauss.* By division of III. by I., and IV. by II., in both sets will be obtained four additional forms known as Napier's Analogies;t viz., V. tan i a tan ( + 7) = os (b 2' *'' cos ~ (b + c) VI. tan a ctan (-)- in (b c) sin 4 (b +c) VII. cot i a tan (b +c) Cos - 7) cos + (/3+ ) VIII. cot a tan - (b-c) - in 4 (f3-) sin -& (,/ + y) These each converted into a proportion or equality of ratios by writing cot -- in the denominator for tan - a in V. and VI., and vice versa, in VII. and VIII., will be as follows: iX. cos - (b + c):cos - (b-c):: cot -a: tan (/3 + y) X. sin - ( 6+ c):sin ( (b-c)::cot I-: tan ( - y).. XI. cos - (/3 + y): cos - ( ~- Y):: tan - a: tan I (b+ c).. XII. sin - (/3 +-):sin - (/3- ):: tan Ia: tan - (b —c).. That is, the cosine of half the sum of two sides of a spherical triangle I; to the cosine of half their diference, as the cotangent of half the included angle is to the tangent of half the sum of the other two angles. The second may be repeated in a similar manner, changing cosine into sine and tangent of the half sum into tangent of the half difference of the other two angles. The 4th may,,be translated into ordinary language thus: The sine of half the sum of two angles of a spherical triangle is to the sine of half their difference as the tangent of half the interjacent side is to. the tangent of half the difference of the other two sides.; Or Gauss equations.. t Analogy is a term synonymous with proportion. The first term bears the same analogy or proportion to the second that the third does to the fourth. TRIANGLES IN GENERAL. 137 The third may be repeated in a similar manner. These proportions were first given by Lord Napier, who is celebrated for many useful inventions of a similar character, but chiefly for that of logarithms. We shall now apply the first set to an EXAMPLE. The latitudes and longitudes of two places on the earth's surface being given to find the angles which the arc of a great circle joining them makes with their meridians, and their distance apart, the earth being supposed an exact sphere. Let P be the pole, s and s' the places, then p PS and Ps' will be their colatitudes, and the angle P will be the difference of their longitudes, since P will be measured by an arc at a quadrant's distance on the equator. (Spher. Geom., Prop. 4.) Let the latitudes of the two places be 51~ 30' and 200; and let their difference of longitude be 310 34' 26". Their colatitudes will be 380 30' and 700. Then we shall know in the above triangle the two. sides opposite s and s' which we will call s and s', and the included angle P. The greater side s' 700, s =380 30' and P - 310 34' 26". Applying forms IX. and X. of Napier's analogies with the use of logarithms, the half sum and half difference of the unknown angles will be obtained, by the addition and subtraction of which the angles themselves may be found. The remaining side p of the triangle may be found by the sine proportion, or to avoid ambiguity, by form XI.* The whole computation is contained in the following table. * For a highly useful practical application of this problem see Great Circle sailing, App. III. C D G ^ Argument. Logarithms for IX. Logarithms for X. Logarithn for XI.'3 ^~ ~ ~ s 38030' s 700 p &pt;c.. ~ (s5'- +) 540 15'* ar. co. log. cos 0'23340 ar. co. log. sin 0*09067 log. tan 10.14273t I I i I31 r O 8 (s -8s) 15~ 45' log. cos 9*98338 log. sin 9.43367 5 p c -o r, 31034'26" 3 o1 0: P 150 47' 13" log. cot 10'54863 log. cot 10*54863 +? % o - (s' +- s)- 800 15' 40" log. tan 1076541 __ log. cos 9.22828 0' 0 ~~ log,- tan490 47~ 98'~ og. t74aI _ I _'+ = | (sl - s) 490 47' 28 log. tan 10'07297 ar. co. log. cos 0-19005 cD' s 130) 3' 8" I I 0 1 I s 300 28' 12" F 1 I | ^ | p 190 59' 58" log. tan 9*56106 0o 0 0 - p 390.59'56" o ~ 0 1 I O - ~- ~ CO c 2. ~ s ~ ^; 5 Had 8 (s' + s) been greater than 900, its cosine must have been negative, and the first term of the proportion being t to ~ 5; 1 negative, the fourth must have been negative also, and I (s' + s) would have been the supplement of the angle found e CD' C M in the tables, since the tangent of the supplement is equal to minus the tangent of an arc. (Art. 36.) It is well 1 to 0 3 ^ always to put a small n against those logarithms belonging to trigonometric functions which are negative;~then the ~ i]} o O sign of the result will be indicated by the rule that an even number of negative factors produce a positive result, and 0*,.. 3 o an uneven number a negative result. TCD - a t This log. need not be found from the tables, but may be obtained by subtracting the ar. comp. of sin % (s' + s) *~" ^^'~' aR sin 6;^~a ~,. above from ar. comp. cos j (s' - s) after adding 10 to the latter; for since tan -, log. tan= 10 + log. sin log. CD: cos o s: ~~ cos. Calling c the ar. comp. log. cos, and s ar. comp. log. sin, this becomes 10 +- (10 - s) - (10-c) - 10 + g g ^~: - -. Q. E. D. TRIANGLES IN GENERAL. 139 site the angle P, that might be found by the proportion (Art'81), the sines of the angles are as the sines of the opposite sides. But to avoid an ambiguity in the result similar to that of Art. 81, and. the trouble of determining which of' the two results corresponds to the other parts of the triangle now fixed, it is better to employ XI. of Napier's analogies, inverting it as seen in the last column above. This gives i p = about 200 and p = about 40~ = 2400 geographical miles, the distance required. EXAMPLE II. Given the moon's R.A. 11 38n 27"'15. Dec. 40 5' 40"'4 N. To find her latitude and longitude, the obliquity of the ecliptic being 203 2' 23"'13. Let P be the pole of the equator, P' that P of the ecliptic, M the place of the moon; / then the angle MpP' 270~ — - R.A. the side <> z p' 23027' 23"2' 2313 the side PM- 90C 40 5 40 "4 To find PP'M = long. of M- 90~ and P'M - 900~: D" latitude* Ans. P" lat. = 10 37' 3"3 N. long. = 1730 25' 57"''t 87. When, in the case considered in Art. 86, the only part required happens to be the side opposite the given angle, the finding of the other two angles then becomes merely a subsidiary operation, and the determination of the required side, by Napier's The summer solstice is 900 from the vernal equinox, and is N of the equator. The pole of the ecliptic r' therefore will be south of r and between it and the 2700 point. The right ascension of the moon being nearly 12 hours is nearly 1800, which fixes its place in the diagram.. t The determination of the latitude and longitude of a heavenly body from its right ascension and declination as above is one of the most useful problems in Astronomy, the right ascension and declination being observed directly with the astronomical instruments as will be explained in a subsequent part of the work, and the latitude and longitude being required for computing the elements of the orbits of the heavenly bodies. 140 SPHERICAL TRIGONOMETRY. analogies, seems unnecessarily long. A shorter method of solution is dedacibie from the fundamental formula, obtained at Art. 82, or cos c = cos a cos b + sin a sin b cos c (l) For substituting cos a tan a for its equal sin a it becomes cos c= cos a (cos b + tan a sin b cos c) Assume cos W COSM tan a cos c =cot W sin; then sin oC cos b + sin b cos 6) os c = os a sin &~ cos a sin (c, -- b) sin C Hence, to find the side c, we must determine a subsidiary angle w from the equation cot w = tan a cos (2 after which c is found by the equation cos a sin (,c +1) cos c = - (3) sin t EXAMPLE. 1. In a spherical triangle are given a 380 30' b= 700, and c 310 34' 28"' to find c. a 380 30' 0' log. tan 9.90061 log. cos 9e89354 c 31 34 28 log. cos 9*93042 t6 55 52 30.5 log. cot 9*83103 ar. co. log. sin 0.08207 b 700 + b 1250 52' 30"*5 log. sin 9-90864 c 400 log cos 9'88425 88. If when two angles and the included side are given, the angle opposite to the given side be the only part required, a similar formula should be employed, deduced as follows. From the fundamental formula (1) above, may be obtained by aid of the polar triangles, the formula cos c -- cos A cos B +- sin A sin B cos Co which becomes when cos A tan A is substituted for sin A, cos c = cos A (tan A sin B COs C - cos B): or assuming COS W tan A cos c = cot,.= sin co sin B cos o.- sin.cos B Cos 0 cos A - - I~ ~ ~ ~ sin ( RBGHT ANGLED TRIANGLESo 141 cos A sin (n- - ) sin en Hence, having found a subsidiary angle A, by the equation cot X - tan A cos c (1) the required iangle is determined by the equation COS A sin (B- o) COs c = - (2) sin X 89. The formulas for the solution of spherical triangles in general, which have now been demonstrated, apply of course to right angled triangles; but if it be recollected that the trigonometrical lines of the right angle or 900 are either it, 0, or oo, it will be evident that these formulas may, when thus applied, be much simplified. The student can easily make the substitutions necessary to change the foregoing formulas into such as apply exclusively to right angled triangles, for himself. We shall not occupy space with them here, but be content with observing that after they have been made, all the formulas which result will be found capable of being expressed in two short rules, or these indeed may be united into a single one.* Amongst all the convenient and useful inventions of mathematicians, none is more ingenious and b.eautiful than this, the author of which is the celebrated. Lord Napier, whose name we already have had occasion repeatedly to mention in connection with the most happy discoveries for facilitating mathematical operations. The rules are known as NAPIER'S RULES FOR THE CIRCULAR PARTS. The circular parts of a right angled spherical triangle ar The two sides including the right angle, called 1. The base. 2. The perpendicular. And 3. The complement of the hypothenuse 4. The complement of the angle at the base. 5. The complement of the angle at the vertex. The right angle being entirely left out of consideration in the solution of triangles of this kind, the angle at the base is that included between The mode of deducing them is given in Appo IL p, 194. 142 SPHERICAL TRIGONOMET'RY. the base and the hypothenuse; and the angle at the vertex is that included between the hypothenuse and perpendicular. The circular parts are then the elements of the triangle itself except the right angle, only that the complements of the hypothenuse and oblique angles are the circular parts, instead of these themselves. The annexed diagram shows of which elements of the triangle the complements are used. There being five of these circular parts, it is evident that any three of them which you choose to select will either be contiguous \ or else two will be contiguous, and one will _Q / be separated from them by a part on each ^, side. In the first case, the part intermediate between the other two is called the middle part, and they are called its adjacent parts. In the second case, the part which is separated from the other two is called the middle part, and'they its opposite parts. By means of this arrangement, all the relations of a right angled triangle may be expressed in the two following rules of Napier: 1. Radius multiplied by the sine of the middle part is equal to the rectangle of the tangents of the adjacent parts. 2. Radius multiplied by the sine of the middle part is equal to the rectangle of the cosines of the opposite parts. Or both rules may be given thus: radius into the sine of the middle part = the rectangle of the tangents of the adjacent parts ==the rectangle of the cosines of the opposite parts.* The memory will be aided by observing that the words tangents and adjacent in the second clause of the above rule both contain the letter a; and that the words cosines and opposite in the last clause both contain the letter o. As the right angle of a right angled spherical triangle is always known any other two parts being given, the rest may be found by the above rules. The method of proceeding is as follows: Take the two given parts and one of the required parts, or if but one of the unknown parts be required, take that, you will thus have under consideration three parts of the triangle. One of these three will be middle, and the other two either adja* The rule may be read without radius which must be understood as entering the resulting formulas, in accordance with the principles of homogeneity. RIGHT ANGLED TRIANGILES. -t43 cent or opposite; apply the rule of Napier, and you will have an equation resulting which will contain the two given parts and the required part; make the required part the unknown quantity in the equation, and resolve it, you will thus obtain the value of the required part in terms of the two which were given. By applying logarithms to this value, you will have it in degrees, minutes, and seconds. EXAMPLE 90. Given the sun's right ascension and declination to find hS longitude. Let the parts of the right angled S spherical triangle EQS represent the same circles of the celestial sphere, as at Art. 80; a and 6 are given, and I is required.' Of these three parts a, 6 and 1, a and 6 are contiguous, and I is separated fiom them by a part on each side; therefore 1 is the middle part and a and 6 are opposite parts. Applying Napier's rule, remembering that the complement of the hypothenuse I is to be employed, we have sin of comp. I = cos a cos S or, cos I - cos a cos 6 91. The same being given, required the obliquity of theecliptic. The required part is the angle E in the figure. Of E, a and 6, since the three are contiguous leaving out the right angle, a is the middle part, hence applying the rule of Napier, sin a = tan 6 cot i: We put cot E instead of tan E, because, according to the directions before given, the complements of the obliqie angles are to be employed Taking the value of cot E from the above equation, we have sin a cot = tan 6 The sun's R.A on 1st of May 1.850 is 2" 33 10'42, and his declination at the same time 150 3' 2"*4 required his longitude and the obliquity of the ecliptic. i44 SPHEREICAL TRIGONOMETRY. M 380 17' 36"'30 log. cos 9'89479 log. sin 9'79217 15~ 3' 2"4 log. cos 9*98484 log. tan 9*42959 1 400 43' 5"'86 log. cos 9'87963 E 230 27' 25'"46 log. cot 10*36259 In the solution of the above triangle it will be observed that we have found each of the unknown parts in terms of the two given, and have not employed one of those first calculated to obtain another. This is agreeable to the principle laid down at Art. 41, of plane trigonometry, and the reason is the same. Such a method of proceeding is always practicable in the solution of right angled spherical triangles. In the examples which we have taken above, we have supposed the base and perpendicular of a right angled spherical triangle given. Any other two parts being given, each of the unknown parts may be calculated by the aid of Napier's rules, in a manner entirely similar to what has been just exhibited. EXER CISEES 1. In the spherical triangle ABC right angled at A, given the hypothenuse a 65~ 5' and the angle c 480 12' to find B, b and c (B 640 46' 141t Ansb 55 7 32 c 42 32 19 2. Given a 1270 12', c 141 1 1' to find b, B and c. b 390 6' 26" Asns. B 52 22 24 c 128 6 26 3. Given the two oblique angles B 1110 l11, c 910 11t to find the three sides. a 890 32' 28"1 Ans. b 111 48 43 91 16 8 N. B. If the given quantities in a right angled triangle be a side, and its opposite angle, there will be legitimate ambiguity in the solution. In all other cases no ambiguity properly exists, but to avoid error it is necessary to observe the two following principles. 1. The greater side is opposite to the greater angle. 2. An angle and the opposite side are of the same affection, i. o., both greater ox both less than 900. DIFFERENT KINDS OF TIME. t14 EXAMPLE 1. 92.( Given the sun's declination to find the time of his rising and setting at any place whose latitude is known. Let n E s Q represent the meridian of the place, z being the zenith, and, Ho the horizon, and let s' s"' be the apparent path of the sun on the proposed day, cutting the horizon in s. Then the arc EZ will be the latitude of 1_ \ the place, and consequently rE, or its equal Qo, will be the colatitude, and \ this measures the angle oAQ; also RS will be the sun's declination,.and AR expressed in time, will express the time of sunrise from 6 o'clock, for nAs is the 6 o'clock hour circle. -ence, in the right angled triangle SAR, we have given Rs, and thi opposite angle A to find AR, the time from 6 o'clock. Required the time of sunrise at latitude 40~ 43', when the sun's deeli: nation is 230~,27' By Napier's rule, Rad. sin AR = cot A tan Rns tan lat. tan dee. 230 27' log. tan 9'63726 400 43' log. tan 9'93482 210 55' 13" log. sin 9'57208 4* 60)870 40 52" An in time 1L 27n' 41' 6 4h 32' 19' time of sun rising. SCHOLIUM. It should be here remarked that the time thus determined is apparent ime, which is that which would be shown by a clock so adjusted as to pass over 24 hours during one apparent revolution of the sun, or from its leaving the meridian to its return to it again, the index pointing to 12, when the sun is on the meridian. But it is impossible that any clock can * Degrees are converted into hours by multiplying by 4 and dividing by 60, which is equivalent to dividing by 15. 10 146 SHERICAL TRIGONOMETRY. be so adjusted, because the interval between the successive returns of the sun to the meridian is continually varying, on account of the unequal motion of the sun in its orbit,;and of the obliquity of the ecliptic; each of these varying intervals is called a true solar day, and it is the mean of these during the year which is measured by the 24 hours of a well regulated clock, this period of time being a mean solar day; hence, at certain.periods of the year, the sun will arrive at the meridian before the clock points to 12, and at other periods the clock will precede the sun; the small interval between the arrival of the index of the clock at 12 and of the sun to the meridian, is called the equation of time, and it is given on pages I. and II. of each month of the Nautical Almanac for every day in the month; this correction, therefore, must always be applied to the apparent time determined by trigonometrical calculation to obtain the mean time or that shown by a well regulated clock or chronometer, or vice versa, and the Nautical Almanac always indicates whether this correction is additive or subtractive.* A third kind of time is called siderial time. A siderial day is the period of revolution of the earth upon its axis with reference to the fixed stars, or it is the time which elapses after a fixed star passes the meridian of any place until the same star comes to that meridian again. Owing to the apparent motion of the sun from west to east along the stars about 3600 daily, occasioned by the real motion of the earth in its annular orbit,'654 the solar day is a little longer than the siderial, because when the meridian of a place has revolved with the earth on its axis from west to east to come under a certain star, the sun which the day before may have been on the meridian with the star having moved a little to the east, the meridian las a little farther to revolve towards the east to come under the sun again, and thus complete the solar day. The difference between the siderial and solar day is about 3'" 57'. The same fixed stars cross the meridian, rise and set about this much earliert every day. The siderial day is divided into 24 siderial hours, the hour into 60 siderial minutes, and these each into 60 siderial seconds. At pages' 584, 585, 586, 587 of the * To solve the above problem very accurately it would be necessary to compute the sun's declination at the time of sunrise as deduced approximately above, and then to go over the calculation again. The Nautical Almanac gives the declination of the sun at noon for every day in the year, and of the process for determining its declination at any other time of the day we shall have numerous examples in Part V. t By the common clock. $ The Nos. of these pages change a little every year. THE- TRANSIT IN'ST'R MENT. 7 Greenwich.Nautical Almanac are what are called tables of time equivalents. On pages 584, 585 will be found for any given interval of mean solar hours, or minutes, or seconds, the equivalent interval in siderial time. And similarly on pages 586, 587, for any given siderial interval will be found the equivalent mean solar interval. EXAMPLE. Required the equivalent of an interval of 7V 28" 30s of mean solai time in siderial time. From page 584 the equivalent of 75 is found to be 7" 1P 8"'99 " 28W" " "6 28' 4'59 30' " " 30'08.Ans. By addition 7' 29' 43"'66 EXAMPLE 31. Required the solar equivalent of 22' "~ 30" 278 (page 586 N. A.) 22" - 21^ 56" 238'75 30 — 29 55 08 27 26 693 Ans. 22 26 45'76 An astronomical clock is one which keeps siderial time. A common clock may be made to do this by shortening a little the pendulum. The weight attached to the pendulum is usually furnished with a screw by which it may be lengthened and shortened at pleasure, and this should be done till the clock goes just 24 hours from the time a star makes its transit over the meridian till the same star makes its meridian transit again, The exact instant of a star's crossing the meridian is observed with a " Transit Instrument." This instrument consists of a telescope a b supported by a horizontal ax is, each half of which c and d is a hollow cone of brass, at the outer ends of which are short solid cylindrical pivots which rest upon stone pillars e'and J; called piers, the latter being imbedded in a mass of masonry extendinmg a few feet below the surface of the MB48 SPHERICAL TRIGONOMETiYo ground, in order that the vibrations occasioned by passing vehicles or the tread of the observers may not be felt. The pivots of the axis do not rest immediately upon the piers, but upon flat pieces of brass about 4 inches square, and an inch in thickness, which are screwed to the top of the stone. These brass pieces have a notch technically called a Y or v from its shape, in which the pivot of the axis rests. The part of the piece of brass having the notch or v is detached and movable, by means of a screw arranged differently at the opposite ends of the horizontal axis c d; so that one end may be moved horizontally, and the other vertically. The' exact line of vision directed to a distant object is marked by two threads of spider's web, technically called wires, crossing each other at rlght angles, at a point in or near the optical axis of the telescope. They are stretched across a ring or diaphragm to which they are fastened with wax, and this ring, which is smaller in diameter than the tube of the telescope, is held in its place by screws passing through the tube, having their heads outside. By loosening the screw on one side of the tube and tightening the other, the diaphragm, and consequently the point in which the wires cross, receives a lateral motion. The diaphragm is placed at the focus of the object glass near the eye end b.f The whole instrument just described has to be so placed that as it turns on the pivots of the horizontal axis the line of vision along the optical axis of the telescope shall describe the plane of the meridian; narrow trap doors in the roof and sides of the transit room serve to expose tlhe meridian to view. As this plane is vertical, if the line of vision above mentioned be exactly perpendicular to the axis c d, whilst at the same time the axis is exactly horizontal, and finally the telescope point due north and south, then will the required position be attained. For this, therefore, three adjustments are requisite. 1. The adjustment of the line of colIimationk in a perpendicular to the supporting axis c d. 2. The adjustment of the supporting axis to a horizontal position. 3. The adjustment of the line of collimation to the meridian. The method of making these several adjustments we shall describe in,their order. 1. To collimate the instrument.-Bring the intersection of the wires upon a well defined point of some distant terrestrial object; take the instrument out of the Y" and reverse the supporting axis end for end.; bring the telescope upon the same distant point, and if the intersection f the wires covers it exactly, the instrument is collimated; if not, move * i. e. the line of vision determined by the intersection of the wires. t For the illumiination of the wires see po 363, TiR ANS I T INXUS T ER V M)N- T. 1 the diaphragm contalling the wires by means of thle screws at the side of the tube, till the intersection of the wires is brought half way back to cover the distant* point; bring the cross wire on the same or some othei point again, by means of the screw in the Y, which gives a horizontal motion to the whole instrument, and repeat the process already describedi; after a few trials the point will be found to be exactly covered by the. intersection of the wires in both positions of the telescopes This indicates that the line of collimation, or line determined by the intersection of the cross wires, and the distant point, is exactly perpendicular to the axis on which the instrument turns as the object end of the telescope is elevated or depressed. 2. To render the supporting axis horizontal.-This is done by meanas. of a spirit level, of which there are two kinds for the purpose, the hanfging level, and the riding or striding level. The former is suspended by hooks fiom the pivots of the supporting axis, so as to hang parallel to it underneath. The latter is sustained above the supporting axis by two long feet with notches at their bottoms, by means of which it stands upon the pivots of the axis. First, to adjust the spirit level itself, place it on the pivots, and by means of the screw in the Y at that extremity of the axis which gives it a vertical mbtion, bring the long air bubble of the level to reach exactly the same distance on either side of the centre marked with a zero on the level. scale above the tube; for which purpose the divisions of this scale are numbered in precisely the same manner on the right and left of the zero. Then reverse the level on the pivots, turning it end for end, and if the bubble still reaches the same distance on both sides of the zero, the leve requires no adjustment. If not, make half the correction by filing away the notch in one of the feet, or by means of a screw sometimes added for shortening the foot, and the other half by the screw in the Y. Repeatl this process till the adjustment is complete. When the level itself is once d * This may be exhibited with the error of collimation exaggerated in the annexed diagram, in which a b represents the supporting axis, c d the true line of collimation, c o the erroneous position of the line of collimation in the first position of the instrument in the direction c o of the object, and c e the position of the line of oollirmation in the reversed position.. 150' SPHiERIA-tL TRIGONOMETRTo adjusted, the supporting axis is made horizontal by placing the level upon it, and turning the screw in the Y till the centre of the air bubble is opposte the zero of the scale. 3. To adjust the instrument to the meridian.-Observe the instant that some circumpolar* star (the pole star is the best, from the slowness of its motion) crosses the vertical wire of the transit instrument both at its superior and inferior transit, that is above and below the pole. If the interval of time between the superior and inferior transit be the same with that between the latter and the next superior transit of the same star again, the instrument is in the meridian. if not, it is on that side of the meridian on which the arc described by the star between the two transits is shortest, and must be moved a little by means of the screw in the Y, which gives horizontal motion, and the same observations repeated.f When the instrument is once fixed in the meridian, a meridian mark about half a mile distant may be made upon some object, set up if necessary, upon which the vertical wire is to be brought whenever afterwards an observation is to be made. Method of observing the meridian transit of a star.-Before describing this we shall observe that for diminishing the error of observation there are inserted on each side of the vertical middle wire, one, two, or three others, making three, five, or seven in all. There is also attached to the.upporting axis near one of the pivots a graduated circle, the plane of which is perpendicular to that axis, and consequently vertical. This circle is graduated so that the index points to zero when the telescope points to the zenith, or else when the telescope is horizontal, so that when the telescope is directed to a star, the index will mark the zenith distance of the star in the former case, and its altitude in the latter. The star's declination being known from the Nautical Almanac, or from a catalogue, and the latitude of the place of observation being also known, the instrument may- be easily set so that when the star makes its meridian transit it will pass through the middle of the field of view of the telescope. For it is only necessary to bear in mind that the declination is the distance of the.tar from the equator and the latitude is the distance of the zenith fiom hde equator,T so that by simple addition or subtraction of these quanti* A circumpolar star is one which never sets, but describes daily a circle round the pole of the heavens, the whole of which is visible above the horizon. f A method of determining the exact deviation fiom the meridian, and the consequent error in the time of meridian transit, will be presently given. { These measures are all made on the same great circle of the heavens, viz. the meridian of the place of observation, upon which the star is supposed to be at the instant of transit. TRANSIT INSTRUIMENT. 151 ties the distance of the star fiom the zenith is obtained. Thus, if the star be south of the zenith and north of the equator its declination must be subtracted from the latitude to obtain the zenith distance. If the star be south of the equator, or its dec. be S., the dec. must be added to the lat. If the star be N. of the zenith, i. e. if its N. dec. exceed the N. lat. of the place, then the lat. must be subtracted from the dec. to obtain the zenith distance. The above definitions for dec. and lat. will always be the best guide. If the instrument begraduated for altitudes instead of zenith distances it is only necessary to recollect that the altitude is the complement of the zenith distance. If an inferior transit, or transit sub polo of a circumpolar star is to be taken, it may be convenient to remember that the altitude of the pole is equal to the latitude of the place.* The instrument being set to the proper altitude, so that when the star crosses the meridian it will be sure to be seen in the field of view of the telescope, it remains now only to know when to look for its arrival at the meridian, and its consequent appearance in the field. This is shown by the astronomical clock, which is supported upon a stone pier in the same room with the transit instrument. This clock, when correctly set, should indicate the zero of time, or 0^ 0" 0 at the exact instant that thevernal equinox is on the meridian; then at the instant any star is on the meridian, the clock would show the distance of that star in time from the vernal equinox or the right ascension of the star. A minute or two therefore before the clock shows a time equal to the right ascension of the star, (for whose zenith distance or altitude the instrument is set,)as given by the Nautical Almanac or by catalogue, place the eye at the telescope, and the star will be seen entering the field of view, and moving in a direction contrary to its real motion, i. e. from west to eastt instead of from east to west, because an astronomical telescope inverts. Bring it near the horizontal wire,' by means of a clamp and tangent screw attached to the vertical circle, and before the star reaches the first vertical wire in its motion across the field, look at the clock and take up the count of the seconds, which keep by the ear, applying the eye again to the telescope, and note the instant the star crosses or is bisected by the first vertical wire; record the second in a blank book, then look * For the zenith being 900 from the horizon, and the pole 900 from the equator, the pole will be just as far from the horizon as the zenith is from the equator. t Unless it be making its transit sub polo. t Or between the two horizontal wires if there are two near together, as i. sometimes the case. 1 52 SPHERICAL TRIGONOMETRY. at the clock, and record the minute; the hour may be left till the observation is finished. Take up the count of the second again, and apply the eye to the instrument; by this time the star will be seen approaching the second vertical wire. Observe and record the instant of its transit over that wire in the same way, and so on for all the vertical wires. The sum of the minutes and seconds divided by the number of wires will give the minutes and seconds of the time of passing the middle wire, with a probable error of 4- or -4 (according to the number of wires) of the error if the observation had been made upon the middle wire alone.* * The star will often be seen bisected by a wire. between two beats of seconds. The eye then notes how far from the wire the star was at the beat before the bisection and how far at the beat after, and estimates the fraction of a second at which the bisection took place. By the aid of electro-magnetism the exact instant, to a very small fraction of a second, may be not only observed, but recorded without the trouble of keeping count. The arrangement for the purpose is a:s follows: A wire is made to communicate fiom one pole of a voltaic battery to the brass work of the clock; the electricity perviates all the brass work, and passes down the pendulum rod; underneath the pendulum a globule of mercury is supported in a little metallic' cup from which a wire passes to a magnet, round which it coils, and then passes on to the other pole of the battery. Every time the pendulum vibrates on reaching the lowest point of its arc, it dips into the globule of mercury for an instant, and thus a communication is made between the two poles of the battery, and the magnet acts, drawing back a little hammer which is armed with a sharp point that pricks a narrow strip of paper made to pass along under it, at a uniform rate, by clockwork. The intervals between the points on the paper will correspond to seconds. Two wires communicate also fiom the opposite poles of a battery (one of them coiling round a magnet close beside the magnet already mentioned) to a wooden block held in the hand of the observer it the instrument, by touching a button in which he connects the wires communicating weith it, and the magnet then acts, causing a small hammer to prick with its sharp point the narrosv strip of paper a little on one side of the line of points which mark the seconds. The precise position in the interval between two even seconds, of the instant of bisection of the star by the wire, is thus indicated with great precision, and by applying a scale with a vernier, the fraction of a second may be obtained to thousandths, the space on the strip of paper corresponding to a second being usually fromr half an inch to an inch. The observer at the end of the observation notes on the clock the minute with which the observation closes, and writes it with the hour on the strip of paper in pencil. The even minutes in the line of dots which marks the seconds on the strip of paper are indicated by the omission of a dot, which is effected as follows. To the axis which carries the second hand of the clock is attached a fork of two prongs, projecting perpendicularly front the axis; when the second hand has made a complete revolution of the clock dial, the two prongs of the fork dip into two globules of mercury communicating by it with the two poles of the battery, one of which wires is the same that communicates the electricity to.the brass work of the clock from which TRANSIT INSTRUMIENT. 153 The interval between the wires ought to be exactly tile same. As this is scarcely attainable in practice, the mean of the times of transit over all the wires obtained as above will be the time of transit over an imaginary wire situated very near the middle wire. If, by sudden cloudiness or any other accident the transits over some or all the wires but one should be lost, the time of transit over the imaginary middle wire may be obtained as follows: Having made a complete observation of the times of transit of some star over all the wires, take the difierence between the mean of all and the time of transit over each wire. Multiply the intervals thus obtained by the cosine of the star's declination (see Spher. Geom., Prop. 2, Cor. 6, and see Navigation, Art. 99*), and the products will be the equatorial interval between that wire and the imaginary middle wire, or the time that would be occupied by a star situated on the equator in traversing the same interval.f The equatorial intervals being once obtained, to know the time of any star's passing the imaginary middle wire from the time of its passing any other wire, divide the equatorial interval between this wire and the middle wire by the cosine of the star's declination. it goes down the pendulum as before described. When the connexion is made by the fork dipping into the two globules of mercury the electricity goes back to the battery by the shortest path instead of taking the course down the pendulum, and that beat is lost on the magnet. A number of rapid strokes of the button, which impress a corresponding number of points on the paper, serve to indicate the commencement of the observation. * The principle alluded to here, which is of frequent use in astronomy, may be stated thus; the arc on a great circle comprehended between two of its secondaries, is to the arc of a small circle parallel to the primary comprehended between the same secondaries, as unity is to the cosine of the distance of the parallel small circle from the primary, this distance being measured on one of the secondaries. The fourth term of the proportion, instead of the cosine of the distance from the primary, may be the sine of the distance from the pole of the primary or the point in which the two secondaries meet. t For as the length of arc passed over between two hour circles in the same time on the equator, and on a parallel of declination, is as the cosine of the declination to 1, so the times of passing over the same length of arc (as for instance that included between the wires) on the equator and on a parallel of declination will be in the same ratio. 154 SPHERICAL TRIGONOMETRY. EXAMPLE. lThe transit of the star fi Ursse Minoris, whose declination was 74~ 46' was observed as follows: Wires. A. M. t. Intervals. I. 14 48 11 I. and III.' + 131'18 IT. 14 49 16'2 II. and III. + 668"16 III. 14 50 22 -' IV. and III. - 65"82 IV. 14 51 28 V. and III. - 131'*02 V. 14 52 33'2 5)70h 250'" 1108.9 14h 50" 228'18'me img. mind. wire, I. and III. II. and III. 1318'18 log. 2'11787 65"98 log. 1*81941 740 46' log. cos 9'41954 9'41954 equat int. 348 47 log. 1' 53741 17'33 log. 1'23895 IV. and III. V. and III. 65'82 log. 1*81836 131*02 log. 211r734 9*41954 9'41954 equat int. 17'29 1i23790 34*42 log. 1'53688 March 8th, 1850. The star e, Canis Majoris, was observed on the Vth wire only. Time of transit over that wire 6a 54m 3"8 equat. interv. 348"42 log. 1'53681 star's dec. (N. Aim.) 28~ 46' 16"'89 log. cos 9*94277 int. on par. of dec. 39"27 log. 159404 Correction -39'27 Transit imaginary middle wire 6A 53" 24'53 To compute the effect of error of level upon the time of meridian transit.It will be first necessary to determine the inclination of the supporting axis to the horizon. For this purpose place the striding level on the pivots, and talpe the readings at both ends; suppose as in the diagram the * III. here and below stand for the imaginary middle wire. TRANSIT N9V' tU'T T.EN 1.55 west end reads 40,and the w east 20. It is evident that each end of the bubble stood at 30 when the level was horizontal, and that each has 40 moved 10 divisions,: which e -- is obtained by taking the dif- _ firence between the readings of the east and west end, and dividing by 2. So that if i denote the inclination of the supporting axis ab to a horizontal, and i' the inclination of the level to ab, i -+ i' will denote the inclination of the level to the horizon, and we have (Jc denoting' the west reading and e the east). - ~-= ~+,'.0(1) 2 Reversing the level, supposing i' to be greater than i, the east end will now be the highest, but the incli- e nation to the horizon will no longer be the sum, but the difference of the inclinations i and i', [ hence i, i - ~ (2) a- -- 2 subtracting (2) from (1), and dividing by 2 we have (w + w')- (e+ e) Ifi be greater than i', after reversing, the west end would still be the highest, and we should have instead of (2) 2 i — (3) 2 Adding (1) and (3), there results after dividing by 2 * The glass tube is a portion of a circle of large radius, so that the movement of the bubble indicates the angular movement of the level. To find the angular value of a division of the level scale, place the level on the telescope of some instrument to which a large vertical circle is attached. Turn the circle till the bubble passes over a number of divisions of the scale, which will be equal to the degrees, minutes, &c., through which the circle has moved; divide this number of degrees and fractions of a degree by the number of divisions of the level scale passed over, and the quotient will be the value of one division of the scale. 9l1~56 SPHERICAL TRIGONOMETRY. (w + J') - (e + e') 4 as before. If the east end of the supporting axis be the highest instead of the west end, as we have supposed above, a similar course of reasoning would produce the formula (e + e') -( + "') I 4 which is the same as the above, changing e and e' for u and &. Both formulas may be expressed in one, thus (e + e') - (w +') (4) -i (4) 4 which is the formula always to be employed for obtaining the inclination of the supporting axis. N. B. If the sum of the east readings exceed the sum of the west readings the east end of the axis is too high, and vice versa. EXAMPLE. First position of the level e 50 w 40 Second " " " e 24 w' 66 e + e' 74 w -+u' 106 74 (e+ e') (c +,') 32 4 i 8 value of I division of level scale 5" i in seconds, 40" As a verification that the level readings have been correctly noted it may be observed that e +- w should be the same in both positions of the level, being the length of the bubble which may be supposed not to change from the effects of temperature during an observation. Thus in the above example 50 + 40 = 24 + 66. To compute now the effect of the inclination of the axis as determined above, upon the time of transit, let nzo be z the vertical circle in which the telescope would play, when the supporting axis was // horizontal, nIo the circle in which it plays / \ when the supporting axis is inclined; then zcz measured by the arc z z is the inclination ~-~0 of the planes of these two circles, and equal TRANSIT INSTRUMENT. 157 to i the inclination of the supporting axis to the horizon. The length of tle arc sa described by a star at s, in passing from the vertical to the inclined circle in which the telescope plays is expressed by z.sin oso (See Spher. Geom., Prop. II., Cor. 6), that is s8 = i sin a a being the altitude of the star. But the length of an arc of the equatoe similar to sa is expressed by s5 - cos 6, 6 being the star's declination. (See note on p. 153, and Spher. Geom., Prop. II. Cor. 6.) This arc of the equator is converted into time by dividing by 15. Hence i sin a i sin a ~- _ - I. 5 or cos 6 l o cos 6 is the difference of time between the passage of the star over the middle wire of the telescope when the supporting axsis is nclined, and its passage over a vertical circle. N. B. 1. The altitude of the star, designated by a in the above formula, is obtained from its declination given by catalogue, and the latitude of the place as at p. 150. 2. If the east end of the supporting axis is too high, the telescope is thrown to the west of its proper position, and the star, moving as it does from east to west by the diurnal motion, passes the wires too late; the correction therefore found as above is then subtractive. If the west end be too high, the star passes the wires too soon; and the correction is additive. To compute the azimurth error of the instrument and its effect upon the,ime of transit. This error arises from the deviation from the meridian of the vertical circle'wich the line of collimation describes, or to which the circle that it describes is reduced as above. The most ready way of determining the amount of deviation and its effect on the time of transit is by " the method of high and low stars," as it is termed, that is by the transit of a star near the zenith, and one remote fiom it or near the horizon, whose right ascension and consequent time of transit does not differ much from the former. To understand this let it first be supposed that one star passes the meridian exactly at the zenith, and that the other passes near the horizon, and let it be supposed also that the stars have the same right ascension. If the instrument were exactly adjusted to the meridian so that the optical axis of.the telescope moved in the plane of that circle, then the two stars would be on the middle wire at the same instant, if the telescope could be brought down instantaneously from the high to the low star. But if the 158 SPHERICAL TRIGONOMETRY. circle in which the telescope plays make a small angle (called the angle of deviation or azimuth error) with the meridian, then the supporting axis being supposed horizontal, when the telescope is vertical it will point to the zenith, in which all vertical circles as well as the meridian intersect, and the zenith star will be on the middle wire at the same instant as before, but the lower down the other star is, the wider will its time of transit over the middle wire differ from that of its transit over the meridian, because the farther will the vertical circle which the telescope describes be from the meridian, the farther we go from the zenith where they intersect. The greater, therefore, the difference between the time of transit of the zenith star and the low star over the wire of the instrument as compared with the difference of their times of transit over the meridian, which in the case supposed is zero, the greater the deviation of the vertical circle described bv the instrument from the meridian. Now it is not necessary that the high star should pass exactly at the zenith, but only near it. If the difference of the observed time of transit of the high and low star be equal to the difference of their right ascensions, that is, of their times of pssing the meridian, the optical axis of the telescope moves in the plane of the meridian; if not, this axis describes a vertical circle which deviates from the meridian, and the amount of this deviation and the consequent error in the time of meridian transit, we proceed now to show how to determine. Let the full circle in the diagram represent the horizon, z the zenith, r the pole, and consequently Pz the meridian. Let zs represent the vertical circle described by the telescope, s the place of a star where it crosses it, and appears on the middle wire, and PS, the declination circle, passing through the stlr. In the spherical triangle Pzs, in which the hour angle P represents the time that has elapsed since the star s passed the meridian, Pz, before it reached the wire of the telescope in the vertical zs, we have by the sine proportion sin P: sin z;' sin zs sin rs From which, taking P in place of sin P, since it is a very small angle, and z the small angle of deviation of the vertical zs from the meridian in place of the sine of its supplement Pzs, which is also its own sine, and representing zs by C and the complement of Ps or the declination of the TRANSIT INSTRUMENT. 159 star by 6, multiplying the means of the proportion and dividing by the last term, we have sin cos C6 1 m Iay be found as at p. 150, by supposing s to be on the meridian, without sensible error, by means of the latitude of the station and declination of the star. Represent the fractional part of the above firmiula by n. It becomes P = z (2) Suppose now another star crossing the meridian nearly at the same time. For this we have P = M'2 r (3) Let now t represent the observed time of transit of the later star, ca its right ascension, t' and a' the same for the other star, and let e denote the error of the clock which may be supposed unknown. For the former star (since a is the time of its passing the meridian), we shall have for the value of the hour angle when it makes the transit of the middle wire, = — t - e-a (4) And for the other star'= t' +e-a' (5) By subtraction of (5) from (4) the error of the clock e is eliminated, and there results P -- P - (t- t') -- (a...') (6) Substituting for P and P' in (6) their values given by (2) and (3), (6) becomes z (n' - n) (t- ) (a — t') (t ---—' (7) The value of the azimuth error or deviation from the meridian z is thus found in time. To convert it into space this value must be multi* In the later catalogues of stars, their north polar distances (N.P.D.) are given instead of their declinations. In the above formula sin (N.P.D.) would of course be in place of cos d. When the declination of the star is south, we have still sin rs=cos 0, for sin (90~ + 6)= sin (90-d.) =-cos 6. (See App. Art. 13.) t N. B. That t + e is the true time of the observed transit. 100 d3O SPIIEUCAL TRIGONOMETRY. plied by 15. The instrument may then be adjusted to the meridian by turning the screw of the Y, which admits of horizontal motion, and which has usually upon it a graduated arc, by means of which the movement of the instrument in azimuth is indicated. To compute the effect of the azimuth error upon the time of transit take the value of z as given by (7) in time, and substitute it before converting it into space, in either (2) or (3), which will give the value of P or,' the hour angle in time, a or n' being an abstract number expressing the ratio of two trigonometrical lines. Tlhe value of P is the correction to be applied to the observed time of transit of the later star, to obtain the time of its meridian transit. That of x,' the same for the other star. If the deviation from the meridian be southwest and northeast, as in the diagrlam where we ehave supposed the zenith to be south of the pole, tlie correction for a star south of the zenith will be subtractive, for one Inorth additive,f unless the latter make an inferior transit or sub polo, in which case the motion being in the opposite direction, the correction is -subtractive. This is evident from an inspection of the diagram. The only remailing correction is for error of collimation. The line of collimation when this error exists describes a cone about the supporting axis as an axis, and tlle point in which it pierces the surface of the celestial sphere, describes a small circle of that sphere parallel to the meridian. and at a very short distance fiom it. The distances between these circles ~measured on parallels of declination may be considered every where the s:ame without sensible error, and the time of traversing this distance by any star will be inversely as the cosine of the star's declination. The equatorial interval between the circles in question may be found. by moving the instrument in azimuth, after reversing, by means of the screw in that Y which gives horizontal motion, till the intersection of the wires is brought back to the terrestrial point on which it was placed before reversing. The degrees and fiactions of a degree passed over on the graduated arc on the Y will indicate double the error of collimation, which, divided * Which will evidently be indicated by the value of z being positive. If the deviation be S.E. and N.w. then z is negative. This may be seen by trying various cases by the diagram, such as one star passing, 1st, N. of the zenith, 2nd, subpolo, &c., first writing the numerator in the value of z in (7) under the form (t- a) - (1' - a'), t The oniy difference in the above diagram for a star north of the zenith would be that the angle of deviation itself instead of its supplement would be the angle of the triangle, but the proportion would be the same. TRANSIT INSTRUMENT. 161 by 15, will give the equatorial value of it in time. This divided by the cosine of any star's declination will give the effect of the error of coltimation on the time of the star's transit. The error of collimation is best measured by means of a movable vertical wire, to which motion is given by a micrometer screw, as described in another place. Should no distant terrestrial object be visible from an observatory, owing to intervening objects near at hand, a small telescope in the building having its object glass turned towards that of the transit instrument may serve as a collimator. The rays of light proceeding from the wires at the focus of the object glass of the small tele, scope strike this object glass, are refracted by it, and emerge in parallel lines; they then strike the object glass of the transit instrument, and are conveyed to the focus of parallel rays, which is the astronomical focus; so that in looking through the eye end of the transit instrument the wires of the small telescope will be distinctly seen. Care should be taken to throw the light of a window or lamp in at the eye end of the small telescope. A similar contrivance may be employed for a meridian mark. But the transit instrument may be made its own collimator, by placing a vessel of mercury underneath, and turning the object end of the telescope downwards. If the axis be horizontal, and the instrument truly collimated, the wires being illuminated by an orifice in the side of the eye piece, the rays of light will pass from them to the object glass, emerge in parallel lines, strike the surface of the mercury vertically, be reflected back in the same lines, and converge to the focus of the object glass at the same points which they left, so that the reflected image of the wires will be seen coinciding with the direct image. If not, there is either error of collimation or of level, or both. If the axis had previously been made horizontal by the striding level, it is the latter, and the diaphragm containing the wires must be moved till there is coincidence between their direct and reflected images; or a movable wire may serve to measure the interval between them. This interval is double the collimation error, because the angle of incidence is equal to the angle of reflection, the former being on one side the vertical, the latter on the other. If, therefore, the direct image of the wire be brought to the vertical by the screws of the diaphragm by a movement over half the distance between the direct and reflected image, the reflected image will be brought there too. The striding level need not be used at all, if the instrument be reversed in the v", in using the collimating eye piece with a basin of mercury; for in one position of the instrument the angle obtained by taking half the distance between the direct and reflected image of the wires is the sum, and in the reverse position is the difference of level error and error of collimation. The well-known algebraic formula, " to half the sum add half the difference for the greater of the two quantities, and from half the sum subtract half the difference for the less," will serve to determine those two errors separately. To know which is the greater, the level or collimation error, we have this rule:-If the reflected image in both positions appears on the same side of the direct, then the level error is the greater of the two, but if on different sides, the collimation error is the greater. All this will appear evident if the student make a diagram with a line to represent the supporting axis with level error exaggerated, a 11 162 SPHERICAL TRIGONOMETRY. line perpendicular to this at the middle, to represent true line of collimation, another line from the same middle point oblique to represent the erroneous line of collimation, a horizontal line below to represent the surface of the mercury, and from the point where the erroneous line of collimation meets it, a vertical and also a line making the same angle with it as does the erroneous line of collimation. In reversing the instrument the only change will be in the erroneous line of collimation, which will now make the same angle on the other side of the true. The following example will serve to illustrate all the foregoing rules for applying the corrections to an observation with the transit instrument. STATION AND DATE. COLLEGE OBSERVATORY. MARCH 8TI, 1850. OBSERVER'S NAME. B. B. STAR'S NAME. a GEMINORUM. a CANIS MINORIS. r I. 7h 11m 11"'5 7T 31m 32-2 I. 7 11 30 7 31 49 5 Wires.m. 7 11 48'8 7 32 7 Iv. 7 12 7 ~5 7 32 24 4 v. 7 12 26 7 32 41-7 SUM. 5)35 59 03 8 37 40 34 8 Mean of Wires. 7 11 48'76 7 32 06 96 Corr. for Collim. Error.* + -05402 + -05024 Corr. for Level Error. t'+.092 + -07397 Corr. for Azim. Error. -- 250 - 1'481 Time of Mer. Transit. 7 11 46 -406 7 32 05 603 Star's R. Ascen. 7 11 10 -20 7 31 27 38 Error of Clock. Sees. 36 -2111 Sees. 37 -22311 e Collimation Error (J Geminorum) thus obtained: Equaterial Error 0.5 log. 2-69897 Dec. N. Aim. March 8th, 220 15' 4"'9 log. cos 9'96639 Collimation Error -05402 log. 2-73258 Collimation Error (a Canis Minoris): Equatorial Error'05 log. 2-69897 Dec. March 8th, 50 36' 6"'5 log. cos 9.99792 -05024 log. 2-70105 I Mean of the two stars 37'. For Notes t and t see next page. -TRANSIT INSTRUMENT. [63 t -lie level error is computed as follows: e 96 w 139 e' 130 a' 105 e -+e' 226 o + c' 244.e - e' t W+ o t 4 =45 Multiplying this result by *3" the known value of a division of the level,scal ~e have 1"*35 as the value of i the inclination of the supporting axis. d GEMINORUM. Dec. 220 15' 5" ar. co. log. cos 0'03360 Lat. of Station, 40 43 Zenith dist. 18 27 55 log. cos 9.97704 i 1"-35 log. 0.13033 15 ar. co. log. 8'82391 *09223 log. 2*96485 a CANIS AMNORIS. Dec. 50 3C' 6" ar. co. log. cos 0'0020Q Lat. of Station, 49 43 Zenith dist. 35 6 54 log. cos 9'9127$ i 0'13033 8*82391 *07397 log. 2*86907 I Azimuth Error obtained by formulas (t t')- (a- a) sin z —= - n/,-P zn, n= —- os n -n' cos t obs'd time transit a Canis Minoris 7* 32", 6*.96 i'... dC Geminorum, 7 11 48'76 t- t 20M 182018K * right asc. a Canis Minoris, 7 31 27*38', "J Geminorum, 7 11 10.20 * — 20*, 17"18 "- *) —(a- =9 10 a CANIS MINORIS, Zenith dist. 350 6' 53".5 log. sin 9*75983 Decli. 5 36 6'5 log. cos 9*99792 a'5779!'7 11. GEMINORUM. Zenith dist. 18027' 55"'1 log. sis 950QG9 Decli. 22 15 4'9 log. cos 9*99639 #' *342 i'53430 164 SPHERICAL TRIGONOMETRY. The correct siderial time being ascertained by the transit of a star of known right ascension, the correct mean solar time may be found from this, as follows: The Nautical Almanac gives on p. XXII. of each month the mean time of transit of the first point of Aries, which is the zero of siderial time, and may be called from analogy the siderial noon.* By means of this and the table of time equivalents, explained at p. 147, the mean solar time corresponding to any given siderial time, may be obtained by the following rule. Mean solar time required = mean time at preceding siderial noon + the equivalent to the given siderial time. EXAMPLE. To convert 7' 11'" 10"20 siderial time (the true time of meridian transit recorded above) March 8th, 1850, into mean solar time for the meridian of New York. Mean time at preceding siderial noon, viz. March 8th, 0O 56" 308'74 7" 0 0 0 1 Corresponding mean 6 58 51 19 For given 11 0 solar intervals by Tab. 10 5820 sideasla timente0ts by9 a.9 sideil time. 10 time equivalents, p. 586, 9 *97. O J0.20J. A. 0 *20 Sum mean solar time required 8 6m 30"'30 Correction for longitude of N. Y.f 48"68 8 5 41 *62 n n' *2357 log. 1X37236 (t- it) - (a- a') 1,02 log. 0o00860 z. in time, 4-9328 log. 0*63624 15 log. 1-17609 z. in space, 6"-491 log. 1*81233 z. 4'328 log. 0.63624 log. 0*63624 n. *5779 log. 1-76191 n''3422 log. 1-53430 r. 2-501 log. 0*39815 P. 1.481 log. 0*17054 - To obtain the mean time of siderial noon at any place having a different longitude from' Greenwich, it is necessary to subtract 95.8565 multiplied by the hours and fractions of an hour, by which the place differs in long. from Greenwich if the place be west, and to add this product if the place be east of Greenwich. As the daily gain of siderial time on solar is about 3" 56", this divided by 24 or 9'*8565 will be the hourly gain or the hourly motion of the sun backward from west to east. t This is obtained by multiplying 9-'856 by 4t'94 the difference of longitude between New York and Greenwich. This correction may be applied here instead CONVERSION OF TIME. 15 The mean solar time may be obtained by direct observation of a meridian transit of the sun, which is made by taling the transit of each limb of the sun, that is to say observing the times when the sun's disc is tangent to the wires, both upon its western and eastern side, and taking a mean of the times as the time of transit of the sun's centre.*'Or mor accurately by applying as a correction additively or subtractively to the bserved time of transit of the limb the time occupied by the sun's semidiameter in passing the meridian, which is given for every day in the year in the Nautical Almanac, p. I. of each month. To convert mean solar time into siderial the rule is as follows. Siderial time required = siderial time at preceding mean moon + the equivalents to the given mean time. EXAMPLE. To convert 8* 5" 41,'62 mean time at New York, into siderial time. Sider. time at preced. mean noon, Gr. viz., March 8th, 23A 3"* 19'*98 Correction for long. N.. + 48'68 23 4 8 866 Fo8 0 0s The tab. p. 518, N. 8 1 18 For mean 5 0 5 0 8, Fora~s imean1}, | A. gives the equiv. sid. 0 8 ntervals. 41 41 4 1 1 in tervals. 062 L 0'62 Sum sid. time required 7" 11" 10O06f — The reasons for the above rules are sufficiently evident. PROBLEM. 93. Given the latitude of the place, and the declination of a heavenly body, to determine its altitude and azimuth when on the six o'clock bOlh circle. of to the mean time of siderial noon. Strictly the equivalent of 4'863 in solar intexvals should be applied, or 48"68 may be applied to the given siderial time additively before taking out the solar equivalents. * A colored glass over the eye piece is necessary in observing the sun. t The sum amounting to more than 24k, of course 24^ must be rejected from it, As after reaching 247 the siderial time begins at zero again. 1G6 SPHERICAL TRI:GONOMETRY. Let HZPO be the meridian of tdhe Z place, z the zenith, no the horizon, s the place of the heavenly body on the six o'clock hour circle Psp, and ZSB the vertical circle passing through it. Then,- E / - O in the right-angled triangle SBA llare lanlown As, the declination, and the ainge SAB, or ar' OP, the latitude of the place, to find the altitude Bs, and AB or the- complement of the azimuth OB. EXAMPLE. L. Required the altitude and azimuth of Arcturus when upon the six oclock hour circle of New York, lat. 400 43' N., on the 1st of Jan. 1800; isdeclination on that day being 190 57"56" N. Ly Napier's rules we have Rad. sin Bs = sin A sin AS Rad, cos A Rad. cos A tan AB cot AS.~. cot BO = - - cot AS 400 43' log. sin 9*81446 log. cos 98'7964 19 57 56 log. sin 9*53334 log. cot 10439'74 aIt. 12 52 12 log. sin 9*34780 a,. 74 38 22 log. cot 9*43990 940 There remains one case in the solution of oblique angled spherical triangles, which we have deferred to this place because we wished to employ in it the rules for the solution of right angled triangles. This is where two sides and the angle oppositef to one of them, or two angles and the side opposite to one of them, are given; or, as it is sometimes expressed, where two of the given parts are a side and its oppoai6 angle. In such a case we may proceed as follows By means of the proportion, the sines of the angles are as the sines of the opposite sides (Art. 81), the unknown part opposite one of the given parts may be found. Four parts of the triangle will then be known, and two will remain unknown; these two will be a side and its opposite angle, to Sid which, from the vertex of the unknown angle let fall an arc perpen-~ The horizon and equator intersect at A, 90~ from the meridian. t The term opposite is used in a more exact sense here than in Napier's rules, of which we have jusat been speaking. OBLIQUE ANGLED TRIANGLES. 167 dicular upon the unknown side opposite, and the given triangle will be divided into two partial triangles, which will be right angled, and in each of which two parts will be known. Applying Napier's rules to the solution of these, the partial angles which compose the unknown angle may be found, and their sum will be the value of the unknown angle; then the unknown side opposite may be found by the proportion, the sines of the angles are as the opposite sides; or this last side may be found by calculating the two parts of which it is composed from the right angled triangles, and adding them together, which is the better method, since it avoids ambiguity. If the perpendicular arc drawn from the vertex of the unknown angle to the unknown side falls without the triangle, of course the difference, instead of the sum of the angles and sides found in the right angled triangles, is to be taken. Thus in the annexed diagram triangle let; the side and angle opposite given be c and c, a~ and the other given angle B. Then first sin c: sin c:: sin B: sin b c by means of which proportion b may be cal- culated and will be legitimately ambiguous. / Then there will be left unknown A and a. - From A let fall a perpendicular AD upon a, which we have not drawn, lest it should confuse the diagram, but which the student can imagine; then in the right angled triangle B A D we know two parts B and c, and also in the right angled triangle c A D we know two parts c and b, the latter having been found by the proportion above. To calculate the partial angles at A, calling that in the first right-angled triangle above mentioned w, and that in the second u', we have by Napier's rules n cos c - cot B COt eO whence R COS c cot = -- cot B and in the same manner R cos b cot c' = cos a then e +- u' = A and sin: c s:: c sinA in a whence 168 SPHERICAL TRIGONOMETRY. sin c sin A sin a. sin c Thus all the parts of the triangle are determined. For an application of this case of solution take the following EXAMPLE. Given the zenith distance, azimuth and polar distance, to find the hour angle and colatitude of the station. Or in the diagram given, z zz, and:s, to find r and Pz. Supposing the declination of a star, as given by catalogue, to be 160 11' N., and its observed altitude and azimuth to be 39~ 10' and 75~ 10' from the south, required the hour angle of the star and latitude of the station. rs = 900-160 11- 730 49' a. c. log. sin 0*01756 log. cot 9.46271 z=1800-750 10'=1040 50' log. sin 9*98528 log.cos 9'40825 zs = 900-390 10'= 500 50' log. sin 9*88948log.cot 9*91095 p,. 3^ 25m 12 -- 51C 18' log. sin 9*89232 log. cos 9'79605 170 26' 46" log. tan 9*49730 650 6t 6" log. tan 10'33334. = —47 39 20"t lat.=42 20 40 We have now demonstrated formulas for the solution of every possible calse of plane and spherical triangles, including the more simple formulas which apply exclusively to the right angled triangles. The examination questions which follow call attention to the most important results of the investigations in the preceding pages. After these, in Appendix II. will be found many useful matters connected with Spherical Trigonometry, for which it was thought not best to interrupt the gleneral train by which the solutions of triangles are deduced. Note.-In assuming hypothetical cases, care must be taken not to suppose such as are impossible. The following are the governing principles to be observed. In plane triangles, 1. One side must be less than the sum of the other two (Geom. Ax. 13 Cor.) 2. The greater side of a triangle is opposite EXAMINATION QUESTIONS. 169 to the greater angle (Geom. Th. 9). 3. The sum of the angles must be exactly two right angles. In spherical triangles, the first two principles also apply (Spher. Geom., App. III. p. 2, and Prop. 7). 4. The sum of the three angles must not be less than two, nor greater than six right angles (Spher. Geom., Prop. 14). 5. The sum of the three sides must be less than a circumference (Spher. Georn., Prop. 8). 6. Each side must be less than a semicircumference (Spher. Geom., Prop. 8, Note). 7. Ench angle must be less than two right angles (Spher. Geom., Prop. 8, Note). EXAMINATION QUESTIONS IN TRIGONOMETRY. What is the object of Trigonometry? How many elements are there in a triangle, and what are they? How many elements must be given in order to determine the rest? In plane triangles what must one element always be? Why? What is the difference between Geometrical and Trigonometrical solutions? Which are most accurate? Are trigonometrical solutions perfectly accurate? Whence arises the very small inaccuracy? Ans. From the decimals neglected in calculating tables of logarithms. How is the circumference of a circle divided for the purposes of Trigonometry' What is the complement of an angle or arc? What is the supplement? What are complements of each other in a right angled triangle? What is the sine of an arc I What is the cosine? The tangent? Cotangent? Secant? Cosecant? What trigonometrical line changes its sign with the sine? Ans. The cosecant. In which quadrant are they negative? What changes with the cosine? Ans. The secant. Where are they negative? What with the tangent? Ans. The cotangent. Where are they negative? In passing through what values do quantities generally change their signs? An. Zero and infinity. What is the least value of the sine? Where is it 0? What is the greatest value of the sine' Where is it radius? How many times does it change its sign in going round the circumference t What is the least value of the tangent? Where is it 0? What is its greatest value? Where is it infinite? How many times does it change in going round the circumference? 170 SPHERICAL TRIGONOMETRY. N. B. Let these questions be repeated for the secant, cosine, cotangent, and cosecant. To what is the sine of 450 equal? The tangent of 450?Why The sine of 300? To what is the sine of a negative arc equal? The cosine of a negative arc? The Tangent? Secant? Cotangent? Cosecant? To what is the sine of the supplement of an arc equal I The cosine of the supplement? The Tangent? Secant Cotangent? Cosecant? To what is the sine of 900 plus an arc equal? The cosine of 900 plus an arc? The Tangent? Secant? Cotangent? Cosecant? What formula expresses the relation between the sine and cosine of an arc? Ans. m- = sin +- cos2. What is the expression for the tangent in terms of the sine' and: cosine sin Als. Tan = cos The expression for the secant? Ans. - Cos Cos For the cotangent? Ans.sin For the cosecant? Ans. - sm How are the tangents of two arcs to each other? To what is the tangent equal in terms of the cotangent? Ans.ot or the reciprocal of the cotangent. What is the formula for the sine of the sum of two arcs? Ans. Sin (a + b) = sin a cos b + sin b cos a or the sum of the rectangles of the alternate sines and cosines. The formula for the sine of the difference? Ans. Sin (a - b) - sin a cos. b - sin b cos a. For the cosine of the sum? Ans. Cos (a + b) = cos a cos b sin a sin b or the difbrence of the rectangles of the cosines and sines. For the cosine of the difference? Ans. Cos (a - b) = cos a cos b + sin a sin b. For the sine of an arc in terms of half the arc? Ans. Sin a = 2 sin a a cos j a, or twice the sine of half the arc into the cosine of half the arc. From what is this formula deduced? The formula for the cosine in terms of half the arc? Ans. Cos a = cosa a - sinl' a. Whence derived? The formula for the sine of half an arc. Ans. Sin -l a V -- -- cos a. For the cosine of half an are? Ans. Cos s a = VA + - cos a. For the sum of the sines? Ans. Sin p + sin q = 2 sin 2 ( + q) cos i (p -), or twice the sine of half the sum into the cosine of half the diffirence. EXAMINATION QUESTIONS. 171 The difference of the sines? Ans. Sin p-sin q-=2 cos J (p+-q) sin J (p - ). The sum of the cosines? Ans. Cos p +- cos q = 2 cos I (p +- q) cos s (p- q). The difference of the cosines? Ans. Cos p — cos q- =2 sin l (q + p) sin ^ q -p).What is the ratio of the sum of the sines to the difference of the sines tank (P ~q) Ans. (_ q or tan of half the sum to tan of half the difference. tan (p —q) How derived? Of the sum of the sines to the sum of the cosines? Of the difference of the sines to the sum of the cosines? Of the sine of the sum to the sum of the sines? 2 - sin A (pq- q) cos ~ (p + q). sin (p -+ q) a cos A (p + q) sin p sin q 2 cos (p-q) sin I (p + q) cos - (p- q) sin ~ (p -+ q) Of the sine of the sum to the difference of the sines? Ans. - sin (p q) tan a What is the formula for the sine in terms of the tangent? Ans. sin a = -- \/l+tane a What is the formula for the tangent of the sum of two arcs? Ans. tan (a b) tan a + tan b 1 —tan a tan b tan a- tan b For the tangent of the difference? Ans. tan (a- b) = + tan a tan b From the tangent of the sum how is the tangent of twice an are found? Of three times an arc? RESOLUTION OF RIG-T ANGLED PLANE TRIANGLES. What are the three formulas for the solution of right angled triangles? Ans. (1) Radius: the hypothenuse:: sine of one of the acute angles: the side opposite: cosine: the side adjacent; or radius being unity, hypothenuse X sine of either acute angle = side opposite and hyp. X cos of either angle = side adjacent. (2) R: either of the perpendicular sides: tangent of the angle adjacent or cotangent of the angle opposite: the other side, or a being 1, one perp. side X tan of adjacent angle side opp. (3) Square of the hypothenuse =sum of the squares of the other two sides. Square of either perp. side - rectangle of sum and dif of the other two sides. In a right angled triangle how many elements must be given? Ans. Two. Why should each required element be found in terms of the two given? In finding each unknown element how many logarithms will be employed? When the logarithms are added what must be rejected from their sum? When one is subtracted from the other what must be added to the latter? When the hypothenuse is given or required with an angle, wiich formula l employed? i72 SPHERICAL TRIGONOMETRY. When the hypothenuse is neither given nor required? When two sides are given to find the third' RESOLUTION OF PLANE TRIANGLES IN GENERAL. Two sides and the included angle of a triangle being given, how are the othel elements determined? Ans. a + b: a -b:: tan ~ (A + B): tan 3 (A — B) A + B = 180 -C i (A + B) + (A- B) = A, ~ (A + B) -- (A - B) B Sin A: A: sin C: c. The three sides being given? Ans. Sin A= \,/( s —) ( s-) or cos A- / s ( s-a) V fbe be. Two angles and the interjacent side being given? Ans. 1800 - (A + B) - C sin C: c:: sin A: a:: sin B: b Two of the three given parts being a side and its opposite angle? Ans. By the sine proportion, or sines of the angles are as the opposite sides. What is the formula for the cosine of an angle in terms of the three sides of a plane b1 q- c1- a2 triangle? Ans. Cos A = - - or the sum of the squares of the sides which 2be contain it, minus the square of the opposite side, divided by twice the rectangle of the containing sides. Which is the fundamental formula in Spherical Trigonometry t RI cos a - R cos b cos c* Ans. Cos A=sin b sinec SOLUTION OF RIGHT ANGLED SPHERICAL TRIANGLES. Upon what are Napier's rules founded? Ans. Upon the formulas in right angled spherical trigonometry. How many parts are considered for the application of his rules 1 What are they? Ans. The base, perpendicular, the complement of the hypotheinuse, and the complements of the two oblique angles. What are the rules? Ans. Sin of the middle part = product of the cosines of.he opposite parts = product of the tangents of the adjacent parts. N. B. Radius must be introduced homogeneously. When two parts are given how are the rules applied 1 SOLUTION OF SPHERICAL TRIANGLES IN GENERAL. Three sides of a spherical triangle being given how are the three angles found? All the formulas for the solution of spherical triangles may be derived from this; for applied to the three angles it gives three equations containing the six elements of the triangle from which any two elements being eliminated, an equation results containing the other four elements. EXAMINATION QUESTIONS. 173 Ans. By the formulas sin 1 A ^ /sin (A s - ) sin (. s- c) sin b sin c ort cos ~ ^/ sin 6 sin (c A -- b) Why is not the formula for the cosine of an angle in terms of the three sides suitable for the application of logarithms? The three angles being given how are the three sides found? Ans. Sin A a = V/cos A Scos (A S- A). Vs ~2 g sin B sin C Two sides and the included angle being given how are the other parts found 1 Ans. By Napier's analogies. Cos A (a + b): cos (a -b):: cot A C: tan A (A — B), sin A (a -+): sin A (a -):: cot C, tai a (A —B). And then A (A + B) - A (A-B) A and ( (A + B) -A (A B) = B, and finally the sine proportion sin A: sin B:: sin C: sin c. Two angles and the interjacent side being given? Ans. Napier's Analogies, 2d set. Cos A (A - B) cos (A - B):: tan - c ta tan (a -+ b) Sin A (A + B): sin A (A - B):: tin c: tan i (a - ) (a +- b) +- (a - b) = a and (a + b) - (a - b) = b, sin a: sin A: sin c: sin C. Two of the three given parts being a side and its opposite angle? Ans. From the vertex of that unknown angle which is opposite the unknown side, let fall a perpena dicular upon this side and apply Napier's rules to the two right angled triangles thus formed. The parts of the given triangle are by this means found either directly, or by adding the parts of the two right angled triangles together. QUESTIONS ON LOGARITHMS. What is a logarithm? What is the constant number which is raised to a power called? To what is the logarithm of the base equal? To what is the logarithm of unity equal? What is the base of the common system? In the common system what is the logarithm of 100? Of 10001 Of all numbers between 100 and 1000? Of all numbers between 1000 and 10,000? What is the entire part of a logarithm called? How does it compare with the number of digits in the number to which the logarithm belongs? How is the logarithm of a number consisting of three figures found from the tables? Of one of four? Of one of more than four? By the tables of Callet How is the number corresponding to any given logarithm found from the tables? What is the rule for multiplication by logarithms'? For division? 174 SPHERICAL TRIGONOMETRY. Raising of powers? Extraction of roots? How is the logarithmic sine, tangent, &c., of any are found when consisting of degrees and minutes only? How the log. sine, tangent, and secant, when of seconds also? How the cosine, cotangent, and cosecant in the last case? How are the logarithmic secant and cosecant computed from the logarithmic siaM and cosine? What is the arithmetical complement of a logarithm? For logarithms entering in what way into formulas are arith. comp'. usedl What must be rejected from the logarithmic sum for each ar. comp. used? QUESTIONS ON THE CIRCLES OF TIE CELESTIAL SPHERE. What is the axis of the earth? The axis of the heavens? What is the celestial equator? The ecliptic What are the equinoxes? What are declination circles? What is the meridian of a place? What is the hour angle of a heavenly body? What is the horizon of a place? What are the poles of the horizon called? What are vertical circles? Which is called the prime vertical? What is the declination of a heavenly body? The right ascension'? What is the celestial latitude? Longitude? What is the altitude of a heavenly body? The azimuth? What are the co-ordinates of a heavenly body? How many sets are there? Which are obtained from observation?'Which of the observed co-ordinates are preferable, and why? Ans. R. A, ind D, because they are the same for every place on the earth, whereas altitude and ai th are different for every place. QUESTIONS ON THE TRANSIT INSTRUMENT. What are the different kinds of time? What is apparent solar time? Mean solar time? What is siderial time? How much longer is a mean solar than a siderial day How is a mean solar interval of time converted into a siderial interval, and the contrary? What instrument is employed for observing the time of transit of stars o'er T;he meridian? Of what parts does the transit instrument consist How many and what are the adjustments' How is the instrument collimated I EXAMINATION QUESTIONS. 175 How is the striding level adjusted? How is the instrument adjusted to the meridian? When the instrument is completely adjusted, in what plane does the line of collimation move? How is an observation made with the transit instrument? How is the equatorial interval of the wires determined'? How the interval between any wire and the middle wire for a particular star? What is the use of knowing the intervals of the wires? How does the probable error of observation compare with the number of wires observed upon? What is the formula for the inclination of the supporting axis? What the formula for the correction of the time* of meridian transit for level error? What the formula for determining error in azimuth or deviation from the meridian t What for consequent error in the time of meridian transit? How is the equatorial error of collimation found? How from this the collimation error for any star? What is used with the transit instrument? If the clock keep true siderial time, what does the time of meridian transit show? If the right ascension of the object observed is known by catalogue, what does the difference between this and the time of meridian transit show? How is siderial time converted into solar and the converse? * The above questions will suffice to show the nature of those which should be put upon the subsequent parts of the work with which we shall not take up further space. APPENDIX II. ON UNLIMITED SPHERICAL TRIANGLES AND THEIR SOLUTIONS OF THE VARIOUS TRIANGLES FORMED BY THE SAME THREE POINTS ON THE SPHERE. 1. If any two points, A and B, be taken upon the surface of the sphere, the arc of a great circle joining them may be considered to be either the arc A B (< 1800), or 3600 - A B; or if we do not limit the arcs to values less than a circumference, we may consider it to have an indefinite number of values expressed generally by the formula 2 n 7r - a, a denoting that value which is less than s or a semicircumference, and n any whole number or zero. 2. If two arcs of great circles intersect in a point A, the angle which they form may be considered to be either the angle A (< 900), or 1800 A, or 1800 + A, or 3600 — A; or, taking the most general view of angular magnitude, the angle will have an indefinite number of values expressed by the formula m w i A, A denoting the value which is less than 7r, and m any whole number or zero. 3. If, therefore, any three points, A, B, c, be taken on the surface of the sphere, and great circles, made to pass through each pair, we shall have an infinite series of triangles whose sides will be generally expressed by 2 n ir, 2n, b, 2 n n c (1) and whose angles will be generally expressed by mfI4 A, m i., tni, c; (2) a, b, c, denoting the arcs less than v joining the pairs of points B c, A, A I, respectively; A, B, c, the angles less than J r formed at those points by the intersection of these arcs; and n and m, any whole numbers, or zero. 4. It is evident, however, that we cannot assume that any three values of the aides from the series (1), combined with any three values of the angles from (2), will form a spherical triangle. Some general relations of the parts composing a triangle must first be established, from which corresponding values of n and m in (1) and (2) Introduced by Gauss. Notwithstanding the elegance and generality thus given, to the solutions of many astronomical problems, nothing is to be found on this subject in our trigonometrical works. The present paper is from Prof. Chauvenet of the U. S. Naval Acad. The explanatory notes are the author's. 12 178 SPHERICAL TRIGONOMETRY, may be deduced. Although these general relations are well known, it may not be out of place to add here a concise demonstration of them. Let the point c, one of the angular points of the spherical triangle A B c, be referred by rectangular co-ordinates to three planes, one of which, the plane of x y, is the plane of the great circle A B; let the axis of x be the diameter of the sphere passing through B, and let the origin be the centre of the sphere. The formulas of transformation from these co-ordinates to polar co-ordinates, the origin being the same, the polar axis being the axis of x, and the fixed plane the plane of x y, are x = aR cos a y = R sin a cos X (3) z = sin a sin B ) where B denotes the angle which the plane passing through the polar axis and the point c makes with the fixed plane; R, the radius-vector in this plane, or distance of the point c from the origin; and a the angle which this radius-vector makes with the polar axis. B is an angle of the spherical triangle, and a is the side opposite the angle A; and, according to the principles of analytical geometry, B and a may be altogether unlimited, due regard being had to the signs of their trigonometric functions, and to those of x, y, and a. Let us now transform from these rectangular co-ordinates to others also rectangular, the origin and the plane of x y remaining the same, but the axis of x in the new system passing through the point A, and therefore making with the first axis the angle c, c also expressing the side of the triangle opposite the angle c. The known formulas of transformation become The rest of Art. 4 implies some knowledge of Analytical Geometry. It may;be readily understood, however, by the mere student of t:igonometry from the annexed diagram, with the follow- C ing explanations. R in formulas (3) is equal to o c in the diagram. The projection of a or of oc,on B which is called the axis of x, that is to say the distance between the foot of a perpendicular from c on o B and the point o 0 -~ B is the value of x in the formulas, the projection of o c on a line called the axis of Y, drawn fiom o in the plane A o B._ perpendicular to o B, is the value of y in the formulas, and the projection of R or o c on o z perpendicular to the plane A o B, is the value of z in the formulas. The first of formulas (3) is now obvious enough; in the second K sin a evidently expresses the value of a perpendicular from c to OB in the plane c o B. and this perpendicular multiplied by the cosine of th6 angle which it makes with its projection on the plane A o B equal to the angle of the two planes or the angle B, expresses the length of the projection on the plane A o B, which is evidently equal to tile projection of R on the axis of Y; or multiplied by the sin B expresses the height of c above the plane A o B, which is equal to the projection of a on o z the axis of z. N. B. The axes of x, y., and z are at right angles each to the plane of the other two. So also are those of x', y', and z'. The student will eadily deduce formulas (4) by the rules of Plane Trigonometry. APPENDIX ix. 79 x x' cos C -- ys sin y = x' sin c +- y' cosc (4 Finally, the formulas of transformation from this last system of rectangular again to polar co-ordinates, the origin being the same, the fixed plane the plane of x1' u and the polar axis the axis of x', are x'- R Cos b ) y' -— R sin b cos A () z'- R sin b sin A Now, if the values of x, y, z, x', y7, z', given by (3) and (5) be substituted in (4), we have at once the following system of equations:cos a - cos c cos b + sin c sin b cos A sin a cos B = sin c cos b- cos c sin b cos A (6 sin a sin B sin b sin A which are the known fundamental formulas of spherical trigonometry, but established without imposing any restrictions upon the values of the parts of the t riangle. From this investigation it appears that these formulas may be regarded as formulas of transformation from one system of polar co-ordinates to another, or rather from one system of spherical co-ordinates to another. For example, the co-ordinates of a star referred to the pole of the equator and the meridian of a place whose colatitude is c, are its polar distance a, and its hour angle B; the co-ordinates of the same star referred to the pole of the horizon and the meridian, are its zenith distance 6, and its azimuth A; and the formulas (6) express the relations by means of which we can pass from one of these systems to the other. (5.) Let us now inquire what are the corresponding values of the sides and angles in the series of triangles expressed by (1) and (2). Let a, b, c, A, B, c, denote the values of the parts of one of these triangles, which, if we please, we may suppose to be the triangle whose parts are less than 7r. Then since sin (2 n ir -" f) = sin p, cos (2 n 9 — ) = cos 9, the equations (6) will be satisfied by the substitution of 2 n 7r + a, 2 n - + b, &t., for a, b, and c; and therefore the triangle (a, b, c, A, B, c) is the first of an infinite series obtained from it by the successive addition of 2 7r to each or all of its parts, every triangle of the series being such, that the relations of its parts are expressed by (6), when ab, b, c, a, c, are assumed to represent those parts. It is evident, also, from the principle of " uniformity of direction " observed bn t:he preceding demonstration in reckoning the sides and angles, that we must be able to satisfy the equations, by making either all tho sides, or all the angles, or all the sides a anangles, negative at the same time,* and, considering each of the triangles * The student will do well to conceive the position of the angular points of the riangles on the surface of the sphere with these variations. N. B. That the sides are all negative together, or the angles together, or both together. The same thing 3tated'n the text may be made evident by referring to equationB I80 SPHERICAL TRIGONOMETRYo thus obtained as the first of a series, as above, we have three more series. We have then the four series following:1st eries. 2d Series. 2 n T7ra, 2 n r+A 2 nr-a, 2 n 7r+A 2 n7r+b,,2nr-+B 2n^r-b,2nir+B 2 n -+c- b, 2 n rr + c 2 n 7r-c, 2 n fr -w 3d Series. 4th Series. 2 n -r+ a, 2 n w~-A 2 n 7r-a, 2 nr - A 2 n 7r r b,2n 2 n -rBb 2 n T b, 2n r-. B 2n.-+- c, 2nr-c 2 n - c, 2 n -c In all the terms of these series, n may have the same or different values; and we thus have all the possible combinations of the values represented by (1) and (2), 8o long as m in (2) is even. But if we substitute 2 n + 1 for m we shall find that the following series will satisfy the equations (6): 5th Series. 6th Series. 2 n r- a, 2 n r+A 2 n r-a, 2 n -+ A 2 n -- b, (2 n + 1) r+B 2 n +. (b, (2 n +1) r+ 2n - c, 2n +1) r+c 2 n 7 c, (2n -l) rc 7th Series. 8th Series. 2 n xr-{ a, 2 n - A 2 n r-a, 2 n r-A 2 n 7r - b, (2 n - 1) n- B 2 n r+- b, (2 n + 1) 7r-B 2n7r- c,((2n- 1)r- c 2n c,(2n 1) r- c the 6th, 7th, and 8th of which series are derived from the 5th, as the Wd, 3d, and 4th were derived from the 1st, in the preceding paragraph. By sFacessively exchanging a for b and c, we find eight more series, namely, 9th Series. 10th Series. 2 n r- a, (2 n + 1) + A 2 n r+a, (2 n + 1) + A 2 n r+ b, 2 n 7r+ B 2 n7- b, 2 n r + B 2nr-c,'(2n+{-1)7r+-c 2n7r+ c,(2n + 1) -c (6), which involve all the relations of the six elements of a spherical triangle, and which will be satisfied by changing simultaneously a, b, and c into - a, -b, - c, or A, B, c, into — A,~B, - c or both; observing the general rule that sin (-) -sia * and cos (-) = cos ^., The student will try these elements given in the 5th series, in eqs., 6) observing datc os {(2n+ l)r+ 0cos(18oO+0)=-os.andsin (2+1l)*+#l ~Theswhni s i, APPENDIX. XtI 181 11th Series 12th Series. 2 n -a, (2 n+ 1) r -A 2 n a, (2n +1) — A 2n,-+b, 2 nr-a B 2nr- -b, 2 nr-B 2 n i —c, (2 n- 1) X —c 2 n {+ c, (2 n + 1) x —c 13th Series. 14th Series. 2n 7r a, (2 n+- 1) r — A 2 n r - a, (2n + ) 7r+A 2n r-b, (2 n+1) r +B 2n -+b,(2n-+-1)7+2 n -{-c, 2 n ~f c 2 nr- c, 2 n. t+-c 15th Series. 16th Series. 2 n 7r-a,(2 n {- 1) — A 2n r + a, (2 n 1) r-A 2 nr- b, (2 n-+ 1)7- B 2n 7rb, (2n+ 1) -- B 2 n 7r + c, 2 n r-c 2 n r-c, ~ 2 n - c 5. Since three great circles by their mutual intersections (provided they have not a common diameter), divide the surface of the whole sphere into eight primitive tri? angles (whose parts are all less than r), the three angular points of each of which give sixteen triangles, whose parts are all less than 2?r,* therefore, three great circles of the sphere form in general one hundred and twenty-eight triangles, each of which may be considered as the first term of an infinite series of triangles formed from it by the successive addition of 2r to each or all of its parts. AMBIGUITY IN THE SOLUTION OF THE GENERAL SPHERICAL TRIANGLE. For the sake of brevity, I shall call the spherical triangle, whose parts are only limited by the condition < 3600, the general spherical triangle. Although any three points of the surface of the sphere may be regarded (in general) as the angular points of sixteen such triangles, yet to the problem " given three parts of the triangle to find the other three," there will in every case be but two solutions, i. e. two triangles containing the same data. From the equations (6), and the consequences that flow from them, we can always obtain expressions for both the sine and cosine of each of the required parts, which would fully determine the triangle, were it not that in every case one of these expressions at least involves a radical of the second degree, and'has either two different numerical values, or two values numerically equal with opposite signs. To avoid this ambiguity it was thought expedient to limit all the parts of the triangles to values less than 1800, or to consider only the simple geometrical triangle. By this means all the cases in which the required quantities can be found by a cosine or tangent, without involving radicals, become fully determined. But this occurs in but four of the six cases, the other two still having two solutions; so that although six conditions were thus imposed, three limiting the data themselves, and three the qunesita, the object of removing all ambiguity was not reached. * i. e., making n = 0, in each of the 16 series of the last art. I82 &13%C ~SPHERICAL TRIGONOMETRY. We shall see from the solutions of the general triangle, that the ambiguity is entirely removed in every case by the imposition of a single condition restricting the sign of either the sine or cosine of but one of the required parts. The general method here, as in many parts of the mathematics, is therefore the simplest. IFORMIULAS REQUIRED FOR THE SOLUTION OF THE GENERAL SPHERICAL TRIANGLE. 1. As the formulas (6) are the same as those deduced in trigonometrical works for limited spherical triangles, we may avail ourselves, for the solution of the general triangle, of all the formulas (found in those works), deduced from them in a general manner. It is not necessary therefore to repeat all those deductions here; but I shall add a demonstration of Gauss's equatiQs, slightly differing from the common one, in order to establish them in their generality. 2. GAUSS's THEOREM. If p.= cos, c sin J (A - B), P = c os C C (a-b) q = cos c c cos g (A +-B), Q= sin ~ c cos 2 (a+ b) r=sin c sin (A- B), R =COs C sin (a -b) s = sin ccos ( (A- B), s cos c sin I (a-b) then the products p q, p r, p s, q r, q s, r s, are respectively equal to the products P Q, P RS, QR, QS, RS. To demonstrate this, we have only to form the following equations, which are easily deduced from the fundamental formulas:-, sin c (sin A I sin B) = sin c (sin a sin b)* sin c (cos A J- COS B) = (1:F COS C) sin (a: b)t (1 cos c) sin (A 4- B) = sin c (cos b - cos a)t which, transformed by the formulas of the trigonometric analysis,~ give respectively sin c sin a sin b * By combining sin A sin si n A sin A t By the second of (6) we have sin c cos A - cos a sin b -sin a cos b cos c sin c cos = cos b sin a — sin b cos a sin c'By addition and subtraction of these we obtain sin c (cos A - cos B) = (sin a cos b: sin b cos a) (1:p cos o) whence the formula in the text.': Thiis obtained from the last by substituting 7r-c &c., for c to produce the supplemental or polar triangle. ~ To wit: formulas which express the factors of the above forms in terms of the sines and cosines of. c, ~ c, I (A: a), ~ (a +b) by means of which we have the following, each of the above forms furnishing two::2 asin c cos ~ c 2 sin ( (A - B) cos I (A - ) = 2 sin J c cos a c 2 sin 4 (a - b) cos (a —b) 2 si a e cos' c 2 sin (A - B) cos ~ (A +- B) = 2 sin co cos I c 2 sin 8 (a b) cos. (a + b) &c., &co APPENDIX II 183 p s 5 P, q r= Q R q s = Q S, p r P R p q = Q, r s i s 3. The same notation being employed, the quantities p2, q2, r2, s8, are respectively equal to the quantities p2, q2Y,R s2. For we have p q X p r = q X aP R and q r R, whence by division p2 - P2, and in the same way q2 = Q2, r == R2 s2 -= Sa. 4. GAUSS'S EQUATIONS. From the preceding paragraph we deduce q = - Q 9' = i R In these equations the positive sign must be taken in all the second members, or the negative sign in all of them. For if we take p = - p, the equations p q = P Q, p r= R, R, p s = s, being divided by this, give q = + q, r =-+ a, s= - s. But if we take p -- P, the same equations divided by this, give q = - Q, r =- = a, s = - s. Hence the two following groups of equations, the first group comprising those commonly known as GAUSS'S equations, which are identical with (I.), (II.), (III.) and (IV.) of Art. 86, by clearing the latter of fractions. cos c c sin (A- B) = Cos c cos (a - b) cos C cos (A + B) = sin c cos (a + b) (7) sin 2 c sin ~ (A-B). = cos, c sin ( (a-b) [ sin ~ c cos (- B) = sin c cos 1 (a - b) cos c c sin % (A - B) =- cos c cos ~ (a-b) ) cos ~ C cosS (A + B) - sin ~ c cos ~ (a+ b) (8) sin j c sin ~ (AA-B) - cos g c sin ~ (a -b) sin ~ c cos A (A - ) = - sin ~ c sin ( (a -+ ) J 5. Now when the parts of the triangle are limited to values less than 1800, the second of these groups is excluded, since cos A c, sin - (A +- B), cos - c, cos ~ (a -b) are then all positive. But when the triangle is unlimited, both groups must be admitted, and the question arises, when are we to employ the positive, and when the negative sign? GAUss himself has remarked (Theoria Cillot. Corp. Cal., Art. 54), that cases occur in practice in which it is necessary to employ the negative sign, and promises elsewhere a fuller explanation, which, however, I have not been able to find. But the nature of these cases and the answer to the question above propounded will be easily inferred from the following considerations. We have seen that the formulas (6) apply not only to the triangle whose parts'r, b, c, A, B, D, are all less than 3600, or 2r, but also to all the triangles whose parts are 2 n r + a, 2 n r + b, 2 n n - c, 2 n 7r - A, 2 n7r + B 2n r +- c, n being The first two of the six that would result are all that we have thought necessary to write. They are identical with p s'= P s r =- Q R 1854 SPHERICAL TRIGONOMETRY. any whole number or zero, and admitting of different. values in each of the pars, Let us, therefore, substitute in (7) the following values of these parts:2 n i - +a, 2 mil -+ A 2no,r+ b, 2m2 r +B 2 n3 3 + c, 2,3 + c We shall have for the factors of the first members values similar to the following: cos (n3 Tr - c) - (- 1)3 COs B ca sin (n3 7r +- C c) = (- 1)3 sin i c* cos [(m,1 + m2) r-+{- (A + B)] (- 1) + m2 cos (A + B) sin [(mI + r) 7r + (A + B)] =(- 1) + 2 sin (A + B) m n CSi [(m, ~ m) T + 3 ( ~ B)] = ( 1) 1- 2 COS ~ (A~-B) sin [( -m) r + (A -B)] ( —1) 2 sin (A - B) Now whatever the values of m, and m2 m1 -j ma and ml - m2 are both even or both odd at the same time, and therefore the above substitution gives the same sign to all the first members of our equations. In the same way it is shown that the second members will all have the same sign; and we may consequently express the result of the substitution thus:cos ~ c sin ~ ( +- B) (- 1) cos g c cos ~ (a- b) cos c cos A (A - B)= (- l) sin c cos c (a - b) sin ~ c sin (A- B) (-1)" cos ~ c sin ~ (a - b) sin C c cos 3 (A -B) (- 1)n sin I c sin e (a + b) which single group involves both (7) and (8). The group (7) will represent one series of triangles, while the group (8) will represent another series, the two differ ing in each of their elements by some multiple of 2n, and the primitive triangle may belong to one or the other of these series. TtVe may dispense, therefore, in practice, with group (8), by deducting 3600 or 2r from the elements found by group (7), till they become less than 3600, as required for use. (See top of this page.) 6. When the parts of the triangle are interchanged in GAUss's equations, it would seem to require proof that the same sign, whether + or -, must continue in these ( quations; i. e. that when the triangle is such as to satisfy the equation, cos I c sin - (A +- B) == COS C COS I (a b) * Apply (3) and (6) of Art. 70 to sin and cos of 2 nsa r + c to obtain these results. Or observe that cos (nwr + ~ c) = cos I c according as n is even or odd. APPENDIX II. 685 it will also satisfy the equations, cos a b sin. (A + c) = cos 4 B Cos a (a- c) (e) cos. a sin 3 (B + c) = os g A cos (b c) (e and that when it is such as to satisfy the equation, cos ~ c sin (A - B) - cos c cos. (a- b) it will also satisfy the equations cos I b sin ~ (A + c) - COs- B cos ( (a - c) cos s a sin A (B + C) -- cos A COS (b — c) ( To demonstrate this, we will show that the groups to which the equations (e) belong may be derived from (7), and those to which (f) belong from (8), by merely linear transformations, and therefore without again introducing the double sign. Let the equations (7) be written thus:sin A (A+- B) cos. (a -b) cos C c cos c cos (A + B) cos t (a + b) sin I c - cos - c ( sin I (A-B) sin - (a- b) cos g c sin c c cos ~ (A B) sin ~ (a + b) sin c - sin -- c The sum and difference of the first two, and the sum and difference of the last two, give* cos (A - B +c) sin j a sin b b sin c - cos c sin c cos - c cos (A - B — ) cos s a sin - b sin c sin c cost (-A+B+C) _ sin I a cos ~ b sin c sin I c By differently combining these four equations, two and two, we may either reproduce the group (g) or the two groups represented by (e). Thus the sum and dif. ference of the first and third, and of the second and fourth, give sin. B sin I (A + c) sin I b cos j (a —c) sin c sin c cos 3 BCos -~ (A + C) sin b cos a (a + c) sc sin c sin c sin - B sin. (A- c) cos - b sin ~ (a- c) sin c sin e cos B cos ~ (A -) cos b sin ~ (a + c) sin c sin c * See the mode of deducing (I), (II), &c., of Art. 86. I S16 SPHERICAL TRIGONOMETRY. These multiplied by sin c sin c sin B sin b give sin I- (A - c) cos ( (a-cc) cos 0 B cos j b cos I (A + C) cos A (a- +c) sin I B cos ~ b sin i (A -) sin A (a- c) cos. B sin 6 b cos - (A-C) sin a (a - c) sin. B sin b Precisely the same transformations applied to (8) would of course give a similar result with the negative sign. Hence In the three groups which GAUSS's equations form by the permutation of the letters, the positive sign must be taken in all the equations, or the negative sign in all of them. AUXILIARY ANGLES. It will be convenient to premise here the following proposition, upon which depends the proper employment of auxiliary angles in preparing our general formulas for logarithmic computation. In the equations k sin d =m (9) kcosp = n whatever the values of m and n, we can always determine k and so as to satisfy at once these equations, and any one of the following conditions arbitrarily imposed. 1st k positive, (I < 3600), 2d k negative, (k < 3600), 3d y> 0 and < 1800, 4th 0 > 1800 and < 3600, 5th ~s < 900 and > -900, 6th 0 > 900 and < 2700. The six conditions above stated are obviously equivalent to the following; 1st, k; 2d, k -; 3d, sin +-; 4th, sin ( -; 5th, cos f +-; 6th, cos 0 —. Of these six conditions, however, we commonly employ only the first, third, or fifth. m The quotient of the equations (9), tan - n, gives two values of' under 3600.* Ifence, also, two values of k, which will be numerically equal with opposite signs, * Because for any arc less than 3600 there is always another are < 360c, having the same tangent. If the former be in the first quadrant, the latter is in the 3d; if the former be in the 2d, the latter is in the 4th quadrant. APPENDIX II. 187 ance the two values of sin 0 will be numerically equal with opposite signs, as also the two values of cos <. If we restrict the sign of any one of the three quantities k, sin 0, cos I, the signs of the other two will become known, and there will be but one value of k and one of p under that restriction. SOLUTION OF TIIE SEVERAL CASES OF THE GENERAL SPHERICAL TRIANGLE. In the solutions of the various cases of spherical triangles, it is of the first importance to have simple and clear precepts, both for removing the ambiguity that occurs in every case and for determining properly the auxiliary angles. Examples might be pointed out, in recent works on trigonometry, of incorrect numerical solutions resulting from an erroneous application of precepts, in themselves correct, but not sufficiently simple or explicit. I have, therefore, given special attention to this point in arranging the following solutions. These solutions have also been carefully verified by the computation of the two triangles following: FIRST TRIANGLE. a c 1250 0 0t A= 264~ 511 30"i4 b 140 0 0 B =231 24 6 *9 c= 46 0 0 c =299 0 0 G6 SECOND TRIANGLE. a = 400 0' 0" A= 3190 211 21"'4 b= 250 0 0 =107 46 57 *6 c = 230 0 0 c= 50 55 9 -3 The first of these triangles requires the positive sign in GAUss's Equations, and the second requires the negative sign. 1. Given b, c, and A, to find a, B, and c. The general relations between the given and required parts are cos a = cos c cos b + sin c sin b cos A sin a cos B = sin c cos b cos c sin b co A (1) sin a sin B = sin B sin A and similar forms to the last two, with c and c interchanged with B and b. The second members being computed, the numerical value and the sign of cos a will be determined from the first equation. From the second and third sin a and s are determined precisely as k and p in the preceding section and are subject to the same ambiguity.* The ambiguity will be removed, therefore, when the sign of either sin a, sin B, or cos B is given, and in like manner when the sign of either sin c or cos c (the other required parts) is given. The solution may be adapted for logarithmic computation, and the condition required for removing the ambiguity may be varied. Let k and 0 be determined by the conditions (9), taking m sin b cos A, and n cos b, and adopting the first arbitrary condition; then these conditions together with equations (1) assume the following form:* For the second members of the 2d and 3d of (1) being computed, they will be fixed quantities like m and n in (9), and sin a occupies the place of k, and s that of t in the same equations (9). I88 FSPHERICAL TRIGONOMETRY. k sin =sin b cos A k cos = cos b (k positive) cos a = k cos (c-) (2) sin a cos B = k sin (c -) sin a sin B = sin b sin A Or, eliminating k and adopting the third condition imposed on eqs. (9) tan =tan b cos A ( < 1800) cos b cos a= cos (c —5) cos - sinacosB cos b (3) cos b sin a cos B - ~ sin (e - 0)* cos 4 sin a sin B - sin b sin A If the quadrant in which a is to be taken is given, then tan =tan b cos A (0 < 1800) tan a cos B = tan (c-) (4) sin 0 tan A tan a sin B= -- ccos (c - ) In (3) and (4) we may also limit 0 to values numerically less than 900, the sign of the tangent being determined according to the fifth arbitrary condition following (9). If both a and b are less than 1800, as not unfrequently happens in the applications of this problem, let k sin a m sin b an k then m and n are both positive (k being positive) and the following form may be employed:-~ m sin m O = cos A m cos 4 = cot b n sin B = sin tan A (5) n cos B = sin (c -~) cot a = cot (c-~p) cos BI Check. —For the purpose of verification we may employ with (4) or (5), the formula sin a sin B = sin b sin A; and with any of the preceding solutions the following check:sin (c-) sin a cos'B tan A sin p sin b cos A tan B The value of k which appears in the 2d and 3d of (3) is obtained from the 2d of (2). t Dividing the 3d by the 2d of (3). t Combining the 1st, 2d, and 4th of (3). ~ Derived from (2). II Dividing the 3d by the 4th of (2). If From 1st, 4th, and 5th of (2). APPENDIX H1. 189 2. Given b, c, and A, to find B and c. We employ GAUSSs equations as follows: cos a sin l (B c) = cos A cos (b- c) cos d a cos L (B + c) = Sill A cos i (b + ) sin s a sin ~ (- c) -cos ~ A sin. (b -c) sin a cos0 (B -) sin A sin (b -c) The first two by eliminating cos 2 a determine ~ (B + c) when the sign of cos @ a is known, and the second two determine ~ (B - c) when the sign of sin. a is known. 1ience, these equations present no ambiguity when the sign of sin a is given; for sin n a is always positive, and cos a has the same sign as sil a according to the formlua, sin a =2 sin a cos - a The equations (6) taken with the positive sign only may give values of B and c exceeding 3600, in which case the required solution will be found by diminishing such values by 3600. 3. Given B, c, and a to find A and b. The general relations between the given and required parts are cos A -cos c cos B + sin c sin B cos a sin A cos b = sin c cos B + Cos C sin B Cco a (7)* sin As sin b i sin a which determine A and b without ambiguity, when the sign of either sin A, sin b, or cos b is given. In like manner the ambiguity is removed when the sign of either sin c or cos c is given. Adapted for logarithms by the method already used, these equations becomet k sin sin B cos a k cos ( = cos B (k positive) cos A -k cos (c + ) (8) sin A cos b k sin (c + )- sn Asb sin sin i a Ort, tan 9 = tan B cos a (p < 1800 always positive; or less than 9)00 with the sign of its tangent,) COS B C08 h=A - ot Cos (C +F) (9) cos a I COS B sin A cos b = sin (c +q- ) cos ain A sin b = sin B sin a WhMen the quadrant in which A is to be taken is known, * Derived from equations (6) p. 179 by polar triangles. t Compare (2). t Compare (3). 190 SPHERICAL TRIGONOMETRY. tan tan B Cos a ( < 1800) tan A Cos b - tan (c +)- ) sin i tan a tan A ill b = - cos (c +-) J IWhen A and B are both less than 1800; m sin =- cos a (mt positive) n cos cot B n sin b = sin p tan a (n positive) (11)1 n cos =B sin (c -+) cot A --- cot (c + p) cos b Check.-With (10) or (11) we may employ sin a sin Bn= sin b sin A, and with any of the solutions (8), (9), (10), (11), the check sin (c - + ) sin A cos b tan a sin -- sin B COS a tan b 4. Given B, c, and a, to find b and c. We employ GAUSS'S equations arranged as follows: sin t A sin (b- ) sin ~ a cos a (B —c) sin 3 A coS j- (b + c) = cos0 a cos3 (B + c) i cos A sinl (b — c)= sin a sin - (B-c) C cos A coS (b -c) = cos - a sin i (B - c) J which present no ambiguity when the sign of cos. A is given; that is, when the sign of, sin A is given, observing that sin ~ A is always positive, and cos j A has the same sign as sin A. As before, when these equations lead to values of b or c greater than 3600, the true values are to be found by subtracting 3600. 5. Given a, b, and A, to find B, c, and c. The general relations between the given and required parts, are sin a sin B - sin b sin A - cos c cos A + sin c sin A cos b = cos B sin c cos A - cosc sin A cos b = sin B cos a (13) cos c cos b + sin c sin b cos A = cos a sin c cos b - cos c sin b cos A sin a cos B The first equation determines B when the sign of cos B is given; and B being known, the remaining equations will fully determine c and c. Thus we find first sin b sin A sin B. (14) sin a Then from the second and third of (13) if we put k sin. =- cos A (k positive)" k cos p sin A cos b k sin <' = cos B (15)1 k cos' sin B Cos a we obtain c =+ 4'~ > Compare (4). t Compare (5). t Compare with former eheck, ~ Because sin (c-'p) - sin 0' and cos (c- ) = cos 0. APPENDIX lI. 191 From the fourth and fifth, (1) k sin 0 = sin b Cos A (k positive) 1 k cos 0 = cos b k sin 0'= sin a cos (16) k cos 0'= cos a c=-0+ o' J In these solutions it may happen that -+;', or 0 - 0' exceeds 3600, in which case c = +' - 3600, or c = 0+0 - 3600. Checks.-One of the following* may be employed when either c or c has alona. been computed:sin c cos A c tan a sin p' cos a cos' tan b sin 0 tan B cos 0 cos b sin 0 tan A cos0' cos a When botn c and c have been computed, the obvious check is sin c sin A sin c sin a 6. Given a, b, and A, to find c and c without finding B. Observing that k is positive in the preceding article we deduce the following forms and conditions, by eliminating B;t k sin. = cos A (k positive) kcos - = sin A cos b cos 0' = cos 0 cot a tan b (0' less (17) than 1800, with the same sign as cos B.)t c= 9 +' J k sin 0 = sin b cos A (k positive) k cos - cos b cos 0 cos a (n ) cos 0' - (0' less cos b ) I than 180S, with the same sign as sin a cos B), Check. sin c sin A sin c sin a (1) The propriety of employing the same factor k in both (15) and (16) will be seen by comparing the values of k deduced from the two groups. We find in both cases k2 = 1 -Sin2 A sin' b. * Deduced from (15) and (16). t By substituting sin A sin b for sin B in the 4th of (15), and for sin A its value -o ~ fronm nd of 15. sin a; See 3d of (15.) The value of cos 0' is obtained by taking the value of k in the Pd of (16), and substituting it in the 4th. I )2. SPIERICAL TRIGONOMETRY. In these solutions, when, -- i', and + 0 + 0' exceed 3600, we must take c -' - 3600, c 0 + 0' - 3600; and when they are negative we must take c - + q- + + 3600, c =- 0+0' +3600. 7. Given A, B, and a, to find b, c, and c. We find b by the formula sin B sin a sin b -. - sin A which determines b when the sign of cos b* is given. The remainder of the solution is by (15) and (16). 8. Given A, B, and a, to find c and c without finding b. We may eliminate b from (15) and (16) in their present form, but the conditions for determining the auxiliary angles will not be so simple as in the following method. Let p and j' in (15) be exchanged for A' - 900, and q + 900 respectively; then after eliminating b, we find k sin,p = -sin B cos a (k positive) k cos ~ = COs B cOS 0 COS A cos'=- — f (t less (19) COS B than 1800, with the sign of sin A cos b.) c = < + ~ Y In a similar manner from (15) we find k sin 0 = - cos a (k positive) k cos 0 = sin a cos B cos 0'= - cos 0 cot A tan Bt (01 less (0) than 1800 with the sign of cos b.) c= 0 -+0' Check. sin c sin A sin c sin a In these formulas, as before, when -+ A', and 0 + 0' exceed 3600, we take c - + -' -3600, c ==0 + 0' - 3600; and when they are negative we take c + -'+- 3600, c = 0 -+ 0 + 3600. 9. Given a, b, and c, to find A, B, or c. The formula (see Art. 82) cos a- cos b cos c COS A = ('21) sin b sin c determines A when the sign of sin A is given; or when the sign of either sin a, sin B, or sin c is given; when the sign of any one of these functions is known, those of -the other two may be discovereid by an inspection of the equation * Which determines the quadrant in which b is, the sign of the sin b only determining whether it is in the first two or last two. t Compare the 3d of (18).' Compare the 3d of (17). APPENDIX It, 193 sin A sin B sin c sin a sinb sin The usual formulas for sin I A, cos. A, tan. A (see Art. 84), derived frcm (21), may be employed, and the ambiguity removed, by the same conditions. Check.-Compute two of the functions sin I A, cos ~ A, tan I A; or one of them in connexion with (21). 10. Given A, B, and c, to find a, b, or e. The formula (see Art. 88) COS A - COS B COS C cos a- -. (22) sin B sin c determines a when the sign of sin a is given; or when the sign of either sin a, sin b, or sin c is given, since when the sign of any one of these functions is known, those of the other two may be discovered by an inspection of the equation sin A sin B sin c sin a sin b sin c the usual formulas for sin ~ a, cos - a, tan 1 a, may be employed, and the ambiguity removed, by the same conditions. Check.-Compute two of the functions sin - a, cos o a, tan I a; or one of then in connexion with (22). 10. From the preceding sketch it appears that for the determinate solution of a spherical triangle generally considered, there are required four data; namely, the numerical values of three of the six parts composing the triangle, and the algebraic sign of one of the functions of a required part. To recapitulate, the triangle is fully determined by the following data:- - 1. b, c, A; and the sign of either sin a, sin B, cos B, sin c, or cos c. 2. a, c, a; and the sign of either sin A, sin b, sis b,, or cos c, 3. a, b, A; and the sign of cos B. 4. A, B a; and the sign of cos b. 5, a, b, c; and the sign of either sin A, sin iB, or sin c. 6. A, B, c; and the sign of either sin a, sin b, or sin c. 11. Since 7r is the symbol which represents the circumference of a circle whose diameter is unity, or the semicircumference whose radius is unity, 7r wrill represent a quadrant of the latter or 900. We may, therefore, for convenience, represent the supplement of an are a by 7r - a, and As complement by - r ~-a. 12. A triangle, one side of which is r or 900, is called a quadrantal triangle; such triangles may be resolved by Napier's rules for the circular parts, if the quadrantal side be neglected, and i - b, i r- c, a, c, and A-"- r be taken for the circular parts. For let A'B'c' be the polar triangle. It will be right angled because A =. wJ Ir. Applying Napier's rule to this triangle we obtain cos a' = COS b' cos c' = cot B' cot c' (1) = sb = sin' sin a = cot c' tan t (2) cos B = cos b sin c = cot a' tan (3) 13 194 SPHERICAL TRIGONOMETRY. These are the expressions for each of the parts of a right angled triangle in terms of two others, because expressions for c' and c' would be exactly like (2) and (3). Substituting — A, — a, &c., for A', a', &c. in the above equations, they become -COS A COS B COS C cot b cot C sin B sin b sin A cot c tan c -cos b — cos B sin c = cot A tan c But these last equations are precisely what would be obtained by the application of Napier's rules, using the complements of b and c, and A — 3 w as the circular parts. 13. Napier's rules may be deduced as follows: The following formulas have been derived in the foregoing pages for oblique angled triangles. Art. 82. cos a =- cos b cos c - sin b sin c os (1) cos a sin B sin c = cos A -p cos B cos c (2)* cos c sin A sin B -= os c +- cos A cos 3 (3)* sin c sin A = sin a sin c (4)t cot c sin b cos b cos A + sin A cot c (5)t cot a sin c =cos c cos B - sin B cot A (6)t Making in the above forams A 900 they become cos a - os b cos c (1) cos a cot B cot c (2) cos C = Cos sin B (3) COsB cos b sin c sin c -sin a sin c (4) sin b = sin a sin B sin b- tan c cot c (5) sin c = tan b cot B cos B= cot a tan c (6) cos c cot a tan b'he above are but expressions of Napier's rules. 14. The case of solution treated at Art. 94, may be solhed by Napier's Analogies. Thus if a, b and A be given, B may be calculated by the sin proportion, and c and c by the formulas sin ( (a- b): sin - (a - b): ~ tan (A - ) cot' c sin (A - ): sin 4 (A - B) tan (a - b): tan. c 15. Napier's rules for the solution of right angled spherical triangles, though applicable to all cases, do not give results of that degree of accuracy which is some* (2) and (3) are the same, and are derived from (1) by polar triangles. t Sine proportion, Art. 81. Appendix II. See formula at top of p. 200. APPENDIX II. 195 times required, when the required part expressed by its sine is very small, or expressed by its cosine is very near 900. The following formulas may in such cases be used. I. By formula (8), Art. 72, 1- cos p =2 sin2 ~ p ) and p. 76, 2d note. (m) I + cos p -- 2 cos p ) whence 1 - cosp 1 -G == tan2 ~ p 1+- cos p But by Napier's rules, r being 1, cos a, = cot B cot c changing p into a, and substituting the value of cos a, given by this last, we have 1 -cotB cot c sin B sin c-cos B cos c tanr 2 a I +- cot i cot c sin B sin c +- cos B Cos c or cos (B + c) tan2 a -= e cos (B- C) which is a formula to be employed, when B and c are given and a required. II. With the same data to find b use the formula tan ~ b = 4 tan [ (B - C) + 450] tan [ (- c)- 450] derived from Napier's rules, which gives COS B cos b = - -, from the formulas preceding (1) and (4) of Art. 12, App. 1. and from sin c formula (42), Art. 15, App. I. III, The hypothenuse a and the side c being given to find the adjacent angle B use the formula tan sin (a — c) tan ) B = / / V sin (a + c) derived in a manner similar to that in case 1. IV. Given a and b to find c;byNapier's rules Cos a COs C = cos b add and subtract 1; by App. I., Art. 15, formula (52), and forms. (m) above tan c = tan ~ (a + b) tan (a - b) SPHERICAL TRIGONOMETRY. V. Finally, to obtain b when the opposite angle B and the hypothenuse a are gh ena we have, by Napier's rules, sin b = sin a sin B 1 - tan x whence, observing that -- tan = tan (450 - x), App. I., Art. 15, form. (43), and making tan x -=sin a sin B, (tan 450 -- b) = /tan (450 x) 16. The part of a spherical triangle determined by the proportion sin a: sin:: sin A sin B admits of a double value, since two arcs answer to the same sine; it becomes necessary, therefore, for us to inquire under what circumstances both these values are admissible, and how we may know which to choose when but one solution exists. Referring to the fundamental formula (Art. 82), we have cos b -cos a cos c cos B.; (2) sin a sin c in which expression we may remark that if cos b is numerically greater than either cos a or cos c, the second member must take the sign of cos b, consequently B and h must be of the same affection if sin b < sin a, or sin b < sin c, that is, an angle,must be of the same species as its opposite side, if the sine of this side is less than the sine of either of the other sides. But if cos b is numerically less than cos a, then whether the right hand member be + or- will depend upon the magnitude of cos c, or cos c will have two values corresponding to -+ cos B, and - cos B; hence an angle has two values, when the sine of its opposite side is greater than the sine of the other given side. In the proportion sin A: sin B: sin a: sin b a being the required part, the nature of the arc b may be discussed, as in the preceding case. By means of the polar triangles, we obtain from (2), in the same manner as at Art. 85, the formula COS B + C COS C C cos b = sin A sin c from which it follows, as in the foregoing case, that if cos B is numerically greater than cos A, B and b will be of the same affection. If cos B is numerically less than cos A, then both the values of b, given by the above proportion, will be admissible, "or c may be determined so as to render cos b positive or negative. Hence any side will be of the same affection as its opposite angle, if the sine of this angle be clss than the sine of either of the other angles; and the affection of the side b will be indeterminate if the sine of its opposite angle B be greater than the sine of the other given angle A. APPENDIX I. 197 17. In practice, the data for the solution of triangles are obtained by observation and measurement, and are liable to error from obvious and inevitable causes. It ia true that from the great excellency of instruments, and the almost inconceivable accuracy of modern observation, these errors are extremely minute, yet in cases where precision is requisite, it becomes necessary to determine the effects which srall errors in the data will produce upon the computed quantities, and to select the data and quasita in such a manner that the given errors in the one shall entail the Esmallest possible on the other. The principles of the Differential Calc.ulus present an easy method for the pur. pose in question, and we shall here indicate the mode of proceeding, for the benefit of the student acquainted with that branch of mathematics. Let us suppose that of the three data (for there are always three in the solution of a. triangle), two have been obtained with sueficient accuracy, but the third x is liable to an error of a given amount, which we shall call h. Let u be the sought quantity. Two of the three data being consid lered constant, the sought quantity u may be considered as a function of the third x. The quantity x becoming x + h, let the quantity u become u', we have by Taylor's theorem, du dau, d(u u' - h- 1 h 3h *+ &c. dx dx^ dx^'' - u is the error sought, and as h is in practice very small, the higher powers of it may be neglected, and we may call du u' - u 1 -T dx hence the following rule: Multiply the given error by the differential coeflicient of the sought quantity considered as a function of the given quantity liable to error, and the product will be the error in the sought quantity. If two of the data be liable to given errors, the eC:3et upon the sought quantity may be computed on similar principles, by considering the sought quantity as a finction of the two data so liable to error, and differenititing it with respect to these two independent variables. It is evident that the same method extends to the case where all the data are liable to given small errors. In this case the sought quantity is to be regarded as a n i'uction of three independent variables, and its diflerential found as before. EXAS3PLE. To determine the relation between the minute variations of the perpendicular side of a plane right angled triangle and the opposite angle, the remaining perpendicular side being considered constant. Let c and c be the side and angle which are subject to variation, and b the con. stant side. Then (Art. 41), c =b tan c and (Dif. Cal.) dc - b sec2 c which is the multiplier of the given small variation in c to obtain that in c. 10.38 SPHERICAL TRIGONOMETRY. The theory of m-aximna and minima as explained in the Calculus, will here admit of an important and easy application, viz., to find under what circumstances ut' - will be least on the supposition of a given variation h in the variable datum, or in du oiler words, under what circumstances the function will be a minimum. dx 17. The effects of small errors may be obtained, but with less brevity and elegance, without the aid of the differential calculus. The following are specimens of the mode of proceeding. PROBLEM I. in a right-angled triangle one of the oblique angles being given, to determine the variation of the opposite side, arising from a small variation in the hypothenuse. Let c be the angle which does not change, c its opposite side, and a the hypothenuse; then 6c denoting the variation of c, and Ja of a sin c -sin c sin a sin (c - dc) =sin c sin (a +- du).'. by subtraction, sin (c + dc) — sin c - sin c {sin (a +- a) -in al, that is (Art. 74, formula 4), 2 cos (c -+ dIc) sin - d6c = 2 sin c cos (a -+ da) sin J da sin c cos (a +- ~ a) sin 4 Za.'. sin -J Cc = — cos (c +- + Ic and if da, Jc be very small, sin c cos a Jc -- ~ ~.5(1' cos C or, substituting for sin c its value from the first equation, sin c cos a dc = — * - - a = tan c cot a sa; cosc sin a which variation will be the least possible when cot a is least, or when a 900. If we restore the 4 Za which has been neglected, and write the above result thus: oc = tan c cot (a +- da) da; then, in the case of' a= 900, the expression becomes dc — tan c tan a dac'a; or, considering the very small arc 4,Ja to be equal to its tangent, we have in the ease snpposed de =- 4 tan c (Ia)2. APPENDIX It. 190 PROBLEM II. In an oblique-angled spherical triangle given two sides to determine the variation produced in the thlrd side by a small variation of the opposite angle. Let a, b be the two given sides, c the included angle, and c the opposite isde. Then cos c = cos a cos b + sin a sin b cos c cos (c +- c) =cos a cos b +- si a sin b cos (c -f- dc);.. by subtraction, cos ( -- Jc) - cos c — sin a sin b g cos (c -F- Jc) - cos c \; that is, 2 sin (c + dc) sin j Jc =- 2 sin a sin b sin (c A.-, Jc) sin c Hence, if Jc be very small, sin c c = sin a sin b sin c dc ~ sin a sin b sin c.*. oc c~ ~ sin c = sin a sin Bad; and Jc is therefore the least possible when sin c is the least possible, that is, when c = 0. To find the expression for c, iu this case, restore what has been rejected, and we shall have sin a sin b sil; (c - + i c) Jc:=- - Jo; sin X which, when c = 0, and dc very small, becomes sin a sin b dc - 2 sin c -(o) PROBLEM III. In an oblique angled spherical triangle given two sides and the included angle, to find the variation in one of the opposite angles corresponding to a small variation in the included angle. Let a, b be the given side, c the included angle, to find what influence a small variation in the value of c will have on A opposite a. Substitute the expression for cos c above, in the corresponding expression for cos a, and I - sin2 b for cos b, th re resalts scs A sin c cos a sin b- sin a cos b cos c; sin a.'. cos A - = cot asin b- cos b cos Bin a But sin c sin c iaa g aisaA 200 APr-ENDIX I?. Then, by substitution, cot A sin c = cot a sin b — cos b cos c cot (A + JA) sill (C +c) = cot a sin b — cos b:c +- ) and, by subtraction, cot (A -+ A) sin (c -+- ) — cot A s'il = cos b {cos C -cos (C + } it) The first member of tlhis equation is equal to cot (A + JA) sin (c -- c) -sin c} + sin c cot (c + dA) -ot A ]; and the quantities within the brackets are respectively equal to sin dA 2 cos (ce - J ) sin.1 05c anr d ~ - -- co (c sm A sin (A + t&A) The second member of (1) is equal to cos b ~ 2 sin (c +- dIc) sin ~ Jc sin e: sin dA..2 cot (A + JA) co? (c - i c) sisin J d- ^ -2cos 6 (c+ i sin A Sin (A + JA) dc) sin I Jc and consequently when Jc and JA are very small, sin c cot A. COs c C - ~ A Cs os b sin c Jo si1i A, ~ (cot A,eo - cos b sin c) 40 MD G ARTJIII i B AV I $7PA T I OI 1 PART ]III WHli, a ship sails fiom any known place, and a correct account is kep4 of her various directions, and rates of sailing, her situation at any time may be determined by the rules of Plane Trigonometry. The processes employed for this purpose constitute what, is called Navigation. But, owing to the imperfection of the instruments with which a ship's course and the distance sailed are observed, it would be unsafe, after a long passage, to compute the place of the ship from the dead reckoning, as the observed direction and distance are called. In such cases recourse must be had to astronomical observations, from which the place of the ship or its latitude and longitude are computed by the rules of Spherical Trigonometry. The;roblem then becomes one of Nautical Astroiomy. Wo sh:all treat successively of each of these important branches. N AY I G A T I 0 N' DEFINITIONS. 96.' 1. For the purposes of Navigation the earth may be considered as spherical. It revolves about one of its diameters, called its axis, in twenty-four hours. This rotation is from west to east, causing the heavenly bodies to have an applarent motion from east to west. 2. The great circle, whose poles are the extremities of the axis, is called the equator. The poles of the equator are called also the poles of the earth; the one being the nor'th pole, and the other the south pole. 3. Great circles passing through the poles are called meridians. Through every place on the surface of the earth such a great circle may be drawn, and will be the meridian of the place. The meridian from which the meridians of other places are estimated is called a first meridian, The English have fixed upon the meridian of Greenwich Observatory for the first'mieridian, which has also been adopted in this country. 204 WA~VIGATION. 4. The longitude of any place is the arc of the equator, intercepted between the meridian of that place and the first meridian; the longitude, therefore, is the measure of the angle between the planes of the two meridians. The longitude is east or west, according as the place is situated cast or west of the first meridian. 5. The difference of longitude between two places is the are of the equator intercepted between the meridians of those places, or the measure of the angle which their planes include; hence, when the longitudes of the places are of the same denomination, that is, either both east or both west, the difference is found by subtracting the one fronm the other; but when they are of contrary denominations the difference is found by adding the one to the other. 6. The latitude of a place is its distance north or south of the equator, mneasured on the meridian of the place. Latitude cannot exceed 90~. T. Small circles parallel to the equator, are called parallels of latitude. The arc of a meridian, intercepted between two such parallels, drawn through any two places, is the difference of latitude of those places; when the latitudes are of the same name, i. e., both'. or both s., the difference of latitude is found by subtraction, but when not, the difierence of latitude is found by addition. 8. The horizon of any place is an imaginary plane, touching the surface of the earth at that place, and. extending to the heavens; such a plane is called the sensible horizon, and one parallel to it, but passing through the centre of the earth, the rational horizon of the place. The line of intersection of the plane of the horizon, and the plane of the meridian of a place, is called a north and south line; the horizontal line throuigll the samne point, and perpendicular to this, is called the east and west line. Besides the NNorth, South, East, and West points, called cardinal points, thus determined on the boundary of the horizon, there are inumerous subdivisions- corresponding to the divisions in the circle of the next page. 9. The course of a ship is the angle which her track makes with the meridians; if this angle continued the same, and the meridians were all parallel, the path of the ship would be a straight line; but as the meridians bend towards the poles, the direction of her path is continually chlanging, and she moves in a curve, called the rhumb line or loxodromic curze. The instrument employed on ship-board to show the course of the ship is called the mariner's compass. 10. The Mariner's Compass consists of a circular card, whose circumference is divided into thirty-two equal parts, called points, and each MARINER S COMPASS. —'LOG AND JLNE. 205 of these is subdivided into four equal parts, called quarter points; across this card, in the direction of a diameter, and fastened to the card, so that they move together, is fixed a slender bar of magnetized steel, called the needle; the extremities of which point to two diametrically opposite divisions of the card. These opposite divisions are marked N. and s., corresponding to the north and south poles, or ends, of ^ N the magnetized bar. The diameter.vw., at rigllt / angles to the diameter VA N.s., points out the east and west points. One point from thle 1 1 north towards the east, is marked NE., and called 1 north by east; two points, N.N.E., and called north north-east; three points, north-east by north; and so on. Each quadrant contains eight points, so that a point is 900 8 1 10 10'. (See Table of Rhumbs, Table XXVIII., at the end.) The card thus furnished being now suspended horizontally, so as to move freely, and allow the needle attached to it, to settle itself, will point out the four cardinal points of the horizon, as also the several intermediate points, provided only that it is the property of the magnetic needle'to point due north and south. Such, however, is not strictly the case, as is found by comparison with astronomical observations. The card rests ii: its centre, on a pivot placed in the vertical plane, cutting the ship f rdm stem to stern, and is held stationary in space by the magnetic forces of the earth, whilst the ship turns under it in changing her course, so that that point of the compass which is directed to the ship's head shows the si p's course, which must be corrected for the slight variation of the compass from the meridian, a variation which is different in difierent parts of the earth; the method of determining it will be hereafter given. 11. A ship's rate of sailing is determined by means of an instrument called the log, and an attached line, called the log-line. The log is a piece of wood in the form of the sector of a circle, the rim of which is so loaded with lead, that when heaved into the sea it assumes a vertica position, having its centre barely above the water. The log-line is so <20G6 Nt tAVIGA TION. attached as to keep the face of the log towards the ship, that it may offer the greater resistance to being dragged after the ship by the logli).e, as it unwinds from a reel on board, by the advancing motion of the ship. The log-line is divided into equal parts, called knots, of which each measures the 120th of a nautical or geographical mile.* A half minute sand glass is used in connexion with the log. When the log is heaved, the instant the first knot on the line passes the hand of a sailor, the half minute glass is turned by a word, and the instant the sand is run out, the line is caught by a word; as half a minute is the 120th of an hour, it follows that the number of knots, and parts of a knot, run in half a minute expresses the -number of miles, and parts of a mile, run in an hour, at the same rate of sailing. ON PLANE SAILING. 97. Let the annexed diagram represent a portion of ~ the earth's surface, P the pole, and E Q the equator. Let A ( be a rhumb line, or path described by a ship in sailing on a single course:fr'OmI A to / Let the rhumb line be divided into portions A,, bc, cd, &c., so small that each may differ insensibly from. a straight line, and draw meridians through these several divisions? as also the parallels of latitude bb', cc', dd', &c.; a series of triangles will thus be described on the surface of the globe, but so small that each may be considered as a plane triangle. These triangles are all similar, for the angles at b, c', d', &c., are right-angles, and the ship's path cuts all the meridians at equal angles; hence (Theorem 63 Geom.), Ab: Ab': b: bce': c: dd:', &c. * The geographical mile is one minute of the earth's circumference. Taking the diameter at 7916 English miles the geographical mile will be about 6079 feeto FP X A NKE -S AR sNG. 207 herefore, since by the theory of proportion the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent, Ab: bab: Ab +- b + -cd + &c., Ab' -+ b' + cd' + &c. But Ab + bc + cd- - &c., is the whole distance sailed, and Ab'- b'-+ cd'- &c. = A/', is the difference of latitude between A and G; consequently, if a right angled triangle ABB', similar to the small triangle Abb' be constructed, that is, one in which the angle A is B' B equal to the course, and the hypothenuse A B is equal to the distance sailed, the side A B' will repre- sent the difference of latitude. Moreover, the other side BB', or that opposite to the course will represent the sum b'b -+ c'c d'd + &c. of all the minute departures which the ship makes from the successive meridians which it crosses; for as the triangle ABB', in this last diagram, is similar to the small triangle A.bb', in the former we have Ab: bb':: an:' (1) bat in the first figure we have Ab: bb': be: cc' ~ cd dd', &c. A. ab: bb': Ab + be + cd + &c.: bb'-' cc'+ dd'+ &c. (2) consequently, since the three first terms of (1) are respectively equal to those of (2), the fourth term nB', of (I), must be equal to the'fourthterm, bb'+ cc'+ dd + of (2), &c. This last quantity is called the depxrture of the ship in sailing from A to 3.* It follows, therefore, that the distance sailed,.the difference of latitude made, and the departure. are correctly represented by the hypothenuse and sides of a right angled plane triangle, in which the angle opposite the departure is the course, so that when any two of these four things are given, the other two may be found simply by the resolution of a right angled plane triangle; so far, therefore, as these particulars are concerned, the results are the same as if the ship were sailing on a plane surface, the meridians being parallelI straight lines, and. the parallels of latitude cutting them at ight angles; and hence that part of Navigation in which only distance sailed, departurei difference of latitude, and course are considered, is called Plane Sailing. * The departure is not to be confounded with,3' in the first diagram. It ir greater than this, because the small departures B6b, cc', &c., whose sum is the whole departure, la.p over each other. 2^~08 H~NAVG rATiON. The two of the four elements which enter into problems of plane sail-. aig usually given are course and distance, being frund from observation.. EXAMPLES. L. A ship from latitude 470 30t N. has sailed S.W. by S. 98 miles, At what latitude has she arrived, and what departure has she made? Let c be the place sailed from, CB the meridian, C the angle c = 3 points = 330 45', see Table of lRhumbs, and cA = 98 miles, the distance sailed; then CB will be the difference of latitude, and BA the departure. Then by the formulas for the solution of right angled triangles (forms (4) and (5) Art. 64) A B Dist 98 log. P199123 log. 1*99123 Course 330 45' log. cos 9091985 log. sin 9,'4474 )iff. of lat. 81'48 log. t191108 D)ep. 54945 log. ]'73597 Latitude left 470 30' No Diff. of lat. = 81'48' minutes 1 22 S. Dep. = 5245 miles W, Latitude in 46 9 N. 2. A ship sails for 24 hours on a direct course, from lat. 38~ 32' Ny ill she arrives at lat. 36c 56' N.; the course is between S. and E., and the rate 5- miles an hour. Required the course, distance and departure. Lat left 380 32' N. 24 X 5j = 132 miles, the distance. Latin 36 56 N. Diff. 1 36 = 96 miles. Dist 132 log. 2*12057 logo 2*12057 Diff lat. 96 log. 1998227 Course 430 20' log. cos 9'86170 log, sin 9*83648 Dep. 90*58 log. 195705 Hence the course is S. 430 20' E., and the departure 90*58 miles E. 3. A ship sails from lat 30 52' S. to lat. 40 30' N., the course being N.W. by W. j W.; required the distance and departure. Distance, 1065 miles; Departure 939*2 miles W, See last note but one. Miles are converted into degrees, &c., by dividing by 60. TRAVERSE SAILING. 209 4. Two ports lie under the same meridian, one in latitude 520 30' N. and the other in latitude 47~ 10' N. A ship from the southernmost sails due east, at the rate of 9 miles an hour, and two cays after meets a sloop which had sailed from the northernmost port; required the sloop's direct course and distance run. Course S. 53~ 28' E., or S.E. & E.; distance run 53t76 mhiles. 5. If a ship from lat. 480 27' S. sail S.W. by W. 7 miles an hour, in what time will she arrive at the parallel of 500 S? In 23'914 hours. 6. If after a ship has sailed from lat. 40~ 21' N. to lat. 46~ 18' N., she be found 216 miles to the eastward of the port left; required her course and distance sailed. Course N. 31~ 11' E., distance 417'3 miles. TRAVERSE SAILING 98. Is where a ship, in going from one place to another, sails on different courses; the determination of the single course and distance from the one place to the other is called working or compounding the traverse. The( method of proceeding is to rule seven columns (see next page), the first to contain the courses, the second the.number of points and quarter points in these courses which may be found from the table of Rhumbs (Tab, XXVIII.), the third the distances sailed on these courses, the fourth and fifth the differences of latitude in two columns entitled N. and S., in the former of which the dif. of lat. for the N. courses is entered, and in the latter of which the dif. of lat. for the S. courses; the sixth and seventh contain the departures, the former those of the E. courses, the latter those of the W. courses, sometimes called eastings and westings. When these several particulars are all inserted, the columns are added up, and the difference of the sums of the N. and S. columns will be the whole difference of the latitude which the ship has made, and the difference of the sums of the E. and W. columns will be her whole departure. The columns appropriated to the difference of latitude and departures are usually filled up frorm a table (Table I.), already computed to every quarter point of the compass, and to all distances from one mile up to 300; so that by entering this table with any given course and distance, the proper difference of' l.titude and departure is found by inspection. The course is found at the top or bottom of the page, and the distance in the first column, the dif. of lat. in the second column, and the departure in the third, if the course be found at top; but if at the bottom, the lat. and 14 210 NAVIGATION. dep. columns are interchanged, as may be seen by the entitling in the bottom of the columns. If the distance sailed be more than 300 miles, it will exceed the limit of, the table; but the difference of latitude and departure may still be determined from it by this simple operation: divide the given distance by any number that will give a quotient not exceeding 300; enter the table with this quotient, and multiply the corresponding diff. of lat. and dep. by the assumed divisor, and there will result the diff of lat. and dep. due to the proposed distance. If there be a remainder add the diff. of lat. and dep. corresponding to this. Or take any two numbers whose sum is equal to the given distance; the sum of their differences of lat. and dep. will be the lat and dep. of the given distance. These rules depend upon the principle that for the same course the differences of latitude and departure are proportional to the distance run, which will be evident if we recollect that dist., diff. of lat. and dep. form a right angled triangle, and that two right angled triangles are similar when an acute angle of one is equal to an acute angle of the other. EXAMPLE. I. A ship sails from lat. 240 32' N., and runs the following courses and distances, viz. 1st, S.W. by W., distance 45 miles; 2d, E. S. E., distance 50 miles; 3d, S.W., distance 30 miles; 4th, S.E. by E., distance 60 miles; 5th, S.W. by S. I W., distance 63 miles: required her present latitude, with te direct course and distance from the place left, to the place arrived at. TRAVERSE TABLE. COURSES. rT8. DIST. DIFFERENCE OF LAT. DEPARTURE.,N sN. S. E. W. S. W. by W. 5 45 25*0 37.4 E. S. E. 6 50 19-1 46*2 S. W. I 30 21'2 21-2 S. E. by E. 5 60 33~3 49.9 S. W 4 by S. 4 W. 3 63 50-6 37.5 149P2 96-1 96*1 It appears from the results of this table that the difference of latitude made by the ship during the traverse is 1492..z —2 29' S. TRAVERSE SAILING. 211 Lat. left 24~ 20' N. Diff. lat. 2 29 S. Lat. in 22 3 N. t appears also that the departures east are equal to the departures west, so that the ship has returned to the meridian she sailed from, consequently the direct course from the place left to that come to is due south, and the distance is equal to the difference of latitude, which is 149*2 miles, There is another mode of finding the direct course and distance, much practised by seamen, viz., by construction. For this purpose the marine's scale is employed, which is a two foot flat rule exhibiting several scalen on each side, by hell) of which and a pair of compasses the usual problems in sailing may be all solved. One of these scales, which is called a scale of rhumbs, is a scale of chords, to every point and quarter point of the compass; and another is a more enlarged scale of chords to every degree. Both these scales are constructed for a circle with the same co~mmon radius, so that the chords on the scale of rhumbs belong to that circle whose radius equals the chord of 600 on the scale of chords; and the method of laying down a traverse from these scales, and a scale of equal parts, and of thence measuring the equivalent single course, and distance made good, will be at once understood from the example. Construction of the traverse for the last example is as follows: With the chord of 60, taken from the line of chords on the mariner's scale, describe the horizon circle, and draw the north and south line N. S. From the line of rhumbs tahe the chords of the several courses, and ( as these are all southerly they must be laid off firom the south point S, those which are westerly to the left, and those which are " C easterly to the right, their extremities being marked 1, 2, 3, &c., in the order of the courses. This done, lay off from any scale of equal parts, and in the direction of Al, the distance AB sailed on the first course; then in the direction parallel to A2, the distance Bn sailed on the second course; in the direction parallel to A3, the distance cD on the third course; in the direction parallel to AII NAVIGATION. A4, the distance DE on the fourth course; and, lastly, in the directon parallel to A5, the distance EF on the fifth course; then F will represent the place of the ship at the end of the traverse; FA, being applied to the acale of equal parts, will show the distance made good, and the chord of the arc included between this distance, and the meridian, being applied to the line of rhumbs, will show the direct course. In the present case the inatercepted are will be 0, showing that F is on the meridian of A. 2. A ship from Cape Clear, in lat 51~ 25' N., sails 1st, S.S.E. 4 E., 16 miles; 2d, E.S.E., 23 miles; 3d, S.W. by W. -- W., 36 miles; 4th, W. N., 12 miles; 5th, S.E. by E. ~ E., 41 miles: required the distance made good, the direct course, and the latitude reached? TRAVERSE TABLE. COURSES. FTS. DIST. DIFFERENCE OF LAT. DEPARTURE. N. S. E. W. S.S.E. i E. 2? 16 I145 16 68 ESE.S.E 6 3 8-8 213 S.W. by W. A W. 5W 36 170 31.8 W. N. 7 12 13 8 11'9 S.E. by E. iE. 51 41 211 3.52 1-8 61P4 63"3 43.7 1*8 43.7 59*6 19*6 Lat. left 510 25' N. Diff. lat 5996 miles 1 0 S. Lat. in 50 25 N. Then by means of the whole dif. of lat. and dep., which are the two perpendicular sides of a right angled triangle, one of the acute angles and hypothenuse, or the direct course and distance, may be computed as fo0llows TRAVERSE SAILXNG. 2 diff. lat. 596 log. 1'77525 departure 19' 6 log. 1*29226 log. 1*29226 course 18~ 12' log. tan 9*51701 logo sin 9'49462 1'79764 distance 62'74 therefore, as the difference of latitude is south, and the departure east, the direct course is S. 18~ 12' E., and the distance made good 6274 miles. To construct this traverse, describe, as before, the horizon circle, with a radius equal to the chord of 600, and taking from the line of rhumbs the chord of the first course, 24 points, apply it from S to 1, to the right of S N, as this course is south-easterly; apply, in like manner, the chord of the second course, six points from S to 2, also to the right of the meridian line; apply the chord of the third course, 5-1 points from S to 3, to the left of the meridian, the chord of the fourth course, 7 from N. to 4, to the left of N S, this course N being north-westerly, and, lastly, apply the chord of the fifth course, 54 points, from S to 5, to the right of SN. In the 4' direction Al, lay off the distance AB = 16 miles from a scale of equal parts; in the direction parallel to A2, lay off the distance S &\ j \ BC = 23 miles; in the direction parallel to A3, lay off CD = 36; \ in the direction parallel to A4 lay off DE = 12 miles; and, lastly, in the direction parallel to A5, lay off EF = 41; then r will be the place of the ship at the E end of the traverse; consequently, AF will be the distance made good, and the angle FAS the direct course; applying, therefore, the distance AF to the scale 1 of equal parts, we shall find it reach from 0 to 621; and applying the distance Sa to the line of chords, we shall find it reach from 0 to 180, o0 by the scale of rhumbs it will be found to measure one point and a halft 21 4 NAiGATION. 3.. A ship from lat. 28~ 32' N, has run the following courses, viz, 1st, S.W. by N., 20 miles; 2d, S.W., 40 miles; 3d, N.E. by E., 60 miles; 4tf4 S. E. 55 miles; 5th, W. by S., 41 miles; 6th, E.N.E., 66 miles. Required her present latitude, the distance made good, and the direct course from the place left. The direct course is due east, and distance 70*2 miles, the ship being in the same latitude at the end as at the beginning of the traverse. 4. A ship from lat. 410 12' N., sails S.W by W. 21 miles; S.W. i S. 31 miles; W. S.W. i S., 16 miles; S. 4 E., 18 miles; S.W. ~ W., 14 miles; and W. i N., 30 miles; required the latitude of the place arrived at, and the direct course and distance. Lat. 400 5' N.; course S. 520 49' W.; distance 111'7 miles. 5. A ship runs the following courses, viz. 1st, S.E., 40 miles; 2d, N.E., 28 miles; 3d, S.W. by W., 52 miles; 4th, N.W. by W., 30 miles; 5th, S. S. E., 36 miles; 6th, S. E. by E., 58 miles; required the direct course and distance made good. Direct course 8, 250 42' E., or S.S. E.; E. nearly; distance 95'69 miles. These examples will sufficiently illustrate the principles of plane sailing, in. which, course, distance, difference of latitude, and departure, are the only quantities which enter into the problem, two of them being always given. The determination of the difference of longitude made on any course, which is the distance between the meridians measured on the equator, cannot be effected by these principles, for this element is not the same as if the meridians were all parallel to each other, as is the case with the other elements. The finding of the difference of longitude is the easiest when the ship sails due east or due west, that is, upon a parallel of latitude; this is called PARALLEL SAILING. 99. The theory of parallel sailing is comprehended in the following proposition, which admits of a variety of other applications. The arc of a great circle comprehended between two of its secondarsi is to the arc of a parallel small circle, comprehended between the same soontlaries, as radius unity is to the cosine of the distance between the great circle and its parallel, measured on one of the secondaries. (See Spherical Geom., Prob. 2, Cor. 6.) Applied to the case under consideration the above proposition would n as follows, viz: The cosine of the latitude of the parallel is to the distance run as the PARALLEL SAILING. 215 radius to the difference of longitude. This may be demonstrated ad abllows: Let IQH represent the equator, and BDA any parallel of latitude; cI will A- c be the radius of the equator, and / CB the radius of the parallel. Let BD be the distance sailed, then the C difference of longitude will be mea- / sured by the arc IQ of the equator, and since similar arcs are to each as the radii of the circles to which they belong, we have cB: c:: dist. B: diff. long. IQ But CB is the cosine of the latitude IB to the radius ci, and as cosine and radius are proportional in different circles, CB: ci: cos lat.: R The first two terms of these proportions being the same, the last are proportional, and we have cos lat.: Rad.: distance: diff. long. (1) Corollary: hence if the distance between any two meridians, measured on. a parallel in latitude L be D, and the distance of the same meridians, measured on a parallel, in latitude L' be D', we shall have (Spher. Geom., Prop. II., Cor. 6), COS L: D:: COS L': D' (2) for both the ratios of (2) will be equal to R: diff. long. By referring to proportion (1) it will be seen that if any one of the legs of a right-angled triangle represent the distance run on any parallel, and the adjacent acute angle be equal to the degrees of lat. of that parallel, then the hypothenuse will represent the difference of longitude, since this hypothenuse will be determined by that proportion. The right angled triangle used in plane sailing may therefore be employed here, changing the names of its elements, viz., course into latitude, difference of latitude into distance, and distance into difference of longitude. And a traverse table computed to degrees and fractions of a degree instead of points and quarter points, may be employed to solve problems in parallel sailing. 216 NAVIGATION. Formula (1) above may be expressed by the following rule. Divide the distance sailed by the cosine of the latitude, and the quotient will be the difference of longitude. EXAMPLES, 1. A ship from latitude 530 56' N., longitude 100 18' E., has sailed due west, 236 miles: required her present longitude. By the rule lat. 53~ 56 log. cos 9*76991 dist. 236 log. 23'7291 dif. long. 409 log. 2*60300 Long. left 100 18' E. 409 Diff. long. = degrees= 6 49 W Long. reached 3 29 E. 2. If a ship sail E. 126 miles, from the North Cape, in lat. 71 10' N., and then due N., till she reaches lat. 730 26' N.; how far must she sail W. to reach the meridian of the North Cape? Here the ship sails on two parallels of latitude, first on the parallel of 710 10', and then on the parallel of 730 26', and makes the same dif ference of longitude on each parallel. Hence by the corollary, As cos. lat. 71~ 10' arith. comp. 0*49104: distance 126 2'10037: cos lat. 73 26 9*45504: distance 111'3 2'04645 3. A ship in latitude 320 N. sails due east, till her difference of lagitude is 384 miles; required the distance run. 325'6 miles. 4. If two ships in latitude 440 30' N., distant from each other 216 miles, should both sail directly south till their distance is 256 miles, what latitude would they arrive at? 320 17 N. 5. Two ships in the parallel of 47~ 54' N., have 90 35' difference of longitude, and they both sail directly south, a distance of 836 miles MIDDLE LATITUDE SAILING. 217 required their distance from each other at the parallel left, and at that reached. 385*5 miles, and 476*9 miles. MIDDLE LATITUDE SAILING. 100. Having seen how the longitude which a ship makes when sailing on a parallel of latitude may be determined, we come now to examine the more general problem, viz., to find the longitude a ship makes when sailing upon any oblique rhumb. There are two methods of solving this problem, the one by what is called middle latitude sailing, and the other by Mercator's sciling. The first of these methods is confined in its application, and is moreover somewhat inaccurate even where applicable; the second is perfectly general, and rigorously true; but still there are cases in which it is advisable to employ the method of middle latitude sailing, in preference to that of Mercator's sailing; it is, therefore, proper that middle latitude sailing should be explained, especially since, by means of a correction to be hereafter noticed, the usual inaccuracy of this method may be rectified. Middle latitude sailing proceeds on the supposition: B that the departure or sum of all the meridional dis- / tances bb', cc, dd', &c. from A to B, is equal to the dis- \ \ ance M'M of the meridians of A and B, measured on the middle parallel of lati- A tude between A and B. This supposition becomes very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance between the extreme meridians will be much greater than the departure, if the track A B cuts the successive meridians at a very small angle. The principle approaches nearer to accuracy as the angle A of the course increases, because then as but little advance is made in latitude, the several component departures lie more in the immediate vicinity of the middle latitude parallel. But since in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance, this principle is not to be recommended in such latitudes if much accuracy s required. ,1 8 NA IGATION. By means, however, of a small table of corrections, constructed by Mr. WORKMAN, the imperfections of the middle latitude method may be removed, and the result of it rendered in all cases accurate. This table we have given at the end of the present volume. The rules for middle latitude sailing may be thus deduced. It has been seen at (Art. 97), that the difference of latitude, departure, and distance sailed on any oblique rhumb, may be A all accurately represented by the sides AB', B'B, An, of a right angled plane triangle. Now, by the present hypothesis, the departure B'B is equal to the middle latitude distance between the meridians of the places sailed from, and arrived at, so that the dif-, dep. ference of longitude of the two places of the ship is the same as if it had sailed the distance B'B on the middle latitude parallel; the determination of this difference of longitude is, therefore, reduced to a case A. of parallel sailing; and since, as we have seen (p. 215), the formula for parallel sailing is a proportion which expresses the relation between the elements of a right angled plane triangle in which the base is the dist. sailed, the angle at the base the lat., and the hypothenuse the diff. of long., let B'BA' be this triangle, in which, according to the theory of mid. lat. sailing, the departure B'B takes the place of the dist. sailed. From these triangles, the two partial ones of which are right angled, and the total one not, we have the following theorems, viz., in the triangle A''B, COS AB': BB"':: radius: BA' that is, i. Cos. mid. lat.: departure:: radius: diff. of long. In the triangle A'BA, which is not right angled, Sill A: AB: sin A: AB; that is, ii. Cos mid. lat.: distance: sin course: diff. long. In the triangle ABB', we have the proportion (Art. 41), R: tan A:: AB' BB' comparing this with the first proportion above, observing that the extremes of this are the means of that, we have A' AB: A:: COS ABBt: tan A; that is, il. Diff. lat.: diff. long.: cos mid. lat.: tan course. MIDDLE LATITUDE SAILING. 219 These three proportions comprise the theory of middle latitude sailing, and when to the middle latitude the proper correction, taken from Mr. Workman's table, is added, these theorems will be rendered strictly accurate. This is Table XXIX; the middle latitude is to be found in the first column to the left; in a horizontal line with which, and under the given difference of latitude, is inserted the proper correction to be added to the middle latitude to obtain the latitude in which the meridian distance is accurately equal to the departure. The formula for constructing this table is obtained as follows' Let d = proper diff. of lat. D. =meridional diff. of lat. m = middle latitude. M = m + correction. L = diff. of longitude. Then (Art. 100, Form III.), COS M X L tan course = -- d But (Art. 101, Rule 1), rad X L tan course = D cos M X L rad X L rad d ~*. -~- ~ === ~.". COS M- d D D rad. d.. correction = cos --.d m D EXAMPLES. 1. A ship, in latitude 51~ 18' N., longitude 220 6' W., has sailed S. 330 5' E., required her latitude and longitude. The required latitude is found by plane sailing, as follows: Course 330 5' log. cos 9'92318 Dist. 1024 log. 3*01030 Diff. Lat. 858 2'93348 * The investigation of this formula should be postponed until after reading the next article, and may be omitted entirely. 220 NAVIGATIONe Lat. left 51~ 18' N. Diff. lat. 14 18 Lat. required 370 N. To find the longitude by mid. lat. sailing. Lat. left 510 18' " reached 370 Sum 88 18 Sum 440 9' mid. lat. Then by the proportion III. above, cos. mid. lat. 440 9' ar. comp log. 0*14418: tan course 33 5 log. 9-81390: diff. lat. 858 log. 2*93349: diff. long. 779 log. 2*89157 In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. To find the proper correction, look for the given middle latitude, viz., 440 9' in the table of corrections, the nearest to which we find to be 44~; against this and under 140 diff. of lat. we find 27', also under 150 we find 31', the difference between the two being 4'; hence corresponding to 14~ 18' the correction will be about 28'. Hence the corrected middle latitude is 440 37', therefore, cos. corrected mid. lat. 440 37' ar. comp. log. 0o14763: tan course 33 5 9*81390: diff. lat. 858 2*93349: diff. long. 785*3 2'89502 therefore, the error in the former result is about 6, miles. Long. left, 220 6' W. Diff. of long. 785 = 13~ 5' Long. required, 9~ 1' W. 2. A ship sails in the N.W. quarter, 248 miles, till her departure is 135 miles, and her difference of longitude 310 miles; required her course, the latitude left, and the latitude come to. Course N. 32~ 59' W.; lat. left 62~ 27' N.; lat. in 65~ 52' N. MERCATOR S SAILINGS 221?. A ship, from latitude 37~ N., longitude 90. 2' W., having sailed between the N. and W., 1027 miles, reckons that she has made 564'miles of departure; what was her direct course, and the latitude and longitude reached, the middle latitude being uncorrected by Workman's table? Course N. 330 19' W. or N.W. by N. nearly; lat. 51~ 18' N.; long. 22~ 8' W. 4. Required the course and distance from a point in lat. 370 48' N., long. 250 13' W., to a point in lat. 500 13' N., long. 30 38' W., the middle latitude being corrected by Workman's Table. Course N. 510 11' E.; distance 1189 miles. MElRCATOR'S SAILING. This is for the determination of difference of longitude when a ship sails on any oblique rhumb. 101. It has already been seen that when a ship sails on any oblique rhunmb, the difference of latitude, the departure, and the distance run, are truly represented by the sides of a right-angled plane triangle. Let AB'B in the annexed diagram be this triangle, A representing the course, An' the diff. of lat., and n'B the departure. Let AC' be a sufficiently greater difference of latitude to make the corresponding departure cc' equal to the difference of longitude required. This increased difference of latitude AC" is called the meridional difference of latitude, AB', being called the proper difference of latitude, by way of distinction. The solution of the triangle AC'C then will serve to determine the difference ^ 11 of longitude c'c. In this triangle we know the course A, and we shall now show how to construct a table for finding the side AC', the meridional difference of latitude. The departure B'B represents the sum of all the very small meridian distances, or elementary departures, b'b, c'c, &c., in the diagram at Art. 100, the difference of latitude J AB' represents the sum of all the corresponding small differences in the figure referred to, and the distance AB the sum of all the corresponding distances Ab, be, cd, &c., and each of these elements is supposed to be taken so exceedingly small as to form on the sphere a series of triangles, differing insensibly from plane triangles. Let Ab'b in the annexed diagram represent one of these elementary triangles, b'b will be one of the elements of the departure, and ab', the cor 222 NAVIGATION. responding difference of latitude; and as b'b is a small portion of a paral lel of latitude, it will be to a similar portion of the equator, or of tbh meridian, as the cosine of its latitude to radius (Art. 99), this similar portion of the equator, or of the meridian, being the difference of longitude between b' and b. Suppose now the distance Ab prolonged to p, till the departure p'p is equal to the difference of longitude of b', and b, then b'b will be to p'p as the cosine of the latitude of b b to the radius; but bb: p'p: xb': ap';hence the proper difference of Ab' is to the increased difference Ap' a the cosine of the latitude of b'b to the radius. Calling, therefore, the proper difference of latitude d, the increased difference of latitude D, the latitude of b'b, 1, and the radius 1, which it is in the table of natural sines, this proportion will be in symbols, d: D::cos 1: 1 d i D - - = d sec I since sec - (Art. 33.) cos I cos The ship, therefore, having made the small departure b'b, and the difference of latitude Ab,'must continue her course till the difference of latitude becomes D, in order that her departure may become equal to the difference of longitude corresponding to b'b. Conceiving all the elementary distances to be in this manner increased, the sum of all the corresponding increased departures will necessarily be the whole difference of longitude made by the ship during the course. The determination of AC' requires the previous determination of all its elementary parts; if d be taken equal to 1', each of these parts will be expressed by D = 1' sec 1, or D sec, that is, sec I expresses the meridional difference of latitude corresponding to a proper difference of latitude of'at the latitude 1; giving I successively the values 1', 2', 3', &c., up to 900, and adding the result of the second substitution to that of the first, and so on, we shall have in succession, the values of the increased latitude corresponding to 1', 2', 3', &c. of proper latitude; these values are called the meridional parts, corresponding to the several proper latitudes, and when registered in a table, form a table of meridional parts, given in all books on Navigation.* The following scheme may serve as aspecimen of the manner in which such a table may be constructed, and, indeed, of the manner in which the * In other words, a table of meridional parts is a table of differences of latitude expressed in geographic miles, each difference of latitude being enough greater than its corresponding proper difference of latitude, to make the departure equal to the difference of longitude. The table gives the meridional difference between th* equator and any given latitude. MERCATOR S SAILING. 223 first table of meridional parts was actually formed by Mr. WRIGHTI the proposer of this ingenious and valuable method. Mer. parts of 1'= nat. sec 1'. Mer. parts of 2'= nat. sec 1'+ nat. sec 2'. Mer. parts of 3'= nat. sec 1'+ nat. sec 2'+ nat. sec 3'. Mer. parts of 4'- nat. sec 1'- nat. see 2'+ nat. sec 3'+ nat. sec 4', &c., &c. Hence, by means of a table of natural secants, wee have Nat. sees. Mer. parts. Mer. parts of' — 1 0000000 1 10000000 Mer. parts of 2'-= 10000000 + 1'0000002 = 200000002 Mer. parts of 3'= 2*0000002 + 1*0000004 _ 3o00000006 Mer. parts of 4'= 3*0000006 + 1P0000007 4'00000013 &c. &c. There are other methods of construction, but this is the most simple and.obvious. The meridional parts thus determined are all expressed in geographical miles, because in the general expression D = 1' sec l, 1' is a geographical mile. Having thus formed a table of meridional parts (Table III. at the end), if we enter it with the latitudes sailed from, and reached, and take the difference of the corresponding parts in the table, the remainder will be the meridional difference of latitude, or the line Ac' in the preceding diagram. The angle A is the given course, so that there are known in a right angled triangle Ac'c, an angle and the side adjacent, to find the side opposite. The following is the rule. 1. The tangent of the course X meridional difference of latitude = the difference of longitude; or if the departure be given instead of the course., then from the similar triangles AB/B, Ac'c, the proportion will be 2. As the proper difference of latitude is to the departure, so is the meridional difference of latitude to the diference of longitude. Other proportions immediately suggest themselves from the preceding figure. * Observe that the meridional parts, or meridional diff. of latitude from the equator to 2' of latitude will be the sum of the meridional parts from the equator to 1', plus the meridional parts from 1' to 2t, which latter is nat. sec. 2'. Again, that the meridional parts from the equator to 3' is the sum of the meridional parts from the equator to 2', and the meridional parts from 2t to 3', which latter is the nat. see. 3,. 224:NAVIGATIONo EXAMPLES. 1. A ship from latitude 420 30' N., and longitude 580 51 W., having sailed S.W. by S. 300 miles, required the latitude and longitude at which she has arrived. To find the diff. of lat. by plane sailing. Dist. 300 log. 2'47712 Course, 330 45' log. cos 9.91985 Proper dif. lat. 249*4 log. 2'39697 To find the diff. of long. by Mercator's sailing. Lat. left, 420 30' N*, mer. parts 2822 Prop. diff. lat. 249 =- 40 9' Lat. reached, 38~ 21' mer. parts 2495 Long. left, 580 51' Mer. diff. lat. 327 log. 2~51455 Diff. long. 219 = 30 39' Course, 330 45' log. tan 9'82489 Long. reached, 620 30' Diff. of long. 218'5 log. 2'33944 2. Required the course and distance from Cape Cod light-house, lat. 42~ 3' N., long 700 4" W., to the island of St. Mary's, lat. 360 59' N., long. 25~ 10' VW. Cape Cod, lat. 420 3' N. Mer. parts, 2786 Long. 70~ 4' W. St. Mary's, lat. 360 59' N. Mer. parts, 2391 Long. 250 10' Wo 50 4 Mer diff. lat. 395 44~ 54' 60 60 Prop. diff. lat. 304 Diff. long. 2694 miles. To FIND THE COURSE (by Rule 1.) To FIND DISTANCE BY PLANE SAILING. Merid. diff. lat. 395 log. 2*59660 Diff. of long. 2694 log. 3'43040 Course 810 40' log. tan 10'83380 log. cos. 9'16116 Proper diff. lat. 304 miles log. 2-48287 Distance 2098 miles log. 3v32171 * Enter tab. of mer. parts with the argument 420 30', and take out the ner. parts corresponding, viz., 2822, which is the meridional diff. of lat. from the equator to 420 30' N. Again enter the tab. with the argument 380 21'; the corresponding mer. parts, or mer. diff. of lat., from the equator to lat. 380 21' N. will be found to be 2495. Subtract the latter from the former, and the remainder will be the meridional diff. of lat. between 380 21' N., and 420 30' N. MERCATOR SAILING. 3. A ship from latitude 3s N., and longitude 320 16' W., has sailed in a north-westerly direction 300 miles, till she has reached the latitude of 41~ N. Required the precise course on which she has sailed, and the longitude at which she has arrived. Ans. Course 360 52'; LIng. 360 8' APPENDIX III. GREAT CIRCLE SAILING. 1. THE shortest path from one point to another on the surface of a sphere'ia ie are of a great circle (Geom., App. III., p. 2). A ship, therefore, sailing on the are of a great circle, joining her point of departure and point of destination on the surface of the earth, will make a shorter voyage than if she sails in the direct course, that is upon the rhumb line joining the same two points. The practical application of great circle sailing will consist in determining as often as the ship's place is found, that is to say her latitude and longitude, which, under ordinary circumstances, occurs daily, the direction which she ought to take, in order to sail on the great circle from the point where the ship is, to the point of destination. This problem is, in effect, solved in the example on p. 137. In the diagram at that place, s denotes the point where the ship is, s' the point of destination, and the angle Pss' the course which the ship ought to steer, in order to sail on the great circle from s to s'. The solution of the problem of Great Circle Sailing, it appears, from the example referred to, requires the application of Napier's Analogies, forms IX. and X. of Art. 86. In a practical treatise on Great Circle Sailing, which appeared in 1846, by S. T. Colt, a table will be found called the " Great Circle Table." It is a table of double entry, in which the logarithm of the ratio of the cosine of the half sum, to the cosine of the half difference, and of the ratio of the sine of the half sum to the sine of the half difference of the colatitudes of s and s', will be found computed for any two latitudes. You enter this table with the less of the given latitudes at top, and the greater at the side; under the former, and on the range of the latter, in the column entitled sine, is found the logarithm of the ratio of the sine of half the sum to the sine of the half difference, and in the column entitled cosine, the ratio of the cosine of the half sum to the cosine of the half difference; if to each of these be added the log. cotangent of the difference of longitude of the point where the ship is, and the point for which she is destined, the results will be the logarithmic tangent of the half sum and half difference of the angles s and s', the former of which, viz., s will be the course upon which the ship should be directed. The example at p. 137 adapted to this place should read as follows: 1. Ashipfromlat200, long. 410 34' 26", is bound to a point in lat. 510 30', long. 100, upon what course must she sail in order to pursue the shortest path to her destination? Anm. 300 28' 12'" 228 NAVIGATION. 2. A ship in lat. 400 301 N., long. 700 W., is bound for a place in lat. 510 22S N., long. 90 37' W., required the course for Great Circle Sailing? Ans. N. 540 3' E. The computation of the third side ss' in the same triangle gives the distance sailed. As a steamer in ordinary weather pursues steadily the course of the great circle from port to port, it may be convenient to calculate beforehand the position of the points in which this great circle intersects the meridians for every five degrees of longitude (five degrees being about the daily progress of a first class steamer), and then if the ship lays upon the direct course for these points successively, it will be sufficient, since the rhumb line differs insensibly from the arc of a great circle for so short a distance. The method of determining these is simple. For when the angle s is calculated, it is only necessary to employ the last two forms, XI. and XII. (Art. 86) of Napier's Analogies, which are used for solving a spherical triangle when two angles and the interjacent side are given. The data will be the colatitude of the point s, the course rss' previously calculated, and the angle which the meridian rs makes with the meridian whose point of inter-: section with the great circle course from s to s' is to be calculated; in other words, the difference of longitude of' these two meridians. The logarithm of the ratio of the cosines and sines of the half sum and half difference of the given angles may be taken from " The Circle Table," entering it with the complements of these angles; the log. tangent of the given side will be the same in the calculation for each meridian; the solution of the triangle for each meridian giving the colatitude of the point in which the great circle path intersects the meridian, and the course which the ship ought to take in departing from that meridian. EXAMPLE. A ship sailing from the port of New York to Havre or Liverpool, by the shortest path, would steer from Sandy Hook, lat. 400 27' 30", long. 740 00' 48" W., E. i S.*running on the rhumb line, and thus give the south shoal of Nantucket a berth of about 15 miles; from this she would sail to a point in lat. 410, long. 680, on southern part of George's shoal, in 25 fathoms of water. From this point she would commence Great Circle Sailing, nothing being gained in taking the great circle rather than the thumb line in the previous short distances. As the great circle from this point to that of destination would pass over Newfoundland, it is recessary to divide the voyage between two great circles, the first terminating at Cape Race, and the second terminating at Cape Clear, the south point of Ireland. Required the course from the south shoal of Nantucket to George's Shoal, and the points at which the great circle from lat. 410 N., long. 680, W., to Cape Race, in lat. 460, 39' 24", long. 530 04' 36", intersects the meridians of 600 and 550 W.; and the points in which the great circle from Cape Race to Cape Clear, in lat. 510 22' N., long. 90 371 W., intersects the meridians of 450, 400, 350, 300, 253, 200, and 150 W. Ans. The course from Nantucket to George's is * This allows for variation of compass. APPENDIX I1i. 220 Tlie first great circle intersects the meridian of 600 at 440 t4' 25", of 550 at 460 5' 28" of N. latitude. The second great circle intersects the meridian of 450 at 480 55' 47";5, of 400 at 490 58' 27/, of 350 at 500 46' 6", of 300 at 510 19t 44", of 250 at 510 40' 3", of 200 at 510 47' 28", and of 150 at 510 42" 7.* The following, from Mr. Coit's work, to which the student is referred for a variety of useful problems, and interesting information, will furnish a number of exercises. TRACK OF THE ARC OF A GREAT CIRCLE FROM CHESAPEAKE BAY TO BORDEAUX. Table showing the longitude of the intersections of the latitudes crossed by the arc of a great circle from Cape Henry's light-house, mouth of Chesapeake Bay, to the Corduan light, near Bordeaux; also showing the courses from the points of intersections of the latitudes crossed, and from intersection to intersection. Long. of the intersec. of the lat. crossed. Courses towards Courses towards latitudes - Corduan from Cape Henry from crossed. From the From the the points the points Meridian o Meridian o of intersection. of intersection. Greenwich. Cordaan. C. Henry. 360~56' 760.04' 740.54' N. 550.40' E. 37 * 75 *55 74'45 - 55 *46 - s. 550-46' w. 38* 74 02 72 52 -56-55 - -56 -5539 - 72 01 70 51 - 58.10 - -58 10 - 40 - 69.52 68 42 - 59 32 - - 59 32 - 41 67 36 66 26 -,6101 - - 61 01 - 42 64 25 63 *15 - 62 41 - -62 *41 - 43. 62 22 61 12 - 94 31 - - 64 61 - 44* 59*19 58 *09 -6637- - 66 *3745 56 *27 55 *17 - 69 -03 - - 69 *0146' 51'52 50 42 - 71 53 - -71.53 47 46 *55 45 45 - 75 29 - -75 29 - 48 - 39-56 38 46 - 80 38 - - 80 38Max. Lat. 48'41 27.5- 26 *15 East. West. 48 * 14 *54 13 *44 s. 80 -38 E. N. 80 38 w. 47 — 7-55 6 45 - 75 29 - -75.2946' 2.58 1*48 - 71 53 - - 71 53 - Corduan..45'35 1 10 0...- 70 -36 _____________ _________ _________ _____ _______ On this track the difference or saving is 110. * The great circle courses and distances are as folows: From George's Shoal to the meridian of 600, 570 14' and 407 miles. From 600 to 550 N., 620 40' E., and 234 miles. From 550 to Cape Race N. 660 13' E., and 87 miles. From Cape Race to the meridian of 450, N. 640 19' E., and 353 miles. From 450 to 400, N. 700 18' E., and 205 miles. From 400 to 350, N. 740 6' E., and 197 miles. From 350 to 300, N. 770 57' E., and 191 miles. From 300 to 250, N. 800 50' E., and 188 miles. From 250 to 200,N. 850 45' E., and 186 miles. From 200 to 150 N. 890 42' E., and 186 miles. From 150 to Cape Clear S. 860 23', and 202 miles, 230 APPENDIX 111. For further information on Great Circle Sailing, see " The Practice of Navigation and Nautical Astronomy, by Lieut. Raper, of the Royal Navy," Art. 336, in which work will also be found a convenient table (tab. 5) called the Spherical Traverse table, for solving problems in this kind of sailing. See also a small collection of. " Tables to facilitate Great Circle Sailing," by John Thomas Towson, published by order of the Lords Commissioners of the Admiralty. SUMNER'S METHOD. This is a method discovered recently by accident, and consists in calculating the ship's longitude by chronometer for two assumed latitudes, the one of which is the next even degree less, the other the next even degree greater (without odd minutes) than the latitude by dead reckoning. The two positions of the ship thus determined from the longitudes found and the assumed latitudes, being projected on a Mercator's chart, the line joining them passes through the true position of the ship, and any land it may happen to pass through in the vicinity, will have the same bearing from the ship that this line makes with the meridian. The theory is, that this line is a small portion of what the author terms a parallel of equal altitude, that is, a small circle of the terrestrial sphere, the pole of which is the point of the earth's surface, at which the sun is vertical or in the zenith at the instant of observation. To all places situated on this circle the sun will have the same altitude at the instant. Now since in the two computations in the above problem the latitude only is different, the altitude and declination, which are the other data, remaining the same, the altitude of the sun is therefore the same at the two positions determined, and they are in the same parallel of equal altitude, and as the observed altitude of the ship is also the same, the ship, too, is upon the same parallel of equal altitude, a small arc of which may be regarded as a straight line. A perpendicular to the line determined as above will be in the direction of the sun's bearing, and the angle it makes with the meridian will be the sun's azimuth. For the perpendicular to an arc will pass through the pole of the arc. If two altitudes of the sun be taken, and two lines projected as above, passing each through the place of the ship, its actual position is determined by their intersection. For the method of allowing for the change of place of the ship between two observations for altitude, and for a variety of problems based upon the principle above explained, see Sumner's work. PART IV. SIURVEYING. PART IV. S UR V EYING 102. HAS for its object to make upon paper an exact delineation called a map, plot, or draft, and sometimes to find the contents of ground. The most common mode of proceeding is to measure a base line upon the ground, as at Art. 10, and take the angles at each extremity with some instrument suitable for the purpose, thus determining the position of a distinct point. This is transferred to paper, by means of a scale of equal parts, and a semicircular protractor, as described in the same article. As many points upon the ground as may be desirable can thus be transferred to the map. If two points thus determined be the extremities of some straight boundary, as -a wall, fence, or side of a building, the boundary itself is drawn by uniting the two points by a straight line. If the boundary be curved or irregular, as the bank of a stream, a coast, the border of a wood, &c., the prominent points should be determined on the map, as above described, and the boundary then traced through them with the hand, by the eye. The positions of points on a map may also be determined by taking their direction and distance from some point used as a basis or base point for the whole survey. Or, when more convenient, some one of the points determined from the first base point may itself become a base point for the determination of others more in advance. The direction of one point from another is expressed by the angle which the line of direction makes with some known line. A very convenient line for this purpose is a meridian or north and south line, for the direction of this latter may always be known by means of a magnetic needle, and the direction of any line from a north and south line by a compass. Allowance must of course be made for what is called the variation of the needle, which is determined by astronomical observations, in a manner to be hereafter explained. Distances are measured upon the ground with a tape or chain. The measuring tape is covered with wax, to prevent the eftects of varying degrees of moisture, in contracting and expanding it. It is divided usually 234 SURVEYING. into feet and inches. The chain is of iron wire, each link being the hundredth part of the whole chain, which is 4 rods, or 66 feet, or 792 inches, so that each link is 7'92 inches in length. Every ten links is, for convenience of counting, marked by a piece of brass, with as many tongues as the brass piece is tens of links from the extremity of the chain. A line is measured on the ground, as follows: Two persons take hold, one of each extremity of the chain, and one going in front, towards the point whose distance is to be measured, carries in his hand a staff and ten marking pins, of iron wire, each about two feet in length, sharpened to stick in the ground at the extremity of the chain measured off. The one behind, by a motion of the hand to the right or left, indicates to the other whether the staff which is held at the end of the stretched chain is on the alignment of the distant point or not. As soon as he discovers it to be so, he makes a motion with his hand downward, and the other places a marking pin. Both then move on, the one behind taking up the marking pins which the other has left in the ground. When all the pins have been passed from one to the other, ten chains have been measured off. For fixing the position of points of ground upon a map, the best instrument, when the survey is of moderate extent, is one which surveys and plots at the same time, called the PLANE TABLE. This consists of a rectangular board, mounted upon a three-legged stand, called a tripod, to which it is attached by a ball and socket movement, that is to say, there is a socket fastened to the tripod and a ball clasped by the socket, which moves in it, the ball being fastened to the underside of the table. This permits the table to be placed exactly horizontal. To ascertain whether it is so or not, a detached spirit level is placed upon the table temporarily, and the table is levelled by means of three screws, which pass through a horizontal circle of wood or brass which projects round the top of the tripod, the screws working against the table underneath. These screws are placed near the outer edge of the circle, and at distances of 1200 from each other. If one of them be screwed in one direction, it lifts the table on that side; if in the opposite direction, it lets it down. In order to level the table by means of these screws the spirit level is placed over the line joining two of them, and by moving them the bubble is brought to the centre; this renders one line of the table horizontal. The spirit level is then placed in a direction perpendicular to its former one, and the bubble brought to the centre by PLANE TABLE. 235 trning the third screw, leaving the others untouched; two lines of the table are then horizontal, and consequently the table itself.* The spirit level, which is a tube of glass inclosed in one of brass, and containing spirits of wine, rests on short feet at the ends, one of which is made movable by a screw, and should be adjusted by reversing the level, end for end, on the table, after the bubble is first brought to the centre, when, if it departs from the centre, it must' be brought back half by the foot screw of the level itself, and half by the levelling screws of the table. This process is to be repeated till in both positions of the level the bubble remains in the centre. A necessary appendage to the plane table is a brass ruler, with a thin edge, upon which are mounted either plain or telescopic sights, the line of vision being parallel to the edge of the ruler. Plain sights consist of two upright flat pieces of brass, one at each end of the ruler, facing each other with a narrow vertical aperture in each, to look through. Sometimes the aperture is made wider in the one towards the object, and a vertical thread or hair stretched down the middle of it, which serves for a sight. Larger orifices are made at some parts, in which to catch sight of the object, which is then brought down to a fine sight:in the narrow part of the aperture. When the sights are telescopic, the telescope may be mounted like that of a transit instrument (if the ruler is very wide) upon the upright piers of brass, by means of a small horizontal axis. Or upon a narrow ruler the telescope is supported at the top of a single column of brass, by a stout axis projecting fiom one side. At the focus of the object glass of the telescope two lines of spiders web cross each other exactly in the optical axis of the telescope. When the ruler is placed upon the table, by turning the table round a vertical axis called the axis of the instrument, the line of sights may be turned in any horizontal direction at pleasure. The telescope has a small vertical play also upon the horizontal axis, for the purpose of directing it to objects somewhat elevated or depressed. To use the instrument, place it over one end of the base line on the ground, so that a line to represent the base line drawn upon paper, stretched tightly and immovably upon the table, may be in the same direction. This is done by placing the edge of the ruler upon the line on the paper, and then turning the table upon the vertical axis of the instrument until a staff placed at the other end of the base line is seen upon the line of sights. There is a convenient arrangement for accomplishing this, consisting of what are termed a clamp and tangent screw, or screw of slow * This operation has generally to be repeated, as the levelling of the second line deranges a little the level of the first. 236 SURVEYING-. motion, an arrangement which is applied to almost all instruments. This consists of a screw working near its head, in a collar in which it has no longitudinal motion, the thread of the screw working in a piece of brass which is at pleasure loose from or screwed against the stand below, by a second screw, called the clamp screw, the first being called the tangent screw. When the instrument is " clamped," the tangent screw being turned, pushes the side of the table to which its collar is attached slowly away from the part of the tripod below, thus giving a slight motion to the table on its vertical axis. The line of sights being thus arranged in the direction of the base line on the ground, whilst the edge of the ruler coincides with the oase line upon the paper, keeping now the edge of the ruler passing through one extremity of the latter, whilst the table is held immovable by the clamp screw, upon the tripod, turn the line of sights accurately towards one of the points to be plotted, and draw a line along the edge of the ruler upon the paper, marking it 1. Turn the line of sights successively to all the points in view to be plotted, drawing lines on the paper from the extremity of the base line on the paper, in the directions of these points, as before, and numbering them 2, 3, and so on, in order. Let the instrument now be taken up and carried to the other extremity of the base line, levelled, and the edge of the ruler being placed upon the base line on the paper, in a reversed position, bring the line of sights in the direction of the station at the first extremity of the base, by turning the table on its vertical axis, using the clamp and tangent screw, as before. Proceed in the manner just described, to draw lines also from this extremity of the base on paper, in the directions of the same objects, by sighting towards them in the same order, and numbering the lines as before. Where 1 meets 1 will be the position of the first object on the map; where 2 meets 2 the position of the second object, and so on. The telescope of the plane table has sometimes an arrangement by which distances can be measured without the use of the tape or chain. Two sets of cross wires, as the spider lines are technically called, are placed at tile focus of fne object glass, the points of intersection of each pair being a small distance apart, the one above the other. If a staff be placed 100 feet from the telescope, and a space of exactly one foot be seen intercepted between the intersection of the wires; if then the staff be removed to a distance of 200 feet, two feet will be seen intercepted on the staff, because, according to an optical law, the size of the image formed at the focus of the object glass is inversely as the distance of the object from the instrument, so that the space between the wires at the focus, where the image is formed2 being occupied by I foot at the distance of THE SURVEYOR S COMPASS. 237 100 feet, the image of this 1 foot at the distance of 200 feet will occupy only half the space, or it will take an image of two feet to occupy the whole space. To ascertain the proportions of the instrument by experiment, place ^ staff at such a distance that 1 foot may be intercepted between the inter, sections of the wires; measure this distance, and it will be the distance to be multiplied by the number of feet and fractions of a foot seen intercepted when the instrument is used for measuring distances. The staff used should be about two inches broad by one in thickness, and painted white, the divisions and numbers upon it being black or deep red, and made very distinct. THE SURVEYOR'S COMPASS. This instrument is a circular box of brass about six inches in diameter, and half an inch deep, mounted upon a tripod with ball and socket motion. The bottom of the box on the interior is silvered, and the circumference of this silvered bottom is graduated. In the centre of the bottom stands up a pivot upon which a long magnetic needle is accurately balanced. The top of the box is of glass, in order that the whole interior may be seen. Upon the box above, in the direction of a diameter, is a line of sights which may be plain or telescopic. The graduation is numbered from each end of the diameter, and runs to 900 each way.* To survey a polygonal field with this instrument place it at one of the corners of the field, and direct the line of sights to the next corner along one of the straight boundaries of the field, and measure with a chain the length of this boundary line. Enter in a field book, ruled in three columns, in the first column the number of the station beginning with station 1; in.the second column the Bearing (which would be, for instance, N. 300 E., if the line of sight were directed to the right of the north end of the needle marked with a cross, and the needle pointed to 300 on the graduated circle;f) and in a third column the distance measured with the tape or chain. Take the instrument now to the corner whose position has just been determined, calling it station number 2, and determine the position of station number 3, a third corner of the field and the bounding line connecting stations 2 and 3 in the same manner, and so proceed quite ound the field. (See p. 233.) To plot this assume the point on the paper at * Sometimes the graduation is numbered from 00 to 3600, the 0 and 180 diameter being the line of sights. t If the graduation of the compass box be from 0 to 360, it will only be necessary to record the number to which the north end of the needle points. 238 SURVEYING. which you will have the first corner of the field, draw through it a line to represent a magnetic meridian, or north and south line, and another line making an angle with this, equal to the first bearing, as 30~ above, which will be laid off with the semi-circular protractor to the right of the meridian line if the bearing be east, and vice versa; then from a scale of equal parts lay down the distance taken from the third column of the field book, and this will determine the second station of the map; through this draw a north and south line parallel to the first drawn, and lay off the second boundary by its bearing and distance, in the same manner, and so proceed till the plot is complete. The accuracy of the work will be tested by the last bearing and distance, reaching exactly to the first station from which the work commenced. The same method may be pursued with the plane table, in an obvious manner. It is only necessary, as the table is moved from corner to corner of the field, to place the line on paper which has just been determined parallel to the corresponding line on the ground, by placing the ruler upon it, and sighting back from the 2d to the 1st station, turning the table on its vertical axis, for the purpose. The line joining the 2d and 3d station may then be drawn by sighting to the 3d station, the edge of the ruler passing through the 2d station in the paper, and by a scale of equal parts the length of this line laid down, and so on. The compass may be employed to survey an irregular line, as a road, the border of a stream, a wood, a coast, &c., by taking stations sufficiently numerous to include portions nearly straight between them. Thus: Another mode is to run a straight line along the irregular boundary, and measure offsets, that is, perpendiculars to the main line, extending to the boundary at points where there are remarkable changes. Thus: The perpendiculars not only are measured with the chain, but the dis THE THEODOLITE. 839 tances between them. To plot such a piece of surveying, the main line having been laid down on paper by its bearing observed with the compass, the measured distances between the perpendiculars are laid off on this line by a scale of equal parts, the perpendiculars drawn and laid off by the same scale, and the boundary traced by the hand through their extremities. An instrument for taking offsets is the surveyor's cross, consisting of two pairs of plain sights, at right angles to each other. Place the cross on the principal line, sight with one pair to a distant staff upon it, the other pair will be directed in a perpendicular to the principal line. Where a survey is extensive, the relative positions of distant points are fixed with an instrument of greater accuracy for taking angles than any we have yet described, by a process similar to that mentioned at p. 236, called triangulation. The instrument used for this purpose, of which that described at Art. 10 is a rude imitation, is called THE THEODOLITE. This consists of a horizontal circle of brass resting upon a tripod by three foot-screws or levelling screws. The circumference of the circle is silvered, and divided into degrees and parts of a degree, numbered from 0 to 360. Directly above a diameter of this circle is supported a small telescope, to which is attached a vertical circle, the plane of which is parallel to its optical axis. There are various modes of supporting the telescope. The best is by means of two upright columns of brass, standing upon a horizontal circle concentric with the horizontal limb, and moving upon or within it, round a vertical axis.* Between these columns the telescope is suspended, by means of two projections from the sides of its tubes, resembling the trunnions of a cannon, the ends of which rest in notches in the tops of the upright columns, called YV. These projections are called the supporting axis, and this passes through,the centre of the vertical circle, which is firmly attached to it at one end, and which revolves with this axis. The circumference of the vertical circle is also graduated, and called the vertical limb, the numbers on this limb running from 0 to 90~ four times. The indices which show the number of degrees passed on the horizontal limb in a horizontal direction, or on the vertical limb in a vertical direction, thus indicating the angular motion of the line of vision, or optical axis of the telescope, are of peculiar construction, and called verniers, from the name of their inventor. As they are used upon many instruments, we shall presently describe them in detail. * This mounting is similar to that of the tratisit instrument. 240 SURVEYING. There are usually three of these upon the horizontal limb, and two upon the vertical, at distances from each other of 1200 in the one, and 1800 on the other. The object of having more than one is to correct for wrong centring of the circle, or excentricity. Those which apply to the horizontal limb are attached to the concentric movable circle which supports the telescope, and move with it. A clamp screw serves to fasten this circle to the limb, and a tangent screw to give it a slow motion along the limb. Those which belong to the vertical limb are attached to the supports of the Y", immovably except for adjustment, the limb moving past them with the telescope about the supporting axis. The line of sights is marked by two wires, one vertical, the other horizontal, crossing each other at the focus of the object glass of the telescope. These wires are fastened across a diaphragm smaller in diameter than the tube of the telescope, and held within it by means of screws passing through from the outside of the tube on the right and left. By tightening the right, and loosening the left, or vice versa, the diaphragm may be moved laterally. The instrument is used for measuring either horizontal or vertical angles. A horizontal angle is one formed by two lines in a horizontal plane; or it is the angle included between two vertical planes which meet. A vertical angle is one formed by two lines in a vertical plane. An angle of elevation is a vertical angle formed by a horizontal line, with an oblique line coming from above to meet it. An angle of depression is a vertical angle formed by a horizontal, with an oblique line meeting it from below. For the measurement of such angles it is evident that the axis about which the verniers of the horizontal limb move, which is called the axis of the instrument, should be truly vertical, that the supporting axis on which the vertical limb turns, should be truly horizontal, and that the line of vision of the telescope should be exactly perpendicular to the latter. The processes of placing them so are called adjustments. The last mentioned, which is the first in order to be made, is called collimation. This is accomplished by placing the intersection of the wires upon some distant, well-defined point, then reversing the supporting axis in the Y", by turning it end for end. If the intersection of the wires passes through the same object, as the telescope is turned round the supporting axis, the instrument is collimated, and no adjustment is necessary. But if not, the intersection of the wires must be brought half way, by estimation, towards the object by means of the screws of the diaphragm, and the other half by the tangent screw of the horizontal limb. This process must be repeated, owing to the difficulty of estimating just half, till in both positions of the supporting axis the intersection of the wires passes through the same THE THIEODOLIT'rE': A. distant point.. The second adjustment in order,. conslt.s in r'ndering the supporting axis horizontal. This is accomplishled by imeans of a spirit level, composed of a glass tube, not quite filled with alcollo, leaving an air buibble at top. This tube is partly encased in a brass one, which rests by its ends upon two feet, notched at the bottom, to stand upon the ends of the supporting axis, so that the tube of the level is above, and parallel t tthe supporting axis, striding over the telescope, and hence the name of striding level, by which it is known. First, to adjust the axis of the level tube into parallelism with the supporting axis,:when the feet of the striding level rest upon its extremities. Bring the bubble to the centre by the levelling screws of the instrument; reverse the level upon the axis, turning it end for end; if the bubble does not continue in the centre, make half the correction by the levelling screws, the other half by filingc away the notch in one of the feet, viz., that nearest to the bubble, as being the highest. Sometimes the foot of the striding level is made capable of being lengthened or shortened, by means of a screw, by turning which the adjustment may be made. The parallelism between the axis of the spirit level and the supporting axis being established, whenever the bubble is at the centre, the latter of these axes is horizontal. To make the supporting axis perpendicular to the vertical axis of the instrument, bring the bubble of the level to the centre, by the levelling screws, turn the instrument round the vertical axis ] 80; if the bubble now departs from the centre, make half the correction by the levelling screws, the other half by screws which elevate or depress one of the -Y. The above are the principal adjustments. To measure an angle, with the instrument thus adjusted, one plane passing through the vertical axis of the instrument, and perpendicular to the supporting axes, is made vertical by bringing the bubble to the centre in one position; then turning the instrument round the vertical axis 900, bring the bubble again to the centre, and the vertical axis will be truly vertical, because it is perpendipular to two horizontal lines which intersect, viz., the lines determined by the spirit level, and consequently to a horizontal plane. The instrument is now prepared for the measure of either a horizontal or vertical angle, having its vertex at the centre of motion of the instrument. For the former, turn the telescope in the direction of one of the sides of the angle, and take the reading of the degrees from one of the three verniers on the horizontal lilmb (which are engraved each with one of the letters A, B, c), and of the mlinutes and seconds from all three. A mean of the minutes and seconds is taken by adding up the three readings for the minutes and * For the theory of this process see transit instrument, p, 149, note. 16 242 SURVEYING. seconds, and dividing the sum by three. The telescope is now turned in the direction of the other side of the angle, by sighting to a station at its extremity, and taking the reading from the three verniers as before. The difference between this reading and the former one will give either the angle required in degrees, minutes, and seconds, or else a number which, subtracted from 300~, will give the required angle. As for instance, if the vernier stood first at 3500, moving a short distance past 360~ to 10^o the angle passed over would be only 20~, which is obtained by subtracting the difference between 350~ and 10o, or 340~ from 360. In-some theodolites the horizontal limb itself as well as the vernier circle, has a motion round the vertical axis of the instrument, and a clamp andtangent screw. This is for the purpose of introducing the repeating process into the measurement of horizontal angles, which is conducted as follows. Sight first to the object in the direction of one of the sides of the angle to be measured, and take the reading; loosen the clamp screw of the vernier circle, and turn this with the telescope round the vertical axs of the instrument to sight to the second object in the direction of the other side of the angle, using the clamp and tangent screw of the vernier circle for making the sight exact. Loosen now the clamp screw of the limb, keeping that of the vernier circle tight, and bring the telescope back t the first object, clamping the limb there, and adjusting the sight by the tangent screw of the limb. Repeat this process several times, ending with a movement of the vernier circle to bring the telescope upon the secondlobject; take now the reading; the difference of the two readings divided by the number of times that the vernier circle has been moved forward, say six times, will be the angle required, with a probable error of observation of w what it would have been without the repeating process. To measure a vertical angle with the theodolite, level the instrument, or make its vertical axis truly vertical as before; elevate or depress the telescdpe to the object, in the direction of the inclined side of the vertical angle, and take the reading by both verniers of the vertical limb. Turn the instrument on its vertical axis 1800, and bring the telescope to bear upon the same point as before, taking the mean of the two verniers again. Half the sum of the mean of the results of the readings in the two positions of the telescope will be the angle of elevation, or depression, as the ase may be corrected for index error.* Half the difference of the same e Because if the index be a little behind the zero on the limb when the telescope horizontal in the one position, it will be a little before the zero in the reverse potioa, after turning the instrument 1800 on its vertical axis. THE THEODOLITE. 243 same results will be the index error which must be applied as a correction if the reading for an observation afterwards is taken only in one position, or with the face of the limb one way, either to the right or left. It will be proper now to describe the vernier, which serves to read accurately smaller divisions of the limb than could be done with a simple index. The vernier is a short are of a circle, in which divisions are cut. as in the limb, but smaller; so little smaller however, that the difference between theni shall be equal to the smallest denomination which the instrument is intended to read. If the first division of the vernier, marked with a crow-foot, and called the zero of the vernier, be made to coincide with a division of the limb, the last division of the vernier will be observed also to coincide with a division of the limb; and, on counting the divisions, there will be found to be one more on the vernier than upon the same length of the limb. Suppose the smallest division of the limb to be denoted by a, and a division of the vernier by x, and suppose tha.t sn'divisions of the vernier are equal to n of the limb, then n'x = na () X= — a (2) a —xa a 1 (3) This last being the difference between a division on the vernier, and a division on the limb, is the expression for the smallest denomination which can be measured with the instrument. A common division of the limb of the theodolite is into degrees or half degrees, or 30' spaces; and of the vernier such that 30 of its divisions cover 29 of the limb. The second member of formulas (3) above, by the substitution of these numbers, becomes 30' -- 29 or 1 \ 30/ The smallest angular space which the instrument is capable of measuring in this case is one minute of a degree. The first dividing line or zero of the vernier coinciding with a dividing line of the limb, if the vernier be moved forward, till the second dividing line of the vernier coincides with the next line of the limb, the zero of the vernier will have moved past the dividing line of the limb, at. which it stood a distance equal to the difference of a space on the vernier, and a space on the limb, or once the smallest denomination which the 244 SURVEYING. instrument measures. If the vernier be moved still farther forward, till its third dividing line, or second from the zero, coincides with the line of the limb, the zero of the vernier will have moved forward again, a distance equal to the difference between a space on the vernier, and a space on the limb, or altogether twice the smallest measure of the instrument. To take a reading, therefore, from a theodolite divided as above mentioned, observe the number of degrees and half degrees pointed out oi the limb by the crow-foot or zero of the vernier, and if this line be a little past one of the dividing lines of the limb, count the number of divisions on the vernier from the zero to a dividing line which coincides exactly with one of the limb; this will be the number of single minutes to be added to the degrees and half degrees indicated by the crow foot. The use of two verniers 180~ apart, in correcting for excentricity, may be thus explained. An excentric angle, or one having its vertex not in the centre of a circle, is measured by half the sum of the opposite arcs. The mean of the two opposite verniers, therefore, gives the true angle moved over by the telescope. If the number of verniers be increased beyond two, error of graduation and of figure in the circle are proximately eliminated, as well as error of excentricity, by taking the mean of all the verniers. TRIANGULATION OF A COUNTRY. This is a process which consists in measuring a base, and taking the angles at its extremities with the theodolite, for the purpose of determining the positions of points, the sides of the triangles thus determined becoming bases for new triangles.* SURVEY OF A LARGE ESTATE. A. good instrument for the purpose is a compass with telescopic sights, graduated fiom 0 to 3600. And a good method of proceeding is to select three stations on elevated ground at remote parts of the estate. Setting out from one of them, go in the direction of one of the others, having previously taken the bearing of the line joining them with the compass. This line may be kept by its bearing in plunging into the low grounds, out of sight of the principal stations. Distances must be measured along the line to points opposite objects on either side of it, which are to be introduced upon the map, and offsets to these objects measured. Oblique instoad of perpendicular offsets will sometimes be found more convenient, 0 This will be fully treated in our chapter on Geodesy. SURVEY OF A LARGE ESTATE. 245 in which case their bearing must be observed with the compass as well as their lengths measured. Or if an object be too inaccessible to measure the offset to it, its bearing with the compass may be taken from two points on the principal line, which will serve to fix its position. The method of keeping the field book is exhibited on the next page but one. The record commences at the bottom of the page, and goes upwards, till one page of the field book is filled, and then commences at the bottom of the next, and so on.* The bearings and distances are entered in the middle column of the page, the offsets on the right and left. Distances are usually measured in chains and links, or decimals of a chain. The objects to which the offsets are measured may have their names written, or still better, may be roughly drawn. The points at which streams, roads, stone walls, hedges, &c. cross the principal line, are indicated in the field book by drawing representations of these in the proper direction on both sides of the central column, and where they cross obliquely, recording their bearings upon them. On reaching the second station, a line is drawn across the page, and a line is run in a similar manner from the second to the third station, and finally from the third to the first. The sum of all the partial measured distances upon any one of these sides of the great triangle will be the length of that side, and the lengths of the three sides being known, the triangle may be plotted. The offsets will be plotted as described at p. 238, or where they are oblique, in an obvious manner. A turn to the right, on reaching a station, may be marked r at the side of the page, and a turn to the lef ". The great triangle with the offsets being completed, other points may be taken within, as the vertices of smaller triangles having the sides of the large triangle for bases, and, if necessary, other points again within these, until all the systems of triangles with the offsets from their sides include every object desirable to be placed upon the map. On p. 246 is a map of ground with the great triangle and offsets, and on p. 247 is the corresponding field book. * This is in order that the book and the ground may lie in corresponding positions before the eye of the surveyor. 2 ~t,~~~~~~~~~~~~~. nR ))/ r-^^^)^^^ A- ~ ~ ~ ~ ~.1 +~~~~~~~~~~~~~~~d "I ts~~~~~~n c~~~~~~~~~-~i~~~~~~~~~~rS ~ ~ ~ "-,' LEVELLING. 247 5300 600 0.75 0''1 D: 49'. ISO. noted, and their bearings taken. LEVELLING. The instrument employed for this purpose is called the level, and conA__ _ _ _^ B B^ sists of a telescope mounted horizontally umn a tripod. At the tup of the tripod, firmly fastened to it, is a smwl horizontal circular plate of brass, parallel to which, and a few inches above it, is another of the same size the two being separated by levelling screws, and called levelling plates. From the centre of the upper levelling plate rises vertically a spindle, o~~o. ^l;. A I~~PO 4$8 BSURVEYING which fis into a socket in the middle of a horizontal bar, about a foot or more in length. At the extremities of this bar are two stout uprights of two or three inches in length, the tops of which are formed as Y', in which she telescope rests, in a position parallel to the horizontal bar. One of these Y- has a vertical motiom. by means of a screw underneath it, which passes up through the end of the horizontal bar. A spirit level is suspended below the telescope and parallel to it, having a horizontal movement by means of a screw at one end, and a vertical movement by the same means, at the other. 1. To collimate the instrument, bring the intersection of the wires upon some well-defined distant point, and then turn the telescope on its optical axis in the Ys, till the spirit level comes at top. If the intersection of the spider lines be not in the axis of motion, it will depart from the object, and will be as much on the opposite side of the axis after this demi-revolution. It must, therefore, be brought back half way to the object, by the screws which move the wires. This experiment must be repeated till the intersection remains on the object, during the revolution of the teleacope on its optical axis. There are three lines of the instrument which ought to be parallel to each other, horizontal, and perpendicular to the vertical axis about which tlhe instrument turns, viz., the line of collimation, the axis of the spirit level, and the horizontal bar. 20 To render the axis of the spirit level parallel to the line of collima-'ion, bring the air bubble to the centre by turning the levelling screws, having previously placed the level in the direction of the line joining two:of them; take the telescope out of the Y", and reverse it, turning it end for end; if the bubble remains in the centre, both lines, that is to say, the line of collimation and the axis of the level, are horizontal; if not, bring the bubble half way back to the centre, by means of the screws at the end of the spirit level, and the other half by the levelling screws. Repeat this process until, in both positions of the telescope, the bubble remains in the centre. The axis of the spirit level may also be oblique to the line of collimation in a lateral direction. To ascertain whether it is so or not, turn the telescope on its optical axis as it rests in the Y", till the. spirit level comes out at one side, not so far, however, as to cause the bubble to disappear; the lateral obliquity will then be converted into a- obliquity partially vertical, whlih the departure of the bubble fromn the centre will render sensible. This obliquity must be corrected by the screws at the other end of the spirit level, which give a lateral motion to its tube. 3, To render the line of collimation and axis of the spirit level now LEVELLING. 249 parallel to each other, both parallel to the horizontal bar, or rather perpendicular to the vertical axis of the instrun.ent, round which the horizontal bar turns, by means of the spindle fitting into the socket at its centre; bring the bubble to the centre, by means of the levelling screws, turn the instrument round the vertical axis 1800,* and if the bubble remains in the centre,tthe axis of the spirit level, and consequently its parallel, the axist about which the telescope revolves, are perpendicular to the vertical axis of the instrument, and the latter, though not vertical, is in a vertical plane,, perpendicular to the line of collimation. To make it vertical, which must be done every time an observation is made with the instrument, the above adjustments being completed, bring the bubble to the centre by means of the levelling screws, then turn the bar 900 on the vertical axis, and bring the bubble to the centre again; the vertical axis will then be truly vertical. In connexion with this instrument two rectangular staves, called levelling staves, are used, divided into feet, tenths and hundredths of a foot. The staves are painted white, and the division lines are made very distinct, in black or red, the feet being numbered with distinct numerals. the hundredths are painted, the whole of each division, alternately white, and black or red. Two or three lengths of 6 feet of staff fit together by joints, so as to make a length of 12 or 18 feet. A plumb line suspended at the side of the staff, or in a groove covered, the lead part with glass so as to be visible, serves to place the staff exactly vertical. In order to find the difference of level between two points of ground or the height of one above the other with these instruments, let an assistant hold one of the levelling staves vertically, standing upon one of the points in question, and another assistant the other staff, at some point in the direction of the second point;the observer turns the telescope, after the vertical axis of the instrument has been made truly vertical, and takes the reading cut by the horizontal wire, first on one of the levelling staffs, -:nd then on the other,~ recording them in separate columns of a field * If there is a horizontal compass box attached to the instrument this may he done by observing the number of degrees to which the needle points. If not, the instrument must be placed on the line of two staves, and between them, sighting first to one, and then to the other. f If not, half the correction must be made by. the screw under the Y, which affects the relative direction of the horizontal bar to the axis of the telescope, the other half by the levelling screws, and the operation repeated. 4 This axis, by the first adjustment, is identical with the line of collimation, or line of vision, when the intersection of the wires is brought upon an object. It is not the optical axis of the telescope, though it should be as nearly as possible, but the axis of the cylindrical parts of its tube, which are in contact with the Y". ~ Not only feet, tenths, and hundredths may be distinctly read, but even thousandths, by estimation. 250 SURVEYING. book, the one in the direction in which he is going, under the title of:" forward readings," the other under that of reverse readings.* The staff on the given point of ground is now taken up and carried round the other (which is simply turned about where it stands), to a point still firther in the direction of the second given point; the level is also removed to a point between the levelling staves, and the process just described is repeated, and so on, till the second given point, between which and the first the difference of level is required, is reached, the last levelling staff being placed there. The difference between the sum of the direct readings and the sum of the reverse readings will be the difference of level of the two given points, or height of the one above the other. To make a section of ground which is the intersection of a vertical plane with the surface of the earth, it is necessary to add three more columns to the field book, a column of differences, a column of heights, and a column of distances. The first contains the difference between each direct and reverse reading;f the second contains the height of each point upon which the levelling staves are placed above a horizontal plane, assumed at pleasure, usually passing through the first point of the survey. Each number in the column of heights will be computed by adding the number of the column of differences to the preceding number in the column of heights; the first number in the column of heights is the * The level is always placed nearly half way between the two staffs, to avoid tle error which would be occasioned by the difference between true and apparent level, the nature of which may be explained by the annexed diagram. Let the circle in the diagram be a vertical section of the earth through the point A, where the instrument is supposed to stand. The arc AM will be the line of true level, A and ni being at equal distances from the centre of the earth c; the line AT tangent at A is the line of apparent level, determined by the optical axis of the instrument. For a point on the opposite side of A, from M, and at the same distance, the difference between apparent and true level will be the same. Hence the advantage of placing the instrument half way between the points whose difference of level is to be observed. t This should be entered with the negative sign, if the direct exceed the reverse Tmadimg. CONTOUR OF GROUNP. 251 same as tnat adjoining in the column of differences. The numbers in the column of heights will be negative for points below the plane of reference. The third column contains the horizontal distances between points at which the levelling staves were placed, supposed to be measured with a tape or chain. To plot the section or profile of ground from such a field book, a horizontal line must be drawn to represent the section of the profile with the plane of reference; on this the numbers from the column of distances must be laid off from a scale of equal parts, and at the points of division ordinates or perpendiculars must be erected and taken from the scale of equal parts, equal to the numbers in the columns of heights; through the tops of these ordinates the section of ground required can then be traced with the hand. The following example of a field book, and the profile constructed from it, will serve to illustrate this subject. D R R R Dif. H Dist. 6*895 2-461 4.434 4*434 1-35 5*321 1 468 3*854 8.287 0.75 8*264 3*812 4.452 12*739 1*00 9*322 2*111 7*211 19.950 1.50 7.444 3*212 4*232 24*280 2*00 4.321 1*211 3*110 27.390 1.25 CONTOUR OF GROUND. The mode of representing this is by means of horizontal sections formed by planes at regular intervals, one above another, these sections being all projected upon one horizontal plane, viz., that represented by the map. To survey a hill for the purpose of drawing its horizontal sections upon a map, place an instrument for taking horizontal angles at the top of the hill, and plant stakes along lines diverging from this point down the hill, and take the horizontal angles formed by the vertical planes in which these lines lie; then level along the lines, taking the difference of level and distance at points where the slope changes. In order now from such a survey to plot the horizontal sections of the hill, draw on the map from the point A, corresponding to the one assumed on the top of the hill (supposed to have been previously determined in position on the map by tri 252 SURVEYING. angulation or otherwise), a system of lines diverging from this point under A angles equal to the observed angles, and supposing, to render the conception definite, that the planes forming the horizontal sections are taken 10 feet apart, the points in which the projections of the sections cut the diverging line thus drawn, may be ascertained by proportion as follows:The difference of level between two points on one of these lines is to the horizontal distance between them, as 10 feet is to the horizontal distance fiom the upper point, at which a plane of section ten feet below this point would cut the diverging line. Commencing with the upper point of all, A, and determining thus the points a, b, c, d, at which the projection of the section 10 feet below cuts each of the diverging lines, it may be traced through them by the hand. In a similar manner another section 10 feet lower, and so on to the bottom of the hill. Where the sections are convex, are the ridges, or back bones of the hill; where the sections are re-entering are the ravines; and in general, the varying forms of the sections present to a practised eye an exact notion of the general configuration of the hill. In topographical maps these sections are drawn in pencil, and the hill shaded with irregular lines, in India ink, perpendicular to the sections, as seen between c and d, which is the direction in which water or alluvion would flow down the hill. These shading lines are best drawn with a pen of short coarse nib, and very short portions of a number of them at a time. The light should be supposed to fall obliquely in a certain direction, and the sides of the ravines towards the source of light will be darker than the opposite sides. The tops of the back bones will be quite light. The summits of the hills being usually more nearly perpendicular, will present deeper shades than the bases, which usually slope gently into the level ground. SURVEY OF ROADS, RAILWAYS, AND CANALS. 203 Water is shaded by making the line of shore quite black, which should be first drawn all round the map, including the islands, then a second and finer line all round, as near the first as possible without touching it, then a third a little more distant, and so on, broader and broader, till the. shadings from opposite shores meet midway.* SURVEY OF ROADS, RAILWAYS, AND CANALS, The problem ordinarily is to ascertain the shortest and cheapest route from one point to another on the surface of the earth, having no greater and more sudden elevations or depressions than:are compatible with the nature of the locomotion to be employed upon it.t The case frequently presented, not only for the whole route, but in minor instances, along its course, is that of a dividing ridge between two valleys, from one to the other of which it is necessary to pass. It becomes necessary then to ascertain the lowest point of the dividing ridge. A good indication may be obtained proximately by a simple inspection of the course of the streams upon a map, inasmuch as the direction ot their flow must be governed by the topography of the country over which they pass. One or two examples will sufficiently illustrate the principles which are to guide such an inspection. )~-\~ ~ If two streams run along two valleys, and k ~' \,g~ ^ tributaries proceed from I{ ~,]> near the same point Ar - _) A of the dividing ridge - t V to etmnpty into them, A then the point A is a low point of the dividing ridge. For the waters accumulating for the formation of the sources of the tributaries at the point A must flow from higher ground on both sides of this point. Again, if two tributaries running at first nearly parallel to the principal streams, turn outward at a point A, this is a low point. For the * For further instruction in topographical drawing see an excellent little work by S. EASTMAN, U.S.A.' t This will of course be modified by various circumstances, such as the greater or less difficulty of working the ground, from the nature of the soil; the vicinity of large towns, to pass through which the construction would turn aside, the existence of mines, or valuable products of any kind, for which it would afford transportation. 254 SURVEYING. tributaries at first descending into a lower country than that at their sources, encounter rising ground, which prevents their progress in that direction, and turns them off towards their principals. These examples will suffice to point out the nature of the investigation to which an ordinary map should be subjected. A good topographical map, exhibiting contour of ground, would of course be a far better guide. Conflicting points of passage of a ridge may be compared by means of an altitude and azimuth instrument or theodolite; by first levelling the instrument, elevating the telescope till the line of sight passes through one of the points in question, and then turning the instrument in azimuth till the line of sight passes by the other point in question; if it passes above, the latter point is lower than the former, and vice versa. Next, a personal reconnoissance of the country should be made, and information sought from the inhabitants as to the nature of the ground, convenience for obtaining constructing materials, the mineral and agricultural wealth of the region, etc., etc. Three or four routes may thus be selected, one or other of which is to be finally decided upon by a rough survey of them all. This survey is conducted with the compass and chain, and level. The former instruments furnish a plot of the route by the method pointed out at p. 238, and the latter, a continuous profile or longitudinal section of the ground along the whole route, as seen at p. 251. A comparison of the compass plots of the different routes will determine which is the shortest, in a horizontal direction, and a comparison of the profiles will show which presents most elevation and depression to be overcome. The route being thus finally selected, it must be surveyed with care, another column being added to the field book, entitled grade, the numbers in which express the height of the roadway or bottom of the canal above the plane of reference at the same points of the route, for which the column of heights expresses those of the natural ground. The numbers in the column of grade will depend upon the elevation or depression of the natural ground, and the slope which the construction permits, that of a common road being greater than that of a railroad, and the latter being greater than that of a canal. The determination of these numbers will require an exercise of eye and judgment. A prime object to be had in view is the equalization of excavation and embankment; i. e., the grade should be so adjusted to the natural slope of the ground as to cut off.as much earth as would be required to fill the depression adjoining, up to the level of the grade. The survey being finished, a double profile must be made, one of the natural ground in black ink, the other of the grade in red, upon the same base line, and with the SURVEY OF ROADS, RAILWAYS, AND CANALS. 255 same abscissas or horizontal distances between the ordinates or lines of heights. An inspection of this, and a computation, if necessary, of areas between the black and red lines of section, will serve to show how well the excavation and embankment have been equalized, and the result may require a modification of the grade to adapt it better for the purpose in question, to the natural ground; cross-sections of the route must also be surveyed at all points of change in the longitudinal or latitudinal slope, and more frequently, if these occur at long intervals. The amount of excavation and embankment may be then obtained with sufficient accuracy for an estimate of expense, by computing the areas of the cross-sections of the work as it will be when completed, and multiplying half the sum of the areas of two of them by the longitudinal distance between them. The cross-sections, wheni the work is in embankment, will be of the form exhibited below, and the same turned upside down when the work is in ~ e. excavation. These are easily drawn from the field book. The cross-section bac being plotted, a corresponding to the point where it intersects the longitudinal section of ground, af will be the difference of the numbers in the columns of height and grade, df, fe, each equal to half the breadth fixed upon for the top of the road or bottom of the canal, dc, eb are then drawn at the proper slopes, for common earth 1 base to 1 in height. These sections may be regarded as quadrilaterals, and each divided into two triangles, for the purpose of obtaining their area.* Where the crosssection of ground is parallel to the top of the road, or bottom of the canal, they are trapezoids, and the area will then be - the sum of the * To compute the area of the cross-section from the numbers in the field book, conceive parallels to be drawn fiom the points b and c in the diagram, to af, meeting de, produced both ways; a trapezoid will be formed, from which, if two right angled triangles be deducted, the area of the sections will be obtained. If d denote the difference of level between b and f, d will be one parallel side of the trapezoid, and the altitude of one of the triangles, the base of which if the slope be 1 will be " d. And if d' denote the difference of level between c andf, similar expressions will be had for the other side and altitude; and the expressions for the area of the section will be, b being the breadth of the roadway, (d+ d') [b+ (d { d')]- d X ) d — 3d' X ~ d'= [(d + d') b + 3dd', 256 SURVEYING. parallel sides de, cb, multiplied by the altitude af. Notes of the nature of the soil are kept in the field book of the compass, and the amount of digging and wheeling or carting that can be done in a day in different kinds of earth, having been ascertained by experiment, and the price of daty labor being known, the data for determining the expense of the work are all known. To the above must be added the accidental expenses of culverts, blasting rocks, construction of tunnels, &c., which are all subjected to the same general rules, and are functions of the price of materials, mechanic labor, and experiments as to relations of time and amount of performance. TO COMPUTE TIlE CONTENTS OF FIELDS. 1. Compute the contents of the figures, whether triangles or trapeziums, &c., by the proper rules for the several figures. If the linear measures be in links, the result is acres, after cutting off five figures on the right for decimals. Then bring these decimals to roods and perches, by multiplying first by 4, and then by 40. 2. In small and separate pieces, it is' usual to cast up their contents from the measures of the lines taken in surveying them, without making a correct plan of them. 3. In pieces bounded by very crooked and winding hedges, measured by offsets, all the parts between the offsets are most accurately measured separately as small trapezoids. 4. But in larger pieces, and whole estates, consisting of many fields, it is the common practice to make a rough plan of the whole, and from it compute the contents quite independent of the measures of the lines and angles that were taken id surveying. For, then, new lines are drawn in the fields in the plan, so as to divide them into trapeziums and triangles, the bases and perpendiculars of which are measured on the plan by means of the scale from which it was drawn, and so multiplied together for the contents. In this way the work is very expeditiously done, and sufficiently correct; for such dimensions are taken as afford the most easy method of calculation; and, among a number of parts thus taken and applied to a scale, it is likely that some of the parts will be taken a small matter too little, and others too great; so that they will, upon the whole, in all probability, very nearly balance one another. After all the fields and particular parts are thus computed separately, and added all together into one sum, calculate the whole estate independently of the fields, by dividing it into large and arbitrary triangles and trapeziums, and add these also together. Then if this sum be equal to the former, or nearly so, the TO COMPUTE THE CONTENTS OF FIELDS. 257 work is right; but if the sums have any considerable difference, it is wrong, and they must be examined and recomputed, till they nearly agree. 5. But the chief secret in computing consists in finding the contents of pieces bounded by curved or very irregular lines, or in reducing such crooked sides of fields or boundaries to straight lines, that shall inclose the same or equal area with those crooked sides, and so obtain the area of the curved figure by means of the right-lined one, which will commonly be a trapezium. Now, this reducing the crooked sides to straight ones, is very easily and accurately performed in this manner:-Apply the straight edge of a thin, clear piece of lantern-horn to the crooked line which is to be reduced, in such a manner that the small parts cut off from the crooked figure by it, riay be equal to those which are taken in; which equality of the parts included and excluded you will presently be able to judge of very nicely by a little practice; then with a pencil or point of a tracer, draw a line by the straight edge of the horn. Do the same by the other side of the field or figure. So shall you have a straight-sided figure equal to the curved one, the content of which, being computed as before directed, will be the content of the curved figure proposed. Or, instead of the straight edge of the horn, a horse-hair may be applied across the crooked sides in the same manner; and the easiest way of using the hair is to string a small slender bow with it, either of wire, or cane, or whalebone, or such like slender or elastic matter; for, the bow keeping it always stretched, it can be easily and neatly applied with one hand, while the other is at liberty to make two marks by the side of it, to draw the straight line by. EXAMPLE. Let it be required to find the contents of the irregular figure below, to a scale of 4 chains to an inch. A B17 258 SURVEYING, Draw the four dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of them, which they do as near as the eye can judge; so is the crooked figure reduced to an equivalent right-lined one of four sides, ABCD. Then draw the diagonal BD, which, by applying a proper scale to it, measures 1256. Also the perpendicular, or-nearest distance, from A to this diagonal measures 456; and the distance of c from it is 428. Then, half the sum of 456 and 428, multiplied by the diagonal 1256, gives 555,152 square links, or 5 acres, 2 roods, & perches, the content of the trapezium, or of the irregular crooked piece, TO FIND TIHE CONTENT OF A FIELD WITHOUT PLOTTING. Take te bearings and lengths of the sides of the field, and enter them in a field book as course and distance, and take out the difference of latitude and departure corresponding to each, and enter them in two double columns, marked N. s, and E. w. as at Art. 98. To obtain these, if the bearings are given in degrees, recourse may be had to a table of difference of latitude and departure for every degree and minute of the quadrant, such as is found in Bowditch's Navigator, or instead of this, the difference of latitude and departure may be calculated for each course and distance.* Another double column must be added, entitled double meridian distances The meridian distance of any line is the distance of its middle point from an. assumed meridian, which should be taken through some corner of the field. The double meridian distance, corresponding to the first course adjoining the assumed meridian, will be equal to the departure of that course. Double the meridian distance of any other course will be equal to the double meridian distance of the preceding course, plus the departure of the preceding course, plus its own departure.t In applying this rule, distances to the right should, be considered -, those to the left -. The double meridian distances east of the meridian * The sum of the numbers in the column marked N. ought to equal that of the numbers in the column marked s. If such be not the case, the difference between the two; sums should be half of it subtracted from the numbers in the column having the greater sum, being distributed among them in proportion to their magnitude; the other half should be added in the same way to the numbers in the column producing the less sum. For in going round a field and returning to the same point, the distance gone north must be equal to that gone south. The same remark applies to the columns marked E. and w. New columns will then be derived which may be,called corrected diff. of lat. and departure. t This may be seen by making and inspecting a diagram. TC FIND THE'CONTENT'OF A FIELD WVil'TItOUT PLOTTING. 25. } should be entered in a column markeld E, an( thlose west in a columnl marked w., the column of double meridianl distances beino n ade double for the purpose. By means of this double columnn of couble meridian distances, and the double colunml of differences of latitude, the content of the field may be computed by the following' rule. The difference between the northi'ngs mtesltiplied by tKe e.sizngas Zpila the southings multiplied by tIe westit'?ngs, cond the rnorthingrs,ultiplied b, the westings, plus the southinsgs multiplied by he eastings, will be eqeel to double the area of the land. The proof of this is left as an exercise for the student., In the following example the bearings were taken w.ith a compass resembling the mariner's in principle. A metallic graduated circle, one diameter of which, that joining the zero and 1800 points, being an attached needle, the graduated circle was held stationary in space by the magnetic force of the earth. The numbering w\as from zero.o 360 in the direction shown in the annexed diagram. The compass sights were plain, and the number on the line of sights towards the extremity next the eye was the one read and recorded in the 1st column p. 261. The equivalents of these readings in bearings of the compass courses \ or sides of the field from the meridian, are recorded in the 2d column. These are ascertained by considering in what cq( \r' part of the circumference in the diagram above the No. in the 1st column would fall; the course would be in the \ direction from this point to the centre of the circle. o The third column contains the lengths of the courses or sides of the field, measured with a chain. Then follow the columns of diffeence of latitude and departiure,5' and the columns of corrected diff, of lat. and departure. The assumed meridian fiom which to estimate the double meridian distances is taken through the poin t at which the survey commenced.t The double meridian distance of the first course then, according to the rule, will be equal to its departure 234'1, and is w. because the departure is w. Double the meridian distance of the second course is equal to that of the preceding course 234*1 + the departure of the preceding course 234.1 + its own departure 50'3. (See rule.) All these numbers being w., their sum in tflo arithmetical sense, 518'5, is taken as the double meridian distance (of the 2d cour-e. For the next course the departure 17*6 is r., and on the general analytic principle that quantities estimated in a contrary sense must have contrary signs, this lma;y be, * The sum of the column N. 264~5 exceeds that of the column s. 263"0 by 1 -; half this, or *8 is subtracted from the numbers of the first column N., -6 fiom the Ist No. in the column, and *1 from each of the other two, to obtain the Nos. in the'3d column N., and *7, in the same manner, is added by distribution among the five numbers of the 1st column s., to produce those of the 2d columtn s., &c. t It would be most simple to assume it through the westernmost point of the lalid, in the present example at the commencement of the 3d cou-rse, where the reading was 1630. Here the courses, which were previously all w., begin to turn E. The advantage of this is that the double meridian distances would be all E. eons-idered positive, if we regard the previous numbers employed in cormputinfg ti h double meridian distances, which are all w. as negative. To add this 17'6 then, in the algebraic sense, to the sum of 518*5, and 50*3, according to the rule, will be in reality to subtract it, which gives 551o2 for the 3d D. M. D. For a similar reason the aurm of 17'6 and 73S2 being both E:., must be subtracted from 551-2, which is w., to produce the 4th D. 1r. D.; and so on till we arrive at the 7th course, marked 800, in the 1st column. Here the sum of the two departures, 96-7 and 59*2, both a., viz., 15559 exceeds the last D. aI. D., 135*9, which is still w., and as they have contrary signs, their algebraic sum will be their difference with the sign of the greater, which is E., and this difference 20'0 must be entered in the column i. of double meridian distances. The D. M. D. of the last course is obtained by adding 59*2 and 20'0 both., and subtracting 40*1, which is w., from their sum. The double meridian distanees E., which have corresponding difference of latitude, N., are now multiplied by them according to the rule; and the double meridian distances w., which have corresponding differences of latitude s., and the products all entered in a column entitled N. X E. + s. X w., and their sum taken. Of the N. X w. + s. X E., which the remaining part of the rule requires to be formed, there is but one product in this example an N X w. 196'2 by 234.], or 45930'42, for which an additional column, which would ordinarily be employed, is not worth while. This product is subtracted from the sum of the former, and the remainder, 68103'90, is by the rule equal to double the area of the land in square links, 10,000" of which make a square chain. Half this will be the area, which is converted into square chains by removing the decimal point 4 places to the left; and this again into acres, by removing the decimal point one place fhrther to the left still, since there are 10 sq. chains in an acre. The decimals of an acre are converted into roods and perches by multiplying by 4 and by 40. The plotting of the above example will be an exercise. A circle should be described on the paper, and points marked on it, according to the compass readings in the first column, the N. and s. line corresponding to the 00 and 1800 points, as in the last diagram. Lines drawn from the points thus marked to the centre of the circle will be parallel to the boundary lines of the survey. For further directions see p. 237 at bottom.' Which is the square of 100, the No. of links in a chain. EXAM PLE, ICompass Bearing froI-m iDf Lai. Dep. Cor. DiT. Lat. Cor. Dep.. A. D. I Compass Bearing from ^ iReading. Meridian. ~Dis. -— __________ N.. E. WV. N. S.. W. E. W. N.XE.+S.XW. 3100 N. 50 w-. 306 196*8 2344 196*2 234l 2534*1 226 s. 46 v. 70 48.6 504 487 503 518*5 25250*95 163 s. 17 E. 60 57.4 175 57*5 176 551*2 31694*00 133 s. 47 E. 100 68*2 73*1 68*3 73*2 461'4 31513*62 1S9 s. 51 E. 100 62*9 77*7 63*1 77*8 310*4 19586*24 105 g. 75 E. 100 25*9 96*6 26*1 96*7 135*9 3546.99 i^ ~ 80 N. 80 E. 60 10*4 59*1 10*3 I 592 20*0 206*00 325 N. 35 w. 70 57-3 40*2 57*2 40*1 39*1 2236~52 ^~~~~~~ ~~2645 263-0 324*0, - 325*0 263 27 263*7 345 324 5 324 114034*32 - ~_-_ ~ 45930*42 2)68103*90 Area in Square Links, 34051*95 The area is therefore I rood, 14*48 perches. 4 Roods, 1 3620780 40 Perches, 14*4831200 SURitV EYING, I-YDROGRAPIIIC SURVEYING, In the survey of harbors, after having surveyed and plotted the outline of the shore, it becomes necessary to set down upon the map the depths of the water in feet or fathoms at a sufficient number of points to serve as a guide to navigators. The depth is ascertained by sounding, and the problem is to fix upon the map the points at which the soundings were made. One method consists in rowing a boat uniformly in a straight line from one point on the shore to another opposite, casting the lead at regular intervals by a watch; this line being drawn on the map and divided into as many equal parts as there were casts, the points of division will be the points required; upon these the numbers obtained by the soundings are to be put down. Another method is to place three signals upon the shore, not in the same straight line, and with a sextant* in the boat to measure the angles subtended by the lines joining these signals; then having these lines plotted upon the map, construct upon each of them a segment capable of containing the observed angle subtended by it (see Plane Geom., Prob. 21), and the intersection of the arcs of these segments 4i will determine the points on the map at which the boat was situated at the time of observation. The sounding of course should be taken at the same point, and recorded at its position (6) thus determined on the map. /~ /) a>~ A third method consists in having two theodolites, and taking the angles with them from the extremities of a base line on the shore, by which means the position of the boat is determined. In this method a system of signals is requisite, by which the observers on shore may know the instant at which the sounding is made. A very perfect one was invented by MR. THiOMAS H. NoRRIS, of New York, and practised in the survey of the mouths of the Mississippi. This consisted in having at one of the two stations on the shore (which in the low lands of the Mississippi were elevated platforms of wood) a flag which could be run up and a chronometer. The boat also carried a chronometer. The intervals of time at which the soundings should be made having been previously agreed upon, about 10 seconds before one of these intervals expired, the flag was run up, and both theodolites brought to bear upon the boat, or rather upon a staff at its bow, from which the * See the instrument of this name described at p. 290, note. HYDROGRAPHIC SURVEYING. 263 sounding was made. The tangent screws served to keep the instruments steadily upon this point, and the instant the 10 seconds were up, the flag was lowered, the lead was cast, and the readings taken from the horizontal limbs of the instruments and recorded. This mode was found to be very rapid and accurate. By way of experiment the boat was frequently made to cross its track, and the agreement was exact. By rowing the boat along at oar's length from the shore, and determining its position at frequent intervals, as above described, the line of shore could be traced upon the map. This was found particularly convenient in the survey of the bayous or inlets of the low muddy banks. Horizontal sections of the bottom of a harbor may be determined, (making the plane of the water a plane of reference, in an obvious manner, and the bottom represented in the same manner as a hill. This, however, is not often practised. Points at great distances out at sea are obtained by triangulating outward with three vessels successively moored at points more and more remote from the shore. PART Ve APPLICATION OF SPHERICAL TRIGONOMETRY TO NAUTICAL AST R ONO M Y. PART VY APPLICATION OF SPHERICAL TRIGONOMETRY TO NAUTICAL AST R0 NO MY. 103. NAVIGATION, as we have seen, is the determination of the place of a ship at sea, that is to say her latitude and longitude, by the " dead reckoning." The dead reckoning proceeds upon the hypothesis that the ship's course and the distance she sails are accurately known; and if this were really the case, her true place might be found by the methods given in Part III. But this is impossible. 1. From the difficulty of steering exactly upon the intended course. 2. From the uncertainty of lee-way. 3. From errors of the log, occasioned by the heaving of the sea, unknown currents, and the rudeness of the instrument itself. The " dead reckoning" is, however, indispensable in determining the ship's place during cloudy weather, and is useful at all times for detecting the existence and velocity of currents. The main reliance must be upon astronomical observations, and the method of determining a ship's place by means of these constitutes the science of nautical astronomy. DEFINITIONS. 104. For the purpose of measuring the angular distances of the heavenly bodies from each other, and from the horizon, it is convenient to suppose them all situated as they really appear to an observer on the earth, viz., in a spherical concave surrounding the earth, and concentric with it. This imaginary concave, which the student may suppose identical with the blue vault of the sky, is called the celestial sphere. The position of a point on the celestial sphere, like the position of a 268 APPLICATION OF SPHERICAL TRIGONOMETRY point on the terrestrial sphere, is fixed by its latitude and longitude. On the celestial sphere the circle of longitude is the ecliptic; and secondaries passing, therefore, through the poles of the ecliptic, are the circles of celestial latitude; the point from which longitude is measured is the Vernal equinoctial point. Commencing at this point, the ecliptic is divided into twelve parts called signs; a sign is, therefore, 300. The twelve signs are named, and symbolically expressed, as follows: 1. T Aries. 4. S Cancer. 7. - Libra. 10. v Capricornus. 2. V Taurus. 5. a Leo. 8. lI Scorpio. I 11. Aquarius. 3. in Gemini. 6., rI Virgo. 9. X Sagittarius. 12. x Pisces. The vernal equinoctial point is called the first point of Aries. The longitude is measured from this point in one direction, viz., in the order of the signs, or from w. to E. Parallels of latitude on the teirestrial sphere correspond to parallels of declination on the celestial. Of these, the two which touch the ecliptic in the first points of Cancer and Capricorn, are called the tropics of Cancer and of Capricorn. These first points of Cancer and Capricorn* are respectively called the summer and winter solstice; because for a day or two before and after the sun enters them he appears to be stationary, and the days to be of equal length, so slowly does his declination at those times change, for his motion is obviously very nearly parallel to the equator. The declination circle, through the solstitial points, is called the solstitial colure, and that through the equinoctial points the equinoctial colure. Secondaries to the equator, we have said (Art. 79), are called declination or hour circles. The declination of a heavenly body is its distance from the equator in degrees, minutes, and seconds, measured on the declination or hour circle which passes through the body. The right ascension of a heavenly body is the number of degrees and fractions of a degree measured on the equator, between the vernal equinox or first of Aries, and the circle of declination which passes through the body. Another definition of right ascension is the angle at the pole of the equator or of the earth, comprehended between the hour or declination circle through the vernal equinox, and the hour circle through the heavenly body. Right ascension is now commonly expressed in hours, minutes, and * At the first of these points the sun, which up to the time of its arrival there had been moving north, begins to move backwards towards the south; at the second from going south he begins to climb upwards towards the north, whence it appears that the points in question are named in allusion to the habits of the animals aftwe which they are called. TO NAUTICAL ASTRONOMY. 269 seconds of ti1me, allowing 15o to the hour, 15' to the minute, and 1" to the second of time. The right lascension is also the difference in the time of transit of the heavenly body and of the first point of Aries over the meridian of any place. The difference of right ascension of two stars is the difference in their times of meridian transit at the same place. Or it is the angle comprehended between the two hour circles which pass through the stars. A catalogue of stars is a list of them with the right ascension and the declination of each annexed.* Having described the principal circle and points of the celestial sphere which are considered as permanent, or which do not alter with the situa tion of the observer on the earth, we come now to describe those which change with his place. The principal of these is the horizon, which has been defined already (Art. 79), and vertical circles, which are secondaries to the horizon, and on which the altitudes of celestial objects are measured. These vertical circles all meet in two points diametrically opposite, viz., the poles of the horizon; one of which is directly over the head of the observer, and called his zenith, and the opposite one his nadir. The ver* The late catalogue of the British Association, the name of which is abbreviatld B. A. C., gives the north polar distances (N. P. D.) of the stars instead of their declinations. The north polar distance of a star is its distance fiom the north pole of the heavens, measured on the circle of declination passing through the star. The right ascension of the star fixes the position of this circle in the heavens, and the nlorth polar distance fixes the place of the star upon the circle, so that its position is completely determined by these two co-ordinates. In the British Catalogue is a. column containing the annual variation in R. A., and four columns marked a, b, c, d, at top; also a column containing the annual variation in N. P. D., and four columns marked a', b', c', d'. The numbers whose logarithms are in these columns may be regarded as constant for a period of about ten years. In the Nautical Almanac, on p. XXII. of each month, will be found four columns marked A, B, c, D, at top, containirg the logs. of numbers, which vary with the time, or are ephemeral. To find the R. A. of a star for any given time, take out its R. A. for the epoch of the catalogue, viz., 1850, to which add the product of the annual variation in a. A., by.the number of years between the given time and 1850. The result will be the mean a.. A. t-h beginning of the given year. Take out from the columns a, b, c, d, the logs. opposite the given star, and from the Nautical Almanac, from the columns A, B,, TD the logs. corresponding to the given date, for which the apparent R. A. is required, and with these logs. compute the following formula: d(a = Aa +- b + cc +- Dd da being the correction to be applied to the result before found, to obtain the R. A. required. This will be the time at which a star ought to make its meridian transit by the siderial clock. The formula for the correction in declination is dJ = a'l + Bb' + cc' + Dd' 270 APPLICATION OF SPIHERICAL TRIGONOMETRY tical circle which passes through the east and west points of the horiz.on is called the prime vertical; it necessarily intersects the meridian of the place (which passes through the north and south points) at right angles. The azimuth of a celestial object has been already defined to be an arc of the horizon, comprised between the meridian of the. observer and the vertical circle through the object, and hence vertical circles are sometilfes called aziemuth circles. f!h3 Tiaslitude of a celestial object is the arc of the horizon comprised between the east point and the point where the object rises, or betweeii the west point and that where it sets; the one is called the rising amplitude, the other the setting amplitude. ON THE CORnECTIONS TO BE APPLIED TO THE OBSERVED ALTITUDES OF CELESTIAL OBJECTS. 105. Th e true altitude of a celestial object is always understood to mean its angular distance from the rational horizon of the observer. This is not obtained directly by observation; but is the result of certain corrections applied to the observed altitude.* These we shall now enumerate and explain. * The observed altitude is obtained by means of an instrument called a quadrant of reflection, or simply a quadrant. This instrument is a frame of wood in the form of a sector of a circle, the arc of which is graduated to degrees and parts of a degree. This frame is suspended so that the plane of the circle shall be vertical. It has an arm, one extremity of which is attached to the centre of the circle, and which is movable about this point; upon this arm is a small mirror, and opposite to it is a plane glass, half of which is mirror, and half transparent. When a heavenly body, seen by double reflection in these two mirrors, is brought by the mnovemert of the arm, upon which one of the mirrors is placed, to coincide with the line of the horizon at sea as seen through the transparent part of the opposite- glass, the outer extremity of the arm points out upon the graduated arc the number of d greos of' altitude cf the heavenly body above the horizon. The construction of this instrument depends upon the optical principle that the angle of incidence is equal to the angle of reflection. The angular movement oi the image of the heavenly body is double the angular movement of the arm, so ihat to measure the greatest altitudes, the limit of which is 900, the graduated arc neei be but the eighth of a circumference; the degrees upon it are however numbered as if it were a quadrant, to save the trouble of doubling them. The instrument takes its name from the amount which it measures, instead of from the magnitude of its arc. There are colored glasses attached, which can be interposed so that the rays of light, coming from the heavenly body to the eye, can be made to pass through them when taking the altitude cf the sun. More compieto instruments of this nature are the sextant and repeating circle, or circle of reflection, for full descriptions of which see p. 290, and p. 299. TO NAUTICAL ASTRONOMY. 271 DIP OR DEPRESSION CP THE IHORIZON. 106. Let E represent the place of the observer's eye, ele- vated to the height EA above the surface of the earth, and s the place of a heavenly body'; ___ _ the first object is to obtain its ^ apparent altitude above the / horizontal line El; that is, the angle SEI. Now, since to the observer, the visible horizon is in the direction EnII', the alti- / tude taken with the instrumel)t tB e. " is the angle SEH'; hence from this observed altitude the angle IHEIl' cllled the P' ir) Depression of the.HLorizon, must be subtracted to obtain the apparent altitude SEn. The angle HEE', or its equal c, is calculated for various elevations, AE, of the eye above the surface of the sea, by resolving the right angled triangle EBO, in which are known oB, the radius of the earth, and hE equal to the radius increased by the height of the eye. The results are registered in a table (Table XXXI.), the argument of which is the height of the eye. The depression thus obtained must be lessened by the amount of terrestrial refraction, which is very uncertain; HO of the whole quantity has been allowed in computing this table. SEMIDIAMETER. 107. The foregoing correction for dip having been applied, the result will be the apparent altitude of the object observed, above the sensible horizon. If this be the upper or lower edge of the disc of the sun or moon, called the upper and lower limb, a further correction will be necessary to obtain the apparent altitude of the centre. The angle at the eye of the observer, subtended by the semidiameter or radius of the sun or moon, must be added to the altitude of the lower limb, and subtracted from that of the -rpper limb. This quantity, which, is continually varying both for the sun And moon, in consequence of the variation of their distance from the 272 APPLICATION OF SPHERICAL TRIGONOMETRY earth, is given in the Nautical Almanac for every day in the year.* But in the case of the moon the semidiameter itself requires a small correction depending upon the observed altitude. For the semidiameter, furnished by the Nautical Almanac, is the apparent horizontal semidiameter, i. e. the apparent semidiameter when the moon is in the horizon, where the distance from the observer is greater than when she is in the zenith by the semidiameter of the earth. Consequently her apparent semidiameter, which is inversely as her distance, will be least in the horizol, and greatest in the,9ei:h; and its value between these limits will vary with the sine of the altitude, as may be easily seen by constructing a diagram. The distance of the moon being about 60 semidiameters of the earth, the rzcon's horizontal seridiameter will be increased about P- part in the zenith. Therefore, if to the logarithm of the sine of,- of the D's horizontal semidiameter or the log. of the arc itself, which is small, we add * It is given for noon of each day for the sun, and for noon and midnight for t.he moon, and is found for any other time of day by the proportion: As 24 or 12 hours: the variation in 24 or 12 hours:: the time after noon or midnight, at Greenwich: the variation in that time, which must be added to the semidiameter given in the Almanac or subtracted, according as the semidiameter is increasing or diminislling from day to day, in order to have the semidiameter at the required time. Proportions of this kind, in which the terms contain two or three denominations, as hours and minutes, minutes and seconds, or hours, minutes, and seconds, degrees and minutes, &c., may be resolved conveniently by means of the table of proportional logarithms, Table XXII. The following example will illustrate the mode of proceeding. 24^: 16t 19.:: S; 2-3'aking the first and third terms one grade lower, we find their proportional logarithms (P. L.) on pp. 134 and 132, writing the arith. comp. of the former, and taking from p. 133 the r. L. of 16' 19", the calculation will be as follows: 24m ar. comp. P. i. 9*1249 161 19" P. r. 1-0426 8m 2' P.. 1*3504 5' 28" P. L. 1*5179 In this as in many other problems of Nautical Astronomy, the time at Greenwich at the instant of observation is required, and may be found by adding or subtracting the difference of longitude in time, according as the place is w. or E. of Greenwich. Thus the time at Greenwich, corresponding to any given time at New York, is found by adding 4* 56m 48 (the difference of longitide between the two places) to the latter. TO NAUTICAL ASTRONOMY. 273 the log. sine of the D's altitude, the result will be the log. of the apparent semndiameter at the given altitude. In this way is formed the Table at the end, entitled Augmentation of the Moon's Semidiameter (Table XXXIII.), which contains the proper correction to be added to the given horizontal semidiameter, to obtain the true seridiameter. On account of the great distance of the sun, no such correction of his semidiameter is necessary. The corrections for dip and semidiameter being thus applied, the results is called the app'rent altitude of the centre. In the case of the starsr the only correction for the apparent altitude is the dip~ To obtain the true altitude requires two other corrections, viz. fir refraction and for parallax. The former of these has indeed an effect upon the two preceding corrections, dip and semidiameter, which require certain modifications in consequence, which we shall notice after explaining the nature and effect of REFRACTION' 108. The rays of light coming from a heavenly body, having to pas. through the atmosphere, are bent towards the vertical by refraction. iA the atmosphere grows more and more dense in approaching the surface of the earth, the light bending continually towards the vertical pursuie; a curvilinear path in a vertical plane, and enters the eye in the last direction of its motion, which prolonged is a tangent to the curve, and it ii. in the direction of this tangent that the object emitting the light appears, The curve being convex upward, the tangent lies above it, and the effect of refraction is therefore to elevate the object, or to make the apparent place above the true place. The correction for refiaction, therefore, like the correction for dip, is always subtractive; it decreases from the hori-A zon, where it is greatest, to the zenith, where it vanishes (as the rays from objects in the zenith enter the atmosphere perpendicularly) in accordance with the optical law that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. At the end of the volume we have given a table of refractions eonl taining the correction for refraction to be applied to every altitude, from the horizon to the zenith,* and adapted to the mean state of the atmo* It will be observed that there is in the table a column of differences for -1 of altitude. The number in this opposite the degrees in the given altitude must be multiplied by the given minutes and the result subtracted from the correction, or added to the altitude. 18 274 APPLICATION OF SPHER[ICAL TRIGONOMETRY sphere (Table XXX.).* When the temperature of the atmosphere i raised, which is indicated by the thermometer, the refraction decreases; and when the density of the atmosphere is increased (indicated by the rising of the mercury in the barometer), its refractive power increases. The change in refraction for a difference of 10 of Fahrenheit, and of I inch in the barometer from the mean state, is given in separate columns, and must be multiplied tile one by the number of degrees which the thermometer differs from 500, and the other by the number of inches and fractions of an inch which the barometer differs from 30~, and the result added or subtracted, as the case may require. It should be observed that below 4~ the refraction is very variable and uncertain, and such low altitudes should be avoided as much as possible at sea. It will be unnecessary to use the correction for the state of the barometer and thermometer, when the latitude of the ship is the only object of the observation, as this could seldom make a difference so great as half a mile in the resulting latitude; but, in determining the longitude by the Lunar Observations, the neglect of these small corrections would sometimes introduce an error in the resulting longitude of more than thirty miles. When the foregoing corrections have been applied to the observed altitude, the result will be the true altitude of the centre above the sensible horizon, and it now remains to apply the correction necessary to reduce this to the true altitude of the centre above the rational horizon; that is, to the altitude which the body would have if the observer were situated at the centre of the earth instead of on its surface. This last correction is called PARALLAX. 109. In order to explain the nature:and effect of parallax, let s represent the place of the object observed fronm the surface of the earth, at E; then the angle SEI, that is, the observed angle, when corrected for dip, semidiameter, and refraction, will be the true altitude of the object, in reference to the. i1 observer's sensible horizon EH; and the angle -sc will be the true altitude * Such a table might be formed by comparing the observed altitude of a star with its altitude computed from the declination or N.r.D. hour angle and latitude. TO NAUTICAL ASTRONOMY. S,5 in reference to the rational horizon CR; and thle difference of these angles is the parallax called parallax in altitude when the object is above the horizon as at s, and horizontal parallax when it is in the horizon as at ni. Since the angle SE'H is equal to the angle scR, Ei and CR being parallel by definition, we have for the parallax in altitude SE'H - SEH _= ESi (Geom. Th. 15), that is, the parallax is the angle which the semidiameter of the earth subtends at the object;* it is obviously greatest in the hor;zon, and nothing in the zenith, and is the quantity which must be added to the true altitude above the sensible horizon to obtain the true altitude above the rational horizon. The sun's parallax in altitude is given in a Table at the end (Table XXXIV.), his horizontal parallax being nearly constant; and the mooons horizontal parallax is given for the noon and midnight at Greenwich, of every day of the year, in the Nautical Almanac, and from the horizontal parallax thus obtained, parallax in altitude must be calculated. This is easy; for since in the triangle sic we have the proportion sc: EC: sin EC s= s in sEZ = cos SLER: ~sin ESC; it follows (since sc, the distance of the heavenly body, as well as Ec, the semidiameter of the earth, may be regarded as constant for a single day), that the sine of the parallax in altitude varies as the cosine of the altitude; but when the altitude = 0, as in the case of horizontal parallax, cos. altitude = 1, and the constant ratio sc to EC, the above proportion shows to be equal to the sine of the horizontal parallax. But from the proportion itself we see that it is necessary to multiply this ratio by the cosine of the altitude, to have the sine of the parallax in altitude; but as the parallax is always a very small angle, it is usual to substitute the are for its sine, or par. in alt. = hor. par. X cos. alt., so that log. hor. par. in seconds + log. cos. alt.- 10 log. par. in alt. in seconds. We must observe here that the horizontal parallax, given in the Nautical Almanac, is calculated to the equatorial radius of the earth; * This result might be arrived at much more simply by means of our definition ot an angle (Geometry, def. 10); viz. " the difference of direction of two lines," and a definition of parallax, viz. the difference of direction in which an object is seen from the centre and surface of the earth, or in a more enlarged sense of the term, from any two points. This in the diagram will be the difference of direction of the two lines cs and ES, i. e. the angle csE, or the angle subtended by the line joining the two points of observation. 3716 APPLICATION OF1 SPHERICAL TRIGONOMETRY and, therefore, except at the equator, a small subtractive correction of the horizontal parallax will be necessary, on account of the spheroidal figure of the earth, in consequence of which the radius of the earth is smaller everywhere else than at the equator, and consequently subtends a smaller parallax. A table of such corrections is given at the end. (See Table XXXV.) It must evidently be a table of double entry, the two arguments being the equatorial horizontal parallax and the latitude, upon which two quantities the correction depends. 110. Such are the corrections necessary to be applied to the observed altitudes of celestial objects, in order to obtain their true altitudes. A few other preliminary, but very simple and'obvious operations, must also be performed upon the several quantities taken out of the Nautical Almanac, in order to reduce them to their proper value at the time and place of observation; for the elements furnished by the Nautical Almanac are computed for certain stated epochs, and their values for any intermediate epoch must be found by proportion. But ample directions for these preparatory operations are contained in the "Explanation of the Articles in the Nautical Almanac,"* to be found in the last pages of that work. * It may be well, however, to give here some general account of the arrangemlent of the Nautical Almanac. The first twenty-two pages contain the right ascension, declination, semidiameter, and a variety of other elements relating to the sun and moon for every day of the month of January, the right ascension.of the sun at mean' noon and at apparent noon, that of the moon at the beginning of every hour of mean time throughout the day at Greenwich. The next twenty-two pages contain the same elements for the month of February, and so on, each month occupying twenty-two pages, marked with the Roman numerals, I. II., &c. The year thus being gone through, after a few pages containing the sun's co-ordinates, follows the ephemeris of the planets, beginning with Mercury, the one nearest the sun. This contains the semidiameter and declination, apparent right ascension, as affected by aberration of light, and some other elements of the planet for every day in the year of mean noon at Greenwich, and also at the time of the planet's meridian transit at Greenwich, each month occupying two pages. This Ephemeris extends from p. 275 to p. 455, in the almanac of 1850. The next three pages contain the mean places or right ascension and declination on the 1st of Jan. of 100 principal fixed stars, with their annual variations in right ascension and decimfation, marked + or. The latter multiplied by the fraction of the year which has elapsed, which is given in the last column of p. XXII. of each month, will be the quantity to be added or subtracted, in order to have the mean R. A. and Dec. at the time. To obtain the true places, corrected for nutation, &c., recourse must be had to formulas and tables given in the next three pages of the Almanac, except that of a number of the prineipal stars, the true a. A. and Dec. are given for every ten days from p. 468 to p. 501, mi the edition of 1850. The remaining matters contained in the Nautical Almanac will be noticed as occasion requires. T'O NAUTICAL AKST RONIXOMY. 277 EXAMPLES OF THE CORRECTIONS. 1. On the 14th of July, 1833, suppose the observed altitude of the sun's lower limb* to be 160 36' 4'", the observers eye to be 18 feet above the level of the sea, the barometer to stand at 29 inches, and the thermometer at 58~; required the true altitude of the sun's centre. Observed alt. e's L. L.... 16~ 36' 4" Depression of the horizon (Tab. XXXI.) - 4'4 App. alt. of L. L.. 16 32 0 Refraction.. - 3 1 Correction for Barometer.. + 6*5t Correction for Thermometer 3. - - 3*2 True altitude of L. L. above the visible horizon 16 28 55*7 Sun's semidiameter (Naut. Alm.), o + 15 45*4 Parallax in altitude, o., + 8'41] True altitude of the sun's centre. 16 44 49*5 2. On the 23d of June, 1850, in longitude 4' 56" 4V W., latitude about 40~ 43' N., at IIA 44' 55 mean time, the double altitude of the moonr's upper limb was observed by reflection fronl Mercury to be 58~ 14'; the index error of the sextant was 15" subtractive; the barometer stood at 30*74 in., and the thermometer at 760, required the true altitude of the moon's centre. The object in this example being the moon, it is necessary to cor — pute her semidiameter and parallax in altitude at the instlant of obser* The limb of the sun or moon is the edge or border of the disc. t Take out the refractions for 160 30' of altitude from the table, then the difl. for 2' of altitude in the column adjoining, multiplying the latter by 2, and subtracting the product from the refraction for 160 30'; the result will be that for 160 32', when the barometer is at 30 in. and the thermometer at 500. The correction for refraction is always subtractive. X The barometer standing at 29 in. the number taken from the column entitled cor. for - 1 must be subtracted from the refraction or added to the altitude, the atmosphere being less dense than in its medium state. ~ The thermometer standing at 80 above its medium state, the atmosphere is more rare, and the number taken from the column Diff. for 10 Fah., after being multiplied by 8, must be subtracted from the refraction, or added to the altitude. II Table XXXIV., the parallax in alt. for 100 is 9", and for 200 is 8". Therefore for 160 by proportion it is 8''4. This correction for par. in alt. is always additive. 9!3T8 BAPPLICATION OF SPtHERICAL TRIGONOMETRY vation since these elements, for the moon changes sensibly in a very short time. The semidiameter of the moon at noon and midnight is given in the Nautical Almanac for every day in the year, at page III. of each month, Rand the difference between these will be the variation of the semidiameter in 192 hours. Therefore we must say as 12: the variation in 12::';he interval between the preceding noon or midnight and the instant of observation: the variation of the semidiameter in that interval; the fourth term of this proportion added to or subtracted fiom the semidiameter at the preceding noon or midnight, according as the semidiameter is observed from the numbers in the almanac to be increasing or decreasing, will give the semidiameter at the instant of observation. In a similar manner must the moon's horizontal parallax, which is given for every noon and midnight on the same page of the Nautical Almanac, e reduced by proportion to the time of observation. The computation of these elements is as follows: Mean time of observation at the station 11 44" 55' Add longitude of the place of observation 4 56 4 Corresponding mean time at Greenwich 16 40 59 Time after midnight Gr. June 23d 4 40 59 Semidiameter previous mid- Horizontal parallax preceding night, June 23d (Naut. midnight, June 23d (Naut. Aim.). 14' 50"'3 Aim.) 54' 271"3 ie'midiar. noon (June 24th), 14 48 *1 Hor. par. noon (24th), 54 19'1 Variat. in 127" 2 *2 Var. in 12. 8 -2 2.. 12: 2"2:: 4A 40" 59* 0'8.. 12: 8tt2: 4h 407 59: 3 *1 Semidiam. at midnight (23d), 14 50 *3 Hor. par. at midnight, 54 27 *3 Semidiam. at time of obs., 14 49.5 Hor. par. at time of obs., 54 24 *2 Augmentation for 290 of alt.t + 7.8 Ditto in seconds, 3264'P*2 Apparent semidiam. to obs., 14 57 *3 Diminut. of par. for lat. 410~ - 4 *7 Subtract contraction,~ - 1 1 Hor. par. at station, 3259"*5 True semidiam. to observer, 14 56'2 * This is the interval from midnight at Greenwich to the instant of observation. t Table XXXIII. This augmentation is in consequence of the moon being nearer to the observer, as it approaches the zenith. See p. 272. t This is occasioned by the effect of refraction, whici. is to make every vertical are, such as the vertical semidiameter of the sun or moon, appear shorter in the heavens than it really is. This will obviously be the case, because the lower extremity of the are is more elevated by refraction than the higher, and consequently the two extremities are brought nearer together, and thus the arc is shortened. The contraction is obtained from Tab. XXXII. ~ Table XXXV., see p. 275, last paragraph of Art. 109. TO NAUTICAL ASTRONOMY. 279 Observed double altitude D's U. L.. 58~ 14' 00 Index error subtractive,... 15" Double altitude corrected for index error,. o 58 13 45 Half this is the obs'd altitude of D's U.L. 29 6 52'5 Corrected semidiameter,.. 14 56'2 Apparent altitude t's centre,.. 28 51 56'3 FOR TIHE PARALLAX IN ALTITUDE.* App. alt. D's centre,... 280 51' 56"'3 cos 9*94238 Horizontal parallax at station,.. 3259"5 log. 3'51315 Parallax in altitude,.. 2854'"5 3*45553 App. alt. D's centre,... 28~ 51' 56"'3 Refraction,.... -- 1 44'9 Barometer,.. — 2'7 Thermometer,.. e..- + *5' Parallax in altitude,... + 47 34'5 True alt. of cent. from cent. of the earth,.. 29 37 48 *' These two examples will serve for specimens of the corrections to lx>: applied to an observed altitude, in order to deduce from it the true altitude of the body's centre. In the case of the moon, the corrections, when the utmost accuracy is sought, are rather numerous, as the last example shows. But in finding the latitude at sea, it is usual to dispense with sonim of these, more especially with the corrections for temperature, for the contraction of the moon's semidiameter, and for the spheroidal figure of the earth; because an error of a few seconds in the true altitude will introduce no error worth noticing in the resulting latitude. When, ho:wever, the object of the observer is to deduce the longitude of the ship, all the data, furnished by observation, should be as accurate as possible: for the problem is one of such delicacy that by neglecting to allow for th(influence of temperature would alone introduce in some cases an error of from 30 to 40 miles in the longitude. When the object observed is a star, several of the foregoing corrections vanish; the only corrections in this case requisite are those for dip and refraction, modified as usual for the temperature. Ill. To determine the latitude at sea from the meridian altitude of any celestial object whose declination is known. The determination of the latitude, by a meridian altitude, is the most * See p. 275. 280 - APPLICATION OF SPHERICAL TRIGONOMETRY easy and safe method of finding that element; the observations and subsequent calculations being few, are readily performed, and with but little liability to error in the result; this method, therefore, is always to be preferred at sea, unless clouds obscure the meridian whilst other portions of the heavens are left visible. The declination of the object observed is supposed to be given in the Nautical Almanac, when it culminates or makes its meridian transit at G4reenwich; its declination when it culminates at the meridian of a ship, may be found by means of the longitude by account,* which will always be sufficiently accurate for this purpose, although it should differ very contsiderably from the true longitude, because declination changes so slowly that even an error of an hour in the longitude would cause an error in the declination too small to deserve notice. The declination being the distance of the object fiom the equator, and the observed altitude, properly corrected, being the distance of the same object from the ship's zenith, the distance of the zenith from the equator, that is, the latitude, immediately becomes known. Let the full circle in the diagram be the meridian. 1. Let s be the object observed, the zenith z being to the north of it, M:nd the object itself north of the equator, EQ, then the latitude EZ is equal to the zenith distance, or co-altitude zs + the declination ES, and it is north. 2. Let s' be the object, still north of the equator, but so posited that the zenith is south of it, then the latitude EZ is equal to the difference between the zenith distance s'z, and declination S'E, and is still north. 3. Let now the object be at s", south of the equator, and the zenith to the north of the object, then the latitude EZ is equal to the difference \ between the zenith distance s"z and' declination S'E, and it is north..____ We have here assumed the north \ to be the elevated pole, but if the south be the elevated pole, then we must write south for north, and north for south. Hence the following rule for all cases. * For this purpose the variation of the declination in I hour, which is given in the Nautical Almanac for the sun, must be multiplied by the longitude in hours and fractions of an hour, and the product added or subtracted will produce the declinae'ion at the time of meridian transit at thle ship. TO NAUTICAL ASTRONOMY. 281 Call the zenith distance north or south, according as the zenith is north or south of the object. If the zenith distance and declination be of the same name, that is, both north or both south, their sum will be the latitude; but if of differ. ent names, their dif'erence will be the latitude, of the samne name as the greater. EIXAMPLE. 1. Ship Admiral, from New York to Havre, at sea Jan. 4th, 1850. Longitude 250 W. Observed merid. alt. 0's lower limb,... 18~ 54' 20" Dip (for height of 17 feet).. 3 57 App. alt. o's L. L.... 18 50 23 Refraction,.. - 2 49 Parallax in alt... e + 8 Semidiameter,.... 16 173 C's true alt...... 19 3 59'3 0's zenith dist. N.... 70 56 00'7 0's declination* S.... 22 43 37'9 Latitude, N...... 48 12 22*8 2. At sea Jan. 11th, 1850. Long. 20 W. O's Dec. 21~ 46' 2" S. Observed alt. 0's L.L.... 18 4' 00 Allowing for semidiameter (Dip 3' 26") parallax, &c. + 12 29 Required the latitude. Ans. 490 51' 29" N. 3. On the 1st of January, 1850, tie meridian altitude of Capella was 27~ 35', the zenith being south of the star, and the height of the eye 22 feet; required the latitude. * This is obtained by taking out from the Nautical Almanac, the declination for apparent noon, p. I., which is 220 44' 5"'2. Then computing the change in decli-.nation for 1l hours, the time in which the sun is passing fiom the meridian of Greenwich to that of the ship in long. 250 W., by multiplying the number 16"*38, found in the column in the Almanac entitled Diff. for 1 hour by 1i. The product 27"*3, subtracted from 220 44' 5"-2, because the declination is decreasing will give the declination at the meridian transit of the sun at the ship. 282 APPLICATION OF SPHEIlIlCAL TRiCONOMETRY Observed altitude,. 27o 35' 0" )ip,. -.... 4 30 Apparent altitude,... 27 30 30 IRefraction,..... 51 True altitude,. 27 28 39 Zenith distance,.... 62 31 21 S, Star's dec. (Nautical Almanac},..45 50 20 N. Latitude,...... 16 41 1 S. 4. Suppose that the altitude of the moon, as given in Example 2, p. 277, was observed when the moon was upon the meridian, required the latitude of the place of observation. The true altitude of the mooon's centre being known, after applying the corrections as at p. 278, it remains to find her declination at the instant of observation. The Nautical Almanac gives the moon's declination for every even hour of the day of every day, on pages V. to XII. of each month, and the variation in declination for 10" of time. The required declination would therefore be computed as follows: D's dec. June 23d, at 16^ (Nautical Almanac), 190 36' 48"'8 )iff. dec. for 41',*.... 1 35 I)ec. at the inst. of observation,... 19 38 23'8 S. D's zenith dist. (90- 290 37' 49"'8f). 60 22 10'2 N. Latitude required,.... 40 43 46 *4 If the time of observation were not known, it could be computed from the fact that the moon is on the meridian. The moon passed the Meridian of Greenwich June 23d (Nautical Alnanac, p. IV.), at. 11 33" 0' June 24th, "... 12 21 7 The interval between the two transits is.. 24 48 7 That is in 24h 48' 7s the moon is retarded in comiing to the meridian, by her proper motion from W. to E. 48'" 7'.. 24" 48"' 7': 48"7 7: 4 56"' 4:t: 9' 39" ~ The time of obs. was 40'"2 59' past 6^ or nearly 41"'. The Naut. Aim. gives 233'"17 diff of dec. for 10".'. 10."': 23/.17:: 41.: 1' 34" the change in dec. in 41'", which as dec. is increasing must be added. Second differences are not used. t This is the true alt. of the D's centre from the centre of the earth, p. 279o I This is the longitude of the place of observation. TO NAUTICAL ASTRONOMY. 28^ This last number is the retardation of the moon in passing from the meridian of Greenwich to that of the place of observation. The moon leaving crossed the meridian at Greenwich at 11 33" on the 23d, will cross that of the station 9" 389 later, so that the time of meridian transit at the station will be ll 36" 39'.* It saves trouble to note the time of meridian transit by a watch, or still better by a chronometer, keeping Greenwich time. SCHOLIUM. These examples will, no doubt, be found sufficient to put the student in possession of the method of applying the various corrections to the observed meridian altitude of a celestial object, in order to deduce from it the latitude of the ship. But it should be remarked, that in most works )n Nautical Astronomy, subsidiary tables are inserted for the purpose of abridging some of the foregoing corrective operations; such tables, there tore, offer very acceptable aid to the practical Navigator. Bowditch's Navigator is the most complete work of the kind. It should also be observed here, that in the preceding examples the celestial object is supposed to be on the meridian above the pole; that is, to be higher than the elevated pole. But, if a meridian altitude be taken below the pole, which may be done if the object is circcmpolar, or so near to the elevated pole as to perform its apparent daily revolution about it without passing below the horizon, then the latitude of the place will be equal to the sunt of the true altitude, and the codeclination or polar distance of the object; for this sum will obviously measure the elevation of the pole above the horizon, which is equal to the latitude.t 112. To determiine the latitude at sea, by means of two altitudes of the stez, aznd the time between the observations. In the preceding article we have shown how to determine the latitude of the ship by the meridian altitude of the sun, or of any other heavenlybody, whose declination may be found. But, as already remarked, the object we wish to observe may be obscured when it comes to the meridian, and this may happen for many days together, although it may be freo.unntly visible at other times of the day. As therefore the opportunity * This differs slightly from the time of observation given. The moon changes so rapidly in declination that her greatest altitude is not always the meridian altitude. t That the elevation of the pole above the horizon is equal to the latitude of the place is evident from the fact that the zenith is 900 from the horizon, and the pol. 90( from the equator. 284 APPLICATION OF SPHERICAL TRIGONOMETRY for a meridian observation cannot be depended upon, it becomes an important problem to determine the latitude at sea, by observations made out of the meridian; and considerable attention has accordingly been paid, by scientific persons, to the method of finding' the latitude by " double altitudes," and various tables have been computed to facilitate the operation. But the direct method, by spherical trigonometry, though rather long, involving three spherical triangles, will be more readily remembered, and more easily applied by persons familiar with the rules and formulas of trigonometry than any indirect or approximate process; we shall therefore explain the direct method. Let P be the elevated pole, z the zenith of the ship, and s, s' the two places of the sun, when the altitudes / p are taken. Then, drawing the great circle arcs as in the figure, we shall have these given quantities, viz., the co-' declinations Ps, Ps'; the coaltitudes zs, zs', and the hour angle sps', which measures the interval between the observations; and the quantity sought is the colatitude zp. Now, in the triangle Pss', we have given two sides and the included angle to find the third side ss', and one of the remaining angles, say the angle Pss'. In the triangle zss' we have given the three sides to find the angle s'sz; having then the angles Pss', s'sz, the angle zsP equal to their difference, becomes known, so that we have, lastly, two sides and the included angle in the triangle zsP, to find the third side zP. Before the application of the trigonometrical process, the observed altitudes must, of course, be reduced to the true altitudes, as in the preceding examples. Moreover, as the ship most probably sails during the interval of the observation, an additional reduction becomes necessary, as follows: Let z be the zenith of the ship, and s the place of the sun, at Z the first observation z' and s' the same at the second. Then the angle z'zs will represent the bearing of the ship's path from the sun, which may be observed s/ with the compass; considering TO NAUTICAL ASTRONOMY. 285 this angle as a course, and the distance sailed, zz', as the corresponding distance, find by the table (or by the formula zz' cos z'zt) zt which subtracted from zs will give z's nearly, which, instead of zs, should be used with z's' in the solution before given. This must be subtracted from the first zen. dist. if the angle z'zt is less than 90~; but it must be added when the angle exceeds 90~. If the angle is 90~, no correction for the ship's change of place will be necessary. Where great accuracy is aimed at, account should be taken of the ship's change of longitude during the interval of the observations; when converted into time it must be added to, the interval of time between the observations when the ship has sailed eastward, and subtracted when she has sailed westward. *This correction is very easily applied. Having thus mentioned the necessary preparative corrections, we shall now give an example of the trigonometrical operation. EXAMPLE. Let the two zenith distances corrected be (see last figure but one), zs = 730 54' 13", zs' - 47~ 45' 51", the corresponding declinations 8' 18' and 8~ 15' north, and the interval of time three hours; to determine the latitude. Considering ss' to be the base of an isosceles spherical triangle, of which one of the equal sides is ~ (Ps + Ps')t = 81~ 43' 30", and the vertical angle equal to 3h or 450, let the perpendicular PM be drawn, then we have in the triangle PMS right angled at M, Ps = 810 43' 30", and 450 v 45o 22~ 30', given,to find SM 2= ss' as follows. 22 1. TO FIND SS' FROM THE TRIANGLE PMSo sin Ps 81~ 43' 30" 9'99545 sin P 22 30 0 958284 sin SM 22 15 1.14 9'57829 2 ss' = 4 30 22*8 * If the student will conceive an addition to the first diagram on the preceding page, to wit, the arc of another great circle, different from Pz, drawn through P, to represent the new meridian of the ship, calling this pz', then the hour angle or time of the first observation would be zPs, as before, but that of the second observation would be z'Ps', and the difference ZPS - z'ps', or the diff. of the times of obs., would evidently be equal to sps' -+ zPz', or srs' - za'Z, according as Pz' is in fiont or behind in the diagram, i. e., east or west of rz. But zpz' is the difference of longitude of the two meridians. t Which we may, without sensible error, where the base is so small 286 APPLICATION Of SPHERICAL TRIGONOMETRY IT. TO FIND PSS FROM THE I LIANGLE PSS. sin ss' 44~ 30' 22'8" arith. comp. 0'15429 sin PS' 81 45 0... 999548 sin sPs' 45 0 0. 9*84948 sin Pss' 86 38 53.. 999925 This angle is acute like its opposite side (see p. 196). II, TO FIND ZSS IN TIE TRIANGLE ZSS'. zs' 470 45' 51" sin zs 73 54 13. arith. comp. 0'01737 sin ss' 44 30 22*8 arith. comp. 0*15429 2)166 10 26e8 - sum = 83 5 13'4 sin ( sum zs) 9 11 0*4, 0*20302 sin (- sum-ss') 38 34 50*6. 9'79492 2)19'16960 sin - zss' 22 36 264.. 9'58480. zss' = 45~ 12' 5298" Pss' = 86 38 53 Psz = 41 26 0*2 iV. TO FIND THE TWO UNKNOWN ANGLES OF THE TRIANGLE ZSP. cos I (zs — rs) 770 48' 521ar. comp. 0'67555 ar. comp. sin 0-00990 *cos (zs,rs) 7049t17"... 9-99594 sin 9-13381 cot ~ PSX 200 43' 3... 10-42228...... 10-42228 tan (zrs+rzs) 850 23 35".. 11-09377 tan r (rP ) 200 12 32t 9.56599 200 12132" srz = 1050 36t 7/ zrs= 65011 3P " This sign is employed to express the difference between two quantities, whichever may be the greater, TO NAUTICAL ASTRONOMY. 287 V. TO FIND ZP IN THE TRIANGLIE ZSP. cos - (z —P) 20~ 12' 32' ar. comp.' 0'02593 cos -I (z + P) 850 23/ 31" 8*904822 tan t (zs + PS) 77 48' 52" 10*665658 tan -i zp 21~ 37' 14" 09598073 z- = 430 14' 28" Upon the same principles may the latitude be determined from the altitudes of two fixed stars, taken at the same time; in this case s, s', in the preceding figure, will represent the two stars: Ps, Ps/, their known polar distances, and the angle sps', the difference of their right ascensions; the same quantities are therefore given as in the case of the sun, but, as in the case of two stars, PS, PS', may differ very considerably, ss' cannot be considered as the base of an isosceles triangle, but must be computed from the other two sides and their included angle. For other modes of determining the latitude, see the next Appendix. ON FINDING THE LONGITUDE. The determination of the longitude of a place always requires the solution of these two problems, viz.: 1st, to determine the time at the place at any instant; and, 2d, to determine the time at the first meridian, or that fiom which the longitude is estimated, at the same instant; for the difference of the times converted into degrees, at the rate of 150 to an hour, will obviously give the longitude. When the latitude of the place is known (and it may be found by the methods already explained), the time may be computed from the altitude of any celestial object whose declination is known; for the coaltitude, codeclination, and colatitude, will be three sides of a spherical triangle given to find the hour angle, comprised between the codeclination and the colatitude. (See Art. 84.) The following example will illustrate the mode of proceeding. At Columbia College, January 13th, 1850, tie double altitude of the sun's lower limb was observed by reflexion from mercury to be 42~ 29' Thermometer 400, and Barometer 30 in, Time by the watch, 10 6"' 10' A.M. Index error of the sextant, 52" additive. Latitude of station 40~ 42' 40". Longitude from Greenwich in time, 4* 56"' 4~. Required correct time of Observation and error of the watch. 288 APPLICATION OF SPHERICAL TRIGONOMETRY Equation of time at ap. noon, January 13th, 1850, 9 00" 27. Difference per hour =0"'926.* Sun's declination at ap. noon, January 13th, 1850, 210 29' 19"'5 S. Difference per hour = 26"'1. Observed double altitude,. 42~ 29' Index error of sextant. Additive (see 2d note, p. 290). 52' Double altitude corrected for index error,. 42 29 52 Altitude,.. e.. 21 14 56 Refraction (Th. 40), (B. 30), Table XXX.,. -- 2 35 Sun's Parallax in Altitude, Table XXXIV.,. 8 Semidiameter (Nautical Almanac)... 16 16'8's true altitude corrected for refraction, parallax, and semidiameter 21 28 45'3 e's zenith distance (900-Altitude).. 68 31 14 7 Approx. time at station,.. 10A 6' 10" A.M. Longitude from Greenwich in time, 4 56 4 Time at Greenwich,.. 3 2 14 Equation of time, subtractive,.. 9 3 08 Time after. apparent noon at Greenwich, 2 53 10'92 a's declination at time of observationf, 21 28 4'18 S. z * This must be multiplied by the time after apparent noon at Greenwich, found below, reduced to hours and decimals of an hour, and the product added to the equation of time at noon above, to obtain the equation of time at the instant of observation. 1 This is computed from the data in the third and fourth lines from the top of tho page, in the same manner as the equation of timel TO NAUTICAL ASTRONOMY. 289 Psor Z -c-'s N.P.D (90-t-D)=111~ 28' 4''18ar.co. og.sin 0'03122 Pz or s = Colatitude 49 17 20 ar. co. log. sin 0*12033 zs orp = Zenith distance 68 31 14'7 s 229 16 38 *88 - 1114 38 19 e44. s —z 3 10 15 ~26 log. sin 8'74285 ~s-~s 65 20 59 *44 log. sin 9*95850 2)18'85290 ~ iP 15 29' 1" log. sin 9942645 Hour angle = 30 58" 2'. -= Hour angle in time, or apparent time before noon, 21 3" 52* ~I Subtract hour angle from.. 12 Apparent ti-me, A.M. by observation, o. 9 56 7 9 Equation of time, additive,. 9 3 *08 Mean time by observation,.. 10 5 10 98 Mean time by watch,... 10 6 10 Error of watch (too fast)..'. 59 *02 2. Ship Admiral at sea, March 5th, 1850, at 10 o'clock, A. M. Observed altitude sun's lower limb,.. 27~ 44 Height of eye above the level sea, 16 feet. Time at Greenwich, by mean of three chronometers 12A 41' 12* Lat. at time of obs. north,.'. 490 54' 00" EXTRACTS FROM' NAUTICAL ALMANAC. At mean noon. March 6th, o's semidiameter, 16' 7"V9; eq. of time, - 11" 30~'98e 7th, ", 6 7 *6 " 11 16*56 o's dec. at apparent noon Gr. (6th)... 50 40' 48"'3 S. Diff. for 1. 58"-20t Required the time of observation. Ans. 91 19' 29 *4 3. March 12th, at 4 P.M. Astronomical Account. Alt. of the sun's lower limb,... 18~ 42' 40" Time at Greenwich by mean of chronometers,.. 7 51 30"' Lat. of ship at time of obs...... 410 41' 30" ~ To be subtracted from mean time. f The dec. is of course diminishing till the equinox March 1laoh 19 290 APPLICATION OF SPHEIRCAL TRIGONOMETRY At mean noon. March 12th, eq. of time — 9' 59"^14 semidiam. 16' 6"'3* " 13th, " - 9 42 71 " 16 6 o's dec. at app. noon, March 12th (Gr.) 30 20 11~ 7 S. Var. of dec. in 1A is 59''"02. Required the time. An s. ^' To find the time at Greenwich requires the aid of additional data, besides those furnished by observations made at the place. The Greenwich time may, indeed, be obtained at once, independently of any observations at the place, by means of a chronometer, carefully regulated to Greenwich time, provided it be subject to no irregularities after having been once properly adjusted. A ship furnished with such a timepiece always carries the Greenwich time with her, and the longitude then becomes reduced to the problem of finding the time at the place. EXAMPLE. Time computed by an altitude of the sun, as at p. 288, was....... 9 45" 10 Chronometer showed Greenwich time at the instant of observation to be... 0 50 20 Difference of longitude of the place of observation fronm Gr. in time,.... 3 5 10 To convert time to space multiply by.. 15 Longitude of the ship west of Greenwich,.. 460 17' 30" The same method applies to examples 2 and 3, p. 289. Still, however, as the most perfect contrivance of human art is subject to accident, and the more delicate the machine the more liable is it to disarrangement, from causes which we may not be able to control, it becomes highly desirable, in so important a matter as finding the place of a ship at sea, to be possessed of methods altogether beyond the influence of terrestrial vicissitudes, and such methods the celestial motions alone can supply. The angular motion of the moon in her orbit is more rapid than that of any other celestial body, and sufficiently great to render the portion of her path passed over in so short a time as two or three seconds, a measurable q aantity even with a small portable instrument (the sextant).f * In computing the D's semi-diameter anid parallax, second differences need not be used, the consequent error being less than Or'*1. t THE SEXTANT ls constructed upon the same principles as the quadrant. It conssts of a graduats 'TO N AUTCA-L A-TtON' OMG. IY. It is'obvious, thereli)re, that if the dist,1 snce of the miooil S centre from ~any celestial body, in or near her path, be co.mpniatel for any Greenwich trime, and this distacee be found the same as that gwien by aetual observation at any place, then the difference between, the time of observing brazen arc of 60., numbered double, however, -Ior the same reason as in the qoadrant, or to 1200, called the limb, upon which moves a vernier attached to one end of an index, the other end of which is at the centre of the are. Upon the latter, in'a direction parallel with it, and perpendicular to the plano of the limb, is a nirsor called the index glass, adjustable by three screwsid to perpendicularity with the plati of the limb. Opposite the index glass, and parallel with its plie when the index is,lt zero, is another glass, half mirror Ad half transparent, called the horizon glass. A small telescope parallel to the plane of the limb is placed before the horizon glass, and directed so as to look through the latter. There are three adjustments. 1. T2o nma.ae th.e index g/lasa perpelndicu' ar io 6ie plane of the limb.'This is done by moving forward the index to ths middle, of the limb, then looking with the naked eye into the index glass; if the part of the Jimb seen by reflection appear in the same plane with the part seen direct, the index glass is perpendicular to the plane of the limb; if not, it must be adjusted. 2. To mnake the horizon glass perpendicular to thee pla of the linmb.,- The ind'x glass having been adjusted, hold the instrument in a vertical position,;and brinf g tho direct and reflected images of the same object to coincide; if this can be do:e exactly, no adjustment is required, but if one image appear at the right or left of the other, the horizon glass must be adjusted by a screw or screws att-ahecld to it'for t!'e purpose. 3. To mak/e the axis of the telescope parallel ti the plane of ihe lizmb..-Brin the images of two objects which are more than 900 apart, to coincide upon one of the parallel wires in the telescope, and then by turnitng the instrument in the hand a little, make the objects appear on the other wire. If the coincidence remains, the position of the telescope is correct; if not, it must be adjusted by thle screws, of he ring into which the telescope is screwed. N. B. There are usually' two telescopes accompanying the sextant, the one an inverting or astronomical telescope, a;:d ihe other not. There is also a plane tube without glasses, and iter fthe three may be screwed into the same ring. There are darkening glasses to be used in observing the sun, four near the indie.x glass, and three before the object glass. They are red and green, of different EhIadeso The latter color is particularly good to take off the glare of the moon. The paler one before the horizon glass may sometimes be used with advantaCge to take off tihe glare of the horizon below the sun, occasioned by the reflection of that luiinary from the small rippling waves. N. B. The parallelism of the surfiice3 of the darkening glasses should be tested by inverting them, and observing if the coincidence of objects be preserved. When the index stands at zero the direct and reflected inages of the sanie object ojught to coincide. If not, there is index error, the amount of which is determined by observing howT far the index stands frolm zero when the direct and reflected images coincide. The best mode of determining the index error is by measuring tho diame 29S2 APPLICATION OF SPHERICAL TRIGONOMETRY ile plhenomienoni and the time at Greenwich, when it was predicted to happen, will give the longitude of the place of observation. Now, in the Nautical Almanac the distances of the moon from the sun, and from several of the fixed stars near her path, are given for every three hours of apparent Greenwich time, and for several years to come, and the Greenwich time, corresponding to any intermediate distance, is obtainable by sirmple proportion; so that by means of the Nautical Almanac we may always determine the time at Greenwich when any distance observed at sea was taken." (See Nautical Almanac, pp. XIII. to XVIII. inclusive.) The distances inserted in the Nautical Almanac are the true angular distances between the centres of the bodies, the observer being considered as at the centre of the earth, and to the true distance therefore every observed distance must be reduced; it is this reduction which constitutes the trigonomnetrical difficulties of this problem. And it consists in clearing the lunar distance from the efcts of parallax and refraction; hosw to do this, it is now our business to explain. ter of the sun by moving the index both forward and backward, the limb being graduated a short distance behind the zero for the purpose. Half this difference of the two measures will be the index error. To measure an angle with the sextant, bring the two objects, the line joining which subtends the angle, the one as seen direct, and the other by reflection, to coincide, by holding the instrument so that the plane of the limb passes through them, and moving the index forward; the reading shown by the index will be the angle required.' The proportion would run thus: As the difference between two lunar distances given in the Almanac, the one greater, the other less than that observed with the sextant, is to 3 hours, so is the difference between the distance observed with the sextant corrected for refraction, &c., and one of those in the Alm., to the difference between the time of observation and the time given by the Almanac corresponding to the latter lunar distance. The Nautical Almanac gives the proportional logarithm of the quotient of the first term of the above proportion, divided by the second term which is constant, viz., three hours. If the distance between the moon and a star increased or decreased uniformly, the Greenwich time corresponding to a given distance would be strictly correct; but an inspection of the columns of proportional iogarithms in the ephemeris, will show that this is not the case. A correction for second differences, or for the irregularity in the lunar distances, must therefore be applied. At page 602 of the Nautical Almanac for 1850 is given, besides the ordinary rule and an example under it, a full explanation of the method of employing a table contained in the Almanac for computing the correction on account of second differncges in finding the Greenwich time. The theory of second differences has been given (Algebra, Art. 235). At the scientific meetingin New Haven, August, 1850. Prof. Chauvenet of the U. S. Naval Academy presented improvements in the formulal and tables for lunars, which it is to be hoped will be perfected and published. TO NAUTICAL ASTRONOMY. 29 Let m, s, be the observed places of the moon and sun, or of the moon and a fixed star, and let iM, s, be their true places. M will be above m, because the moon is depressed by parallax more than it is elevated by refraction; but s will be below s, because the sun | is more elevated by refraction than it is depressed by parallax. Ob- -g i servation gives the apparent distapee ms, and the apparent zenith' distances zm, zs:by applying the proper corrections to these latter, we also deduce the true zenith distances ZM, zs, and with these data we are to determine the true distance, Ms, by computation. Put d for the apparent distance.* D true distance. a, a apparent altitudes.* A, A' true altitudes. Then in the triangle izs we have (Art. 82), R being I, cos D-sin A sin A' cos z == COS A COS A and in the triangle mzs cos d-sin a sin a' cos z - - cos a cos a' hence, for the determination of D, we have this equation, viz., cos D-sin A sin A' cos d sin a sin a' COS A COS A' cos a cos a' from which we immediately get COs A cos A'S I cos D = (cos d —si a sin a') + sin. sin A cos a cos a * In observing d with the sextant, it is the nearest point of the limb of the moon which is made to coincide with the other heavenly body, and in observing a with the quadrant, it is the limb also which is made to coincide with the horizon; so that d and a must be corrected for the semidiameter of the moon; similar remarks apply to the sun, if he be the other heavenly body. t Observe that A and A' are the complements of zAl and zs. ~9':)4 APPLICATION OF SPHERICAL'RIGONOMETRT.:>'t oa cos a' -- sin a sin a' -= cos (a + a') Art. 70; transposing cos a cos a', amd substituting the value of - sin a sin a' thus obtained, wet...h! ave cos c -I- cos (a + a') - cos a cos a' C D5.B.0 ~ — ~ — ~ - A'-s - - -S cos A cos - sin A sin' cos a cos a Dividing the last term of the numerator by the denominator, the quotient. is -- I; then observing, that -cos A COS A'- sin A sin A' ~~ cos (A + A') and that cos d + cos (a + a') = 2 cos - (a -j- a' + cl) cos - (a- -a' a d) Art. 83, we have 2 cos i (a_ + a'+ d) cos 4- (a -t a''d) cos A cos A. CC i — COSO(A+A')(i) cos a COS a EXAMPLE. i. Suppose the apparent distance between the centres of the sun and moon to be 830 57' 33", the apparent altitude of the moon's centre 270 34' 5', the apparent altitude of the sun's centre 48~ 27' 32", the true aititude of the moon's centre 28~ 20' 48". and the true altitude of tlhe sitn's cent;re 480 26' 49" then we have d 830 57' 33", a - 2 7 34' 5", a' = 480 27' 32 A 2 280 20' 48", A' =- 48 26" 49'; and the colmputation for D, by formula (1), is as follows: (d 83~ 57' 33" a 27 34 5 ar. comp. cos 052339 a' 48 27 32 ar. comp. cos 178383 2)159 59 10 log. 2 301030 sum 79 59C 35 cos 9*239969 - Suim 57 d 3 57 58 cos 09998959 A 28 20 48 cos 9'944527 A' 48 26 49 cos 9'821719 (Reject 40 from index) 1*536926 g log. 344292 - A + A' 76 47 37 nat. cos'228460 True distance 830 20' 54" nat. cos *115832 By glancing at the formula (1), we see that 30 must oe rejected from the sum of the aboie column of logarithms, to wit, 20 for the two ar. TO NAUTICAL ASTRONOMY. 295 comp'. and 10 for l, which must be introduced into the denominator, in order to render the expression homogeneous, so that the logarithmic line resulting from the process is 90536926. Now, as in the table of log. sines, log. cosines, &c., the radius is -supposed to be 10"~, of which the loTg. is 10, and in the table of natural sines, cosines, &c., the rad. is 1, of which the log. is 0; it follows that when we wish to find, by help of a table of the logarithms of numbers, the natural trigonometrical line corresponding to any logarithmic one, we must diminish this latter by 10, and enter the t;able with the remainder. Hence the sum of the foregoing column of logarithms must be diminished by 40, and the remainder will be truly the logarithmn of the natural number represented by the first ternt in the se, ond number of the equation (1). If this natural number be less than nat. cos (A + A'), which is to be subtracted from it, the remainder will be negative, in which case D will be obtuse. VARIATION OF THE COMPASS. 114. We shall conclude this part of our subject by briefly considering the methods of finding the variation of the compass, or the quantity by which the north point, as shown by the compass, varies easterly or westerly from the north point of the horizon. The solution of this problem merely requires that we find by computation, or by some nmeans independent of the compass, the bearing of a celestial olject, that we observe the bearing by the compass, and then take the difference of the two. The problem resolves itself, therefore, into two cases, the object whose bearing is sought being either in the horizon or above it: in the one case we have to compute its amilitude, and in the other its azimuth. The computation of the amplitude is simply determining the hypothe- nuse of a right angled triangle MISN, of / which one side is given, viz. the declination -xs of the object, as also the angle opposite to it, viz. the i colatitude M. The computation of the azimuth requires the solution of an oblique spherical triangle, the three sides being given to find an angle; the three given sides are the colatitude PZ, the zenith distance of the object zs,.and its polar 296 APPLICATION OF SPHERICAL TRIGONOMETRY distance ias, and the azimuth being measured by the angle at the zenith z, Z opposite the polar distance, this is the angle sought. We shall give an exampie in each of these eases of the p>roblem. EXAMPLES. i. In January, 1830, at latitude 270 36' N., the rising amplitude of Aldebaran was by the compass* E. 230 30' N.; required the variation. From the Nautical Almanac, it appears that the declination of Aldebaran at the given time was 16~ 9' 37" N., therefore since, by Napier's rule, Rad. X sin dec. = sin. amp. X cos lat., the computation is as follows sin declination 160 9' 37" 9*44455 cos. latitude 27 36 9'94753 sin amplitude E. 188 1 17 N. 9*49702 Magnetic amplitude E. 23 30 0 N. Variation 5 11 43 all As the object is farther from the magnetic east than from the true east the magnetic -east has therefore advanced towards the south, and therefore the magnetic north towards the east; hence the variation is 50 11 43 " E. 2. In latitude 48~ 50' north, the true altitude of the sun's centre was 220 2', the declination at the time was 100 12' S., and its magnetic bearing 1610 32' East. Required the variation. * The compass amplitude must be taken when the apparent altitude of the object is equal to the depression of the horizon. TO NAUTICAL ASTRONOMY. 297 O's polar distance 1000 12' sill zenith distance 67 58 ar. comp. 0*03294 sin colatitude 41 10 ar. comp. 0*18161 2)209 20 is. 104 40 sin ( s. -zen. dist.) 36 42.. 9'77643 sin ( s. - colat.) 63 30... 995179 2)19.94277 sin i azirn. 690 25' 40".. 9'97138 2 9~s true azimuth N. 138 51 20 E. observed azimuth N. 161 32 0 E. Variation 22 40 40 West. The variation is west, because the sun's observed distance from the north, measured easterly, being greater than its true distance, intimates that the north point of the compass has approached towards the west. 3. In latitude 480 20' north, the star Rigel was observed to set 90 50' to the northward of the west point of the compass; required the vtariation, the declination of Rigel being 8~ 25' S. Variation, 220 33' West. APPENDIX TO PART V. A VERY portable and accurate instrument for the measurement of altitudes of the'heavenly bodies is THE CIRCLE OF REFLECTION.* This instrument is constructed on the same principles as the sextant, the only difference being that the circle, as its name imports, has a limb which is a complete circumfrerence, measuring on the doubling principle of optics already explained 7200 instead of 3600. By having two- verniers 1800 apart, the instrument corrects its own eccentricity and by having three, as in Troughton's construction, for error of figure and division, to some extent also. By reversing the face of the instrument the angle mi ay be takern on what is called the off arc, that is, on the other side of the zero. This, by taking the mean, corrects fbr index error, so that with three verniers six readings may be obtained for one angle. Doalland's circle has an inner movable circle, with a vernier upon it, in consequence oi' which it admits of the repeating process explained for the theodolite, at p. 242. The horizon glass and telescope are attached firmly to the inner movable circle, which has a clamp screw and screw of slow motion. The index glass is attached to an arm which moves freely around the centre, and is unconnected with the inner circle, telescope and horizon glass. This arm has a vernier and screw for clamping it to the outer circle, and. a screw of slow motion. The repeating process is conducted as follows: Place the zero of the vernier of the inner circle clamped accurately at 720( on the outer, and move the free index carrying the other vernier forward until the two objects are brought in contact, as in observing with the sextant. Leaving this index fast to the limb, unclamp the inner circle, which carries the telescope and horizon glass, and move it forward also, not nI;-ely by the same amount, which would bring back the horizon glass to parallelism again with the index glass, but move it twice the angular distance necessary for this purpose; the horizon glass will now be inclined to the index glass, just as much as when the free index was first moved forward to produce the contact of the objects, btl the inclination will be the other way. The contact may again be produced;twseen the objects by the tangent screw of the inner circle, the telescope being, Angles in any plane may be observed with the Circle of Reflection as with the' Sextant. 300 ArPEN,-DIX V. presented to the other object if the face of the circle be continued one way, but io the same object if the face of the circle be reversed. The reading now of the vernier attached to the inner circle would be twice the angle between the objects. If now the free index be moved forward a distance equal to the angle between the objects, the horizon and index glasses will be parallel again; and if it be moved forward still further by the same amount, the glasses will be inclined to each other, exactly as at first, and the contact of the objects may be made with the screw of slow motion attached to the free index, the face of the circle being as at first. Again, the inner circle is to be moved forward as before, over twice the angle between the objects, the face of the circle being inverted, the telescope directed always to one object, and the contact made with the screw of slow motion attached to the inner circle. The process above described is to be repeated until the vernier of the inner circle approaches near, or is a little past the 720 point again. The reading at which it stands, divided by twice the number of times that the inner circle has been moved forward, will give the angle subtended by the two objects corrected for all errors of division, centering and observation. If the angle between the objects be changing, as is often the case with celestial objects, the times of each contact of the object should be noted. The sum of the times, divided by the number of contacts, will be the mean time corresponding to the mean angle obtained, as already described. The Sextant and Reflecting Circle are used for taking altitudes on land, by the aid of a basin of mercury, called an artificial horizon. The telescope is presented to the reflected image of the sun or other heavenly body, seen in the mercury, and the angle between this and the sun in the heavens measured by moving the index forward. This angle will be double the altitude of the sun. The following example of an observation of the altitude of the sun for time will illustrate the mode of using this instrument. Observation with Repeating and Reflecting Circle, and Box Chronometer, of the sun's lower limb. 2a 29m 33, " 30 00-5 " 30 19 " 30 43-5 31 06.5 Timesofcontacts. " 31 34*5 1170 36' 10" Vernier reading. " 31 59*5 7200 " 32 15 " 32 36 10)837 36 10 32 36 " 32 55*5 2)83 45 37 10)313 03 41 52 48*5 2^ 31, 18-.3 The mean of the angles being 410 52' 48".5, and of the times, 2^ 31-' 18,.3, the former is the apparent altitude at the instant expressed by the latter, with no error of excentricity or graduation, and a probable error of observation I that which would have been obtained without the repeating process. LATITUDE. BY A SINGLE ALTITUDE. 301 VARIOUS MODES OF DETERMINING LATITUDE. LATITUDE BY A SINGLE ALTITUDE. The data for this problem are the declination, the altitude, and the hour angle. We have thus three elements of the triangle sPz (p. 296), viz., sr, sz, and the angle r, known, to calculate a fourth. This method requires an accurate knowledge of the time. If the latitude be known nearly, which it generally is, by means of the dead reckoning, it may be accurately found as follows. From (7) of Art. 72 we have cos A=- 2 sin2 A (1) which substituted in the formula deduced at Art. 82, written thus cos a =cos b cos c + sin b sin c cos A (2) gives by (8) of Art. 70, K:- cos a - cos (b- c)- 2 sin b sin c sin'" A (3) substituting in (3) the sides and angles of zrs, calling the polar dist. ir, the zenith dist. $, and the colatitude A, that equation becomes cos ~ =os s (r -X) -2 sin r sin X sinz 2 p (4) But if (' denote the meridian zenith distance of the heavenly body I' =: A: (7 — x) Hence (4) becomes cos = cos 5' -2 sin 7r sin A sin2 f or, by transposition, cos i' cos -+ 2 sin 7r sin X sin2 ~ P (6) the formula for use in which id is meridian zenith dist. g is observed zenith dist. Tr is polar distance of the object observed. A is the approximate colatitude. P is the hour angle. The use of formula (6) is facilitated by Table XXIII., commencing p. 154 of Bowditch's Navigator, entitled log. rising, in which log. 2 sin2 I r is calculated for every value of the hour angle r. To this it is only necessary to add the logarithms of the sines of the polar distance and approximate colatitude, and we have the difference between the observed and meridian altitudes, or the correction to be applied to the former to obtain the latter, from which the latitude is calculated as at p. 280. A formula may be derived from either of the forms (B) of Art. 86. Applied after clearing of fractions to the present triangle it becomes sin2 ~ P sin,r sin sin ( X - -7) sin ( + 7- X) or from (5) above sin n r sin = sin -sin ( --') sin ( — sin (e) sin r sin r sin X cosec - (-+ i) (7) 302, APPENDIX V. which will serve to determine - -', the correction for the zenith distance, if in thi second member the value of (' be used, which would be given by the approximate value of the latitude. CIRCUM-MERIDIAN ALTITUDES. If the heavenly body be near the meridian, r and —' will be very small, and writing the small arcs in place of their sines, to which they are sensibly equal (7) becomes (considering. (' in the second member) ( ( 4 ) i=, sin r sin X cosec' (8) in which the value of the first member is expressed in terms of the radius as unity. To express it in seconds of arc, its value, as well as that of r must be divided by the sine of 1", which may be regarded as the length of 1", in terms of radius as unity. (8) thus becomes, striking out at the same time the common factor _... sino 2 sin Ii' pa ~ S == sin- lI sin Xr sin X cosec I (9) in which - (' is expressed in seconds of arc, as is also r the hour angle. If' denote the hour angle in time, 15p must be substituted for:, and (9 becomes 225p2 - ~ 2 sin 1I Sin 7r sin A cosec (10) The value of -' for the same station and star being proportional to p%, if it were calculated for a value of p = 11, it might be found for any other value of p, by multiplying this by pg in minutes of time. Table XXXII. of Bowditch's Navigator contains values of - (' for p = 1"L for all latitudes and all declinations less than 24~, so that entering this table with the declination of the star and proximate latitude of the station as arguments, the number taken from the table multiplied by the square of the hour angle, in minutes and decimals of time, which is given in Tab. XXXIII. of Bowditch, for every value of p up to 13n, will produce the correction to be applied to the observed altitude, to obtain the meridian altitude. ~-' may be calculated more accurately by means of a table (Tab. XXXV.>, p2 adapted to formula (9) above, in which - is equal to the versed sine of P, as may be seen by referring to (2) on p. 94, from which, using only two terms on account of the smallness of P, we have p2 2 cos P - -2.. vers p l — cos Table XXXVI. gives vers P and the arithmetical complement of the logarithm of the sin 1" is 903144. N. B, The correction' - - is subtractive from the observed zen. dist. to obtain' the merid. zen. dist. or it is additive to the observed altitude to obtain meridian altitude. If several altitudes or zenith distances were observed near the meridian, and each reduced to the meridian by the above formula, the mean of the latitudes thence CIRCUM-.11 -ItDItA N ATI'ITUDIS, 303 derived would be the true latituide more nearly in proportion to the number of observations. But the mean of the values for' --', obtained from (10), since for each the qua,;tities entering into (10) are the same except p2, may be obtained by taking the mean of the values of p)', and multiplying this by the constant factors. And if the mean of the values of -' be subtracted from the mean of the values of 3, the same latitude will result as if the latitude were calculated for ea.ch observed zenith ditance, and tIhe mean of all the latitudes taken, but with munch less computation. EXASMP IE. Observed circum-meridian altitudes of O's lower limb, at head of Upper Mistigougiche Lake, latitude about 480, July 24th, 1841, with Repeating and Reflecting Circle, Artificial Horizon, and Box Chronometer. SUN'S CE.ITTRES ON THE MERIDIAN BY CHRONOME'TE'I AT 111. 1.2)1 48'1.*, BAROMETEI'i 3289, THERnIOMIETER 69. Times of obs. Hour Angles, p. p" Tu;b. XXXrII. Tab. XXXVI. 11! 09 55*5 2'n 52-.6'r.3 78 1031 2 17 71 5 2 50 1047 2 1 1 4 1 39 1109 1 39 1 2 7 26 11 30 1 18*1 1 7 16 11 51*5 0 56'6 0*9 8 12 13.5 0 34'6 0 3 3 12 36'5 0 11*6 0 0 0 1253 0 4 9 0 0 0 1311'5 0 23 4 0*1 1 1324 0 35 9 0 4 3 13 44 0 5 559 0 9 8 12)24'6 12)232 Vernier Reading 190 26' 20" mean 2'*)5 19 - (' forp = 1" in lat. 480, dec. 190, Table XXXII. 2-/*; (-' for mean of hour angles = product 5' 3 The times in the first column above are th; irn:tants of contact of the images as observed with the reflecting circle, according to tlhe method described at p. 300. The numbers in the 2d column are obtained by subtracting those of the first from 11I, 129' 4S8'1, the time at which the sun is on the mneridian. The numbers in the 3d column are the squares of those in the 2d, expressed in minutes and tenths. The mean of the values of p2 is 2it.05, which, multiplied by 2'*5f, the value of -' lat. 480 and dec. 190, gives 5'*3, the correction required, by which the mean of the observed altitudes is to be increased, to produce the meridian altitude. The fourth column is for another mode of computation, the numbers in it being * This of course depends on the equation i)f time, and the error of ttie hronometer. 304 APPENDIX V. taken from Tab. XXXVI. The computation of formula (9) by the aid of this column is as follows: Mean of vers. 24 log. 1.2787 Latitude, 480 8' log. cos 9.8243 Declination, 19 10 log. cos 9'9752 Zenith Dist., 29 12 log. cosec 0*3116 Constant 1" ar. co. log. sin 903144 Correction 5- 5"1106 log. 0*7042 The reading of the vernier at the end of the observations 190 26' 20"t was after two complete revolutions of the circle; therefore the whole angle passed over is obtained by adding this to 14400 = twice 7200, and the double altitude is obtained by dividing this by twice the number of times that the inner circle has been moved forward, or by the whole number of contacts of images, in this case 12. 190 26' 20" 1440 12)1459 26 20 Double altitude, 121 37 117 Altitude, 60 48 35X85 Reduction to Meridian, 5*06 Semidiameter, 15 46'4 61 4 27-31 Refraction-Parallax, - 25*40 61 4 1.91 Declination, 19 10 47*3 Colatitude, 41 53 14*61 Latitude, 48 6 45*39 The following method of determining the latitude by means of three altitude taken near the meridian, has been given by Mr. Littrow, of Vienna. Let A be the first observed altitude corrected for dip, refraction, parallax, and semidiameter, and T the time. A + a the second observed altitude, corrected also, and T + t the corresponding time. A + a' the third altitude, and T + t' the time. Let At be the meridian altitude, and T - T' the corresponding time. Then tha variations of the altitude near the meridian being as the squares of the variations of the hour angle nearly, we have Ag o A: A -(: T2: (T f)a'. =T~=- (A' - A) (2Ttt 1 t) (1) Similarly a'T'2 (A - A) (2T't - t~t) a 2 Tt -- t at2 ~ a'tt By division a 2 T and T' (2) e^a' 2 T't' - 2 a t' - a rt aT12 And from (1) A' A 2 Tt -t (3) * For a table which gives the value of the whole fraction in (9) and (10), 8 Lee's Tables and Formula, p. 64, Part IIL TO FIND THE LATITUDE BY ALTITUDE OF THE POLE STAR. 305 From (2, imay 06 obtained the value of Tt, which added to T gives T + T', the time when the object observed was on the meridian, and from (3) may be obtained the value of A' —A, and by adding A, the value of A' the meridian altitude becomes known, from which the latitude is to be computed as at Art. 111. Determination of Latitude by an Altitude of the Pole Star out oJ the Meridian. Let r be the pole, z the zenith, and consequently rz the meridian; A the place of the pole star at the time that its altitude a is P observed. Denoting its polar distance PA by or, drawing the arc Ao perpendicular to rz, expressing Po by y, and the hour angle by p, in the right angled triangle PAO, which may be regarded as plane from being so small, we have / AO = T? sin p In the right angled triangle ZAO, ZA and zo are nearly equal; Z denoting their difference by x, we have zo =90 - a - x, and by Napier's rules, C0o ZA - COS z Cos AO or, sin a sin xa + a) cos (7r sin p) (1) But sin ( x a) =sin x cos a +e cos sin a ssin a +x cos a (2) since cos x = 1, and sin x = x because x is so small. Also [p. 94 (2)] cos (Ir sin p) = 1- 7rs sin2 p (3) Uniting (2) and (3), (1) becomes sin a (sin a + x cos a) (1 - 7r2 sinp) Performing the multiplication indicated in the second member, and omitting the term ~ xr2 cos a sin p on account of the smallness of the quantities x, 7r, and p we deduce xw -r rtan a sins p But in the right angled triangle PAO tan y tan ir cos p And (p. 94, note), tan r= 7f - 7r3 Also, (p. 95, note), y =tan y -X tans y (.'. ~y (T - i r3) cos p -- r3 c03 p, or y = f cos p + i A3 sin' p cos pd omitting the powers of n higher than 03. * Since sin2 p 1 - cos' p. 20 3Q0 APPENDIX V. Denoting the latitude required by X, it is evident (see second note, p. 283), that X\ - a x- y.'. - a - Cos p +- C2 tan a sin2 p - (r cos p) (7r sin p)2 (4) To have X in seconds of arc we must substitute for X, X sin 1", and for r, f sin 1". Thus (4) becomes, after dividing by sin 1" X a- cos p — + sin l' 7re tan a sin2 p- ~ sin2 1" ( op) (r sin p)2 (5) log. A sin 1" -= 6'3845449 log. - sin2 1" = 12*8940285 Formula (5) is that upon which the rule p. 619 of the Naut. Aim. of 1850 is founded. The argument for the first correction - r cos p, in the formula, of the observed altitude, necessary to be applied to obtain the altitude of the pole equal to the latitude of the place, is the siderial time, upon which p or the hour angle evidently depends, the declination being regarded as constant. There will be two arguments of the second correction ~ 7r2 tan a sin2 p - (r cos p) (7r sin p)2 to wit, the siderial time and the altitude. The third correction is the change in the value of the first, -7r cos p, which alone is of sufficient magnitude to be sensibly affected by the change in the right ascension and declination of the pole star. Denoting the changes by da and dZ, the change in -- cos p occasioned by the latter will evidently be dd cos p and that occasioned by the former da will be - r (cos (p - da) - cos p) — sin da sin p - da sin 1i sin p as will be seen by developing cos (p - da), and making cos da = 1 on account of the smallness of da. Or the above expression may be written 5380 X 0*000073 da sin p = 0'4 da sin p The third correction is sometimes positive, sometimes negative, but always less than 1', and therefore 1' + the third correction, which is the quantity given in the table in the Naut. Aim., is always positive, and hence the necessity of subtracting 1' according to the rule given in that work.* The instrument employed for measuring altitudes in a fixed observatory is THE MURAL CIRCLE, so called because it is sustained against a wall about four feet thick, and wide and * The above method of finding the latitude is so convenient as to be worthy of particular attention. It is not confined to any precise hour, as is the case with meridian altitudes, but may be employed during the whole night, whenever the pole star is visible. An officer of the British Navy has, by means of it and a quadrant with a spirit level attached, the horizon being invisible, been able to run his frigate boldly up the English Channel, at the rate of eleven knots an hour, in a dark squally night, which allowed only occasional glimpses of the pole star between flying clouds. Captain Porter, U.S.N., is in the habit of running his steamer along the Florida Reefs by the same means. THE MURAL CIRCLE. 307 iigh enough to allow ample space for the circle, which is usually five or six -eet in diameter. The graduation is not where it usually is, but upon the outer rim, the surface of which is normal to the plane of the circle. The degrees ruin from zero to 360. Exterior to the circle, and fastened against the wall, are six microscopes, pointing towards the centre of the circle for reading the graduated rim. They are placed at equal distances, viz. i60 apart. The circle turns round past these microscopes, hlich are stationary, upon a very;strong axi;, fastened solidly to one side of the circle at the centre, where the axis is about six inches thick. This axis passes entirely through the stone wall, tapering to aboul three inches or less at the other side of the wall. Against this small end of the axis, screws work both laterally and vertically, by means of which the circle is adjusted to the plane of the meridian. This adjustment is effected by the aid of what is called a ghost apparatus. Before describing this it will be necessary to mention that the telescope of the instrument is attached firmly to the outer face of the circle, in the direction of the diameter, to which it is equal in length, and turns with the circle on its axis. The ghost apparatus consists of two small microscopes, Iplaced one at each end of the telescope, on the outside of its tube, and pointing horizontally and parallel to the plane of the circle, when the telescope is vertical. From a bracket at the top of the wall, moving outward and inward by a screw, is suspended a plumb line, which passes down directly before the object glasses of the microscopes, between them and a small mother of pearl disc, on which is a black spot, called a ghost, which is brought exactly on the line of vision of the microscopes, so as to be covered by the plumb line. This being done, if now the plumb line be moved out by means of the screw in the bracket, so as to clear everything, and the telescope be reversed by turning the circle 1800 on its axis, then, if the plumb line, when brought back to cover one of the ghosts covers also the black spot in the other, the axis of the circle is horizontal If not, half the correction must be made by the adjusting screws of the axis at the rear of the wall, and the other half by moving one of the ghosts. This latter movement is effected by turning on its axis the tub( containing the mother of pearl disc, which tube is in the prolongation of the tubu of the mniQroscope, the connexion between them being broken, to permit the plumb line to pass in. The black spot of the ghost is at one side of the centre of the, mother of pearl disc, so that as the tube containing the disc turns, the black spot is carried fiom side to side, through a small space, sufficient to bring it into coincidence with the plumb line, the latter remaining stationary, and covering the black spot of the other ghost. The process above described must be repeated till the plumb line coincides with the black spot of the ghosts in both positions of the telescope. The rationale of this adjustment may be made evident by the following diagram. a p Suppose the plane of the diagram to be a vertical plane through the centre of the mural c-rcle. and perpendicular to its plane; ab its section with the plane of the circle, cd perpendicular to ab, the axis on which the circle turns, ae the first position of the plumb line coin- 3068 APPENDIX V, ciding with the black spots on the ghosts a and e. When the telescope is reversed by turning the circle, on its axis cd, the ghost e will take the position f, and the ghost a will come to by and the plumb line suspended from f will be in the direction fg, The lower ghost b will thus be the whole distance bg away from the plumb line fg. Half the correction necessary to bring it into coincidence with the plumb line, must therefore be made by lowering the end c of the axis, which is firmly fixed to the circle ab, till the point b comes to e,* which will render the axis cd horizontal, and the other half by moving b, now at e, to g, or the ghost f, together with the plumb line, to a. The axis cd is thus made horizontal, and the circle ab vertical. It being difficult in practice to lower the point c just enough to make half the colrection which would render cd horizontal, the process has to be repeated. Each trial will bring the axis nearer to the horizontal position required, so long as only part of the correction necessary to bring the ghosts into alignment with the plumb line is made by the motion of the point c of the axis cd, by means of the screws which work against this point. The axis being thus rendered horizontal, the instrument may be collimated by the method described at p. 161, and may be adjusted to the meridian, or its deviation from the meridi.n determined by the method of high and low stars, given at p. 157. To measure the polar distance of a star with this instrument, the reading must be first taken as indicated by an index or pointer, fastened to the wall a little outside of the circle, when the telescope points to the pole of the heavens, and afterwards taken when the telescope points to the star, the circle having been moved on its axis past the index, which remains stationary. The difference between these two readings will be the polar distance of the star. The first of the above mentioned readings is called the polar zero of the instrument. It may be found by taking the readings with the telescope pointing to a circumpolar star, at both its superior and inferior transits.,The middle point between these readings or half their sum will be the polar zero. In taking the observation the horizontal wire in the axis of the telescope is made to thread the star with great exactitude. This wire does not require collimating, but the vertical wires do, for the reason stated at p. 160. Each degree of the graduated limb is divided into halves and quarters, or 15'; these again into 3 parts, or 5' spaces. The reading of the degree is taken by the index or pointer above mentioned, the minutes and seconds with one of the microscopes, and the seconds with all six microscopes, the mean of the seconds given by the whole six being the number of seconds adopted. The construction and mode of using these microscopes must now be described. The wires of one of these microscopes, which are of the finest spider's web, cross each other at a very acute angle, with a horizontal line, and are placed in a frame in the tube of the microscope at the focus of its object glass. This frame is movable up and down, by means of a screw placed under the tube of the microscope, and working into it. A single turn of the screw moves the point of intersection of the wires over of one of the 5' spaces, and the screw head is made large, and its circumference divided into 60 parts, so that the * Or rather till e and b come together, the motion being about the point d, a moing half the distance be to the left, and b half the same distance to the right. THE MURAL CIRCLE. 0 number of turns and fractions of a turn of the screw necessary to bring the wires to one of the dividing lines of the 5' spaces will show how many minuter and seconds the intersection of the wires stood past this division. To save the trouble of counting the number of turns of the screw, a notched scale is placed vertically a little on one side, at the focus of the microscope, so as to be magnified by its eye-glass. A pin, which points horizontally past the scale, towards the intersection of the wires of the microscope, moves over one notch at each turn of the screw. The zero of the notched scale at which the pin ought to stand at the commencement of the observation, is marked by a hole at the middle of the scale. To recapitulate, the reading of the degrees, minutes, and seconds will be taken as follows: the degrees by the index or pointer, then looking into the microscope, the degree nearest which the pointer stands will be recognised by a round indentation in the metal, near a long dividing line; the half degree is marked by a dividing line of the same length, without any indentation; the quarter degree by a shorter line, rlnd the 5' spaces, of which there are three in each quarter degree, by shorter lines still. If the intersection of the wires stand exactly upon one oft these dividing lines, the reading will be so many degrees, halves, quarters, and five minutes; but if the intersection of the wires of the microscope be a little past one of the dividing lines on the limb, the intersection must be brought back to the dividing line by turning the screw which moves the fiame containing the wires. The number of turns of the screw necessary for this purpose, indicated by the number of notches passed over on the scale by the pin within, will be so many additional minutes to be added to the reading as it now stands, and the seconds will be obtained from the screw head by noting at what numbers upon it its index stands. The mural being used in the same observatory ordinarily with a transit instrument, by observing the time of transit of the star over the wires of the mural, and comparing the result with the transit observation corrected as explained under the head of that instrument, the hour angle of the star when it crosses the imaginary middle wire will be known; or it can be computed from a high and lowstar observed with the mural itself, as explained for the transit instrument. From this hour angle the error in the observed altitude of the star arising from the deviation of the instrument firom the meridian, may be obtained as follows: REDUCTION TO MERIDIAN OF AN OBSERVATON T MADE WITH THE MURAL CIRCLE. Let the full circle be the meridian of the p station, s'ss" the diurnal path described by the star under observation, ms the arc of a f great circle of which the horizontal wire is a /\ portion, s the place of the star when observed. Then in the triangle mSP, right angled at im, (since the meridian is a vertical circle) sin rmt cos s cos r = tan pm I cot. - s == i-.- cOS P0n Sin PS q IO APPENDIX VT siln Pm * co' rs Icos r =I —c - --- cosPm * sin rs cos im Sin PS - sin pm co Ps ssin (es-?m) cos Pm sin is cos i'm sin Is But 1- cos = 2 sin2(l r), see (8) Art. 72.' Since rs = P's, PSs = Pm very nearly, whence Ps - rm = s'm, and the ho-r angle srs' is generally very small. s'm sin 1". sin (1, a) -2 (sin 11)2 f P2 si n2whence sin = Reduction = -4 sin 2's, p' sin 1" which is expressed in arc. To express it in time make p = 15 p, whence 225. Reduction =- sin 2 p i' s sin 1-; its log=6435694+log sin 2 pspi in which p is the hour angle in seconds of time, and Ps the polar distance of the star. The correction or reduction does not become of any sensible magnitude, till the hour angle exceeds 5 seconds. The correction or reduction is always to be added to the apparent polar distance, if the star has a N.P. distance less than 90c, but subtracted if greater than 900. This may be seen from the diagram, and still better on a globe; the great circle of which the horizontal wire is a portion, intersecting the meridian N. of the diurnal circle of the star, when its N.P.D. is less tlan 900, and s. when it is greater. Instead of the polar zero, a horizontal zero has been more commonly used. It is obtained by taking the reading with the horizontal wire bisecting the star, as sean direct on one night, and as seen by reflexion from mercury on the following night; half the difference of the two readings will be the meridian altitude of the star, from which and the latitude the polar dist. or declination may be obtained as explained at p. 130. Within a few years past a third zero, which may be called the nadir zero, has * See (7) Art. 70. l.;: - t The sine of a very small are differs insensibly in length from the are itself, and its value may be found by multiplying the sin of 1", taken from the tables, by the number of seconds in the arc. This serves to express the length of the arc in the same terms as the radius, sine, cosine, &c., are expressed, so that these quantities can enter the same formula. Sin 1'0=0'000004848137; log. sin 1"t=- 16855749. J See form. (1) Art.'1.': " ~ ~ Or both, the direct and reflected observation may be made on the same night, lhe star for one or both being a little off the meridian, and such observation reduced to the meridian by the formula above. A movable vertical micrometric wire serves to measure the distance of the star from the meridian. This may be still better done by observing the time of transit over vertical wires, and comparing this with the true time at mieridian transit. THE MURAL CIRCLE. 311 been employed. This is obtained by means of a collimating eye-piece and basin of mercury, placed under the object glass of the telescope, turned vertically downward, as described at p. 161, note. When the direct image of the horizontal wire coincides with that seen by reflexion, the reading will be the nadir zero, and the difference between this and the reading when the telescope points to a star crossing the meridian, will be the distance of the star from the nadir, which is the supplement of its zenith distance, from which the polar dist. or dec. may be obtained, as already explained. The circle and transit instrument are combined in one, in the German observatories. The instrument which results is called a transit circle or meridian circle. Both observations, to wit, that for the right ascension and that for the declination of the object, are taken simultaneously. TRIAL MTETHOD OF DETERMINING BOTH LATITUDE AND TIME. Supposing neither the chronometer error nor the latitude to be exactly known, but the altitudes of two stars A and B, differing 900 in azimuth be observed, and assume successively two trial latitudes, computing the chronometer error by each latitude, and by the altitude of each of the stars. Suppose the first assumed latitude for example to be 450 15', and the resulting chronometer error by star A to be 46'2, and by star B, 50"82; the discordance is 4s. Suppose the second assumed latitude to be 450 25', and the resulting chronometer errorby A to be 53s4, by B 37s-4; the discord now is 16, the opposite way. Then 10' of latitude has changed the discord 20s; therefore a change of 2' of latitude will remove the first discord of 4, and the latitude is therefore 450 17'; and since 10' altered the error by A 7'T2, and by B 12"~8, 2' will alter the former by 1 -44., and the latter by 2"56, and bota agree in giving the error 47d'640 PART VIt GEODES Y PART Vl. GEOD ESY. GEODESY is a higher kind of surveying, which takes into account the curvature of the earth's surface. It has for its object to determine, with the utmost possible accuracy, the geographical positions of points on the earth's surface by the process of triangulation already repeatedly described ill this work, but requiring for the present purpose certain modifications, which it will be the object of the following pages to unfold. The triangles composing the chain best fulfil their destination when the iargest possible, and nearly equilateral. The sictes are ordinarily from ten to fifty miles, and limited only by the want of distinct vision with the instruments, or interruptions from the nature of the ground. The primary chain being finished, a secondary chain of smaller triangles, having their vertices within the larger, is surveyed, and a still smaller chain of tertiary triangles. A great number of small bases and points of reference are thus determined, from which the surveys with the plane table or compass irlay originate, to complete the map in all its details. One of the chief problems, after the triangulation is finished, is the determination of the difference of latitude and longitude between the vertices of the triangles, so that when the absolute latitude and longitude of some of the vertices shall have been determined by astronomical observations, that of all the others may be known by diferentiation. Previous to the explanation of the method employed for this purpose, it will be necessary to give some account of the mode of conducting the triangulation and, first, of the MEASUREMENT OF BASES. The measured base, from which the triangulation commences, should be selected upon ground which will admit of its extending several miles. Its length should be ascertained with the greatest care, and for this purpose it has been customary to use metallic rods, allowing for their expansion and contraction from changes of temperature, the amount of which 316 GEODESY. had been previously determined by experiment, with a thermometer which was always observed when making the measurements.* For greater accuracy in placing the ends of the rods together, an optical contact has been employed. This is produced by placing horizontally upon tressels the rods, one a little above, and its end projecting over that of the next, a notch being cut out of the ends of the rods, across wlich a thread is horizontally stretched; a microscope supported on a stand, from which an arm holding the microscope projects over the ends of the rods so that the microscope can be placed vertically over the threads, looking down upon them; the rods are then moved by screws in the tressels, till the threads are seen to coincide in the microscope, or till each coincides with a mark on the stand below. The optical contact is then complete. The best base apparatus now in existence, probably, is that employed upon the coast survey of the United States, and of this we shall attempt a more particular description. It is made self-adjusting under changes of temperature. The measuring rods consist each of two bars, the one above the other; the lower of brass and the upper of iron, which is less expansible. The two bars are connected firmly by a cross-piece at one end, but allowed each to expand freely, being unconnected at the other, at which, however, there is a short lever of the second order placed vertically, having its fulcrum or extremity at the end of the lower bar, on which it works as a hinge, and its point of resistance at the end of the upperbar, the lever continuing above this, and having the power applied horizontally to its upper end, through a steel rod, which projects horizontally from the next measuring rod, above which it is sustained by a short support. The contact between this steel rod and the upper end of the lever is that of a blunt knife edge against a plane of agate. The other end of the steel rod works freely against the lower end of a very short vertical lever of the first order, the upper end of which supports a spirit level. The end of the spirit level farthest from the steel rod has a tendency to fall, occasioned by a counterpoise weight, which projects from it, but this is prevented by pressure of the steel rod against the lower end of the short lever, on the top of which the spirit level rests. This delicate mode of contact, first suggested by Bessel, depends on the sensibility of the spirit level.t The tressels on which the rods rest, at,two * For each degree of the centigrade thermometer, platina expands 0.000008565 of its dimensions in every direction, iron 0*000010666, and brass 00000017843. For o1 Fah. the expansion is for brass, 0.00001050903, iron 0-000006963535. t The effect of this arrangement is evident; the whole rod, composed of the two bars, lengthens by the effect of heat, but the lower more than the upper bar, so that MEASUREMENT OF BASES. 317 points only, are strongly trussed, and their upper parts at least metallic, and admit, by means of screws, of the various motions required in placing the apparatus. In measuring the base line, the measuring rods need not be placed exactly in a horizontal position, but the measure they give can easily be reduced to a horizontal one, if only the inclination of the rod to the horizon be known*. The contrivance to indicate this consists of a sector, the plane of which is vertical, attached to the rods and graduated; a brass radius of this sector, moving about its centre, and supporting a spirit level, stands at the zero of the sector when the rod and spirit level are both horizontal. When the rod is inclined, the number of degrees passed over on the sector, in making the spirit level horizontal, shows the inclination of the rod. the upper end of the vertical lever, which makes the contact, is thrown back. The ratio of the length of the arms of the lever will depend on the ratio of expansibility of brass and iron. * The formula would be evidently B3 I cos a B being the horizontal measure required, I the oblique measure given by the apparatus, and a its inclination shown by the sector. But a being very small, it is better to compute the correction I —B to be subtracted from I to obtain B. It is (1 - cos a) =-2 sin2 2 a = I a2 sin2 1'. or, I - B - 000000004231 a2 1 or, log. (I -B) =2'626422 + 2 log. a - log. 1. a being very small. In practice it is customary to tabulate this formula. The base must next be reduced to the level of the neighboring sea. Let p be the radius of the earth (or better the normal) for the level of the sea; p + h the radius for the level of the base, h being the mean height of the ground on which the base is measured above the level of the sea, and b the reduced base. Then, since similar arcs are as their radii, we have Bh p - h Bh B 1.- = - = - P h )- l ~' = +-(1 I Developing the last expression by the binominal formula aBh h2 h3 B-b= —B- + B - &C, P PI P3 Which is the correction, always subtractive. But p being very great in comparison of h, it is sufficient for all ordinary values of B to take aIt P 318 GEODESY. The horizontal alignment is made by means of a small transit, altitudeand-azimuth instrument, or theodolite, and pickets with cards of pasteboard in their tops, placed along the line. The telescope of the instrument is directed. to the distant extremity of the base, and then brought down on its horizontal or supporting axis, to indicate the proper position of each picket. The correctness of the ratio of the length of the arms of the vertical lever, to prevent changes in the distances between the contacts from changes of temperature, is tested by a pyrometer, invented by Mr. Saxton, of the Coast Survey. The rod is placed level upon two marble piers, sunk in the ground, which support its ends. The rod is then subjected to changes of temperature, artificially created, and being prevented fiom expanding in one direction by an upright stancheon, against which it abuts, its expansion, if any, in the opposite direction, acts against the vertical axis of a small plane mirror, giving the mirror an angular rotation about its vertical axis, the plane of the mirror always continuing vertical. At the opposite side of the room is a telescope placed horizontally on a pier, and directed to this mirror, and directly under the object glass of the telescope, is a horizontal scale of three feet in length, the divisions of which are 4 of an inch. The movement of one of these divisions over the vertical wire of the telescope, occasioned by a motion of the small mirror in which it is reflected, corresponds to a change in length of the rod of I -o of an inch. The distance of the piers, which is liable to change, by hygrometric changes, is tested by a standard bar at the temperature of freezing. The standard bar is compared with a bar from France, the only one in the country. Its length is an exact French metre, which is the ten millionth part of a quarter of the meridian, the value of which in units already in use has been found by means of two measured degrees in distant latitudes. (See Appendix VI. p. 368.) A bar of brass and iron exposed to the same temperature will not heat equally in equal times; this is well known to depend upon the different conducting powers of the two metals, their different specific heats, and the different powers of their surfaces to absorb heat. The bars, then, if of equal sections, when the temperature is rising or falling, will not have the same temperature, and the system is not compensating. By adapting the sections according to rules deduced, partly by theory and partly by experiment, a small residual quantity remains to be corrected, which is detected only by the delicate tests of the Saxton pyrometer, or the lever of contact and level of Bessel. This correction is made by applying a covering more absorbent of heat to one bar than to the other. The rods are surrounded by a tin covering, and of about twenty THE GREAT THEODOLITE. 319 feet (six metres) in length, the whole being easily transported by four men. TO REDUCE A BROKEN BASE TO A STRAIGHT LINE. Let a and b be the given sides, and c the included angle, which is nearly 1800. Put c = 180~^. 6 The formula is -1 c = a + b-log. (2'6264222 4- loa log. b - ar. co. log. (a + b) + 2 log. 0.) The angles of the triangle of the great or primary chain are observed in England and this country with an instrument called THE GREAT THEODOLITE. That employed on the Coast Survey of the United States was made by Troughton and Simms, of London, under the direction of the late Mr. Hassler. It consists of a horizontal limb thirty inches in diameter, made narrow and light, with a vertical rim below, to strengthen it, supported by conical arms from beneath a central drum, upon which the telescope is mounted on pillars, as in the transit instrument. The telescope is of four feet focal length. The limb is graduated to 5' spaces, and numbered to 360~. Three reading microscopes, reading to single seconds at intervals of 1200, mae sustained above the limb, by horizontal arms projecting from the central drum. Three of the six horizontal arms which support the limb extend beyond the limb, and are provided with foot screws, upon which the instrument rests, on short pillars or legs, which fit each into one of three holes 1200 apart, in a wooden frame strongly trussed. The method of observing is as follows. The instrument is directed to the signal at one of the distant vertices of the triangle whose angles are to be measured, upon which the middle vertical wire is adjusted with oreat accuracy, and the reading taken by the three microscopes, the degrees for only one of them. The telescope is then, inverted by turning it on its horizontal or supporting axis, and the instrument turned on its vertical axis 1800, till the middle vertical wire again coincides with the signal, when the reading of the three microscopes is again taken. The legs 0* being very small, the formula preceding (2) Art. 21, App. I., putting for sin its value in (1), becomes, putting sin ~ 0 or ~ 0 for its equal cos ~ c, neglecting 0, &c., abo' c = a +- b- sin' 1' ab = a + b 0'00000004231 -+02 Whence the formula in the text, in which 0 is expressed in minutes 3 0 GEODESY. are then shifted in the holes of the wooden frame, which changes the position of the limb 1200, and the same operation gone through as before, and so on.* As there are three holes for the legs, the whole number of readings on a single signal will thus amount to 18, the indiscriminate mean of which will involve a compensation for all instrumental errors.f SELECTION OF STATIONS AND SIGNALS. Eminences are usually chosen, and, if possible, such as to permit the signals, which serve to mark the stations, being seen against the sky. A very small object is thus made visible at a very great distance. Permanent monuments of masonry or pottery are sunk below the surface of the ground far enough to be undisturbed by ploughing, with an orifice in the top, in which to insert a signal staff. The top of this staff supports some object to which the telescope is directed. Spheres made of barrel hoops, covered with white muslin, have been found to answer well in.mountainous reoions, being visible fifty miles, but not so well in low grounds, and near the sea. Tin cones of an angle adapted to reflect advantageously the light of the morning and evening sun, were used by Mr. l-assler,who gives (Trans. Am. Phil. Soc., 1825), the mode of reducing in a simple manner the observation to the axis of the cone, a reduction depending on the relative position of the cone to the sun and the observer.4 Steeples, circular or polygonal towers, windmills, &c., htave been employed, and the methods of correcting the observations upon these are given by Puissant.~ At night, signals have been used formed of * This is for the purpose of measuring the same angle upon different parts of the limb. t The French employ a large repeating circle, with which the angles are observed, in oblique planes, passing through the objects and the eye. These requi-:e to be reduced to the horizon by methods given by Puissant (Trait6 de Geodesie). The theodolites employed on the Coast Survey, besides the 30 inch, are one of 24 inches, and one of 18 inches, by Troughton and Simms, and a number of 14, 12, 10, and G inch repeating circles by Troughton and Simms, and Gambey, of Paris.. t A formula easily deducible from Hasler's is r cos I z Correction in seconds =: sin 1tt In which r = mean radius of signal cone, z - diff. of azimuth between ( and signal, D = distance. ~ Traite de Geodesie, a standard work on this subject, to which we shall have froquent occasion to refer. REDUCTION TO TIlE CENTRE OF THE STATION. 321 lamps, placed at the focus of a parabolic reflector, or behind a lens at the focus of parallel rays. The most perfect is the Drummond light, which may be seen at the distance of seventy miles. The best signal when the sun shines is one employed at present on the Coast Survey of the United States, called a heliotrope. It consists of a common telescope, mounted upon a three-legged stand, horizontally. It is accompanied by a man called a heliotroper, who directs the telescope to the tent in which the great theodolite is placed. Upon the eye end of the telescope is supported a small plane mirror, which has motion round both a vertical and horizontal axis, so as to be capable of being placed in a position to reflect the sun in any direction, at pleasure. The heliotroper attends and turns the mirror continually, so as to reflect the sun in a direction parallel to the axis of the telescope, which he accomplishes bv causing the rays to pass through two perforated discs, supported like the mirror, on the top of the telescope tube, one being near the object end, and the other between it and the mirror.* The signal pole is supported by a wooden tripod at least i its height, and the heliotrope is placed at a short distance from,it, in a line with the theodolite station. All the signals visible fiom the station at which the great theodolite is placed, are observed every day for several weeks. The instrument is then moved to a new station, and by. this means all the angles of every triangle are repeatedly observed. REDUCTION TO THE CENTRE OF THE STATION. It sometimes happens from the nature of the signal employed, that the theodolite cannot be placed at the axis of the signal called the centre of - the station. The process of determining what the observed angle would have been with the instrument so placed, from observation with the instrument placed at a short oneasured distancet from the proper point, is called reducing to the centre of the station. Let c in the diagram be the centre of the station, o the place of the instrument. From the observed angle AOB required the angle ACt. Make AOB = C, BC =q, AC = d, ACG =x,OC = r, con = y. Then AIB = t + TAO, and AIB = x + cBO * The heliotroper is on duty till 10 A.M. and after 3 P.m., the atmosphere in the middle of the day being too unsteady near the earth's surface for good observations. t If the centre of the station be inaccessible, this distance must be calculated from measurements which can be made by methods which the student will easily devise for any given case. 21 322 GEODESY. Equating these two values of AIB we obtain. x~ — J IAO-CBO B But, i ) sin(y + v). rsin \ \ sin CAO d -- sin CBO = \ \ / / d g Substituting these values of the sines of CAO, CBO, for the angles themselves, which are very small, we have c o r sin (w + y) r sin y ) = dL sin L e03 L very nearly, because - sin 2 L = sin L cos L. The quantity neglected here is quite insensible in practice. R -Rl a (1 - e2) e dL sin r cOS (1 - e2 sinl' L ) We have then -' s(1 inL)X~ n, a( (-' si- R) (I sin L) (1 — a e2) e dCL sin L cos L R - R,, e sin L COS L -(e 3sin) and - d (dL)2 - (C (Ir -ee sin'D L' " m (1 - e sin2 K) Lc As this quantity never amounts to o' of dI, the third term of dLa in (a) need not * This dL is the sum of the first two terms a+ -. 1 See Appendix VI. p. 366. 330 GEODESY. For. secondary work the 4th term 7h2 sin' Z.E may always be omitted. The 3d term very frequently is of no sensible value, and a' may always be written in the place of diL, when K does not exceed 86-500 metres (54 miles), or log. K = 4*937, which comprise the vast majority of cases. When K is less than 34000 metres, two terms are sufficient. The best rule for the omission of the third term is that it need not be used unless log, a is greater than 2*31..., log. D (which scarcely varies), being about 2*38..., we shall, in that case, have lo g D-= 700.. =log. of 0"001. It appears that there are hardly any cases in which the second term may be omitted. The 1st term gives the distance on the meridian of the point of departure fiom that point to the foot of the perpendicular from the second point, the second term gives the reduction to the parallel. It is only at a very small azimuth then, that the 2d term may be neglected even for a very short line. The 4th term may be omitted between latitude 450 and 400, when K is not over 17000 metres, or log. K = 4P2304. Between latitude 400 and 35~, when K is not over 18500 metres, or log. K = 4P2671, and between latitude 35~ and 300, when IK is less than 20,000 metres, or log. K 4*301. In computing carry log. B to 7 places, 6 "~ c to 5 places. e" 6 " D and E to 4 places. The formula for difference of longitude is dM-A'Kssim (a) COS.L be used in making the substitution of di, in (c), and the second term only when it is over 180. If in (c) we introduce dL expressed in seconds obtained by (a), we must of course multiply by arc 1", and we have, finally, K COS z I2 sin2 z tan L K3 sin- z cos z -d, - - N ar 1 1 + 3 tana L) + (dL) R are ib, -2 N a are 1 } e sin L Cos L are 1" (1. ea sin2 L) (A) 1 tan rJ e2 sin L Cos L arc 1+r 1 - 3 tan' L iR arc- t u NRare (1-er sin~,) 2 D 6 2 2 Pnd call the first term h, we may write - dL = - + cos z'B + IC sinl zec + (daL)~ n - h Ki' sin2 ZE, in which E may be taken out at the same time and from the same page as n andc; h could be copied fiono the bottom, or sum of the logs. of the 1st term, and K' sin2 z, by taking the sum of the first two logs. of the 2d term. h The formula for the difference of longitude is obtained in an obvious manner, by applying the sine proportion (Art. 81), to the spherical triangle rss', which gives, writing dx for sin dMi = sin P and i for sin K, GEODETIC DETERMINATION OF POSITIONS. 331 In which i _ (t -e' sin' L') N arc 1" a sin 1" L/ = new latitude, computed from the formula for - dL. r sin z dM = — COS LI And the value of dM in seconds of arc is obtained by converting lc into seconds, by dividing K in metres by N sin 1", N being the normal, and the length of the radius used at that part of the earth in metres. The above formula thus becomes the one 1 already given (G). Lee's tables and formula gives a table for log. N, log. s 1 and log. (1 +- e2 cos 1i) for any latitude between 20 and 50 degrees. (G) is founded on the supposition that cos L: sin z:: K dat, whereas, in reality, cos L': sin z n: sin i di. The error committed by the former supposiiibn is expressed by- 6 ensit - 7 I. -sin z alNon is expressed by -~ I~ 7 7are 1" ~, ~ sin z [see (1) p. 94], or for i/ 6 COS i are I" cos2 Li Ka 45 - Nar 1 (2 sin z - sin z). This is a maximum when z -240 06', 3 v2 N3 arc 1" nnd if we substitute this z in the latter expression, and make it equal to 0-001", wo find the corresponding log. i to be 4*4315 =log. of 27000. For any line over 2.7000 metres, then a correction ought to be applied to diM, or if we will allow an error of 0.002, ftr any line over 34000 metres = about 21 miles. In the annexed table, the column headed dy contains the log. of the seconds in a given are; the column headed diff. contains the diff. between the log. of that arc and the log. of its sine (to the seventh place); the column headed K containing the log. of the length of that arc in metres. To apply the correction in question after having first computed dE by the formula (G), enter the table with the given log. K, and take out the corresponding diff.; again enter the table with the computed log. dm, and take out the corresponding diff, and lastly, subtract the difference between the two quantities thus obtained from log. di, the result will be the corrected log. dm.1 K diff. d diff. contains diff. contains diff. contains between log. arc contains log. between log. of log. K and log. sin to d in seconds ar and log. sine in metres. seven places of of are. to seven places de als of decimals. of decials. I For denoting the difference between the log. arc and log. sin by 8, the formula (a) should be after the application of logarithms, K KI sin z log. dMw - d log. dt = log. - + log. - + log. -. Whence the rule is obvious. 0' N COS L 332 GEODESY. Log. A' should be carried to eight places. S\e en places of logarithms should be used for dmt. For azimuth we have dm sin X -dz- = (D)* cos ~ dL X= -= (L +L') dM sin X z'- 180 -+ z - cos I dL For any line less than 340000 metres (21 miles) cos - dL may be omitted, being regarded as 1. In computing dz, sin [X = (L + L')] is taken out to five places for main chain of triangles, and to four for the others, carrying forward dz in tenths of seconds in the first, and in whole seconds in the second. * The formula for the difference of azimuth is deduced as follows:-In the triangle Psst we have, by Napier's analogies, calling a, w' the polar distance of s, s^, cot I r cos (T - r') tan.~ (s' +- s) =- coS ~ (I - 7- 7r') 1 Or recollecting that tan = ~ (Art. 37). tan r cos. (, +- ir') tan dm& sin 4 (L + L') cot i (s' + s) = - (n) os ( (- a') cos A d( ~') But s'-= 1800 - z', and s = z, and cot - (s' + s)= tan [900- ~ (s -+ s)]... n becomes tan j dM sin X tan j (z'-) =Z cs) cos j dL which is the formula (D) above, if we write. dz, for tan 4 (z - z) and j daM for tan ~ dm. The formula for dz requires some amendment within the same limits, within which we obtain dL and de; we have sin > tan j (z' - z) =tan j dM i (1) cos I dr( for which we have hitherto used sin, dz = dm - cos d CL By transformation of (1) we get (see note p. 94, and make cos2 and cos3 of d dL =1) sin X dz=. d5 O - d- + a dA13 sin X cos' X sin' 1" cos 3 dL We write the second term thus, + diM3 F where log. F is to be taken from the tables, into which it can easily be inserted, as only one value will be required for every half degee of Lo It is 7e8324 for 250, and 7'8404 for 450; diff. for 30' 0'0002. Tho term dxaa i can never exceed 0".1. GEODETIC DETERMINATION OF POSITIONS. 333 Whenever the log. of any term is not over 700.. the corresponding number need not be taken out. Azimuths are reckoned from south round by west, and from 0) to 3600, the sipns of sin z and cos z varying accordingly. Y The following form, filled up with an example, is that at present used on the Coast Survey of the United States for the computation of the above formulaefor difference of latitude, longitude, and azimuth. I^..' ~z A to s 101 23 16-757 /Z. B and c 83 36 43'243 z- A to c 185 00 00*000 -dz. + 05 47.413 180 Yz c to A 5 t 05 47J413 1, 45 O0 00'000 A M 70 31 50-000 -dL 1 04 32-264 d M - 08 06'748' 46 04. 32-264 C M' 70 23 43.252 K 5'0791812'5 l 0-15836 h 3*5880 cos z 9-9983442.2 sin2. 7*88059 JL2 721760 K2 sin' z 8-0389 B 8*5105124*1 c 140411 D 2*3872 E 6*2132 -h 3*5880379 9*44306 9*5632 7*8401 1st term 3872*914- 3d term 0'366 2d term 0.277 + 4th termn 0007 3872*637 - -249 3d&4thterms 0.373 + 1^ 8*5090285 Arg.i — dL 3872'264- d M 2'C873039 IK 5*0791812 K -255 X 450 32' 16" sin X 9*8535235 sin z 89402960 d- + 6 dLr 32 16 cosAdLI 191 1COS'L' 0-1588231 -249 ar. co. ____- lar. co. 25408465 2*6873039 -d z 347"*413 d M 486"'748 * In this form the first horizontal line expresses the azimuth of the line joining the two stations A and B; the second the angle formed by a line from A to a third station c, with AB; the third is found by the addition of these, and is the azimuth of AC; the fourth the excess of the difference of azimuth betweenAc and CA over 1800, computed below at tne bottom of the form; the fifth the azimuth of CA required, formed by the addition of the two above, and 1800. The sixth horizontal line contains the latitude L and longitude M of A; the seventh the difference of latitude and longitude of A 334 GEODESY. PROJECTION Ob MAPS. The geographical positions of the vertices of the triangles having been determined by calculation, as above explained, blank maps are prqpared with lines upon them representing meridians and parallels of latitude, upon which these points are accurately put down in their true positions, and the maps thus prepared being placed in the hands of the plane table parties, are filled up with the details of the ground which they represent, in the manner described at p. 235 et seq., the points marked upon them, and identified upon the ground by the sunk masonry or pottery of the signals employed in the triangulation becoming the base points or points of departure for the operations of the plane table. The mode of preparing these maps in practice, it will be now proper to explain. A spherical or spheroidal surface like that of the earth not being developable, it is impossible to represent upon a plane the positions of places without changing more or less their distances from one another.* When a small portion of the earth's surface is to be represented, the best mode is to conceive the earth to be enveloped by a tangent cone, the andc, computed below; the eighth the latitude LI and longitude M' of c, found by taking the:algebraic sum of the two above. The next four horizontal lines of the form contain the computation of the difference of latitude dL between A and c, the first column being the computation of the logarithm of the first term, the second that of the second term, and so on, of the formula dL, at p. 325. The next two lines contain the four terms themselves,land the next two their combination to form - dL. The first two columns of the remainder of the form contain the computation of - dz; the third column that of the difference of longitude of A and c, viz. diM, and the fourth the correction of this, which is sometimes employed. Applied here the log. dM becomes 2-6873039, and dam 486-748, differing only 0"'012 from what it was without the correction. A correction is also sometimes applied to dz, as has been already stated at p. 332, the formula for which is FdMa, in which.F -,a sin X cos'2 X sin' I ". The computation of this correction in the present example would be as follows:dM3 = 8*0619 Fr= 7'8404 5'9023 In which F may e:taken from a table previously prepared. The last number is the logarithm of the correction to be. applied.to - dz. * For the ordinary modes of projecting the hemisphere, see "spherical projections" in Davies' Descriptive Geometry, and for the analytical investigations of the. same, ee Franecour (Geod6sie, 3.09, et scq.) 1 z being between 90~ and 2700, cos z is negative, and.'. h is negative. PROJECTION OF MAPS. 335 circle of contact being the middle parallel of the region to be embraced, and to suppose the surface of this cone to coincide with that of the earth over the whole extent between the northern and southern parallels of the map. This cone, when developed, becomes the sector of a circle, the portion of which between the two extreme parallels which we have sup posed to embrace the surface of contact, will represent the surface of the map. Supposing the earth to be spherical, which may always be done in the projection of maps, its oblateness being so small, and representing the latitude of the middle parallel of the map by X, and the number of degrees of longitude to be contained in the map by D, it is evident that the absolute length of the middle parallel of the map will be' expressed by (see 1st note, p. 153) D ~- I cos X (1) 1800 In the above expression the radius of the earth is unity, and this being the case, the slant height or length of the element of the cone from the vertex to the circle of contact will evidently be the cotangent ofthe latitude. The arc of the sector, which is expressed by (1), divided by its radius cot X, gives the length of the arc which measures the angle of the sector to radius unity. The result is D __ %sin s. 180~ and this, which is the absolute length of the arc, must be multiplied by 1800 1 * to have its value, or the measure of the angle of the sector in degrees, D sin X (2) then will be the formula for this angle, and the construction of the map will be very simple. It will only be necessary to draw two lines forming the angle expressed by (2), and with a radius equal to cos X and the vertex of the angle as a centre, the arc representing the mean parallel may ird be described. If the map is to contain d degrees of latitude, then will express the distance between the extreme parallels, and by describing arcs from the vertex of the sector with radii greater and less, than 1800 * - - = 590, 29573 33437, 74677 = 206264", 80625. t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 8334~6 GGEODESY. cot X by the half of this expression, the extreme parallels of the map will be constructed. The distance between the parallels is then divided into any number of equal parts at pleasure, and arcs described with the vertex as a centre, and passing through the points of division. As to the meridians, they are drawn as straight lines through the vertex, and through points of division equally distant from one another upon the arc of the middle parallel. This construction is so simple, that it is generally preferred to any other, and the greater part of maps of kingdoms and states are drawn upon this system. For greater precision, the cone, instead of being taken tangent to the sphere, is partially inscribed in it by making it pass through the two extreme circles of latitude, so that these circles shall be sections of the cone perpendicular to its axis. Imagine a meridian section of the cone and sphere, the angle a formed by the element of this. section with the axis will be measured by half the diffirence of the arcs included between its sides (Geom. Ex. 30, p. 48). Supposing a and a' to be the points in which the element intersects the meridian section, and X and X' their latitudes, N being the place of the north pole, and s the south, the expression for the measure of the above angle will be a (sa Na') But sa=90~ ++X, and a'= 900-X.~. a = (X + ) Now in the right angled triangle formed by the element of the cone, the axis and the radius of the parallel, which last is equal to cos X, we have for the length of the element terminating at a cos X sin (X + X) and for the length of the element terminating at a', cos A s iX+ sil ~ (x + X) The lengths of the elements of the developed sector being thus known, the rest is as above. Still better, the cone may be made to pass through two parallels, situated at half distance between the middle parallel and the extremes; the cone would then be partly internal and partly external to the sphere' * It was in this way that Delisle constructed the great map of Russia. PROJECTION OF FLAMSTEED. 337 PROJECTION OF FLAMSTEED. This consists in drawing a straight line vertically to represent the central meridian of the map, laying off upon it equal distances say 10, and through the points of division drawing perpendiculars to this meridian line, which represent parallels of latitude; then laying off upon these parallels distances bearing the same proportion to the distances on the meridian as the cosine of each latitude does to radius unity; finally, drawing. through' he points of the same graduation, thus determined, curved lines which will represent the other meridians, The oblateness of the earth may be taken into the account in this method, by lvying off in the central meridian not equal distances, but increasing towards the poles in the same proportion as the degrees of the meridian increase. For the demonstration of the formula see App. VI., p. 367. The formula itself is (1 - I c 3- o cos 2X) —" In. which a 637739O7'15 metres log 6e8046434637; log J -'5233789824; X latitude. The objection to the method of Flamsteed is that it distorts somewhat the regions distant from the central meridian. METHOD OF THE FRENCH DtPOT DE LA GUER.RE. This is a modification of the conic projection already given, and is that now in use on the coast survey of the United States. The radii of the arcs of circles representing the parallels upon the map being too long to be conveniently described from a centre, /' they are determined by points. Let there be drawn in the middle of the sheet the perpendiculars CA, NN; NAN' represents the middle Nf~- LN parallel of the map. Then is known the A radius Fr = CA = R cot X, R being the radius of the earth. Suppose that the map is to ~ For a table which gives the length of a meridional arc in any latitude in yards and a table which gives the length of a parallel, see Lee's Tables and Formula, Part II., p. 84. 21 338 GEODESY, embrace D degrees of longitude, the angle c is then known = D sin X Representing half the chord NN' by c i, cI by i, we have in the triangle CNI a= r sin iJ c, = r cos -} c, A= f (1 -cos c) = 2r sin2 c [ee (7) p. 100]. The extreme points N and N' of the arc to be described Ns', are thus determined, and the point A, in which it intersects the meridian. Now for other points, such as M, a distance IP = y is laid off from I, and a perpendicular PM is drawn in length equal to x, the value of x being expressed by the following formula = /(r' +y) —.y) - p * The demonstration of this formula, which requires a knowledge of Analytical Geometry, is as follows:-The equation of the circle, the origin of co-ordinates being at c, is x2 + y2 = r2. Transferring the origin to i, the formula for transforma. tion will be x -= x +- t, and the equation of the circle becomes (x: - +)2 -, r~ -,.. x- (r + y) (r-y)-f The formula in the text. The above method does not take into account the earth's oblateness; the following is the generalization of the theory. Let c be the centre of the projection, AK the middle parallel, the latitude of which represent by I; BM another parallel, whose latitude is X; m the point in question, whose co-ordinates are Ar = x, PM = y, AX being tangent at A, and perpendicular to the principal meridian ca. We have AB= s, the distance in latitude between the two parallels, this length a being known by equation (5) p. 367, App. VI. The radius cA- = r is also known, being equal to xcx Q in the diagram on p. 365, which, in the right B angled triangle xMN, where MN is the normal N, has for its value r =N cot 1. Representing the angle Acn by 0, and ca by e, we have A r X QM = =x p sin 0, CQ = p cos 0 = P M = - - s 3 BC -CQ =s - -+ p -,CoN 0 =S + (1 - cos 0) = s + 2 p sin2. 0 [see (8) Art. 72]. We may eliminate p from this last by means of the first x= p sin 0. It becomes -- cos!J$ + — x - in 0 -= 8 + x tan i 0 [se App. I.,(74) p. 103.1 METHOD OF THE FRENCH DtPOT DE LA GUERE 339 Dividing therefore N into equal parts, and for each point of division finding the corresponding value of x from the above formula, so many points in the arc NN' will be determined. p is known since p r - s. It remains only to find 0 in order to have for each point like a the values of the co-ordinates x and y, viz: x = p sin 0, y- s + x tan ~ e N. B.-That s must be taken negative when A < 1. The longitude of m estimated from the central meridian suppose to be A. This will also be the number of degrees in the arc of the parallel. But N being the normal at the point M, terminating at the polar axis, the radius of this parallel will be (see diagram, p. 365, App_.VL), N cos A Moreover, the arc BM of the projection is of the same length with the arc of the parallel, but the number of degrees in two arcs of the same length will be in the inverse ratio of their radii,'. A: 0:: p: N cos; N cos > O=-A which formula serves to determine 0 in the same denominations that A is given. l is known in terms of A from formula a (1- e sinA) ) (see App. VI., p. 368), in which e = 0.0816967, log. e = 8X9122052271. It is easy to perceive now how a map would be drawn according to the projection under consideration. Two lines AC and AX are first drawn at right angles to each other, intersecting at the middle of the sheet A. Setting out from A, we lay of above and below distances such as AB, respectively equal to the values of s, that is to say to arcs of the meridian corresponding to 10, 20, 30,..of distance from a, arcs which go on increasing towards the pole. Next we compute the values of the normals M, N.... from degree to degree, the radii p of the projected parallel, and finally the amplitudes of the angles 0, which correspond to values of A and A, varying also by degrees, whence result the co-ordinates x and y of the vertices of quadrilaterals in which meridians and parallels of latitude distant from each other, respectively the space of 1~ intersect. It remains only to lay off these co-ordinatio by a scale of equal parts. The sides of the quadrilaterals joining these vertices thus determined may be drawn without sensible error as straight lines. The territory to be represented by the map is ordinarily too extended to be placed For their values see p. 367, App. VI., also Lee's Tables and Formulas, p. 84, Part II. 340 GE0OES' o LATITUDE BY ASTRONOMIC OBSERVATIONS. OBSERVATIONS FOR LATITUDE WITH ZENITH SECTOR. The zenith sector employed on the coast survey of the United States for determniing latitude astronomically, is the same as the mural circle already described, p. 306, except that only two small portions of the limb, the one above, the other below the centre, are retained, the rest being conceived to be cut away, to render the instrument more portable. The limb and telescope, instead of being sustained by a wall, are attached to a vertical flat beam of iron, which is capable of reversal about a vertical axis, and also end for end. Long spirit levels can be attached to the upon a single sheet. It is customary to form the map by the union, border to border, of a series of sheets, the dimensions of which are 8 decimetres by 5. To find the positions of the vertices of the quadrilaterals upon these sheets, the origin of co-ordinates is transferred to one of the corners of the sheet, an operation which consists simply in adding or subtracting 1, 2, 3,... times 8 decimetres in the direction of the xS, and as many times 5 decimetres in the direction of the Yr, according to the place which the sheet ought to occupy in the assemblage. The order of the sheets is marked upon them. Thus the sheet E3 is the one which is second in the horizontal direction, and third in the vertical, estimating from A the intersection of the middle meridian and parallel. As to the inverse problem, to find the latitude and longitude of a point given upon the map, it will be sufficient to draw through the point lines parallel to the sides of the quadrilateral within which it falls, and to determine upon the scale of equal parts, the values of the fractions which the lines represent. The following formulas are used on the United States Coast Survey, when thi extent of the map is not more than 40 of latitude and longitude. p = (n') p sin I z m = (n') p cos A z In which Jp is the ordinate y = BQ. i' am " abscissa x = QM. " (n) p" length of the parallel. (n') p' sin I Z 2T T N COt L. a (1 - e2 sin2 L) 1,= latitude differentiated from. Here the cone, instead of being assumed tangent to one of the parallels of the waap, is supposed successively tangent to each, that it may be required to draw upon OBSERVATIONS FOR LATITUDE WITH ZENITH SECTOR.,' t bacl of this bar, in addition to the ghost apparatus in front*. Zenith telescopes are also employed, of similar but less elaborate constructiol. The practice on the coast survey with these instruments is to observe two stars near the zenith, one north of it, the other south, and differing so little in' zenith distance that the difference may be measured with a micrometer.t The following is a very full exposition of the theory of this nmethod a Z /- I X/ s B In the above figure let s represent the station occupied, z the Zenith,. the Pole, liORns the Horizon. it. The map thus becomes the developement of the surfaces of several successive cones. To make the projection, a central meridian is drawn upon the map, along which the lengths of the required minutes are laid off; perpendicular lines are drawn at each of these points, and the values of Sp and inm are laid off successively, along and from each of these lines. Lee's Tables give the values of 6p and 3m for parallels 80o apart. The manascript tables in use on the Coast Survey are computed to every minute. In the. diagram MB, which is very small, is regarded as a straight line, and the angle QMB c is - z; cM is equal to MK =- T in the diagram on p. 365, App. VI., N = M, and L = MNV in that diagram. With these explanations the student will readily deduce the above formulas for Sp and Sm. * The mode of observing for latitude is similar to that employed with the mural, except that the readings of three levels, one above the other, at the back of the bar, are taken, and the observation is repeated upon the same star in reversed positions of the instrument. The correction for the state of the levels and the reduction to the meridian are made on the principles indicated at pp. 343, 344. ~I' For description of micrometer see p. 362, App. VI. In Silliman's Journal -of Sept., 1852, Prof. Bache says, "The chief rodifications of the instrument since its introduction in the Survey, have been in securing stability by a bras1 arc', instead oft' a rod in increasing the facility of reaching the zenith, by raising the centr;Al' The brass arc is attached to the bottom of the strand, and the eye end of the telescope slides upon it. CB, Q C"),'E G EODES > aIn the foowving investigation let FOR STAR S OR'I o ZENITH. FOR STAR SOUTR OF ZEIO ENITH. = North Polar Distance. A" = North Polar Distance. -- Zenith Distance. z" = Zenith Distance. n -= Reading of north end of N = North End of Level Level Scale. Reading. s" = Reading of south end of s" = South End of Level Level Scale. Reading. W"= Reduction to Meridian, ME = Reduction to Meridian. = Refraction for Star. - " = Refraction for Star. X = Latitude. Now if the observation of a star were not affected by refraction, and it were observed at the moment it passed the meridian, and the instrument at the same time were perfect as to level, then the latitude resulting from tle observation of a star north of the zenith would be expressed by X - 900 -(A+ z) (1) and for a star south of the zenith by X = 900 - (A - z) (2) lBut every observation on a star is affected more or less by refraction, according to its distance from the zenith, and the instrument is constantly changing (as indicated by the level) during the observations. Suppose a star to be observed north of the zenith, and its north polar distance to be equal to NA in the diagram. If there were no refraction, the star would be observed at A, and the zenith distance would be AZ. But by the effect of refraction, which elevates an object, the star is seen and observed at a, consequently the observed zenith distance of the star is equal to az, which is too small, since NA is the true N. P. D. of the star. The measured z. D. must therefore be increased by Aa, or the amount of refraction. Hence we have, as a result for latitude by observing a star north of the zenith, X = 900o-(A" + z")- r' (3) Having observed the star north of the zenith, the telescope is turned column and somewhat diminishing the diameter of the azimuth circle; in substituting a single for a double micrometer; in providing a parallactic eye-piece; in illuminating by a lamp not resting on the instrument; in bringing the divisions of the level and micrometer into just relations; in an adjustment for verticality of the axis (adapted by Mr. Simms, of London); and in providing stops to set the imtrument in azimuth," OBSERVATIONS FOR LATITUDE WITH ZENITH TUBE. 343 in azimuth 1800, for the purpose of observing the star south of the zenith, the N. P. D. of which is equal to NB. On account of refraction, the star is observed at b, and the measured zenith distance is zb, which is evidently too small, and must be increased by Eb, the amount of refraction. For a star south of the zenith we have, therefore, X - 900- (rA'- z) + r" (4) If a star north of the zenith be observed at a, while the instrument is perfectly level, but by a sudden change in the temperature of the atmosphere or any other cause, the instrument is thrown out of level, and the vertical axis takes the direction of the line sc; inclining the vertical axis towards the south, the north end of the level will become elevated, and the south end depressed; and therefore the north end of the level scale will read greater than the south end. Suppose another star of the same north polar distance as the one just observed, to come into the field of the telescope, it will be seen at a, and consequently the distance ac must be measured with the micrometer, thereby making the measured z. B. greater than the true by'the quantity ae = zc. Since this distance zc is also measured by half the difference (in arc) of the readings of the level scale (see p. 155), it follows that the level correction will always equal the difference between the measured z. D., and the true z. D. of the star. Hence we have for latitude, by observation of a star north of the zenith, 900 -(Z.n + z,) + (x - ls) (5) The telescope is next turned in a:zirnuthl 1803, for the purpose of observing a star south of the zenith. The vertical axis still being in the line sc, the telescope will take the line sb, but the star will be seen at d, therefore the measured z. D. will be too small by the quantity bd - zc. The distance zc being measured (as I have before stated), by half the difference of the north and south readings of the level scale, we have for latitude by observing a star south of the zenith, X =90o-(E A-z) +.+' +N ( 6) If a star be observed before it reaches, or after it has passed the 344 GEODESY. meridian, the observation will require a third correction, called the Reduction to Meridian."* In the figure suppose the star seen at B to be observed on the meridian, and after it has passed the meridian, to be again observed at B', then the difference between ZB' and zB = BD'"Reduction to Meridian." If the vertical wire is in the meridian, and the star is observed before or after culmination, then the correction is Ha m = — sin 2 A 4 In1 which m is the correction, Hi the hour angle, and A the polar distance,f the star. It is evident, without demonstration, that the zenith distance of any ita:r will be the smallest when it is on the meridian (the telescope being moved in azimuth), therefore the algebraic sign of the correction for " Reduction to Meridian " to the latitude resulting from an observation of a star north of the zenith will be -i, and for the latitude resulting fiom the observation of a star south of the zenith, the algebraic sign of the correction will be -. Hence for a star north of zenith x D-.oo (A" + zn) - ( -- s) + Mn (7) and for a star south of the zenith X=i~ 900 _A ZB~) ++ (,. ~)_+ (8) By adding (7) and (8) we have for twice the latitude, 2X = 8 00o - + ) - (A +A) - (z ) - ( -~.) +4 ( — ) +orI, I0o-(A" + a.)-(z"- z) a (N" + Nu)- (S + 8) + (M,,- ~,) + (k'-t:es {b) 2 4 2 * A single observation on the meridian has, in the experience of the Coast Survey, been found preferable to several circum-meridian observations. When there is a recorder to assist the observer, he calls out the time for setting the instrument for each pair of stars and when each star enters the field; after that, every ten seconds, the observer in the mean time following the star with the horizontal wire and moving the instrument in azimuth, if necessary, as the instant of culmination approaches, in order to have it take place in the centre of the field. At ten seconds before the culmination, the recorder calls each second till the instant of culmination previously computed. The level and micrometer are then read and recorded, and the instrument turned 1800 to observe the opposite star of the pair. OBSERVATIONS FOR LATITUDE WITH ZENITH TUBES. 345 in which a represents the arc value of one division of micrometer, and b the arc value of one division of the level scale. The value of the micrometer measures is obtained by turning it at right angles to its ordinary position and noticing the number of divisions passed over by polaris in a given time near its culmination or elongation; in the latter case, preserving the ordinary position: when a theodolite is at hand, by the apparent diameter of a distant object measured by the two instruments. From these observations, or a mean of many, a table is made, which, by simple inspection, shows the angular value of any number of turns of the screw and parts of a turn. By suitably selecting pairs of stars, any effect of inaccuracy in this determination may be avoided, by making the sum of the zenith distances of all the pairs, N and s of the zenith, zero, as nearly as may be. The value of the level divisions is found by fixing the telescope on a distant mark, or, better, on a collimating telescope, moving the instrument till the bubble traverses the whole length of the level tube, and measuring the distance passed over by the micrometer. The value of the level divisions is then converted into arc, and a table made which shows the correction of twice the latitude for the difference between the sum of N and sum of s end readings of the level. One of the great advantages of Talcott's method is, that the correction for refraction is very small, being for the difference merely of the two refractions on each side the zenith. The correction for variation of temperature and pressure of the air from the mean state is insignificant, amounting for 25~ zenith distance and a difference of 20' of z. D. in a variation of 2 inches of the barometer and 50~ of the thermometer, to only 0'"02. The correction for refraction may be obtained by the formula,i sin I" diff. zen. dist., log. correction = logc _-.._ — ~.~ ~ coss (less zen. dist.) for demonstration of which, see App. VI., p. 372. A table may be computed from this, or by differences from ordinary refraction tables.'The following directions are given by Prof. Bache: 1. The latitude of the place is assumed to within 2' or 3' of are. 2. The zenith distances should be as small as possible, and not extended beyond 25~. 3. The difference of zenith distance should be small, and in no case exceed a convenient range of the micrometer, in the instruments used on the Coast Survey say 10', including about 13 turns of the micrometer screw. 4. The interval of time between the culmination of the stars should be not less than I", so as to give time to read the micrometer and to turn the instrument in azimuth for the second observation, and should not exceed 20", to avoid changes in the instruments. 346 GEODESY. EXAMPLE. SURVEY OF THE COAST 8Ec. I. Latitude, Station Mt. Independence, Cumberland Co., Me. Micrometer. Diff Z. D. Levels. N. __ ____ Twice Date. No. or Polar Approx. B.A.C. S. Turns Divisi's. By Microm. In arc. distances Lat'de. N. S. 1849.' " n0 Oct. 13 819 N. 3'32.0 37 07 13*525 Q110 43.2 877 S. 19 6U8 55 33 51.71 870 44.0 59.5 -f-1i 30C8 -1-12 14*18 92 41 0523 30 68*95 1(040 102.7 87 ]8 54*77 14 819 N. 3 37,0 37 07 13*27 55*0 49*5 877 S. 19 66*6 55 33 51.55 49*5 55-0 +l; 29.6 -1-12 13-64 92 41 04-82 68*82 104.5 104.5 87 18 55*18 " 15 819 N. 4 060 37 07 13*01 51.0 48.5 877 S. 20 34-6 55 33 51*39 46*5 53*5 +16 28~6 +12 13*20 92 41 04.40 68-0O 97.5 102.0 87 18 55.60 18 819 N. 3 10*4 37 07 12*241 42*0 55*2 877 S.' 19 41*2 55 33 50901 55*0 42.5 +16 30*8 +12 14*19 92 41 03.14 71*05 97.0 97-7 87 18 56.86 19 819 N. 2 06.2 37 07 11*99 55.0 47.5 877 S. 18 32.-6 55 33 50*74 49*0 53.fi +1I6 26.4 +12 12-20 92 41 02.73 69*47 104.0 101*1 87 18 57271 " 20 819 N. 3 15.0 37 0711*73 55*0 50.0 877 S. 19 40*2 55 33 50.58' 51*3 53.8 +16 25*2 +12 11*66 192 41 02*311 69*35 1063 103*8 87 18 57*69 LATITUDE W'ITH ZRNITII TELESCOPE. 347 OF THE UNITED STATES. Observations witn Zenith Telescope, No. 3, U. S. C. S. I State of Corrections for twice the Leovel. e v Latitude. -—'-:- ~ ~.. TTwice Latitude REMARKs. I NS-. -. Level. Merid'n. Refract. Lat'de. I 1 i. _. 870 +-t3 4-058 -4-0-21 30 69-74 43 45 34.87 Approx. Latitude= 43 45 30 cos =9.85870 I)^-0 +0.00 -0-21 69.03 34,51 N.P.D. 37 07 14 sin -9 67 4 -mn=0-31 X 2-75=0'8 9. -4-! 5 -2.02 +0.86 +0.21 67.85 33*92 — 07 — 032 +0.'21 70*94 35*47 -+29 -. +130 +0.21 70-98 35.49 +2-5 +1-.12 +0-21 70.68 35.34 Mean=34-93 348 GEODESY. in the preceding example the 1st column contains the date of the observation, the2d column the numbers of the stars observed in the British Association Catalogue, the 3d column indicates whether the star is north or south of the zenith; the 4th and 5th columns give the number of turns and fractions of a turn of the micrometer screw, necessary to bring the wire from zero to coincide with the star; the 6th column gives the difference of the micrometer reading, the 7th the value of the same in arc depending, of course, on the value of one turn of the micrometer screw;* the 8th column the polar distances of two stars observed on each day, their sum n -A - A', and 1308( (A" - + A'), according the iormula, fiom which subtractingit the number in the 7th column.r which is the difference of their zenith distances in arc, or z.^ -- z", according to the formula, p. 345, the remainder will be by the formula equal to twice the approxi-mate latitude which is written in the 9th column. The 10th and 11tho columnsa contain the level readings at the north and south ends of the scale, in both positions of the instrument, together with the values of N" -{- N, and sn" S', the 12th column shovv the difference of these results, or the value of (N" +- N") -- (s" -~ sS), according to the formula; the 13th column contains the hour angle of the star when not observed exactly on the meridian in minutes and seconds of time; the 14th column contains the correction for error of level, which is obtained by dividing the result in the 1Sth by 4, and multiplying by the value of one division of the level scale, according to the formula; the 15th column contains the result obtained by the computation in the 19th column, of the correction.for the star's not being observed exactly upon the meridian by a method similar to that at top of p. 304, using a more accurate table, in which the constant sin 1" is incorporated; the 16th column contains the correction for refraction; the 17th column, the double latitude after the corrections in the 16th and 14th have been applied to the 9th; the 18th column, the latitude which is half the result contained in the 17th; the 19th column is for miscellaneous purposes, used in this example for the computation already mentioned. LONGITUDE BY CELESTIAL OBSERVATIONS. The best mode of determining longitude ordinarily is by means of moon culminations. The Nautical Almanac gives p. 504 et seq. in the edition of 1850, the apparent N. A. of the bright limb of the moon at the instant of its transit at Greenwich, both for the upper culmination marked u * A good way of finding the value of one division of the screw head of the micrometer is to note the time by chronometer of the transit of Polaris over the movable wire placed vertically, and set successively to every division of the screw head. Representing by x the angular distance fiom the meridian at which any reading was taken, by p the hour angle, and by A the polar distance of the star, we have sin x = sin A sin p The value of x being computed for each reading, the difference of these values, divided by the difference of the corresponding micrometer readings, gives the value of one division. f Really adding in this example, because z" is less than z". LONGITUDE BY CELESTIAL OBS ERVATIONS. 349 in the almanac, and for the lower marked L. If now the siderial time of transit be observed at any other station, this will be the'D R. A. at the instant of observation (see p. 151), and the difference will be her variation in R. A. during the interval between the two transits, viz. that over the meridian of Greenwich and that over the meridian of the station. The meridian of the station (supposing it to be w. of Greenwich), has in this interval of time revolved by the diurnal rotation through an angular space equal to the longitude of the station from Greenwich, plus the distance which the moon has moved in R. A. towards the east. To know this angular space, we have only to compute the time occupied by the moon in changing her right ascension by the difference above mentioned. This may be done by means of the variation of the moon's a. A. for one hour, given at the same page of the Almanac* by proportion. EXAMPLE. Oct. 8th, 1840, sid. time transit D)'. limb, 23A 1" 9D2 Error of clock, too slow, 7'75 True time of transit, 23 1 16 95 t. A. D' I. limb (Nautical Almanac, D culm. stars), 23 1 23'31 Difference = var. in Ra. A6,36 Nautical Almanac gives DI' var. in R. A. in 1, 122*59, 122"59: 1: 6 "36: 3" 6"8 which last term is the time occupied by the meridian of the station in revolving to that of Greenwich and 6"'36 further, the last being the angular motion of the D in a. A., since it made its transit at Greenwich, 3"' 6"8 - 6"36 =- 3" 0"'44.is the longitude of the place of observation. The above method requires an exact knowledge of the siderial time. To obviate this necessity, the Almanac also gives the right ascensions of some stars which make their transit nearly at the same time with the moon, and differ little in declination from her, so as to be conveniently observed * If the distance of the station in long. from Greenwich be great, the variation in R. A. corresponding to the middle interval between the two transits should be used, which may be obtained by interpolation. The numbers in this column of the Almanac include the change of the semidiameter of the limb. 350 GEODESY. in connexion with the moon. If; the moon had no motion the difference of her right ascension from that of the star would be constant at all meridians; but in the interval of her transit over two different meridians, her right ascension will have varied, and the difference between the two compared differences will exhibit the amount of this variation,* which added to the difference of the meridians shows the angle through which the westerly meridian mustrevolve before it comes up with the moon. This angle, as before, will be the time in which the moon is undergoing the observed variation in R. A., which may be computed by means of her hourly variation in R. A. Oiven in the Almanac. The variation of A. A. being subtracted from this result, the remainder will be the difference of longitude required. EXAMPLE. Oct. 8th, 1840, were observed the transits, Of the D) I. limb, 234 1- 9"2 I R.A. D'i. limb (N. A.)23^ 1 23"31 Of the star.Piscium, 23 18 27'6 R.A. x Pisciumr (do.) 23 18 47'98 Difference, 17 18 4 Difference, 17 24 67 17 18' 4 The diff. of the two diffs. D" var. in R. A. 6 *27 By proportion as before, 122"29: 1: 6"27: 3= 48. Long. required = 3" 4 - 6" 27 - 2" 57'*73. When the meridian to be determined is distant from Greenwich, a very simple and unexceptionable way of proceeding is to assume the longitude which is supposed to be known approximately, and from the culminations of the moon's limb, as given in the Nautical Almanac, to find by interpolation the time of culmination of the limb at the assumed meridian. The difference between this and the observed time of culmination will be the interval of time occupied by the moon's limb in passing from the assumed mneridian to the true meridian of the station. The motion of the limb in * For the determination of this variation with great accuracy, observations should be taken simultaneously at the different meridians to be compared. Errors in the computed places of the moon or stars are thereby avoided. The results given in the Almanac may be considered as a very near approximation to what would have been the indication of the Greenwich instruments, had the observations actually been made with them. The traveller has thus the opportunity of rendering his observation immediately available for determining his longitude with considerable accuracy. LONGITUDE BY ECLIPSES OF JUPITZR'S SATELLITES. 351 right ascension during this interval must be computed by first determining the hourly motion in right ascension, by interpolation, for the instant of passing the assumed meridian, and proceeding by proportion, as in the examples above. The result thus obtained being subtracted fiom the interval, the remainder will be the difference of longitude between the assumed meridian and the meridian of the station. The formula for interpolation is y=A +- x+ is -- f- 1) S 6 + -( + 2)o +&c n n / n In which A is the element for the noon, midnight, or complete hou, preceding the given instant, y is the element required for the given time, m the given number of hours since noon or midnight, or minutes since the even hour (the long. in time of the assumed meridian above). n is 12 hours, 24 hours, or 60 minutes, the interval between the times, for which the element is given in the Nautical Aim. 6 the difference between two consecutive elements in the Naut. Alm. 62 the difference between the successive values of 6, i 6 " " " " " "6, &c. For a convenient mode of proceeding, and an example under it where the meridian is distant from Greenwich, see Lee's Tables and Formula, pp. 69-78, Part III. A.' - LONGITUDE EY ECLIPSES OF JUPITER'S SATELLITES. The eclipses of Jupiter's Satellites, especially the first, afford the readiest mode of obtaining the longitude, both from the frequent occurrence of the phenomena, and the simplicity of the calculation. All that is necessary to be known is the exact time of observation; the difference between this time and the time at Greenwich* shows the difference of longitude, and is east. or west of Greenwich, according as the time of observation is greater or less than the Greenwich time. EXAMPLE. Suppose the emersion of Jupiter's first satellite to be observed August 8th, 1850, at Paris, and the time of observation there to be 14A 30" * This is given at p. XX. of the Nautical Almanac for each month. At p. 605 of the edition of 1850 is a full description of the page and its use. a352 GEODESYo 1 7"3 mean time. The elmersion takes place at Greenwich (Naut. Aim., p. XX.), at 141 20" 55"'8 Greenwich mean time; the difference 9" 21"P5 is the liffeience of longitude between Greenwich and Paris, And because the time at Paris is greater than that at Greenwich, the former is east of the latter. ASTRONOMICAL DETERMINATION OF AZIMUTH. In the previous pages the methods of determining difference of azimuths geodetically, or from the triangulation, have been given. But the usefulness of these methods depends on the implied ability to obtain by astronomic observation the azilmuths of certain lines- from which the others 5are differentiated. The method of proceeding is to determine by observation the difference of azimuth between the sun or a star, and the line whose azimuth is to be determined, then to find by calculation the azimuth of the sun or star; the sum or difference of these results will be the azimuth required. The diffbrence of azimuth between the sun or star and the line whose azimuth is to be observed is obtained with an altitude and azimuth instrument, or theodolite. The middle vertical wire is made to bisect the star, or to touch the limb of the sun, and the siderial time is observed at the same instant; the reading is then taken on the horizontal limb of the instrument, which is afterwards turned to a signal (bearing a larmp, if at night), which is placed upon one of the sides of the triangulation, or upon any other convenient line, the horizontal angle between which, and the line whose azimuth is required, can be subsequently measured, and the reading of the horizontal limb again taken. The difference of the two readings will be the difference of azimuth between the sun's limb, or star and the signal, at the instant of siderial time above mentioned. With the.altitude and azimuth instrument, the transit of both limbs of the sun, or the transit of the star, may be taken over all the wires of the instrument, and the mean of the times taken as the time at which the azimuthal position of the sun's centre, or the star, corresponded to the reading of the horizontal limb. AZIMUTH OF THE SUN OR A STAR. The determination of this requires merely the solution of the triangle zPs, in which PZ the colatitude of the place of observation, PS the polar distance of the sun or star, ancd the hour angle: equal to the difference AZIMUTH BY POLARIS. 353 between the right ascension of the object and the siderial time of observation are Z given to compute the angle z, which is the azimuth required. Where the object is the sun, of course the value \ of the semidiameter at the instant must be computed and applied to the reading for the sun's limb to obtain that for his centre. If the altitude of the object is also observed at the S same instant, or immediately before or after, and reduced to the instant by interpolation, one of the above data, either the hour angle or the latitude, may be replaced by the zenith, distance zs in the triangle. The formule will be cos ~ (T — X) tan (z + s) = cot ~ P - i'cos ( - X) tans (z s)-= cot P p - X)~~~, sin -- (, +X) l,~.^,,... z = ~ (z + S) q-i (z-S) In whlich z =azimuth required. t hour angle. o = polar distance of object. X - colatitude of station. The formula, if the siderial time be unknown, and the altitude observed is /sin 2- s sin (- s-s~) cos - z = /. -~ S V. -sin l sinll X s= + + In which = observed zenith distance. AZIMUTH BY POLARIS. The best mode of obtaining the azimuth of a line upon the surface of the earth is by means of the pole star when at its greatest eastern'or western elongation. With a telescope as powerful as that of the great theodolite, the necessary observations may be conducted in the day time, 23 354 GEODESY, the star being distinctly visible. The mode of proceeding is to commence about 15 minutes before the time of greatest elongation, and place the middle vertical wire alternately upon the star, and upon a signal nearly in the direction of the meridian, a mile or two distant, illuminated if the observation be at night. The readings are taken by the micrometer microscope, on the horizontal limb, both when the middle wire is upor the star,and upon the signal, the difference of azimuth of which will be indicated by the difference of the reading, so that when the azimuth of Polaris, at the instant of each observation upon it, is known, the azimuth of the signal becomes known; the mean of all the results is taken as the true azimuth, and thus a line whose azimuth is fixed becomes determined on the ground,from which other azimuths may be differentiated. The following is the mode of determining, at any instant, the AZIMUTH OF POLARIS. If we suppose a spherical triangle having for its three vertices the zenith, the pole, and the star; this triangle, at the time of the star's greatest elongation, will be right angled at the star; for if a cone be conceived having its vertex at the eye of the observer, and for its base the liurnal circle of the star, the tangent plane to this cone, passing through the star, is perpendicular to the declination circle through the star, which is a meridian plane of the cone; the visual or tangent plane through the rtar at its greatest elongation being a vertical plane, passes through the zenith, and, also passing through the star, determines on the celestial phellee a side zs of the spherical triangle zsP, so that the angle at s is therefore a right angle. In this right angled triangle are known zp,the colatitude of the station, and psthe polar distance of the star, to find the hour angle P, and the azimuth z, at the time of greatest elongation. The former, applied to the time of the star's meridian transit or R. A. will give the time of greatest elongation. The formulas are For the hour angle cos P = tan r cot X. For the azimuth,sin z = sin e cosec X In which ir = polar distance, X = colatitudeo Tf the star be observed within 45'" of the time of the greatest elongalion, the observation may be reduced by the formula 225 c = -- t2 sin 1" tan z wch c is the corection of the zimuth t te siderial tie fro in which c is the correction of the azimuth, t the siderial time from elon TRIGONOMETRIC LEVELLING. 355 gation, and z the greatest azimuth. The correction is deduced in a manner similar to that on p. 302. Table XXXVI. may be made available as explained at the bottom of that page, or the constant log. 112'5 sin 1" = 6C7367274 mav be used with the logs. of t2 and tan z. This correction being applied subtractively to the azimuth at the time of greatest elongation, computed as above, will give the azimuth at the time of observation. If the axis of the telescope be not horizontal, the correction for azimuth is, d being the value of one division of the level scale, d ( + ) (e + e') tan n alto 4 TRIGONOMETRIC LEVELLING. This consists in observing the zenith distances of two stations, and applying the corrections for curvature and refractions, to obtain their difference of level. The theory is simple, and the necessary formulas and table are found at pp. 50 to 54, Part I. of Lee Tables and Formulas..'' The usual mode of observing zenith distances is as follows: the instrun.ent is carefully levelled, i. e., the vertical axis is placed truly vertical; the horizontal wire of the telescope is then pointed at the object, and the vertical circle read off; next the instrument is revolved 180~ ini azimuth, and the telescope being then moved through the double zenith distance of the object, is pointed again. If we now read off, the difference between the two readings will be 2 z. D.; the operation is, however, repeated (generally six times) if the vertical limb has the repeating motion,before it is read off again. The instrument should be levelled for each set of observations. MAGNETIC OBSERVATIONS. These usually accompany the operations of a Geodetic survey. They have for their object to determine, 1. The angle which the magnetic meridian makes with the astronomic meridian, commonly called the variation of the needle, but more properly the Declination. 2. The angle under which a needle suspended by a perfectly flexible thread at its centre of gravity, would be inclined to the horizon, commonly called the dip, but more properly the inclination; and 3. The intensity of the magnetic * Immediately following (p. 55) are formulae and tables for'he barometric measurement of heights. 356.GEODESY. force at any place; with the daily and other periodical variations in these three elements. The instrument for observing the declination is called a declinometer or declination magnetometer. Where only the variation of the declination is to be observed, the instrument consists of a horizontal telescope, firmly supported, pointing towards a magnetic needle bearing a mirror so adjusted as to reflect a horizontal scale placed directly under the object glass of the telescope. The least change in the direction of the needle will be indicated by a change in the reading of the scale marked by the middle vertical wire of the telescope. The best form of instrument for the measurement of absolute declination is a theodolite, or altitude and azimuth instrument, in front of which is suspended a collimator magnet, by fibres of untwisted silk, resting horizontally in a stirrup of gun metal. The collimator magnet is a hollow cylindrical magnet, with a small object glass like a telescope, and a horizontal scale at its focus. The adjustments of this instrument consist in bringing the collimator magnet into the magnetic meridian without torsion of the thread; in determining the zero division of the scale corresponding to the magnetic axis of the collimator magnet; and in bringing the line of collimation of the theodolite telescope into the magnetic meridian, its vertical wire coinciding with this division. These adjustments are all made at once, by putting in a bar first, equal in weight to the collimating magnet, and adjusting the stirrup approximately; then, after restoring the collimating magnet by repeated trials, making half the necessary correction by moving the theodolite in azimuth, and half by turning the torsion screw at the top of the thread, till the same division of the scale is read, with the collimator in two positions, the second position being produced by turning the collimator over, so that it shall have revolved 1800 about its optical axis. There is then no torsion of the thread, the axis of the collimator magnet and of the theodolite are both in the magnetic meridian, and the division read is the zeio of the scale. If in this position the verniers of the azimuth circle of the theodolite be read, and if its telescope then be turned in the direction of some object, whose azimuth is known or can be afterwards determined, the difference of the reading, added to or subtracted from the azimuth of the object, will give the absolute declination. The angular value of one division of the scale is determined by measuring with the theodolite the horizontal angle subtended by a certain number of the divisions, the magnet being temporarily fixed. ABSOLUTE HORIZONTAL INTENSITY. 357 If a denote the angular value of one division of the scale, and - the ratio of the torsion and magnetic forces, the true declination changes are deduced by multiplying the observed differences of reading by a (1 - ). The value of is determined by turning the torsion through two large angles, and noting the corresponding differences of reading. If u denotes the angular value of the former, and u that of the latter, H u F X-U ABSOLUTE HORIZONTAL INTENSITY Requires for its determination, 1, experiments of deflexion, 2, of vibration. The former give the ratio of the magnetic moment of the deflecting magnet to the horizontal intensity, the latter the product of the same quantities, and their separate value is obtained by algebraic elimination. Experiments of deflection consist in placing a magnetic bar, called a deflector, at one side of a freely suspended magnet, in a line drawn horizontally through the centre of the suspended magnet, perpendicularly to the magnetic meridian, its axis coinciding with this line. The deflector should be placed at three different distances from the suspended magnet on this line, in direct and reversed positions, or turned end for end, at each. Experiments of vibration consist in suspending the same magnet which was used as a deflector, and noting the times at which some central division of the scale passes across the vertical wire of the telescope, at the beginning and end of at least 300 horizontal vibrations of the magnet, the magnet vibrating steadily in a very small are. As the time of vibration depends on the form and weight of the suspended mass as well as upon the product of the magnetic moment and horizontal intensity, its moment of inertia must be ascertained by means of a series of vibrations with two cylindrical weights of equal dimensions, whose moment of inertia is known, at opposite ends of the magnet. If m denote the magnetic moment of the deflecting magnet, x the horizortal intensity, the formulas are - i 2X tan u ~358~ GEODESY. Mnx T i the first of which r dist. between centres of deflecting and suspended magnets in feet and decimals. -u angle of deflection obtained by multiplying half the mean of each partial result by the coefficient (see above), a (1 +-) In the second = — 31416 T2 -K= ~, K T - T Tr'= time of vibration with weights. T -= " without " The moment of inertion of the weights is I' (2+ 2r) p Tn which 1 = interval of the points of suspension. 6" r- -radius of the cylinders in decimal of a foot. 2p == their mass in grains. THE INCLINATION is found directly by the dipping needle, which consists of a magnetic needle, suspended at the centre of a graduated vertical circle. The mean must be taken of results with several needles, reversed on their magnetic axes, and reversed as to their poles by remagnetizing. Observations should be made at different azimuths to test the limb of the instrument, which is often magnetic, in which use the formula tan 8 -tan X cosee a In which 6 0 inclination sought " = — " observed. aC azimuth of the vertical circle. The inclination may also be found by means of the horizontal and vertical components of the intensity, as it would be determined by the direction of their resultant. The vertical component is observed by means of a vertical force magnetometer, which is a needle suspended like the dipping needle, but placed in a plane perpendicular to the magnetic meridian, and made to vibrate in this plane. TIHE INCLINATION. 359 On the other hand the vertical component mlay be deduced from tle inclination and the horizontal component. For further information on this subject see Lees Tables and Formula,' and Riddell's Magnetical Instructions. For the theory of magnetism as applicable, see the papers of Gauss, and a late elementary work of Prof. Lamonte, of Munich. Translations of some of the papers of (Gans,s Lanonte, and Weber have been published in Taylor's Scientific Memoirs, Parts 5, 6, 11, and 12. APPENDDIX TO PART VI. INSTRUMENTS FOR EXTRA MERIDIAN OBSERVATIONS. THE principal of these are the equatorial, and the altitude and azimuth instrument. THE EQUATORIAL ts a telescope usually of large size, upon an equatorial mounting. The latter consists, I, of a strong metallic axis, placed in a position parallel to the axis of the earth, so as to point to the pole of the heavens, the lower end of this axis, which is called the polar axis, being enlarged into a circle called the hour circle, the plane of which is parallel to the equator, and the circumference of which is divided into hours and fractions of an hour. 2. Of another axis crossing the upper end of the former at right angles, called the equatorial axis, one end of which is enlarged into a circle called the declination circle, divided into degrees and fractions of a degree, and firmly fastened to the other end of which, at right angles, near the middle of its tube, is the telescope. The instrument must be so adjusted that when the optical axis of the telescope describes the meridian as the instrument moves upon the equatorial axis alone, the index of the hour circle is at the zero, and when the optical axis points to a star in the equator, the index of the declination circle is at zero. Ther if the instrument be turned on its equatorial axis till the index or vernier of the declination circle points to the declination of any, celestial object as given by the Nautical Almanac or by catalogue, and on its polar axis till the index of the horn circle points to the hour angle, which is the difference between the right asdension and the time by the siderial clock, the object will be seen in the centre of the field of view of the telescope. As it passes out of the field of view by the rotation of the earth on which the instrument stands, the telescope is made to follow it by a rotation of the instrument on its polar axis alone. Attached to this axis is a clamp and screw of slow motion for the purpose. Sometimes the polar axis is made to move by clockwork, the velocity being regulated by friction, and the motion becoming uniform when the friction is equal to the accelerating force of the clock-weights. In the Fraunhofer* mounting, a hollow inverted frustum of a cone contains balls, supported at the ends of a flexible bar at right angles to the axis of the frustum, about which it revolves, carrying the balls which rub against the sides of the frustum. The velocity should be different for the sun, moon, each of the planets, and a So called, from the inventor and first manufacturer, Fraunhofer, of Munich. 362 APPENDIX VI. for the fixed stars. If the bar be lowered in the frustum, less velocity will make the friction equal to the accelerating force. If raised, more velocity will be required. For each kind of heavenly body the requisite velocity is determined by experiment, and a permanent mark made where an attached index stands. Approximate adjustment is sufficient for this instrument,which is crdinarily used as a differential instrument, by means of an appendage which we proceed to describe, called THE POSITION MICROMETER. This is an eye-piece which screws on in place of the ordinary eye-piece of the telescope, and consists of a circle of brass about four inches in diameter, the plane of which is perpendicular to the optical axis, which passes through its centre. It is graduated on the outer rim, the graduation being numbered to 360~. A rectangular box, about one inch by four, and the eighth of an inch in thickness, is fitted to the circle in the position of a diameter, at the ends of which are micrometer screws, which move each one of two parallel wires along a notched scale, the wires and ecale being seen (when the eye is applied to the telescope) at the focus of the object glass, where also the image of the heavenly body is formed. The circle carrying the box has also a motion round the optical axis, by means of a screw projectinl perpendicularly to the plane of the circle, which acts as a pinion by cogs, in a cogwheel of less diameter than the circle, attached to the piece which screws into the telescope to which two verniers, 900 apart, marked A and B, are firmly fixed. To determine the right ascension and declination of a new heavenly body, as a comet, for instance, with this instrument in any part of the visible heavens, let the object be brought into the field of view at the same time with some fixed star, one whose place is given by catalogue, if possible. Bring the star to one of the movable micrometer wires, and turn the micrometer in position, i. e. round the optical axis till the wire threads the star in its motion along its diurnal path. The wire is then parallel to the equator. Let the two wires now be separated by turning the micrometer screws till one of them passes through or bisects the star, and the other bisects the comet; the number of turns of the screw shown by the notched scale, and the fractions of a turn by the screw-head, will indicate the difference in declination between the comet and the star.* Let the micrometer now be turned in position 900, and the transits of the comet and star across the two wires be observed by the siderial clock. The difference in the times of transit will be the difference of right ascension of the comet and star. The absolute right ascension and declination of the comet thus becomes known, if that of the star of comparison be known fiom catalogue. If this be not the case, the star must be brought to the centre of thie field of view indicated by the point at which a third wire at right angles to the other two crosses them when they are made to coincide at the zero of the notched scale, and the approximate right ascension and declination of the star must be noted by the declination and hour circles and clock, with sufficient accuracy to identify it, in * It will be found convenient to make the wires coincide before commencing the operation with the screwhead of one of them at the zero, and let this be the only ane moved, if possible, and read by its screw., For methods of exact adjustment, see Lend. Ast. Soc. Memoirs, vol. IV. p. 495. TlE ALTITUDE AND AZIMUTH INSTRUMENT, 363 order that its place may be more exactly determined by observation with the meridian instruments, the transit and mural, at some subsequent time. The micrometer screw head is divided into 100 parts. The value of a single turn of the screw in arc is determined by measuring the diameter of a planet, given by the Nautical Almanac or the known distance apart of a pair of stars, and then by the proportion, as the number of turns and hundredths of a turn of the micrometer screw is to the known distance measured, so is 1 to the value of one turn. MEASUREMENT OF ANGLES OF POSITION AND DISTANCE OF DOUBLE STARS. The angle of position of a pair of stars is the angle which the visual plane passing through both the stars makes with the plane of the declination circle passing through the larger star. It is estimated from the s. round by the w. to 3600. The following is Capt. Smythe's method of observing position and distance. Bring the wires coinciding at the zero of the scale, with the index of one screwlhead at the zero, upon the line of the two stars, so as to bisect both, and read one of the verniers; next turn the instrument 900 in position, and measure the distance of the stars apart; finally turn in position till one of the stars runs along either of the wires, and read one of the verniers again, the difference between the first and last vernier reading, will give the angle which the visual plane of the two stars makes with the equator, from which the angle of position may be obtained in an obvious manner. In the transit instrument and instruments of that class the wires are made visible at night by a lamp placed at one end of the supporting axis, which is left open for the purpose, with a piece of glass over it; the light is received by a small plane mirror in the axis of the telescope, and reflected down the tube to the wires. In the equatorial instrument the horizontal tube bearing the lens (colored red) of a small lamp is inserted in the tube of the telescope, near the wires of the micrometer, and a reflector so arranged as to throw the light on the wires. When the object to be observed is so faint as not to bear illumination, a ring micrometer is used. This is a black circle on a piece of plane glass in the focus of the object glass, with which differences of right ascension and declination are obtained by noting the times occupied by the two objects to be compared in crossing the circle. Half the sum of the times of either object's making the transit of the circumference on opposite sides will be the time of its passing the middle diameter, and the difference of the time of passing the middle diameter by the two objects will be their difference of right ascension. For declination it is necessary to ascertain, by experiment, the time of an equatorial star's passing over a diameter of the ring, by observing the time of any other star, and multiplying by the cosine of the declination. Then the ratio of the time occupied by a star in passing over a chord of the ring to the time which it would occupy in passing over the diameter is the cosine of an angle, the sine of which to the radius of the ring is the difference of declination between the centre of the ring and the star. (For the whole theory see Ast. Nach., Vol. 8.) THE ALTITUDE AND AZIMUTH INSTRUMENT. This is in effect a large theodolite, with two micrometer microscopes, 1800 apart,.3 both the horizontal and vertical limb. It may be used as a theodolite, also as a 364 APPENDIX VI. transit instrument. It may be used like the zenith sector, for determining latitude by a star nearly on the meridian, the instrument having a spirit level parallel to the plane of the vertical circle. The following is a good form. READINGS. i K T -perLevel.. Microscopes. A B 0. a: iE1 I__._ _ _ ^. I ^ 1 ^ 1.I 0. h.m.s. o0 f 0 o i There are five vertical and five horizontal wires, the latter of which are convenient for observing single altitudes, or equal altitudes of the sun for either time or latitude. The mean of the times of the two limbs passing the five horizontal wires is taken as the time of the altitude shown on the vertical limbo A very accurate mode of determining the true time and error of a ti eme keeper is by equal altitudes, morning and afternoon. If the sun did not change his declination in the interval between the two observations, half the interval in time added to the time of the morning observation would express the hour by the time keeper, when the sun was on the meridian. But as the declination does change, a correction of the half interval must be made, the formula for which is d - p = dJ (tan d cot d p -tan X cosec A p) In which d p = the correction of the half interval, d- = the change of declination in the half interval, X =- the latitude ( if south). The correction is + if the declination is increasing, and -if decreasing. The time of the sun's being on the meridian being corrected for the equation of time will give the time of mean noon by the watch, which will show the error of the watch. If a star be used instead of the sun, no correction is requisite for change of declination, the mean between the two times of observation must be compared with the computed mean time at which the star culminates, in order to have the error of the time keeper. If the readings be taken on the horizontal circle at the two times of observation, X This is for error in the divisions of the limb, tested by running the microscope over thom, CONVERSION OF ASTRONOMIC AND GEOCEN rRIC LATITUDES, 365 the reading midway will correspond to the direction of the meridian. This also, in the case of the sun, requires a correction for change of declination in the interval, the formula for which is cos 3 dz= C sin z cos X cos a In which dz = correction of azimuth. a = observed altitude. CONVERSION OF ASTRONOMIC AND GEOCENTRIC LATITUDE. From the nature of the astronomic instruments, the zenith point being determined by a plumb line, basin of mercury, or spirit level, it follows, by the law of gravitation, that the line from the station to the zenith is a normal to the elliptical meridian; and the angle which this line makes with the major axis or equatorial diameter will be the astronomic latitude, or latitude deduced from observation. This will be the angle MBA, in the diagram. Pt C ~ 0. A The expression for the subnormal BQ is 2 X Aa and the ratio of this to cQ x is B2 A But fiom the diagram we have the proportion CQ: BQ: tan CMQ tan BMQ.'. A: B2:: cot MCQ: cot MBQ But if the ratio of A to B be taken as 305 to 304, then B2 - = 9934 Aand the formutl for converting astronomic into geocentric latitude will'b tan \' -= 9934 tan A X - astronomic latitude, X'- geocentric latitude. 366 APPENDIX VIz RADIUS OF CURVATURE IN TERMS OF THE LATITUDE. AVe have had occasion to use al expression for the radius of curvature* of the meridian considered as an ellipse, in terms of the latitude of the point of the meridian, under consideration, in various places in the part of this volume devoted to Geodesy. The following is the mode of deriving it. The ordinary expression for the radius of curvature of the ellipse found in elementarv mathematical works is (a4 yo + b2 x2) 3 P — ~~16~1 (1) P = ~^^~4 b4( il which a and b denote the semi-axes of the ellipse x and y, tee co-ordinates of the point at which p is the radius of curvature. If X denote the latitude of this point, since it is the angle which the normal makes with the major axis, we have a2 y. a4 y2 at y 2 + b4 x2 tan.. ta2 =.. sec = whence, tanl2 A ay4 y2 - sin2 2 sec2 ~ a4 y2+ b4 x = But (e denoting the excentricity of the ellipse), a2 b2 a2 ( a2 b2) yi e -- (2).. e2sin2 a" a*4 y2+ b4 x2 b4 x2 + a2 6b y2 b2 (b2 x + ay2) a b4 1 - e2 sin2 A -- -- 1 e sin = a4 y2 + 6 X~ = a y2 + b4 x2 a4 y2 + b~ x2 (m) siace the part of the numerator in parenthesis is, by the equation of the ellipse, equal to a2 be. From the last equation we obtain a3 b6 +(a4 y 1 -b" x1) a. 1 - e sin2e X) Substituting the second member of the last result for the numerator of (1) thaS formula becomes b2 1 a (1 - esina X) (, But from (2) b 2- a (1 -e2), therefore (3) becomes I -e2 p a (1-e sine X) (4) DETERMINAT"ON OF THE FIGURE AND DIMENSIONS OF THE EARTH FROM THE MEASUREMENT OF TWO DEGREES OF THE MERIDIAN AT TWO DISTANT LATITUDES. The deviation of an oblate spheroid from a sphere is expressed by what is termed * The radius of curvature of a curve at any point is the radius of a circle having the same curvature as the curve at that point. FORM OF TERRESTRIAL MERIDIAN. 367 its compression or oblateness. This is the ratio of the difference between its axis to the major axis in symbols, ts representing the oblateness, a-b --- (1) (gC -(I) c'. b- (1-o) a But (2) p. 366, a2' — b as- - (1 )2 a e -- a 2 =1 (1 - t) =- =_ 1_&2 2W & a a2 omitting "2, which is a very small fraction, in consequence of the smallness oI u, we may write e2 202 (2) But from (4) p. 366, applying either the binominal or McClaurin's theorem to the second member, we have p = a (1 - e2 -+ e" sin2 X + terms too small to affect the result) (3) By formula (5) p. 100, cos 2X - 1 - 2 sine X, sin X 1- cos 2X' 2 4 substituting the second member of this last in place of the first in (3), that equation becomes p= a (1- ea- e2cos2X + &c.) or by (2), p =a (1 - -- c cos2X) (4) If now d denote the length of a degree measured in the latitude X, since p is the radius of the arc of the meridian in that latitude, 2rp rp -~360 - 180 hence substituting the value of p given by (4) 7ra d= — (I -- -- cos 2X (5) If d' denote the length of a degree measured in another latitude t, in a similar manner 7ra I' - 18- (1 -L O- o~ cos 2X') 1- w,(1 + 3 cos 2X)' —1 -, (1 +3 cos 2X') Performing the division in the second member, and neglecting the squares and higher powers of w we have -=1 + C (cos 2' -- cos 2X) 368 APPENDIX VI. Therefore 1 3 -- ~ cos 2X'-cos2A d () Hence this rule. 1. Form a fraction which shall be the ratio of a degree in one latitude to its excess over a degree in another latitude. 2. Form another fraction, which shall be the ratio of unity to the difference between the cosines of the doubles of each latitude. 3. Take ~ of the product of these two fractions. The value of a, being known, that of a may be found from (5)~ EXAMPLE. The length of a degree at the equator is d' = 56*753 toises..". " at 450 =-57*008 cos 2A =0 cos 2A' - I d -''255,& -- 57.008 2'255'510. 255 3 56'753 - 170'259 = 85'129 0299. We have had occasion to use the expression a - (1 — e sin" A) A in which N denoted the y normal, and X the latitude. This formula may be deduced as follows:-The expression for the y normal in the ellipse is ari y~2- b4 x2 y normal = ( b ) X But we have seen (m) p. 366, that a4 y2 + b4 x2 1 2 b4 1 e2 sin.2 > whence the formula required. PRIME VERTICAL TRANSIT. A transit instrument, mounted in the prime vertical, or at right angles to the meridian, affords a very accurate method of determining latitude or declination when either is known, by observing the times of transit of the same star over the prime vertical on both sides of the zenith. Stars near-the zenith are the best for this purpose, as tne interval between the two transits is shortest for them, and there is less opportunity for instrumental changes between the two observations. The triangle rzs, which we have so often used in the preceding pages, which must, for our present purpose, be considered right angled at z, and in which r = the half interval between PRIME VERTICAL TRANSIT. 369 the transits, will produce the requisite formula in a very simple manner. They are as frIollows: cos of I interval reduced to arc X cot dec. = cot lat. sin dec. cot lat. sin alt. l. cos. hour ang. = c - sin, lat. cot. dec. Tle first is for computing the latitude of the station when the declination oi the star observed is known, or vice versa. The other two are for the purpose of determining at what altitude to set the instrument, and at what time to look for the transit of any given star. In the latter two an approximate value of the latitude is to be used. For the description of a large transit instrument, contrived for rapid reversal in the prime vertical, and having the telescope at one end of the axis, so that the striding level need never be removed from the supporting axis, see Struve's accounr of the Pulkova instrument, in 468 of the Astronomische Nachrichten of Schumacher. See also the Washington Astronomical Observations of 1845, introduction p. li. andi p. 131. In these instruments there are 16 wires, 7 on each side of the middle one, The transits are taken over seven, and the instrument being quickly reversed, the transits are taken over the same seven in the reverse order. The following is an example of one ofi Struve's observations. The 2d colaumn reads upward. January 15th, 1842, a Draconis. EAST VERTICAL. Telescope S. Telescope S. Wires. 1. 17 54 30-75 19 42 51'4 II 55 8'65 42 13-65 IIL 55 44*4 41 38'0 TV. 56 22'25 40 59.85 V. 57 0~6 40 21*7 VI. 57 40'9 39 41'4 VII. 17 58 19.5 19 39 2'7 Telescope N. Telescope N. VII. 18 1 4~0 19 36 17'85 VI. 1 455 35 37 V. 2 29'8 34 52-35 IV. 3 12~7 34 9'3 III. 3 57-6 33 24.7 II. 4 39-8 32 42-1 I. 18 5 26-35 19 31 55'6 i = 0t687 i' - 0'923 inme of meridian transit not corrected for azimuth error of the instrument, N. 18h 48"' 41e.13 S. 41'05 Mean 18 48 41'09 A knowledge of the distance of each wire from the optical axis is unnecessary; it if this distance be denoted by c plus when the wire is N. of the optical axis, and 24 379t APPENDIX VAo nTuns when it is south, we have, I and t' being the corresponding hour angles, 4;i declination, and 0 the latitude, -sin c = cos t COS d sin ( - siln cCos 5 + sin c = cos t' cos si -sn - si s s o = (cost' cos t) cos d sin -- sin J cos 4 2 sin c = (cos tV - cos t) cos J sin p If we make (' -- t).( -, and d (t' — ) -i t, we have tan - tan cos s cos u (1 sin c =-sin s sin u cos J sin + Formulal (1t gives the declination, c being eliminated. " (2) " distance of the wire from the optical axis. Thlnc following is the application of (1) to the example above. %r.v re,. T. III. IV. V. V1. VI. W.~l..:= t' 5 1 48 0'*79 47 5 *09 45 53*69 44 37*69 43 21-19 42 0O59 40 43-29 Telescoe. S, ='. 1 26 29.34 28 2.39 9 27.1930) 56.6932 -2.64 33 5159 35 13.94 N.. = (t + 4t') 1 48 40-531 484687 48 50-22 48 53*60 48 55.96 48 58-05 48 59311 A =- (t' - t) 1 5 27*86 4 4 4567 4 6 *62 i 5255 2 44-64 2 2-25 1 222331.i. coS 9(*990] 167 00871 00642 00411 00250 00107 00020! it, toS. os 9.9998765 063 301 516 688 828 c2 Mean.,p ". tln 1023428 23457 2 24 8 2345728 2345728 2345728 2345728 234a728 6o hT. t!an 10|20245660 662 671 655 6661 663 67O0 2245.66:i8 J 15jo 11' 39" 39"'04 39t'*233 38" 9l0 39"1 39"06 3t)'l j590 1 31'.071 If the inclination of the axis be denoted by i, which is the mean of the two incen-ations, telescope N. and telescope s., then + -- i should be used in place of b in tWha abYove formulas, or the correction for the declination should be sin 26 di=.- i sin 23 In the example above 6 590 11t 3911071 Correction for inclination of axis dd = 0 -814 Observed declination, 590 11' 39,885 Thli declination thus lound is exact only on the hypothesis that the azimuth of the axis of rotation is zero. If there be an azimuth a we have -a -- for the angle at the pole sin 0 between the true meridian and the meridian of the instrument, or circle of declination perpendicular to the circle described by the optical axis of the instrument. The instant of transit of the star over the meridian of the instrument is the half sum of thes times of corresponding transits E. and W. Thus for the star o Draconis abos,, we have PRIME VERTICAL TRANSIT. 371 Wires. Telescope S. Telescope N, h In a A m i I. 18 48 4110 18 48 40-93 II. 41-15 41'25 III. 41,20 41^07 IV. 41*05 41 V. 41*15 41,15 VI. 41*15 40~95 VII. 41*10 40'97 Mean 18 48 41*13 41*05 Mean 18. 48i'" 41809 tine of meridian passage. T'Ie instant of meridian passage p requires a small correction for the difierence of inclinations of the axis in the two verticals E. and W. Denoting the former by i, and the latter by i' i - i' sin & dp --- 30s/ sin (+ S ) sin ( -) n In our example, i -= - 0'*687, i' = - 0'*923. i' -^- 0'2336,.-. ==- 0"'08 and hence the true time of meridian passage by the instrument instead of 18.' 48' 41.*09, as above, isp' = 18 48 4101. If a denote the right ascension of the star, and e the error of the clock, let a' == - e denote the time of passage of the star over the true meridian. Then, for the angle of the two meridians w -= p' -a' in time and for the azimuth of the axis of rotation reckoned from the south round by the west, a = 15i7 sin,, in are With this the correction of the observed declination for the axis of rotation becomes dJ = (15r) 2 sin 1'l sin 2J For stars near the zenith 0 may be used instead of S, and the formula becomes dJ = i (15r)2 sin 1" sin;2 The clock error in the above example on 15th January, 18a 48t was e = 8 31 The apparent right ascension of o Draconis a = 18a 48- 50o^ 17.-. a= 18h 483n 4r1-8t.'.. a~ 0'*85 in time, and a - I - 1l'.0 in ar Finally the correction of the declination is, dd -=O"00017 too small to notice. 872 NOTE, The formula, for reftaction, given at p. 845, depends on the well-known larw cf atmospheric refraction, viz., that it is nearly proportional to the tangent of the zenith distance, and therefore for very small differences of zenith distance, proportional to tlhe differential of the tangent. But dzs d tan zs = cos zs being the zenith dristance. Applying logarithms and introducing the constant; )f refraction rf, the formula in the text is obtained. CONCLUDING NOTE. T'he anthor had designed giving the theory of eclipses anac occultations, with their;application to the determination of longitude, and also Bessel's method of measurilng an arc of the meridian; but as longitude by moon culminations is the hetter mode, and as measurements of the meridian are not at present being carried on or immediately contemplated in this country, it was thought expedient not to ncrease tne size of a worlk intended for very general use. As, however, the metl-od by occultations requires ollly L1 con-moii telescope and a good timepiece, the read:ci is referred to the Appendix of the Nautical Almanac for 1836, pp. 134 andt 145, for the necessary forinulas, and an example of tlhe computation of longitudoe by an occultation. (See also Lee's Tables and FormulasI p. 78, Part IIf.) TABLES TABLE I. CPage 1 Difference of Latitude and Departure for ~ Point.?.; N.- E. N. 4W. S.~E. S. W. Dist. Lat. Dep. Dist. Lat. IDep. Di. t. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01.0 00.0 6i 60.9 o3.o i121 120.9 05.9 i8i i8o.8 0o8.9 241 240.7 1i.8 2 02.0 00.,I 62 61.9 o3.o 22 121.9 o6.0 82,i8.8 08.9 42 241.7 11.9 3 o3.0 oo. I 63 62.9 0o3.1 23 122.9 06.0 83 182.8 o9.o 43 242.7 II -9 4 o4.oo 00.2 64 63.9 o3.1 24 123.9 o6.I 84 i83.8 09.0 44 243.7 12.0 5 o5.o 00.2 65 64.9 03.2 25 124.8 06.1 85 i84.8 og9. 45 244.7 12.0 6 o060 00.3 66 65.9 03.2 26 125.8 06.2 86 85.8 o09.1I 46 245.7 12.1 7!07.0 00oo.3 67 66.9 o3.3 27 126.8 06.2 87 186.8 09.2 47 246.7 12.1 8 o8.0 00.4 68 67.9 0o3.3 28 127.8 o6.3 88 187.8 09.2 48 247.7 12.2 9 0o9.o 00.4 69 68.9 o3.4 29 28.8 o6.3 89 i88.8 09.3 49 248.7 12.2 o io.o oo00.5 70 69.9 o3.4 30 1I29.8 0 6.4 90 189.8 09.3 50 249.7 12.3 11 11.0 o00.5 71 70.9 03.5 i31 130.8 o6.4 191 190.8 09.4 251 250.7 12.3 212.0 oo00.6 72 71.9 03.5 32 131.8 06.5 92 191g.8 09.4 52 251.7 12.4 13 r3.o oo.6 73 72.9 o3.6 33 132.8 o6.5 93 192.8 09.5 53 252.7 12.4 i4 14.0 00.7 74 73.9 o3.6 34 i33.8 o6.6 94 1i93.8 09.5 54 253.7 12.5 15 15.o 00.7 75 74.9 03-7 35 134.8 o6.6 95 194.8 09.6 55 254.7 12.5 16 i6.o oo00.8 76 75.9 03.7 36 i35.8 06.7 96 195.8 09.6 56 255.7 12.6 17 17-090.8 77 76.9 o03.8 37 i36.8 06.7,97 196.8 09.7 57 256.7 12.6 i8 18.o 00.9 78 77.9 03.8 38 137.8 o06.8 98 197.8 09.7 58 257.7 12.7 19 19.0 00.9 79 78.9 03.9 39 138.8 o6.8 99 198.8 09.8 59 258.7 12.7 20 20.0 0.0 80 799 03.9 4o 139.8 06.9 200 199.8.8 6o 259.7 12.8 21 21.0 01.0 8i 80.9 o4.o 141 i40.8 06.9 201 200.8 09.9 261 260.7 12.8 22 22.0 01.1 82 81.9 o4.0 42 141.8 07.0 02 201.8 09.9 62 261.7 12.9 23 23.0 oi.1 83 82.9 04.1 43 142.8 07.0 03 202.8 io0.0 63 262.71 2.49 24 24.0 I01.2 84 83.9 o4.i 44 43.8 07.1 04 203.8 io.o 64 263.7 13.0 25 25.0 01.2 85 874.9 04.2 45 I44.8 07.1 05 204.8 io.i 65 264.71i3.0 26 26.0 oi.3 86 85.9 04.2 46 i45.8 07.2 06 205.8 io.i 66 265.7 i3.i 27 27.0 oi.3 87 86.9 o4.3 47 146.8 07.2 07 206.8 10-2 67 266.7 13. 28 28-0 o1.4 88 87.9 o4.3 48 147.8 07.3 o8 207.7 10.2 68 267.7 13.2 29 29.0 o0I.4 89 88.9 o04.4 49 148.8 07.3 09 208.7 o10.3 69 268.7 13.2 30 30.0 01oi.5 go90 899 4.4 50 149.8 07.4 io 209.7 o10.3 70 269.7 13.2 31 31.0 01.5 91 90.9 04.5 151 i50.8 07.4 211 210.7 10.4 271 270.7 i3.3 32 32.0 oi.6 92 91.9 o4.5 52 151.8 07.5 12 211.7 10.4 72 271.7 13.3 33 33.0 oi.6 93 92.9 04.6 53 152.8 07.5 13 212.7 o10.5 73 272.7 i3.4 34 34.o 017 94 93.9 o04.6 54 53.8 07.6 14 91~3,7 io.5 74 273.7 13.4 35 35.0 01.7 95 94.9 04.7 55 154.8 07.6 i5 214.7 io.5 75 274.71 3.5 36 36.0 oi.8 96 95.9 04.7 56./55.8 07.7 i6 2T-o 1io.6 76 275.7 513.5 37 37.0 oi.8 97 96.9 o4.8 57 i56.8 07.7 17 216.7 10.6 77 276.7 i3.6 38 38.0 01.9 98 97.9 o4.8 58 157.8 07.8 18 217.7 10.7 78 277.7 13.6 39 39.0 01.9 99 98.9 04.9 59 1i58.8 07.8 19 218.7 10.7 79 278.7 13.7 40 40.0 02.0 100 99.9 04.9 6o 159.8 07.9 20 219.7 io.8 80o 279.7 13.7 4i. 41.o 02.0 o101 [oo00.9 05.0 161 i60o.8 07.9 221 220.7 10.8 281 280.7 13.8 42 41.9 02.1 02 101.9 05.0 62 161i.8 07.9 22 221.7 10.9 82 281.7 i3.8 43 42.9 02.1 03 102.9 05.i 63 162.8 o8.0 23 222.7 o10.9 83 282.7 13.9 44 43.9 02.2 o4 10io3.9 05.i 64 163.8 08.0 24 223.7 11.0 84 283.7 13.9 45 44.9 02.2 0o5 104.9 05.2 65 164.8 08.i1 25 224.7 ii.o 85 284.7`4.0 46 45.9 02.3 o6 105.9 05.2 66 165.8 o8.i 26 225.7 11.1 86 285.7 14.o 47 46.9 02.3 07 106.9 o5.3 67 i66.8 08.2 27 226.7 ii.i 87 286.7 i4.i 48 47.9 02.4 o8 107.9 o5.3 68 167.8 08.2 28 227.7 11.2 88 287.7 14.1 49 48.9 02.4 09 108.9 o5.3 69 168.8 08.3 29228.7 11.2 89 288.7 14.2 50 49.9 02.5 Ioio109.9 o5.4 70 169.8 08.3 30 229.7 11.3 90 289.7 14.2 51 50.9 02.5 III 110ii.9 o5.4 171 170.8 o8.4 231 230o.7 11.3 291 290.6 i4.3 52 51.9 02.6 12 111.9 o05.5 72 171.8 o8.4 32 231.7 11.4 92 291.6 i4.3 53 52.9 02.6 13 112.9 05.5 73 172.8 08.5 33 232.7 11.4 93 292.6 14.4 54 53.9 02.6 14 113.9 o05.6 74 173.8 08.5 34 233.7 11.5 94 293.6 I4.4 55 54.9 02.7 15 114.9 05.6 75 174. 8 o08.6 35 234.7 11.5 95 294.6 14.5 56 55.9 02.7 i6 115.9 05.7 76 175.8 o8.6 36 235.7 ii.6 96 295.6 14.5 57 56.9 02.8 17 116.9 05.7 77 176.8 08.7 37 236.7 ii.6 97 296.6 i4.6 58 57.9 02.8 i8 17.9 o05.8 78 177.8 08.7 38 237.7 11-7 98 297.6 14.6 59 58.9 02.9 19 118.9 o5.8 79 178.8 o8.8 39 238.7 11.7 99 298.6 14.7 60 59.9 02.9 20 119.9.9 17908.8 4 239.7 11.8 300 299.6 14.7 Iist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep Lat. Dist. Dep. ILat. E. jN. E. jS. W. iN.' W. oS. [For 71 Points. 9I1~~ 3 3~./o 3~=7~. Page 2] TABLE I. Difference of Latitude and Departure for j Point.. -7 - N. E. N.AW. S.^ E. S. W.. Dist Lat. Dep.Dist. Lat. Dep.. L t. Lat. Dep. Dist. Lat. Dep. I OI.o oo.i 61 60.7 o06. 12 I20I.4 I1.9 I I80. II7 -7 2 239.8 23.6 2 02.0 00.2 62 6I.7 o6.I 22 I21.4 II2.0 82 181. 17.8 42 240.8 23.7 3 03.0 o0.3 63 62.7 06.2 23 122.4 12.1 83 182.1 17.9 43 241.8 23.8 4 o04. 00.4 64 63.7 o6.3 24 I23.4 12.2 84 I83.I 18.0 44 242.8 23.9 5 5.0 00oo.5 65 64.7 o6.4 25 124.4 12.3 85 I84. I 8.1 45 243.8 24.0 6 6.o oo.6 66 65.7 o6.5 26 I25.4 12.4 86 I85.I I8.2 46 244.8 24.1 7 07-0 00.7 67 66.7 o6.6 27 I26.4 12.4 87 I86. I8.3 47 245.8 24. 2 8 8.o oo00.8 68 67.7 o6.7 28 127.4 12.5 88 187-. 18.4 48 246.8 24.3 9 og.o 00.9 69 68.7 o6.8 29 128.4 12.6 89 188.i I8.5 49 247.8 24.4 10 IO.0 OI.O 70 69.7 06.9 30 129.4 I2.7 90 I89.I 18.6 50 248.8 24.5 II I0.9 o.l 71 70.7 07.0 I3 130.4 12.8 IgI 190o. 18.7 251 249.8 24.6 12 11.9 01.2 72 71.7 o7.1 32 i3i.4 12.9 92 191.1 i8.8 52 250.8 24.7 13 I2.9 01.3 73 72.6 07.2 33 i32.4 i3.0 93 192.I 18.9 53 251.8 24.8 14 13.9 oi.4 74 73.6 07.3 34 i33.4 I3.i 94 I93.1 Ig.0 54 252.8 24.9 15 I4.9 OI.5 75 74.6 07.4 35 I34.3 I3.2 95 194.1 I9g. 55 253.8 25.0 16 I5.9 o0.6 76 75.6 07.4 36 I35.3 13.3 96 9I5.i 19.2 56 254.8 25.1 17 16.9 01.7 77 76.6 07.5 37 i36.3 13.4 97 196-. 19.3 57 255.8 25.2 i8 17-9 o0.8 78 77.6 07.6 38 137.3 3.5 98 197.0 19.4 58 256.8 25.3 I9 18.9 OI.9 79 78.6 07.7 39 i38.3 i3.6 99 I98.0 19.5 59 257.8 25.4 20 19.9 02.0 80 79.6 07.8 40 139.3 13.7 200 199g0 19.6 60 258.7 25.5 21 20.9 02.1 8i 80.6 07.9 I4I 140.3 i3.8 20I 200.0 19.7 26I 259.7 25.6 22 2I.9 02.2 82 8I.6 o8.o 42 I4I.3 13.9 02 201.0 19.8 62 260.7 25.7 23 22.9 02.3 83 82.6 08.I 43 I42.3 I4.0 03 202.0 19.9 63 26I.7 25.8 24 23.9 O2.4 84 83.6 08.2 44 143.3 i4.i o4 3. 2 03.0.0 64 262.7 25.9 25 24.9 02.5 85 84.6 08.3 45 I44.3 4.2 o05 204.0 20.I 65 263.7 26.0 26 25.9 02.5 86 85.6 o8.4 46 145.3 i4.3 06 205.0 20.2 66 264.7 26.1 27 26.9 02.6 87 86.6 08.5 47 i46.3 I4.4 07 206.0 20.3 67 265.7 26.2 28 27.9 02.7 88 87.6 o8.6 48 147.3 I4.5 08 207.0 20.4 68 266.7 26.3 29 28.9 02.8 89 88.6 08.7 49 i48.3 I4.6 09 208.0 20.5 69 267-7 26.4 30 29.9 02.9 90 89.6 o8.8 50 149.3 I4.7 10 209.0 20.6 70 268.7 26.5 31 30.9 03.0 91 90.6 08.9 i51 150.3 I4.8 211 210.0 20.7 271 269.7 26.6 32 3I.8 03.i 92 9g.6 09.0 52 151.3 14.9 12 21I.0 20.8 72 270.7 26.7 33 32.8 03.2 93 92.6 09.1 53 152.3 15.0 i3 212.0 20.9 73 271.7 26.8 34 33.8 03.3 94 93.5 09.2 54 153.3 15.1 14 213.0 21.0 74 272.7 26.9 35 34.8 o3.4 95 94.5 09.3 55 i54.3 15.2 15 214.0 21.1 75 273.7 27-0 36 35.8 o3.5 96 95.5 09.4 56 155.2 I5.3 I6 215.0 2I.2 76 274.7 27.1 37 36.8 03.6 97 96.5 og.5 57 156.2 15.4 17 216.0 21.3 77 275.7 27.2 38 37.8 03.7 98 97.5 09.6 58 157.2 i5.5 I8 217.0 2I.4 78 276.7 27.2 39 38.8 o3.8 99 98.5 09.7 59 158.2 i5.6 19 217.9 21.5 79 277-7 27.3 40 39.8 03.9 Ioo 99.5 09.8 60 159.2 I5.7 20 218.9 21.6 80 278.7 27.4 41 40.8 o4.o IOI 1oo.5 09.9 I6I I60.2 I5.8 221 2I9.9 21.7 281 279.6 27.5 42 4I.8 o4. I 02 1oi.5 10.0 62 I6I.2, i5.9 22 220.9 2I.8 82 280.6 27.6 43 42.8 04.2 o3 1 02.5 To. 63 162.2 i6.0 23 22I.9 21.9 83 281.6 27.7 44 43.8 04.3 04 Io3.5 10.2 64 163.2 i6.I 24 222.9 22.0 84 282.6 27.8 45 44.8 04.4 o5 1o4.5 o1.3 65 164.2 16.2 25 223.9 22.I 85 283.6 27.9 46 45.8 o4.5 06 io5.5 o0.4 66 165.2 i6.3 26 224.9 22.2 86 284.6 28.0 47 46.8 o4.6 07 io6.5 io.5 67 166.2 I6.4 27 225.9 22.2 87 285.6 28.1 48 47.8 04.7 o8 107.5 IO.6 68 i67.2 i6.5 28 226.9 22.3 88 286.6 28.2 49 48.8 104.8 09 08.5 10.7 69 168.2 i6.6 29 227.9 22.4 89 287.6. 28.3 50 49.8 04.9 I I09.5 Io.8 70 169.2 16.7 30 228.9 22.5 90 288.6 28.4 5I 50.8 0o5.o IIl IO.5 10.9 171 170.2 16.8 23I 229.9 22.6 291 289.6 28.5 52 51.7 05.I 12 111.5 1I.O 72 171.2 16.9 32 230.9 22.7 92 290.6 28.6 53 52.7 o5.2 13 112.5 Ii.1 73 172.2 17.0 33 23I.9 22.8 93 291.6 28.7 54 53.7 05.3 4 74 173.2 17.1 34 "32.9 22.9 94 292.6 28.8 55 54.7 05.4 i5 I4.4 ii.3 75 174.2 17.2 35 233.9 23.0 95 293.6 28.9 56 55.7 o05.5 i6 iI5.4 1.4 76 175.2 17.3 36 234.9 23.1 96 294.6 29.o 57 56.7 o05.6 I7 ii6.4 11.5 77 176.1 17.3 37 235.9 23.2 97 295.6 29.1 58 57.7 o5.7 I8 117. 4 II.6 78 177.- 17.4 38 236.9 23 3 98 296.6 29.2 59 58.7 05.8 19 8.-4 I11.7 79 178.1 17.5 39 237.8 23.4 99 297.6 29.3 60 59.7 05.9 20 II9.-4 80 79 40 238.8 23.5 300 298.6 29.4 Dist. I)ep. ILat. Dist.:Dep. Lat. Dist., Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. E.... S. W.: N. W. S. [For 7A Points. TABLE 1. [Page 3 Difference of Latitude and Departure for. Point.-; — N.IE. N. IW. S. E. S. W. Dist. Lat. Dep. Dist. Lat. Dep. Dist.Lat. Dep. Dist. Lat Dep.. Dist. Lat. e ep. I o.o oo00. 61 60.3 09.o I21 II9.7 17.8 181 179.0 26.6 24 238.4 3.4 2 02.0 00.3 62 61.3 o9.i 22 I20.7 I7.9 82 180.0 26.7 42 239.4 35.5 3 o3.0 00.4 63 62.3 09.2 23 121.7 i8.0 83 I8I.0 26.9 43 240.4 35.7 4 04.0 00.6 64 63.3 09.4 24 122.7 18.2 84 I82.0 27.0 44 24T.4 35.8 5 04.9 00.7 65 64.3 09.5 25 123.6 18.3 85 I83.o 27.1 45 242.3 35.9 6 05.9 00.9 66 65.3 09.7 26 I24.6 I8.5 86 I84.0 27.3 46 243.3 36.I 7 o6.g9 oi. 67 66.3 09.8 27 I25.6 I8.6 87 I85.o 27.4 47 244.3 36.2 8 07.9 oi.2 68 67.3 Io.o 28 I26.6 18.8 88 I86.0 27.6 48 245.3 36.4 9 o8.9 oi.3 69 68.3 IO.I 29 I27.6 I8.9 89 187.0 27.7 49 246.3 36.5 io 09.9 oi.5 70. 69.2 Io.3 30 I28.6 I9. 90 I87.9 27.9 50 247.3 36.7 ii IO.9 oi.6 71 70.2 10.4 i3I I29.6 19.2 I9I I88.9 28.0 251 248.3 36.8 12 11.9 oi.8 72 71.2 io.6 32 i30.6 19.4 92 I89.9 28.2 52 249.3 37.0 13 12.9 01.9 73 72.2 10.7 33 i31.6 19.5 93 I90.9 28.3 53 250.3 37.1 I4 I3.8 02.1 74 73.2 10.9 34 I32.5 I9-7 94 I9I.9 28.5 54 251.3 37.3 15 i4.8 02.2 75 74.2 II.0 35 I33.5 I9.8 95 192.9 28.6 55 252.2 37.4 16 I5.8 o2.3 76 75.2 I].2 36 I34.5 20.0 96 I93.9 28.8 56 253.2 37.6 17 I6.8 02.5 77 76.2 I.3 37 35.5 20. I 97 194.9 28.9 57 254.2 37.7 18 17.8 02.6 78 77.2 I1.4 38 I36.5 20.2 98 I95.9 29.- 58 255.2 b37.9 19 i8.8 02.8 79 78. I I.6 39 I37.5 20.4 99 I96.8 29.2 59 256.2 38.0 20 19.8 02.9 80 79-1 11-.7 40 I38.5 20.5 200 197-8 29.3 60 257.2 38. 21 20.8 o3. 8o 80.I 1.9 I41 I39.5 20.7 20I I98.8 29.5 26I 258.2 38.3 22 21.8 03.2 82 81i.1 2.0 42 I40.5 20.8 02 99.8 29.6 62 259.2 38.4 23 22.8 03.4 83 82.1 12.2 43 i4i.5 21.0 o3 200.8 29.8 63 260.2 38.6 24 23.7 o3.5 84 83.I 12.3 44 I42.4 2I.1 04 201.8 29.9 64 26I. 38.7 25 24.-7 037 85 84.I I2.5 45 I43.4 21.3 05 202.8 30.I 65 262.1 38.9 26 25.7 03.8 86 85. I 2.6 46 I44.4 21.4 06 203.8 30.2 66 263.I 39.0 27 26.7 04.0 87 86.i I2.8 47 i45.4 21.6 07 204.8 30' 4 67 264.2 39.2 28 27 7 o4.I 88 87.0 12.9 48 i46.4 21.7 08 205.7 30.5 68 265.2 39.3 29 28.7 04.3 89 88.o I3.1 49 147.4 21.9 09 206.7 30.7 69 266.1 39.5 30 29.7 o4.4 90 89.0 I3.2 50 I48.4 22.0 10 207.7 30.8 70 267. 39.6 31 30.7 04.5 91 90.0 13.4 I5I I49.4 22.2 211 208.7 3i.0 271 268.1 39.8 32 31.7 04.7 92 9I.0 13.5 52 i50.4 22.3 12 209.7 3i.1 72 269.1 39.9 33 32.6 o4.8 93 92.0 I3.6 53 15I.3 22.4 I3 210.7 31.3 73 270.0 401i 34 33;6 05.0 94 93.0 13.8 54 152.3 22.6 I4 2II.7 31.4 74 271.0 40.2 35 34.6 o5.i 95 94.0 13.9 55 I53.3 22.7 15 212.7 31.5 75 272.0 40.4 36 35.6 05.3 96 95.0 14.1 56 i54.3 22.9 i6 2I3.7 31.7 76 273.0 40.5 37 36.6 o5.4 97 96.o I4.2 57 155.3 23.0 17 2I4.7 31.8 77 274.0 40.6 38 37.6 o5.6 98 96.9 I4.4 58 i56.3 23.2 i8 215.6 32.0 78 275.0 40.8 39 38.6 05.7 99 97.9 i4.5 59 I57.3 23.3 19 216.6 32.1 79 276.0 40.9 40 39.6 05.9 Ioo 98.9 24.7 60 158.3 23.5 20 217'.6 32.3 80 277-0 4I1. 4I 40.6 o6.0 ioi 99.9 14.8 I6I 159.3 23.6 221 218.6 32.4 281 278.0 41.2 42 41.5 06.2 02 o00.9 15.o 62 I60.2 23.8 22 2I9.6 32.6 82 278.9 41.4 43 42.5 o6.3 03 101.9 i5.i 63 161.2. 23.9 23 220.6 32.7 83 279.9 41.5 44 43.5 o6.5 04 I02.9 i5.3 64 162.2 24.1 24 221.6 32.9 84 280.9 41.7 45 44.5 16.6 05 I03.9 I5.4 65 I63.2 24.2 25 222.6 33.0 85 28I.9 41.8 46 45.5 06.7 o6 I04.9 i5.6 66 264.2 24.4 26 223.6 33.2 86 282.9 42 o 47 46.5 06.9 07 io5.8 15.7 67 I65.2 24.5 27 224.5 33.3 87 283.9 42.1 48 47.5 07.0 o8 io6.8 i5.8 68 i66.2 124.7 28 225.5 33.5 88 284.9 42.3 49 48.5 07.2 09 107.8 6.o 69 I67.2 24.8 29 226.5 33.6 89 285.9 42.4 5o 49.5 07.3 io Io8.8 16.1 70 I68.2 24.9 30 227.5 33.7 9~ 286.9 42:6 51 50.4 07.5 III I09.8 I6.3 17I1 69.I 25.1 231 228.5 33.9 291 287.9 42.7 52 Si.4 07.6 12 10.8 16.4 72 170.1 25.2 32 229.5 34.o 92 288.8 42.8 53 52.4 07.8 13 11i.8 16.6 73 17I.1 25.4 33 230.5 34.2 93 289.8 43.0 54 53.4 07.9 14 II2.8 16.7 74 I72.I 25.5 34 231.5 34.3 94 290.8 43,. 55 54.4 o08.1 I5 I3.8 I6.9 75 173.I 25.7 35 232.5 34.5 95 291.8 43.3 56 55.4 08.2 16 II4.7 17.0 76 174.- 25.8 36 233.4 34.6 96 292.8 43.4 57 56.4 o8.4 17 1I5.7 17.2 77 175.- 26.0 37 234.4 34.8 97 293.8 43.6 58 57.4 08.5 i8 I16.7 17-3 78 I76.I 26.I 38 235.4 34 9'98 294.8 43.7 59 58.4 o8.7 19 117.7 17.5 79 177.- 26.3 39 236.4 35.1 99 295.8 43.9 60 59.4 08.8 20 118.7 17.6 80 178. 26.4 40 237.4 35.2 300 296.8 44.o 1)ist. Dep. Lat. Dist Dep. Lat. Dist. Dep. Lat. Dist.. Dep. Lat. Dist. Dep. I Lat. E. i N. E. S. W. N. W. a S. [For 71 Poinfs. Page 4] TABLE 1. Difference of Latitude and Departure for 1 Point.- //: /' N byE. N.byW. S.byE. S.byW. Dist. Lat. Dep. Dist Lat. Dis t Lat. Dep. Dist. Lat. Dep. Dist Lat. Dep. 0o.0 o00.2 6I 59.8 11.9 12 1 18.7 23.6 18I 177.5 35.3 241 236.4 47.0 2 02.o 00.4 62 60.8 12.1 22 19.7 23.8 82 178.5 35.5 42 237.4 47.2 3 o2.9 oo.6 63 6I.8 12.3 23 120.6 24.0 83 I79.5 35.7 43 238.3 47.4 4 o3.9 oo.8 64 62.8 12.5 24 12I.6 24.2 84 I80.5 35.9 44 239.3 47.6 5 04.9 oI.o 65 63.8 2.7 25 122.6 24.4 85 I81.4 36.i 45 240.3 47.8 6 o5.9 01.2 66 64.7 12.9 26 123.6 24.6 86 I82.4 36.3 46 241.3 48.o 7 o6.9 o0.4 67 65.7 13.1 27 124.6 24.8 87 I83.4 36.5 47 242.3 48.2 8 07.8 oi.6 68 66.7 i3.3 28 125.5 25.o 88 I84.4 36.7 48 243.2 48.4 9 08.8 oi.8 69 67-7 I3.5 29 126.5 25.2 89 I85.4 36.9 49 244.2 48.6 io o9.8 02.0 70 68.7 13.7 30 127.5 25.4 90 i86.3 37. 50 245.2 48.8 II 0i.8 02. 1 71 69.6 13.9 13I I28.5 25.6 191 I87.3 37.3 251 246.2 49.0 12 11.8 02.3 72 70.6 i4.0 32 129.5 25.8 92 I88.3 37.5 52 247.2 49.2 13 12.8 02.5 73 71.6 14.2 33 130.4 25.9 93 189.3 37.7 53 248.1 49.4 14 13.7 02.7 74 72.6 14.4 34 I31.4 26.1 94 19o.3 37.8 54 249.1 49.6 1 5 I4.7 02.9 75 73.6 14.6 35 132.4 26.3 95 191.3 38.o 55 250.1 49.7 16 15.7 03.1 76 74.5 14.8 36 I33.4 26.5 96 192.2 38.2 56 251.1 49.9 17 16.7 o3.3 77 75.5 15.o 37 34.4 26.7 97 193.2 38.4 57 252.1 50. I8 I7.7 o3.5 78 76.5 15.2 38 I35.3 26.9 98 194-2 38 6 58 253.0 50.3 I9 18.603.7 79 77.5 i5.4 39 I36.3 27.1 99 195.2 38.8 59 254.0 50.5 20 19.6 03.9 80 78.5 I5.6 4o 137.3 27.3 200 196.2 39.0 60 255.0 50.7 21 20.6 04.i 81 79.4 15.8 i4I 138.3 27.5 201 197-1 39.2 261 256.0 50.9 22 21.6 04.3 82 80.4 I6.o 42 I39.3 27.7 02 I98.1 39.4 62 257.0 51.1 23 22.6 04.5 83 8i.4 16.2 43 140.3 27.9 o3 I99.I 39.6 63 257.9 51.3 24 23.5 04.7 84 82.4 I6.4 44 141.2 28.1 04 200.1 39.8 64 258.9 51.5 25 24.5 5 0~4.9 85 83.4 16.6 45 142.2 28.3 05 201.1 40.0 65 259.9 51.7 26 25.5 05.I 86 84.3 16.8 46 143.2 28.5 o6 202.0 40.2 66 260.9 51.9 27 26.5 o5.3 87 85.3 17.0 47 144.2 28.7 07 203.0 40.4 67 261.9 52.1 28 27.5 o5.5 88 86.3 17.2 48 145.2 28.9 08 204.0 40.6 68 262.9 52.3 29 28.4 05. 7 89 87.3 17.4 49 i46. 29.1 09 205.o 40.8 69 263.8 52.5 30 29.4 05.9 g9 88.3 17.6 50 147-1 29.3 IO 206.0 4.0o 70 264.8 52.7 31 30.4 06.o 91 89.3 17.8 151 i48.i 29.5 21I 206.9 41.2 271 265.8 52.9 32 31.4 o6.2 92 90.2 17.9 52 149-1 29.7 12 207.9 41.4 72 266.8 53. 33 32.4 06.4 93 91.2 i8.i 53 150.1 29.8 13 208.9 4i.6 73 267.8 53.3 34 33.3 o6.6 94 92.2 i8.3 54 15i.0 30.0 14 209.9 41.7 74 268.7 53.5 35 34.3 06.8 95 93.2 i8.5 55 152.0 30.2 15 2I0.9 41.9 75 269.7 53.6 36 35.3 07.0 96 94.2 18.7 56 153.0 30.4 I6 211.8 42.1 76 270.7 53.8 37 36.3 07.2 97 95.1 18.9 57 I54.o 30.6 17 212.8 42.3 77 271.7 54.0 38 37.3 07.4 98 96.1 19. 58 i55.o 30.8 I8 213.8 42.5 78 272.7 54.2 39 38.3 07.6 99 97.1 19.3 59 155.9 3i.0 19 214.8 42 7 79 273.6 54.4 4o 39.2 07.8 1oo 98.1 19.5 60 156.9 31.2 20 215.8 42.9 80 274.6 54.6 41 40.2 o 8.0 Io 99.- 19-7 16i 157.9 31.4 221 216.8 43.1 281 275.6 54.8 42 41.2 08.2 02 100.0 19.9 62 158.9 3i.6 22 217.7 43.3 82 276.6 55.o 43 42.2 o8.4 o3 0oi.o 20.I 63 159.9 3i.8 23 218.7 43.5 83 277.6 55.2 44 43.20 8.6 04 102.0 20.3 64 i60.8 32.0 24 219.7 43.7 84 278.5 55.4 45 44.1 o8.8 05 103.0 20.5 65 i61.8 32.2 25 220.7 43.9 85 279.5 55.6 46 45.1 09.0 06 104.0 20.7 66 162.8 32.4 26 221.7 44.1 86 280.5 55.8 47 46.1 09.2 07 104.9 20.9 67 i63.8 32.6 27 222.6 44.3 87 281.5 56.o 48 47.1 09.4 o8 1o5.9 21.i 68 164.8 32.8 28 223.6 44.5 88 282.5 56.2 49 48.i og.6 09 106.9 21.3 69 165.8 33.0 29 224.6 44.7 89 283.4 56.4 50 49.o o9.8 Io 107.9 21.5 70 166.7 33.2 30 225.6 44.9 9~ 284.4 56.6 5i 50.0 og.9 III 108.9 21-7 171 I67.7 33.4 231 226.6 45.I 291 285.4 56.8 52 51.o o.I 12 109.8 21.9 72 168.7 33.6 32 227.5 45.3 92 286.4 57.o 53 52.0 10.3 13 1io.8 22.0 73 169.7 33.8 33 228.5 45.5 93 287.4 57.2 54 53.0 1o.5 14 111.8 22.2 74 170.7 33.9 34 229.5 45.7 94 288.4 57.4 55 53.9 10.7 5 112.8 22.4 75 171.6 34.i 35 230.5 45.8 95 289.3 57.6 56 54.9 10.9 i6 II3.8 22.6 76 172.6 34.3 36 23I.5 46.0 96 290.3 57.7 57 55.9 11.1 17 114.8 22.8 77 I73.6 34.5 37 232.4 46.2 97 291.3 57.9 58 56.9 113 i8 115.7 23.0 78 174.6 34.7 38 233.4 46.4 98 292.3 58.i 59 57.9 I.5 19 116.7 23.2 79 175.6 34.9 39 234.4 46.6 99 293.3 58.3 6c 58.8 11.7 20 117.7 23.4 80 176.5 35.1 40 235.4 46.8 300 294.2 58.5 DEist. Dep. Lat. Dist. Dep. j Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. E.byN. E.byS. W.byN. W.byS. [For 7 Points. TABLE I. [Page 5 Difference of Latitude and Departure for 1~ Points.,/4',,.. N.byE.jE. N.byW.JW. S.byE.AE. S byW.jW Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. i o0.o 00.2 61 59.2 i4.8 121 117 —.4 29.4 i8i 175.6 -44.0 241 233.8 58.6 2 1o.9 oo.5 62 60.1 15.1 22 II8.3 29.6 82 176.5 44.2 42 234.7 58.8 3 02.9 00o7 63 6i.i 15.3 23 119.3 29.9 83 177.5 44.5 43 235.7 59.0 4 03.9 01.0 64 62.I 15.6 24 120.3 30.I 84 178.5 44.7 44 236.7 59.3 5 04.9 01.2 65 63. I15.8 25 121.3 30.4 85 179.5 45.0 45 237.7 59.5 6 05.8 o01.5 66 64.0 i6.o 26 I22.2 30.6 86 80o.4 45.2 46 238.6 59.8 7 06.8 01.7 67 65.0 i6.3 27 123.2 30.9 87 i8i.4 45.4 47 239.6 60.o 8 07.8 01.9 68 66.0 i6.5 28 124.2 31.1 88 182.4 45.7 48 240.6 6o.3 9 08.7 02.2 69 66.9 i6.8 29 I25.I 3i.3 89 i83.3 45.9 49 241.5 60. o Io 09.7 02.4 70 67.9 17.0 30 I26.1 3i.6 90 i84.3 46.2 50 242.5 6o.7 II 10.IO7 02.7 71 68.9 17.3 131 127.1 3i.8 191 185.3 46.4 251 243.5 6i.o 12 11.6 02.9 72 69.8 17.5 32 128.0 32.1 92 186.2 46.7 52 244.4 61.2 13 12.6 03.2 73 70.8 17.7 33 I29.0 32.3 93 187.2 46.9 53 245.4 6i.5 I4 i3.6 o3.4 74 71.8 i8.o 34 130.o 32.6 94 188.2 47.i 54 246.4 61.7 i5 i4.6 03.6 75 72.8 18.2 35 i3i.o 32.8 95 189.2 47.4 55 247.4 62.0 i6 I5.5 03.9 76 73.7 18.5 36 131.9 33.o 96 190.1 47.6 56 248.3 62.2 17 i6.5 o4.i 77 74.7 18.7 37 132.9 33.3 97 191.1 47.9 57 249.3 62.4 18 17.5 o4.4 78 75.7 19.0 38 133.9 33.5 98 192.1 48.1 58 250.3 62.7 19 i8.4 o4.6 79 76.6 19.2 39 i34.8 33.8 99 193.0 48.4 59 251.2 62.9 20 19.4 04.9 80 77.6 19.4 40 i35.8 34.0 200 194.0 48.6 6o 252.2 63.2 21 20.4 05.1 81 78.6 19.7 141 136.8 34.3 201 195.0 48.8 261 253.2 63.4 22 21.3 o5.3 3 82 79.5 19.9 42 137.7 34.5 02 195.9 49.' 62 254.1 63.7 23 22 3 05.6' 83 8o.5 20.2 43 138.7 34 7 03 196.9 49.3 63 255.1 63.9 24 23.3 05.8 84 81.5 20.4 44 139.7 35.0 04 197.9 49'6 64 256., 64., 25 24.3 o6.I 85 82.5 20.7 45 i4o.7 35.2 05 198.9 49.8 65 257.1 64.4 26 25.2 06.3 -86 83.4 20.9 46 i4i.6 35.5 06 199.8 50o. 66 258.0 64.6 27 26.2 06.6 87 84.4 21.1 47 142.6 35.7 07 200.8 50.3 67 259.0 64.9 28 27.2 06.8 88 85.4 21.4 48 i43.6 36.0 o8 201.8 50.5 68 260.0 65.I 29 28.1 07.0 89 86.3 21.6 49 144.5 36.2 09 202.7 50.8 69 260.9 65.4 30 29.1 07.3 90 87.3 21.9 50 145.5 36.4 io 203.7 51.0 70 261.9 65.6 31 30.1 07.5 91 88.3 22.1 151 i46.5 36.7 211 204.7 51.3 271 262.9 65.8 32 31.o 07.8 92 89.2 22.4 52 147.4 36.9 12 205.6 5i.5 72 263.8 66.i 33 32.0 o8.0 93 90.2 22.6 53 I48.4 37.2 13 206.6 51.8 73 264.8 66.3 34 33.o o8.3 94 91.2 22.8 54 149.4 37.4 14 207.6 52.0 74 v65.8 66.6 35 34.0o 08.5 95 92.2 23.1 55 i5o.4 37.7 i5 208.6 52.2 75 266.8 66.8 36 34.9 08.7 96 93.1 23.3 56 151.3 37.9 i6 209.5 52.5 76 267.7 67.1 37 35.9 09.0 97 94.1 23.6 57 152.3 38.I 17 210.5 52.7 77 268.7 67.3 38 36.9 09.2 98 95.1 23.8 58 i53.3 38.4 I8 211.5 53.0 78 269.7 67.5 39 37.8 09.5 99 96.0 24.1 59 154.2 38.6 19 212.4 53.2 79 270.6 67.8 40 38.8 09.7 ioo 97.0 24.3 60 155.2 38.9 20 213.4 53.5 80 271.6 68.o 41 39.8 1o.o0 io 98.0 24.5 i6i 156.2 39.1 221 214.4 53.7 281 272.6 68.3 42 40.7 10. 2 02 98.9 24.8 62 157.1 39.4 22 215.3 53.9 82 273.5 68.5 43 41.7 1o.4 03 99.9 25.0 63 i58.i 39.6 23 216.3 54.2 83 274.5 68.8 44 42'..7 10.7 o4 10oo0.9 25.3 64 159.1 39.8 24 217.3 54.4 84 275.5 69.o 45 43.7 10.9 05 o101.9 25.5 65 i6o0. 40.I 25 218.3 54.7 85 276.5 69.:~ 46 44.6 11.2 06 102.8 25.8 66 I6I.o 40.3 26 219.2 54.9 86 277.4 69.5 47 45.6 ii.4 07 ro3.8 26.0 67 162.0 4o0.6 27 220.2 55.2 87 278.4 69.7 48 46.6 11.7 08 io4.8 26.2 68 163.o 40.8 28 221.2 55.4 88 279.4 70.0 49 47.5 11.9 09 o105.7 26.5 69 163.9 41. 29 222.1 55.6 89 280.3 70.2 50 48.5 12.1 io 106.7 26.7 70 164.9 4i.3 30 223.1 55.9 90 281.3 70.5 51 49.5 12.4 Ill 107.7 27.0 171 165.9 4i.5 231 224.1 56.I 291 282.3 70.7 52 50o.4 12.6 12 io8.6 27.2 72 i66.8 4i.8 32 225.0 56.4 92 283.2 71.0 53 5i.4 12.9 13 109.6 27.5 73 167.8 42.0 33 226.o 56.6 93 284.2 71.2 54 52.4 13.1 14 iio.6 27.7 74 i68.8 42.3 34 227.0 56.9 94 285.2 71.4 55 53.4 I3.4 I 5 11.6 27.9 75 169.8 42.5 35 228.0 57.1 95 286.2 71.7 56 54.3 i3.6.3 6 6 2.5 28.2 76 170.7 42.8 36 228.9 57.3 96 287.1 71.9 57 55.3 13.8 17 i13.5 28.4 77 171.7 43.o 37 229.9 57.6 97 288.1 72.2 58 56.3 14.1 i8 114.5 28.7 78 172.7 43.3 38 230.9 57.8 98 289.1 72.4 59 57.2 i4.3 19 iI5.4 28.9 79 173.6 43.5 39 231.8 58.I 99 290.0 72.7 6o 58.2 I4.6 20 Iz 6.4 29.2 80 174.6 43.7 40o 232.8 58.3 300 291.0 72.9 Dit. ep.Lat Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep., Lat. Dit. Dep. Lat. Dist. Dep. Lat. E.N.E.JE. E.S.E.}E. W.N.W.aW. W.S.W.}W. [For 61 Points. Page 6] TABLE I. Difference of Latitude and Departure for 1~ Points. ~:'-: N.byEE.E N.byW.JW. S.byE.^ E S.byW.JW. Dist......... L at.Dep.Dist. at Dep. Dist. Lat. Dep. Dist Lat Dep. Dist. Lat. Dep. I oi.o 00.3 61 58.4 I7-7 I2I II5.8 35.1 181 173.2 52.5 241 230.6 70.o 2 oi.9 oo.6 62 59.3 i8.o 22 I16.7 35.4 82 174.2 52.8 42 231.6 70.2 3 02.9 oo.9 63 60 3 I8.3 23 I I7.7 35.7 83 175.1 53.I 43 232.5 70.5 4 o3.8 01.2 64 61.2 I8.6 24 II8.7 36.o 84 176. I53.4 44 233.5 70.8 5 o4.8 oI.5 65 62.2 18.9 25 Ig9.6 36.3 85 177.0 53.7 45 234.5 71.1 6 05.7 o I.7 66 63.2 19.2 26 I20.6 36.6 86 178.o 54.o 46 235.4 71-4 7 o6.7 02.0 67 64.' 19.4 27 121.5 36.9 87 178.9 54.3 47 236.4 71.7 8 07.7 o0.3 68 65.I 19.7 28 122.5 37.2 88'79-9 54.6 48 237.3 72.0 9 o8.6 o2.6 69 66.0 20.0 29 123.4 37.4 89 180.9 54.9 49 238.3 72.3 io og.6 0o.9 70 67.0 20.3 30 124.4 37.7 go I8i.8 55.2 50 239.2 72.6 II io.5 03.2 7I 67.9 20.6 I3I 125.4 38.o 91 182.8 55.4 251 240.2 72.9 12 11.5 03.5 72 68.9 20.9 32 126.3 38.3 92 183.7 55.7 52 241.1 73.2 13 I2.4 o3.8 73 69.9 21.2 33 127.3 38.6 93 184.7 56.o 53 242.1 73.4 4 13.4 04.1 74 70.8 21.5 34 128.2 38.9 94 I85.6 56.3 54 243.1 73.7 15 I4~4 04.4 75 71.8 21.8 35 I29.2 39.2 95 i86.6 56.6 55 244.0 74.0 16 I5.3 o4.6 76 72.7 22.I 36 30.o. 39.5 96 I87.6 56.9 56 245.0 74.3 17 I6.3 04.9 77 73.7 22.4 37 131.1 39.8 97 188.5 57.2 57 245.9 74.6 I8 17.2 05.2 78 74.6 22.6 38 132.1 40., 98 I89.5 57.5 58 246.9 74.9 I9 I8.2 o5.5 79 75.6 22.9 39 133.0 40.3 99 I9g.4 57.8 59 247.8 75.2 20 19.1 o5.8 80 76.6 23.2 4o 134.0 40.6 200 191.4 581i 60 248.8 75.5 2I 20.1 o6.I 8I 77.5 23.5 I4I 134.9 40.9 201 I92.3 58.3 261 249.8 75.8 22 21.1 o6.4 82 78.5 23.8 42 35.9 41.2 02 193.3 58.6 62 250.7 76.1 23 22.0 06.7 83 79.4 24.I 43 I36.8 41.5 o3 I94.3 58.9 63 251.7 76.3 24 23.0 07.0 84 80.4 24.4 44 137.8 41.8 04 195.2 59.2 64 252.6 76.6 25 23.9 07.3 85 8I.3 24.7 45 i38.8 42.1 05 I96.2 59.5 65 253.6 76.9 26 24.9 07.5 86 82.3 25.0 46 139.7 42.4 o6 197 I 59.8 66 254.5 77.2 27 25.8 07.8 87 83.3 25.3 47 I40.7 42.7 07 198. 60.i 67 255.5 77.5 28 26.8 o8.i 88 84.2 25.5 48 i4I.6 43.o 08 199.0 6o.4 68 256.5 77.8 29 27.8 o8.4 89 85.2 25.8 49 I42.6 43.3 09 200.0 60.7 69 257.4 78.1 30 28.7 08.7 90 86.1 26.I 50 i43.5 43.5 o1 201.0 6I.0 70 258.4 78.4 31 29.7 09.0 9g 87.1 26.4 i51 144.5 43.8 211 201.9 61.3 27I 259.3 78.7 32 30.6 09.3 92 88.0 26.7 52 I45.5 44.I 12 202.9 6i.5 72 260.3 79.0 33 3i.6 09.6 93 89.0 27.0 53 i46.4 44.4 13 203.8 6i.8 73 26i.2 79.2 34 32.5 o9.9 94 90.0 27.3 54 i47.4 44.7 I4 204.8 62.1 74 262.2 79.5 35 33.5 0.2 95 90.9 27.6 55 i48.3 45.o I5 205.7 62.4 75 263.2 79.8 36 34.4 io.5 96 91.9 27.9 56 I49.3 45.3 I6 206.7 62.7 76 264.1 8o.1 37 35.4 10.7 97 9928 28.2 57 150.2 45.6 17 207.7 63.0 77 265.1 80.4 38 36.4 ii.o 98 93.8 28.4 58 151.2 45.9 i8 208.6 63.3 78 266.0 80.7 39 37.3 11.3 99 94.7 28.7 59 152.2 46.2 19 209.6 63.6 79 267.0 81.0 40 38.3 1i.6 IOo 95.7 29.0 60 I53.i 46.4 20 210.5 63.9 80 267.9 81.3 41 39.2 11.9 io 96.7 29.3 i61 154.i 46.7 221 211.5 64.2 281 268.9 8I.6 42 40.2 12.2 02 97.6 29.6 62 I55.o 47.0 22 22.4 64.4 82 269.9 81.9 43 4I.I 12.5 03 98.6 29.9 63 156.0 47.3 23 213.4 64.7 83 270.8 82.2 44 42.1 I2.8 o4 99.5 30.2 64 I56.9 47.6 24 2I4.4 65.o 84 271.8 82.4 45 43.I 13.1 05 Ioo.5 30.5 65 157.9 47.9 25 215.3 65.3 85 272.7 82.7 46 44.o i3.4 o6 oi.4 30.8 66 158.9 48.2 26 216.3 65.6 86 273.7 83.( 47 45.o I3.6 07 I02.4 31.i 67 159.8 48.5 27 217.2 65.9 87 274.6 83.3 48 45..9 39 o8 03.3 3I.4 68 I60.8 48.8 28 218.2 66.2 88 275.6 83.6 49 46.9 14.2 09 io4.3 31.6 69 I6I.7 49. 29 2I9.1 66.5 89 276.6 83.9 50 47.8 14.5 io io5.3 31.9 70 162.7 49.3 30 220.1 66.8 90 277.5 84.2 5I 48.8 I4.8 oil 106.2 32.2 171 163.6.6 49 231 22.1 67- 1 291 278.5 84.5 52 49.8 i5.I 12 107.2 32.5 72 164.6 49.9 32 222.0 67.3 92 279.4 84.8 53 50.7 i5.4 13 io8.I 32.8 73 165.6 50.2 33 223.0 67.6 93 280.4 85.1 54 51.7 i5.7 I4 109.1 33.1 74 166,5 50.5 34 223.9 67.9 94 28t.3 85.3 55 52.6 i6.0 i5 iio.o 33.4 75 167.5 50.8 35 224.9 68.2 95 282.3 85.6 56 53.6 I6.3 i6 iii.o 33.7 76 168.4 51. 36 225.8 68.5 96 283.3 85.9 57 54.5 i6.5 17 112.0 34.0 77 169.4 51.4 37 226.8 68.8 97 284.2186.2 58 55.5 16.8 i8 II2.9 34.3 78 170.3 51.7 38 227.8 69.1 98 285.2 86.5 59 56.5 17.1 19 113.9 34.5 79 171.3 52.o 39 228.7 69.4 99 286.1 86.8 Dis. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. E.N.E.E. E.S.E.4E. W.N.W.AW. W.S.W.4W. [For 6j Points. TABLE 1. [Page 7 Difference of Latitude and Departure for l3 Points..-.',. N.byE.PE. N.byW.}W. S.byE.IE. S.byW.aW. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 00.9 oCo.3 61i 57.4 20.6 121 113.9140.8 181 I 70.4 6.0o 241 226.9 81.2 2 01.9 00.7 62 58.4 20.9 22'14.9 4r.1 82 I71.4 6t.3 42 227.9 8i.5 3'02.8 oi.o 63 59.3 21.2 23 ii5.8 4i.4 83 172.3 61.7 43 228.t 81.9 4 o3.8 I 01.3 64 60.3 21.6 24 116.8 4i.8 84 173.2 62.0 44 229.7 82.2 5 04.7 01.7 65 61.2 21.9 25 I7.71 42.1 85 174.2 62.3 45 230.7 82.5 6 o05.6 02.o 66 62.1 22.2 26 118.6142.4 86 175.1 62.7 46 231.6 82.9 7 06.6 02.4 67 63.1 22.6 27 11i9.6 42.8 87 176.1 63.o 47 232.6 83.2 8 07.5 02.7 68 64.1 22.9 28 120.5 43.1 88 177-0 63.3 48 233.5 83.5 9 o8.5 o3.0 69 65.0 23.2 29 121.5 43.5 89 178.0 63.7 49 234.4 83.9 io 09.4 03.4 70 65.9 23.6 30 122.4 43.8 90 178.9 64.o 50 235.4 84.2 ii 1io.4 037 71 66.8 23.9 131 123.3 44.1 191 179.8 64.3 251 236.3 84 6 12 11.3 1o4.0 72 67.8 24.3 32 124.3 44.5 92 18o.8 64.7 52 237.3 84.9 3 12.2 0o4.4 73 68.7 24.6 33 125.2 44.8 93 181.7 65.0o 53 238.2 85.2 i4 13.2 04.7 74 69.7 24.9 34 126.2 45.i 94 182.7 65.4 54 239.2 85.6 15 4.1 0o5.I 75 70.6 25.3 35 127.1 45.5 95 183.6 65.7 55 240.1 85.9 i6 15.1 05.4 76 71.6 25.6 36 128.0 45.8 96 i84.5 66.o 56 241.o 86.2 17 6.o o5.7 77 72.5 25.9 37 129.o0 46.2. 97 185.5 66.4 57 242.0 86.6 i8 16.9 06.1 78 73.4 26.3 38 129.9 46.5 98 i86.4 66.7 58 242.9 86.9 19 17-9 o 6.4 79 74.4 26.6 39 130.9 46.8 99 187.4 67.0 59 243.9 87.3 20 i8.8 06.7 80 75.3 27.0 40o i3.8 47.2 200 i88.3 67.4 60 244.8 87.6 21 19.8 07.1 81 76.3 27.3 141 1i32.8 47.5 201 189.3 67.7 261 245.7 87.9 22 20.7 07.4 82 77.2 27.6 42 133.7 47.8 02 190.2 68.I 62 246.7 88.3 23 21.7 07.7 83 78.1 28.0 43 i34.6 48.2 03 191.1 68.4 63 247.6 88.6 24 22.6 o8.1 84 79.' 28.3 44 135.6 48.5 04 192.1 i68.7 64 248.6 88.9 25 23.5 o8.4 85 8o.o 28.6 45 136.5 48.8 o5 193.o0 69.1 65 249.5 89.3 26 24.5 o8.8 86 81.o 29.0 46 137.5 49.2 o6 194.0 69.4 66 250.5 89.6 27 25.4 o9.1J 87 81.9 29.3 471 38.4 49.5 07 194.9 J69.7 67 251.4 89.9 28 26.4 09.4 88 82.9 29.6 48 139.3 49.9 o8 195.8 70.1 68 252.3 90.3 29 27.3 09.8l 89 83.8 3o.o 49 14o.3 50.2 09 196.8 70.4 69 253.3 90.6 30 28.2 ro.i 90[ 84.7 3o.3 5o 141.2 50o.5 o 197.7 70.7 70 254.2 91.0 31 29.2 10.4 91 85.7 30.7 151 142.2 50.9 211 198.7 71.1 271 255.2 91.3 32 3o.1 io.8 92 86.6 3i.o 52 143.1 51.2 12 1g99.6 71.4 72 256.1 91.6 33 31.1 ii.i 93 87.6 3i.3 53 144.i 51.5 13 200.5 71.8 73 257.0 92.0 34 32.0 11.5 94 88.5 31.7 54 i45.o 51.9 14 201.5 72.1 74 258.o 92.3 35 33.0 1o1.8 95 89.4 32.0 55 145.9 52.2 15 202.4 72.4 75 258.9 92.6 36 33.9 12.1 96 90.4 32.3 56 146.9 52.6 16 203.4 72.8 76 259.9 93.0 37 34.8 12.5 97 91.3 32.7 57 147.8 52.9 17 204.3 73.1 77 260.8 93.3 38 35.8 12.8 98 92.3 33.o 58 148.8 53.2 i8 205.3 73.4 78 261.7 93.7 39 36.7 13.1 99 93.2 33.4 59 149.7 53.6 19 206.2 73.8 79 262.7 94.0 4o 37.7 13.5 ioo 94.2 33.7 6o0 5o.6 53.9 20 207.1 74.1 80 263.6 94.3 41 38.6 i3.8 1oi 95.1 34.0 161 151.6 54.2 221 208.1 74.5 28I 264.6 94.7 42 39.5 i4.i 02 96.0 34.4 62 152.5 54.6 22 209.0 74.8 82 265.5 95.0 43 4o.5 1i4.5 o3 97.0 34.7 63 1i53.5 54.9 23 210.0 75.i 83 266.5 95.3 44 4i.4 14.8 04 97.9 35.o 64 i54.4 55.2 24 210.9 75.5 84 267.4 95.7 45 42.4 15.2 o05 98.9 35.4 65 155.4 55.6 25 211.8 75.8 85 268.3 96.0 46 43.3 15.5 o6 99.8 35.7 66 i56.3 55.9 26 212.8 76.1 86 269.3 96.4 47 44.3 1i5.8 07 100.7 36.o 67 157.2 56.3 27 213.7 76.5 87 270.2 96.7 48 45.2 16.2 o8 101.7 36.4 68 158.2 56.6 28 214.7 76.8 88 271.2 97.0 49 46.1i 6.5 09 102.6 36.7 69 159.1 56.9 29 125.6 77-1 89 272.1 97.4 50 47.1 16.8 io io3.6 37.1 70 I6o.1 57.3 3o 216.6 77.5 90 273.0 97-7 51 48. 17.2 III I04.5 37.4 171 6i.0o 57.6 231 217.5 77.8 291 274.0 98.0 52 49.0 17.5 12 io5.5 37.7 72 161.9 57.9 32 218.4 78.2 92 274.9 98.4 53 49.9 179 1/ 3 3o6.4 38.I 73 162.9 58.3 33 219.4 78.5 93 275.9 98.7 54 5o.8 18.2 14 107.3 38.4 74 i63.8 58.6 34 220.3 78.8 94 276.8 99.0 55 51.8 18.5 5 o108.3 38.7 75 164.8 59.0 35 221.3 79.2 95 277.8 99.4 56 52.7 18.9 i6 109.2 39.1 76 165.7 59.3 36 222.2 79.5 96 278.-7 99.7 57 53.7 19.2 17 110.2 39.4 77' 166.7 59.6 37 223.1 79.8 97 279.6 1o00o58 54.6 19.5 18 iii.i 39.8 78 167.6 6o.o 38 224.1 8o.2 98 280.6 100oo.4 59 55.6 19.9 19 112.0 40o. 79 168.5 6o.3 39 225.0 8o.5 99 281.5 100.7 6o 56.5 20.2 20 1i3.o 4o.4 80 169.5 6o.6 4o 226.o 80.9 3oo00 282.5 1o. Dist. p. st. Dep. Lat. IDist Dsep. 1 Lat. Dist. Dep. Lat.. Dist. Dep. Lat. Ds. IPep.I Lat. E.N.E.iE. E.S.E.dE. W.N.W.4W. W.S.W.W. [For 61 Points. 'age 8] TABLE I. Difference of Latitude and Departure for 2 Points.:-., "" N.N.E. N.N.W. S.S.E. S.S.W. )ist. Lat. Dep. Dist. Lat. Dep. Dist. Lt. ep.D Lat. Dep. Dist. Lat. Dep. r oo0.9 oo.4 6I 56.4 23.3 I21 III.8 46.3 181 167.2 69.3 241 222.7 92.2 2.8 oo00.8 62 57.3 23.7 22 f 12.7 46.7 82 I68.1 69.6 42 223.6 92.6 3 02.8 oi.I 63 58.2 24.1 23 1I3.6 47.- 83 I69.1 70.0 43 224.5 93.0 4 03.7 oi.5 64 59.I 24.5 24 II4.6 47.5 84 I70.0 70.4 44 225.4 93.4 5 04.6 01.9 65 6o. 24.9 25 115.5 47.8 85 170.9 70.8 45 226.4 93.8 6 05.5 02.3 66 6.0o 25.3 26 116.4 48.2 86 171.8 71.2 46 227.3 94.I 7 06.5 02.7 67 61.9 25.6 27 117.3 48.6 87 I72.8 71.6 47 228.2 94.5 8 07.4 o3.I 68 62.8 26.0 28 II8.3 49.0 88 173.7 71.9 48 229.I 94.9 9 08.3 03.4 69 63.7 26.4 2 19.29.4 89 174.6 72.3 49 230.0 95.3 IO 09.2 03.8 70 64.7 26.8 30 I20. I 49.7 90 175.5 72.7 50 231.0 95.7 II 10.2 04.2 71 65.6 27.2 131 121.0 50.I 19I I76.5 73.1 251 231.9 96.1 r II.I o4.6 72 66.5 27.6 32 I22.O 50.5 92 I77.4 73.5 52 232.8 96.4 13 12.0 o5.0 73 67.4 27.9 33 122.9 50.9 93 I78.3 73.9 53 233.7 96.8 14 12.9 05.4 74 68.4 28.3 34 123.8 51.3 94 I79.2 74.2 54 234.7 97.2 I5 13.9 05.7 75 69.3 28.7 35 124.7 51.7 95 180.2 74.6 55 235.6 97.6 16 I4.8 o6.1 76 70.2 29.I 36 125.6 52.0 96 181. 75.0 56 236.5 98.0 17 I5.7 06.5 77 7I.1 29.5 37 126.6 52.4 97 182.0 75.4 57 237.4 98.3 18 16.6 06.9 78 72.1 29.8 38 127.5 52.8 98 I82.9 75.8 58 238.4 98.7 I9 17.6 07.3 79 73.0 30.2 39 I28.4 53.2 99 I83.9 76.2 59 239.3 99.1 20 18.5 07.7 80 73.9 30.6 40 129.3 53.6 200 I84.8 76.5 60 240.2 99.5 21 19.4 o8.o 81 74.8 31.0 141 130.3 54.0 201 185.7 76.9 26 241.1 99-9 22 20.3 08.4 82 75.8 3I.4 42 131.2 54.3 02 i86.6 77. 62 242.I I00.3 23 21.2 08.8 83 76.7 31.8 43 132.1 547 3 87.5 777 363 243.0 o00.6 24 22.2 09.2 84 77.6 32. 44 133. 55. 04 r88.5 78.1 64 243.9 Io.o 25 23.1 o9.6 85 78.5 32.5 45 i34.o 55.5 05 189.4 78.5 65 244.8 ioi.4 26 24.0 09.9 86 79.-5 32.9 46 134.9 55.9 o6 90o.3 78.8 66 245.8 o10.8 27 24.9 10.3 87 80.4 33.3 47 i35.8 56.3 07 I9I.2 79.2 67 246.7 102.2 28 25.9 10.7 88 81.3 33.7 48 136.7 56.6 08 192.2 79.6 68 247.6 102.6 29 26.8 ii.i 89 82.2 34.I 49 I37.7 57-. 09 193.1 80.o 69 248.5 I02.9 30 27.7 11.5 90 83. 34.4 50 I38.6 57.4 io I94.0 80.4 70 249.4 I03.3 3r 28.6 11.9 91 84.1 34.8 151 139.5 57.8 2II I94.9 80.7 271 250.4 103.7 32 29.6 12.2 92 85.0 35.2 52 40o.4 58.2 12 I95.9 8i.I 72 25I.3 104.i 33 30.5 12.6 93 85.9 35.6 53 14I.4 58.6 13 196.8 8I.5 73 252.2 I04.5 34 3r.4 3.o 94 86.8 36.0 54 142.3 58.9 I4 197.7 8I.9 74 253.1 104.9 35 32.3 13.4 95 87.8 36.4 55 143.2 59. 15 198.6 82.3 75 254.1 I05.2 36 33.3 i3.8 96 88.7 36.7 56 I44.I 59.7 I6 199.6 82.7 76 255.0 1o5.6 37 34.2 14.2 97 89.6 37.I 57 I45.o 60. 17 200.5 83.0 77 255.9 1o6.o 38 35.1 14.5 98 90.5 37.5 58 i46.0 60.5 18 201.4 83.4 78 256.8 106.4 39 36.o 14.9 99 91.5 37.9 59 I46.9 60.8 19 202.3 83.8 79 257.8 o16.8 40 37.0 15.3 Ioo 92.4 38.3 60 147.8 61.2 20 203.3 84.2 80 258.7 107.2 41 37.9 15.7 o10 93.3 38.7 161 148.7 6I.6 221 204.2 84.6 281259.6 107.5 4-2 38.8 i6. 02 94.2 39.0 62 149.7 62.0 22 205.1I 85.o 82 260.5 107.9 43 39.7 I6.5 03 95.2 39.4 63 150.6 62.4 23 206.0 85.3 83 261.5 108.3 44 40.7 16.8 o4 96.1 39.8 64 151.5 62.8 24 206.9 85.7 84 262.4 108.7 45 4I.6 17.2 05 97-0 40.2 65 52.4 63.I 25 207.9 86.I 85 263.3 I09.1 46 42.5 I7.6 06 97.9 4o.6 66 53.4 63.5 26 208.8 86.5 86 264.2 109.4 47 43.4 I8.o 07 98.9 40.9 67 154.3 63.9 27 209.7 86.9 87 265.2 19.8 48 44.3 I8.4 08 99.8 4i.3 68 155.2 64.3 28 210.6 87.3 88 266.1 110.2 49 45.3 18.8 09 100.7 41.7 69 I56. 64.7 29 211.687.6 89 267.0 0o.6 50 46.2 19.1 io io1.6 42.1 70 i57. 65.1 30 212.5 88.o 90 267.9 I1I.0 SI 47.I 19.5 III 102.6 42.5 I71 I58.0o 65.4 231 2I3.4 88.4 291 268.8 II.4 52 48.o 19.9 I2 Io3.5 42.9 72 158.9 65.8 32 2 4.3 88.8 92 269.8 111.7 53 49.0 20.3 13 o04.4 43.2 73 59.8 66.2 33 215.3 89.2 93 270.7 112.1 54 49.9 20.7 i4 1o5.3 43.6 74 160.8 66.6 34 216.2 89.5 94 271.6 I12.5 55 50.8 21.0 o 5 106.2 44.0 75 161.7 67.o 35 217.1 89.9 95 272.5 112.9 56 5I.7 21.4 i6 107.2 44.4 76 162.6 67.4 36 218.o0 96 273..3 3.3 57 152.7 21.8 17 xo8.i 44.8 77 63.5 67.7 37 29.o 90.7 97 274.4 113.7 58'53.6 z22.2 8 109.0 45.2 78 i64.5 68.r 38 219.9 91.1 98 275.3 II4.0 59 54.5 22.6 I9 I09.9 45.5 79 165.4 68.5 39 220.8 9 9.5 276.2 I4.4 6o 55.4 23.0 20 Io.9 45.9 80 i66.3 68.9 40 221.7 91.8 300 277-2 14.8 Dlist. Dep. ILat. Dist. Dep. Lat.Di Dep. t. ist.i Dep. Lat. Dis t.l Dep. Lat. i EN.E. E.S.E. W.N.W. W.S.W. [For 6 Points. TABLE 1. [Page 9 Diferenoe of Latitade and Departure for f2 Points...'.' N.N.E.~E. N.N.Wi.W. S.S.E.~E. S.S.W.~W. Dist., Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. i oo. 00o.4 61 55.1 26.1 121 109.4 51.7 181 163.6 77-4 241 217-9 iz3c 2 oI.8 00.9 62 56.0 26.5 22 110o.3 52.2 82 i64.5 77.8 42 28.8 o103.5 3 o2.7 oi.3 63 57.0 26.9 23 111.2 52.6 83 i65.4 78.2 43 219.7 103.9 4 03.6 01.7 64 57-9 27.4 24 II112.1 53.o 84 ~66.3 78.7 44 220.6 10o4.3 5 o04.5 02.1 65 58.8 27.8 25 113.0 53.4 85 167.2 79. 1 45 22.5 o104.8 6 o5.4 02.6 66 59.7 28.2 26 113.9 53.9 86 168.1 79.5 46 222.4 105.2 7 o6.3 03 o 67 6o.6 28.6 27 114.8 54.3 87 169.o 80.0 47 223.3 10o5.6 8 07.2 03.4 68 6i.5 29.1 28 115.7 54.7 88 169.9 80o.4 48 224.2 o106.0 9 o8.1 o3.8 69 62.4 29.5 29 Ii6.6 55.2 89 170.9 80o.8 49 225.1 106.5 io o9.o 04.3 70 63.3 29.9 30 117.5 55.6 90 171.8 81.2 50 226.0 106.9 II 09.9 (04.7 7I 64.2 30.4 i3i ii8.4 56.0 191 172.7 81.7 251 226.9 107.3 12 10.8 05.1 72 65.i 30.8 32 1i19.3 56.4 92 173.6 82.1 52 227.8 107.7 i3 11.8 o5.6 73 66.o 31.2 33 120.2 56.9 93 174.5 82.5 53 228.7 108.2 14 12.7 o6.0 74 66.9 3i.6 3,-4 121,1 57.3 94 175.4 82.9 54 229.6 108.6 15 i3.6 o6.4 75 67.8 32.1 35 122.0 57-7 95 176.3 83.4 55 230.5 109.0 i6 I4.5 06.8 76 68.7 32.5 36 122.9 58.1 96 177.2 83.8 56 231.4 109.5 17 i5.4 07.3 77 69.6 32.9 37 123.8 58.6 97 178.1 84.2 57 232.3 o109. i8 i6.3 07.7 78 70.5 33.3 38 124.8 59.0 98 179.0 84.7 58 233.2 110.3 19 17.2 o8.1 79 71.4 33.8 39 125.7 59.4 99 179.9 85.1 59 234. 1 110.7 20 18.i 08.6 80 72.3 34.2 40 126.6 59.9 200 i8o. 85.5 6o 235.0 111o.2 21 19.o 09.0 81 73.2 34.6 141 127.5 60.3 201 181.7 85.9 261 235.9 iii.6 22 i9.9 09.4 82 74.1 35.i 42 128.4 60.7 02 182.6 86.4 62 236.8 112.0 23 20o.8 09.8 83 75.0 35.5 43 129.3 61.1 03 i83.5 86.8 63 237.7 112-. 24 21.7 1 o.3 84 75.9 35.9 44 130.2 61.6 o4 184.4 87.2 64 238.7 112.9 25 22.6 10.7 85 76.8 36.3 45 131.1 62.0 05 i85.3 87.6 65 239.6 113.3 26 23.5 ii.i 86 77.7 36.8 46 132.0 62.4 06 186.2 88.1 66 240.5 113.7 27 24.4 11.5 87 78.6 37.2 47 132.9 62.9 07 187.1 88.5 67 241.4 114.2 28 25.3 12.o 88 79.6 37.6 48 i33.8 63.3 08 i88.o 88.9 68 242.3 114.6 29 26.2 12.4 89 80.5 38.1 49 134.7 63.7 09 188.9 89.4 69 243.2 115.o 30 27.1 12.8 90 81.4 38.5 50 i35.6 64.1i i 189.8 89.8 70 244.1 1i5.4 31 28.0 13.3 91 82.3 38.9 151 136.5 64.6 211 190.7 90.2 271 245.0 115.9 32 28.9 13-7 92 83.2 39.3 52 137.4 65.o 12 191.6 90.6 72 245.9 ii6.3 33 29.8 14.1 93 84-.i 39.8 53 138.3 65.4 13 192.5 91.1 73 246.8 116.7 34 30.7 14.5 94 85.0 40.2 54 139.2 65.8 i4 193.5 91.5 74 247.7 117.2 35 31.6 15.0 95 85.9 4o.6 55 140.1i 66.3 15 194-4 91.9 75 248.6 117.6 36 32.5 i5.4 96 86.8 410.o 56 141.0o 66.7 16 195.3 92.4 76 249.5 ji8.o 37 33.4 15.8- 97 87.7 41.5 57 141.9 67-1 17 196.2 92.8 77 250.4 1i8.4 38 34.4 16.2 98 88.6 41.9 58 142.8 67.6 18 197-1 93.2 78 251.3 118.9 39 35.3 16.7 99 89.5 423 59 143.7 68.0 19 198.0 93.6 79 252.2 119.3 40 36.2 17.1 ioo 90.4 42.8 60 i44.6 68.4 20 198.9 94.1 80 253.1 119.7 41 -37.1 17.5 oi 91.3 43.2.161 145.5 68.8 221 199.8 94.5 281 254.0 120.T 42 38.0 18.o 02 92.2 43.6 62 146.4 69.3 22 200.7 94.9 82 254.9 120.6 43 38.9 18.4 03 93.1 44.o 63 147.4 69.7 23 201.6 95.3 83 255.8 121.0 44 39.8 i8.8 04 94.0 44.5 64 i48.3 70.1 24 202.5 95.8 84 256.7 121.4 45 40.7 19.2 05 94.9 44.9 65 149.2 70.5 25 203.4 96.2 85 257.6 121.9 46 41.6 19-7 o6 95.8 45.3 66 i5o.1 71.0 26 204.3 96.6 86 258.5 122.3 47 42.5 20.1 07 96.7 45-7 67 i51.0 71-4 27 205.2 97-1 87 259.4 122.7 48 43.4 20.5 08 97.6 46.2 68 151.9 71.8 28 206.1 97.5 88 260.3 123.1 49 44.3 21.0 09 98.5 46.6 69 i52.t 72.3 29 207.0 97.9 89 261.3 123.6 50 45.2 21.4 10 99.4 47.0 70 153.7 72-7 30 207.9 98.3 90 262.2 124.0 51 46.1 21.8 - 100oo.3 47.5 171 I54.6 73.1 231 208.8 98.8 291 263.1 124-4 52 47.0 22.2 12 101.2 47-9 72 155.5 73.5 32 209.7 99-.2 92 264.0 124.8 53 47-9 22.7 13 102.2 48.3 73 i56.4 74.0 33 210.6 99.6 93 264.9 125.3 54 48.8 23.1 14 o103.1 48.7 74 157.3 74.4 34 211.5 oo.o 94' 265.8 125.7 55 49.7 23.5 15 o104.o 49.2 75 158.2 74.8 35 212.4 oo00.5 95 266.7 126.1 56 50.6 23.9 6 o104.9 49.6 76 159.1 75.2 36 213.3 1oo.9 96 267.6 126.6 57 51.5 24.4 17 o105.8 50.0 77 i6o.o 75.7 37 214.2 o101.3 97 268.5 127.0 58 5-.4 24.8 i8 106.7 50.5 78 160.9 76.1 38 215. 101o.8 98 269.4 127.4 59 53 3 25.2 19 107.6 50.9 79 161.8 76.5 39 216.1 102.2 99 270.3 127.8 60o 54.2 25.7 20 108.5 51.3 80 162.7 77-0 40 217.0 102.6 300 271.2 128.3 I)ist. ___________________________Ct D t __a. i._ D ist.Dp. Lat. Dist. Dep. Lat. Dist. Dep L Dist Dep. La t. Dist. Dep.. Lat. N.E.byE.aE. S.E.byE.aE. N.W.byW.4W. S.W.byW.IW. [For 51 Points. 2 Page 10] TABLE I. Difference of Latitude and Departure for 2{ Points. -'' N.N.E.1E. N.N.W.AW. S.S.E.4E. S.S.W.4W. Dist. Lat. Dep. IDist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. i. L0.9 00.5 61 53.8 28.8 121 106.7 57.0 I81 159.6 85.3 24I 2I2.5 113.6 2 01.8 oo09 62 54.7 29.2 22 107.6 57.5 82 i6o.5 85.8 42 213.4 1II4. 3 02.6 1I.4 63 55.6 29.7 23 1o8.5 58.o 83 16i.4 86.3 43 214.3 114:5 4 o3.5 01.9 64 56.4 30.2 24 109.4 58.5 84 162.3 86.7 44 215.2 15.0 5 o044 02.4 65 57.3 3o.6 25 10o.2 58.9 85 163.2 87.2 45 216.1 I5.5 6 05.3 02.8 66 58.2 31.1 26 iii.i 59.4 86 164.o 87.7 46 217.o 116.0 7 6. 2 o3.3 67 59. I 3.6 27 11 2.0 59.9 87 164.9 88.2 47 217.8 116.4 8 07.1 o3.8 68 6o.0 32.1 28 112.9 6o.3 88 165.8 88.6 48 218.7 116.9 07.9 04.2 69 60.9 32.5 29 113.8 60.8 89 166.7 89.1 49 219.6 11 7.4 rO o8.8 04.7 70 61.7 33.o 3o0 4.6 61.3 90 1 67.6 89.6 50 220.5 117.8 i1 (19.7 0(5.2 71 62.6 33.5 131 115.5 6i.8 191 168.A 90.0 251 221.4 118.3 12 1o.6 05.7 72 63.5 33.9 32 I6.4 62.2 92 169.3 90.5 52 222.2 118.8 13 T1.5 o6.1 73 64.4 34.4 33 11 7.3 62.', 93 z70.2 91.0 53 22 3.1 I I 9.3 14 12.3 o6.6 74 65.3 34.9 34 118.2 63.2 94 17I.1 91.5 54 224.0 119.7 15 I3.2 07.1 751 66.1 35.4 35 119.i 63.6 95 172.0 91.9 55 224.9 120.2 i6 14.I 07.5 76 67.0 35.8 36 11 9. 9 64. 96 172.9 92.4 56 225.8 120.7 17 15. * o8.o 77| 67.9 36.3 37 120.8 64.6 97 173.7 92.9 57 226.7 121.1 18 15.9 o8.5 78 68.8 36.8 38 121.7 65.I 98 174.6 93.3 58 227.5 121.6 19 I6.8 og.0 79 69-7 37.2 39 122.6 65.5 99 175.5 93.8 59 228.4 122.I 20 r17.6 09.4 80o 70.6 37.7 40 123.5 66.o 200 I76.4 94.3 60 229.3 122.6.21 18.5 09.9 8i| 71.4 38.2 141 124.4 66.5 201 177.3 94.8 261 230.2 I23.0 22 19.4 ro.4 82 72.3 38.7 42 125.2 66.9 02 178.I 95.2 62 231.1 123.5 23 20.3 io.8 83 73.2 39.1 43 126.1 67.4 03 I79-0 95-7 63 23I.9 124.0 24 21.2 11.3 84- 74.1 39.6 44 I27.0 67.9 o4 179.9 96.2 64 232.8 124.4 25 22.0 iT.8 85 75.0 40. 45 127.9 68. 4 05 180.8 96.6 65 233.7 124.9 26 22.9 12.3 86 75.8 40).5 46 128.8 68.8 o6 I181.7 97. 66 234.6 125.4 27 23.8 12.7 87 76.7 41.0 47 129.6 69.3 07 182.6 97.6 67 235.5 125.9 98.4.-7 13.2 88 77.6 41.5 48 i30.5 69.8 o8 I83.4 98.1 68 236.4 1 26.3 29 25.6 13.7 89 78.5 42.0 49 131.4 70.2 09 I84.3 98.5 69 237.2 1 26.8 30 26.5 14.1 90 79-4 42.4 50 132.3 70.7 IO 185.2 99.0 70 238.I 127.3 31 27.3 14.6 91 80.3 42.9 151 133.2 71.2 211 186.I 99.5 271 239.0 127.7 32 28.2 i5.i 92 81.1 43.4 52 i34.I 71.7 I2 187-0 99.9 72 239.9 128.2 33 29.1 15.6 93 82.0 43.8 53 134-9 72.1 I3 187.8 100.4 73 240.8 128.7 34 3o0.o 6.0 94 82.9 44.3 54 i35.8 72.6 14 188.7 100.9 74 241.6 129.2 35 30.9 16.5 95 83.8 44.8 55 136.7 73.1 I5 189.6 101.4 75 242.5 129.6 36 31.7 17.0 96 84-7 45.3 56 137.6 73.5 16 190.5 1io.8 76 243.4 130.i 37 32.6 17.4 97 85.5 45.7 57 138.5 74.0 17 19.-4 102.3 77 244.3 30o.6 38 33.5 17.9 98 86.4 46.2 58 i39.3 74.5 I8 192.3 102.8 78 245.2 131.o 39 34.4 18.4 99 87.3 46.7 59 140.2 75.0 19 193.1 103.2 79 246.1 131.5 40 35.3 18.9 ioo 88.2 47-I 6o 141.1 75.4 20 194.0 1I03.7 8o 246.9 1 32.0 41 36.2 19.3 ioi 89.I 47-6 161 142.0 75.9 221 194-9 104.2 281 247.8 132.5 42 37.0 19.8 02 90.0 48.I 62 142.9 76.4 22 195.8 104.7 82 248.7 1 32.9 43 37.9 20.3 o3 90.8 48.6 63 143.8 76.8 23 I96.7 I05.I 83 249.6 133.4 44 38.8 20.7 04 91-7 49.0 64 144.6 77.3 24 197.6 o15.6 84 250.5 33:9 45 39.7 21.2 05 92.6 49.5 65 r45.5 77.8 25 198.4 1o6.I 85 251.3 134.3 46 40.6 21.7 o6 93.5 50.0 66 146.4 78.3 26 199.3 106.5 86 252.2 34.8 47 4.5 22.2 07 94.4 50.4 67 147.3 78.7 27 200.2 107.0 87 253.1 135.3 48 42.3 22.6 o8 95.2 50.9 68 148.- 92 792 28 201.1 107.5 88 254.0 135.8 49 143.2 123.1 09 96.I 5I.4 69 149.0 79.7 29 202.0 107.9 89 254.9 136.2 50o 4.4I 23.6 io 97.0 51.9 70 I499 80o. 30 202.8 o18.4 90 255.8 36.7 51 45.024.o III 97-9 52.3 17I i50.8 80.6 231 203.7 108.9 29I 256.6 137.2 52 45| "V - 4.5 12 98.8 52.8 72 151.7 81.1 32 204.6 109.4 92 257.5 137.6 53 46.7.o 13 99.7 53.3 73 152.6 8I.6 33 205.5 109.8 93 258.4 138.I 54 47.6 25.5 I4 Ioo.5 53.7 74 i53.5 82.0 34 206.4 11o.3 94 259.3 1 38.6 55 48.5 25.9 i5 Ioi.4 54.2 75 i54.3 82.5 35 207.3 10.8 95 260.2 139.1 56 49.4 26.4 16 102.3 54.7 76 155.2 83.o 36 208.I 111.2 96 261.0 1 39.5 57 5.3 26.9 17 103.2 55.2 77 156.1 83.4 37 209.0 111.7 97 261.9 140.0 58 51.2 27.3 18 o14.1 55.6 78 157.0 83.9 38 209.9 112.2 98 262.8 1405 59 52.0 27.8 19 104.9 56.1 79 157-9 84.4 39 210.8 112.7 99 263.7 1409 60 52.9 28.3 20 o05.8 56.6 80 158.7 84.9 40 211.7 13.1 300 264.6 141.4 Dist. IDep. L. D. Dist. Dep. Lat. Dist. Dep.. Lal.. lst Dep. at. Dist. Dep. at. N.E.byE.4E. S.E.byE4.E. N.W.byW.4W. S.W.byW.W. [For 5A Points. TABLE I. [Page ii Difference of Latitude and Departure for 23 Points.:' N.N.E.j' E. N.N,.WW. S.S.E.PE. S.S.W./W. ist.I~at. )ep-. IDist. Lat. Pep. D)ist.j Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 1-100. 00.5 61 52.3 31,.4 121 io3.8 62.2 18I 155.2 93.1 24I 206.7 123.9 2 01.7101.0 62 53.21 1.9 22 io4.6 62.7 82 i56.I 93.6 42 207.6 124.4 3 o02.6 oi.5 63 54.0 32.4 23 io5.5 63.2 83 157.0 94.1 43 208.41124.9 4 03.4 o02.1 64 54.9 32.9 24 106.4 637 84 157.8 94.6 44 209.3 125.4 5 o4.3 02.6 65 55.8 33.4 25 107.2 64.3 85 158.7 95-1 45 2101 126.0 6 05.1 03.r1 661 5 6.6133.9 26 io8.i 64.8 86 159.5 95.6 46 211.0 126.5 7 o6.0 03.6 67 57.5 34.4 27 108.9 65.3 87 160.4 96.1 47 211.9 127.0 8 06.9 04.1 68 58.3 35.0 28 109.8 65.8 88 161.3 96.7 48 212.7 127.5 9107.7 o4.6 69 59.2 35.5 29 iio.6 66.3 89 162.1 97.2 49 213.6 128.0 io 08.6 05.1 70 6o.0o 36.0 30 111.5 66.8 90 163.0 97.7 50 214.4 128.5 1o 09.4 05.7 71 60.9 36.5 131 112.4 67.3 191 163.8 98.2 251 215.3 129.0 I2 10.3 o6.2 72 61.83 3. 113.2 67.9 92 164.7 98.7 52 216.1 129.6 ~ 13 11. o06.7 73 62.6 37.5 33 114.1 68.4 93 165.5 99.2 53 217.0 130.1 1 A 12.o 07.2 74 63.5 38.0 34 114.9 68.9 94 166.4 99-7 54 217.9 I30.6 5 12.9 07.7 75 64.3 38.6 35 ii5.8 69.4 95 167.3 ioo.3 55 218.7 131.1 i6 13.7 08.2 76 65.2 39.1 36 116.7 69.9 96 168.1 100ioo.8 56 219.6 131.6 17 14.6 08.7 77 66.o 39.6 37 117.5 70.4 97 169.0 ioi.3 57 220.4 132.1 i8 15.4 09.3 78 66.9 4o0. 38 ii8.4 70.9 98 169.8 101.8 58 221.3 132.6 19 16.3 09.8 79 67.8 40.6 39 119.2 71.5 99 170.7 102.3 59 222.2 133.2 20 17.2 o0.3 8o 68.6 4/.i 40 120.1 72.0 200 171.5 102.8 6o 223.0 133.7 21 18.0 io.8 81 69.5 41.6 141 120.9 72.5 201 172.4 103.3 261 223.9 134.2 22 18.9 11.3 82 70.3 42.2 42 121.8 73.0 02 173.3 103.8 62 224.7 134.7 23 19.7 11.8 83 71.2 42.7 43 122.7 73.5 o3 174.1 i04.4 63 225.6 135.2 24 20.6 12.3 84 72.0 43.2 44 123.5 74.0 04 175.0 104.9 64 226.4 135.7 25 21.4 12.9 85 72.9 43.7 45 24.4 74.5 05 175.8 105.4 65 227.3 136.2 26 22.3 i3.4 86 73.8 44.2 46 125.2 75.1 o6 176.7 105.9 66 228.2 i36.8 27 23.2 13.9 87 74.6 44.7 47 126.1 75.6 07 177.5 i0 6.4 67 229.0 137.3 28 24.0 14.4 88 75.5 45.2 48 126.9 76.1 08 178.4 io6.9 68 229.9 137.8 29 24.9 14.9 89 76.3 45.8 49 127.8 76.6 09 179.3 107.4 69 23.7 1138.3 30 25.7 15.4 90 77.2 46.3 50 128.7 77.1 io 180.1 108.0 70 231.6 i38.8 31 26.6 15.9 91 78.1 46.8 151 129.5 77.6 211 i8i.o 108.5 271 232.4 139.3 32 27.4 16.5 92 78.9 47.3 52 130.4 78.1 12 181.8 109.0 72 233.3 139.8 33 28.3 17.0 93 79.8 47.8 53 131.2 78.7 13 182.7 o109.5 73 234.2 i40.4 34 29.2 17.5 94 80.6 48.3 54 132.1 79.2 14 183.6 iio.o 74 235.0 140.9 35 30.0 18.o 95 8i.5 48.8 55 1329 79-7 15 i84.4 110.5 75 23.5.9 1641.4 36 30.9 i8.5 96 82.3 49.4 56 133.8 80.2 i6 i85.3 Iii.o 76 236.7 141-9 37 31.7 19.0 97 83.2 49.9 57 134.7 80.7 17 186.1 111ii.6 77 237.6 142.4 38 32.6 19.5 98 84.1 50.4 58 i35.5 81.2 i8 187.0 112.1 78 238.4 142.9 39 33.5 20.1 99 84.9 50.9 59 i136.4 81.7 19 187.8 112.6 79 239.3 i143.4 4o 34.3 20.6 ioo 85.8 51.4 60 137.2 82.3 20 I88.7 ii3.1 80 240.2 143.9 41 35.2 21.1 I ioi 86.6 5.9 i6i i38. 82.8 221 189.6113.6 281 241.0 o 44.5 42 36.0 21.6 02 87.5 52.4 62 i39.o 83.3 22 190.4 114.1 82 241.9 145.o 43 36.9 22.1 03 88.3 53.o 63 139.8 83.8 23 191.3 114.6 83 242.7 145.5 44 37.7 22.6 04 89.2 53.5 64 140.7 84.3 24 192.1 115.2 84 243.6 146.o 45 38.6 23.1 5 go90.1 54. 65 14i.5 84.8 25 193.0 115.7 85 244.5 i46.5 46 39.5 23.6 06 go90.9 54.5 66 142.4 85.3 26 193.8 116.2 86 245.3 147.0 47 4o. 3 24.2 07 91.8 55.0 67 143.2 85.9 27 194.7 116.7 87 246.2 147.5 48 41.2 24.7 o8 92.6 55.5 68 i44.i 86.4 28 195.6 117.2 88 247.0 148.1 49 42.0 25.2 09 93.5 56.0 69 1i45.o 86.9 29 196.4 117.7 89 247.9 11Z48.6 50 42.9 25.7 i0 94.4 56.6 70 145.8 87.4 30 197.3 ii8.3 90 248.7 149.1 51 43.7 26.2 III 95.2 57.1 171 146.7 87.9 231 198.1 118.8 291 249.6 149.6 52 44.6 26.7 12 96.1 57.6 72 147.5 88.4 32 199.0 119.3 92 250.5 150.1 53 45.5 27.2 13 96.9 58. 8. 73 48.4 88.9 33 1999 9.8 93 251.3 15.6 54 46.3 27.8 i4 97.8 58.6 74 149-2 89.5 34 200.7 120.3 94 252.2 151.1 55 47-2 28.3 15 98.6 59.1 75 150o.1i 90.0 35 201 6 120.8 95 253.0 151.7 56 48.0o 28.8 i6 99.5 59.6 76 151.0 90.5 36 202.4 121.3 96 253.9 152.2 57 48.9 29.3 17 1oo.4 60.2 77 151.8 91.0 37 203.3 121.8 97 1254.7 152.7 58 49.7 29.8 18 101.2 60.7 78 152.7 91.5 38 204.1 122.4 98 255.6 153.2 59 50.6 30.3 19 102.1 61.2 79 i53.5 92.0 39 205.0 122.9 99 256.5 153.7 60 51.5 30.8 20 102.9 61.7 8o0 154.4 92.5 41 205.9 123.4 300 257.3 154.2 ist. Pep.,Lat. Dist. Dep. bat. L Dist. Dep. -at. Dist. Dep. Lat. Dist. )cp. Lat. N.E.byE.pE. S.E.byE.-E. N.W.byW./ W. S.W.A1byW.W. [For 5- Points. 45~~/8o 45 4,548 4513.8[8. Page 12 TABLE I. Difference of Latitude and Departure for 3 Points.- - N.E.byN. N.W.byN. S.E.byS. S.W.byS. Dist. Dist. Lat Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist.at. epDis Lat. Dep. 0oo.8 oo.6 61 50.7 33.9 12I.5 00.6 672 8.5 oo6 24 200.4 133.9 2 01.7 01.1 62 51.6 34-4 22 io1.4 67.8 82 I51.3 IO1. 42 201.2 I34.4 3 02.5 01.7 63 52.4 35.0 23 1o2.3 68.3 83 152.2 101.7 43 202.0 I35.o 4 03.3 02.2 64 53.2 35.6 24 Io3.I 68.9 84 I53.o 102.2 44 202.9 1356 5 04.2 02.8 65 54. 36.1 25 103.9 69.4 85 I53.8 I02.8 45 203.7 i36.1 6 05. o3.3 66 54.9 36.7 26 i04.8 70.0 86 154.7 0o3.3 46 204.5 136.7 7 o5.8 ~03.9 67 55.7 37.2 27 Io5.6 70.6 87 155.5 103.9 47 205.4 137.2 8 06.7 04.4 68 56.5 37.8 28 Io6.4 71.1 88 I56.3 104.4 48 206.2 137.8 9 07.5 05.0 69 57.4 38.3 29 107.3 71.7 89 157-1 io5.o 49 207.0 138.3 1o o8.3 o5.6 70 58.2 38.9 30 Io8.1 72.2 9o 158.0 1o5.6 50 207.9 138.9 JI 09.1 o6. I 71 59.0 39.4 I3I 108.9 72.8 I9g 158.8 106.i 251 208.7 139.4 12 io.O 06.7 72 59.9 40.0 32 109.8 73.3 92 159.6 106.7 52 209.5 I40.0 13 10.8 07.2 73 60.7 40.6 33 11O.6 73.9 93 I6o.5 107.2 53 2IO.4 40o.6 14 11.6 07.8 74 61.5 41.1 34 111.4 74.4 94 I6I.3 107.8 54 211.2 141.1 15 12.5 o8.3 75 62.4 41.7 35 112.2 75.0 95 162. 10 8.3 55 212.0 141.7 16 13.3 08.9 76 63.2 42.2 36 113.1 75.6 96 163.0 108.9 56 212.9 142.2 17 14.1 09.4 77 64.o 42.8 37 113.9 76.I 97 163.8 109.4 57 213.7 142.8 i8 15.0 1o0o 78 64.9 43.3 38 114.7 76.7 98 I64.6 11O.O 58 214.5 143.3 19 I5.8 io.6 79 65.7 43.9 39 I5.6 77.2 99 65.5 o10.6 59 215.4 143.9 2,0 i 6.6 ii.i 80 66.5 44.4 4o 116.4 77.8 200 I66.3 iii.i 60 216.2 144.4 21 17.5 11.7 81 67.3 45.o 141 117.2 78.3 201 167.1 111.7 261 217.0 145.0 22 18.3 12.2 82 68.2 45.6 42 J18.1 78.9 02 168.0 112.2 62 217.8 r45.6 23 19.I I2.8 83 69.0 46.1 43 118.9 79.4 03 i68.8 112.8 63 218.7 i46.1 24 20.0 i3.3 84 69.8 46.7 44 119.7 80.o o4 I69.6 113.3 64 219.5 146.7 25 20.8 13.9 85 70.7 47.2 45 120.6 80.6 05 170.5 113.9 65 220.3 147.2 26 21.6 14.4 86 71.5 47.8 46 121.4 8I.] 06 171.3 II4.4 66 221.2 147.8 27 22.4 I5.0 87 72.3 48.3 47 122.2 81.7 07 172.1 Ii5.o 67 222.0 148.3 28 23.3 15.6 88 73.2 48.9 43 123.I 82.2 o8 172.9 i5.6 68 222.8 148.9 29 24.I 16.I 89 74.0 49.4 49 23.9 82.8 09 173.'8 1 6.1 69 223.7 149.4 30 24.9 16.7 90 74.8 50o. 5o 1I24.7 83.3 Io 174.6 116.7 70 224.5 150.0 31 25.8 17.2 91 75.7 50.6 I51 125.6 83.9 211 175.4 117.2 271 225.3 15o.6 32 26.6 17.8 92 76.5 51.I 52 126.4 84.4 12 176.3 117.8 72 226.2 15I1. 33 27.4 i8.3 93 77.3 51.7 53 127.2 85.0 13 177.I i18.3 73 227.0 I51.7 34 28.3 I8.9 94 78.2 52.2 54 128.0 85.6 14 177.9 118.9 74 227.8 152.2 35 29.I 19-4 95 79.0 52.8 55 128.9 86.1 15 178.8 119.4 75 228.7 152.8 36 29.9 20.0 96 79.8 53.3 56 129.7 86.7 16 179.6 20.0 76 22'9.5 153.3 37 30.8 20.6 97 80.7 53.9 57 130.5 87.2 17 I8o.4 I20.6 77 230.3 153.9 38 3i.6 21.1 98 8I.5 54.4 58 I31.4 87.8 18 I8I.3 I21.I 78 231.1 I54.4 39 32.4 12I.7 99 82.3 55.o 59 I32.2 88.3 19 182.1 121.7 79 232.0 I55.0 40 33.3 22.2 100 83. 55.6 60 133.0 88.9 20 I82.9 122.2 80 232.8 155.6 4i 34.1 22.8 ioi 84.0 56..1 61 i33.9 89.4 22I 183.8 122.8 8281 233.6 156.I 42 34.9 23.3 02 84.8 56.7 62 134.7 90.0 22 84.6 123.3 82 234.5 156.7 43 35.8 23.9 03 85.6 57.2 63 135.5 90.6 23 185.4 123.9 83 235.3 157.2 44 36.6 24.4 04 86.5 57.8 64 I36.4 91I. 24 186.2 124.4 84 236.1 1578 45 37.4 25.0 05 87.3 58.3 65 i37.2 91.7 25 187. 125.0 85 237.0 158.3 46 38.2 25.6 o6 88.1 58.9 66 138.0 92.2 26 187.9 25.6 86 237.8 158.9 47 39.] 26. 07 89.o 59.4 67 138.9 92.8 27 188.7 126.1 87 238.6 159.4 48 39.9 26.7 o8 89.8 60.0 68 139.7 93.3 28 189.6 126.7 88 239.5 i60.0 49 40.7 27.2 99 90.6 60.6 69 I40.5 93.9 29 190.4 127.2 89 240.3 160.6 50 4i.6 27.8 O1 91.5 61.I 70 I4I.3 94.4 30 19I.2 127.8 90 241.I 161.1 -5 42.4 28.3 III 92.3 61.7 171 142.2 95.0 231 192.1 128.3 291 242.0 161.7 52 43.2 28.9 12 93.1 62.2 72 143.o 95.6 32 192.9 128.9 92 242.8 162.2 53 44.1 29.4 13 94.0 62.8 73 143.8 96.1 33 193.7 129-4 93 243.6 162.8 54 44.9 30.o 14 94.8 63.3 74 144.7 96.7 34 194.6 i3o.o 94 244.5 163.3 55 45.7 3. 95.6 95. 63.9 75 145.5 97.2 35 195.4' i30.6 95 245.3 163.9 56 46.6 31. 1 6 96.5 64.4 76 146. 3 97.8 36 96.2 131.. 96 246.1 I64.4 57 47.4 3I.7 17 97.3 65.0 77 147.2 98.3 37 I97-1 131.7 97 246.9 165.0 58 48.2 32.2 I8 98.1 65.6 78 148.o 98.9 38 197.9 132.2 98 247.8 i65.6 59 49.1 32.8 19 98.9 66.1 79 I48.8 99-4 39 98.7 132.8 99 248.6 166.1 60 49.9 33.3 20 99.8 66.7 80 149-7 100o0 40 199.6 133.3 300 249.4 166.7 Dist.l Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. N.E.byE. S.E.byE. N.W.byW. S.W.byW. [For 5 Points. TABLE I. Page 13 Difference of Latitude and Departure for 3-k Points.'.. N.E.aN. N.W.aN. S.E.1S. S.W.,S. Dist. Lat. Dep. Dist. Lat. Dep. Dist.l Lat. Dep. Dist.l Lat. Dep. Dist. Lat Dep. i oo.8 oc.6 61 49-0 36.3 121 97.2 72.I 181 145.4 107.8 241 193.6 I43.6 2 o0.6 01.2 62 49.8 36.9 22 98.0 72.7 82 146.2 108.4 42 194.4 I44.2 3 02.4 oi.6 63 50.6 37.5 23 98.8 73.3 83 147.0 109Io 43 195.2 I44.8 4 o3.2 02.4 64 51.4 38.1 24 99.6 73.9 84 147-8 109.6 44 i96.o 145.4 5 o4.o o3. o 65 52.2 38.7 25 o00.4 74.5 85 148.6 I10.2 45 196.8 145.9 6 o4.8 o3.6 66 53.0 39.3 26 101.2 75.1 86 149-4 110.8 46 197.6 i46.5 7 o5.6 04.2 67 53.8 39.9 27 102.0 75.7 87 150.2 I11.4 47 98.4 I47.8 o6.4 o04.8 68 54.6 40.5 28 102.8 76.2 88 I51.o 112.0 48 199.2 147-7 9 07.2 o5.4 69 55.4 41.I 29 o03.6 76.8 89 I5i.8 112.6 49 200.0 148.3 10 o8.o o6.0 70 56.2 41.7 30 Io4-4 77.4 90 152.6 113.2 50 200.8 148.9 ii o8.8 o6.6 71 57-0 42.3 I13 105.2 78.0 191 I53.4 13.8 251 201.6 149.5 2 g09.6 07.1 72 57.8 42.9 32 106.0 78.6 92 154.2 I 4.4 52 202.4 150o. 13 o1.4 07.7 73 58.6 43.5 33 Io6.8 79.2 93 I55.0 115.o 53 203.2 150.7 14 11.2 o8.3 74 59.4 44.i 34 107.6 79.8 94 i55.8 115.6 54 204.0 I5I.3 15 12.0 08.9 75 60.2 44-7 35 i08.4 80.4 95 I56.6 116.2 55 204.8 I5I.9 i6 12.9 09.5 76 6I.0 45.3 36 109.2 8.o0 96 1574 11 6.8 56 205.6 152.i 17 13.7 1o.1 77 6.8 45.9 37 110.0 8I.6 97 I58.2 117.4 57 206.4 i53.I 18 14.5 10.7 78 62.7 46.5 38 11o.8 82.2 98 159.0 117.9 58 207.2 153.7 9 i5.3 11.3 79 63.5 47.- 39 1 Ii.6 82.8 99 i59.8 118.5 59 208.0 o54.3 20 I6.i 1 I.9 80 64.3 47.7 4o t12.4 83.4 200 60o.6 119.1 60 208.8 154-9 21 16.9 12.5 8i 65. 48.3 I4I 113.3 84.0 20I i6i.4 119-7 261 209.6 i55.5 22 17.7 13.1 82 65.9 48.8 42 114.1 84.6 02 162.2 120.3 62 210.4 156.1 23 18.5 13.7 83 66.7 49.4 43 114.9 85.2 o3 I63. 120. 63 211.2 156.7 24 I9.3 14.3 84 67.5 50.0 44 115.7 85.8 o4 i63.9 I2I.5 64 2/12.0 157.3 25 20.1 1 4.9 85 68.3 50.6 45 I 6.5 86.4 o5 I647 122. 65 212.8 157.9 26 20.9 1s.5 86 69.1 51.2 46 117.3 87.0 o6 i65.5 122.7 66 213.7 I58.5 27 21.7 16.1 87 69.9 51.8 47 118. 87.6 07 I66.3 123.3 67 214.5 159.1 28 22.5 16.7 88 70.7 52.4 48 118.9 88.2 o8 I67.1 123.9 68 215.3 159.6 29 23.3 17.3 89 71.5 53.0 49 119.7 88.8 09 167-9 124.5 69 216.1 160.2 30 24.1 I 7.9 90 72.3 53.6 50 I20.5 89.4 io 168.7 125.I 70 216.9 i60.8 3I 24.9 -8.5 91 73.1 54.2 I5I I21.3 o90.0 211 169.5 125.7 27I 217-7 161.4 32 25.7 19.1 92 73.9 54.8 52 122.1 90.5 12 170.3 126.3 72 218.5 162.0 33 26.5 19.7 93 74.7 55.4 53 122.9 91.1 13 171-1 126.9 73 2I9.3 162.6 34 27.3 20.3 94 75.5 56.0 54 123.7 91-7 14 171.9 127.5 74 220.1 163.2 35 28.I 20.8 95 76.3 56.6 55 124.5 92.3 5 172.7 128. 75 220.9 163.8 36 28.9 21.4 96 77.1 57.2 56 125.3 92.9 i6 I73.5 128.7 76 221.7 I64.4 37 29.7 22.0 97 77.9 57.8 57 126. 93.5 17 174.3 129.3 77 222.5 I65.0 38 30.5 22.6 98 78.7 58.4 58 126.9 94. I 8 175.1 I29.9 78 223.3 i 65.6 39 3I.3 23.2 99 79.5. 59.o 59 127.7 94.7 19 175.9 30.5 79 224. 166.2 4( 32.1 23.8 Ioo 80.3 59.6 6o 128.5 95.3 20 176.7 1i3i. 80 224.9 i66.8 /1 32.9 24.4 oi1 8r.I 60.2 I61 129.3 95.9 221 177.5 i3r.6 28 225.7 167.4 42 33.7 25.0 02o 81.9 60.8 62 130.1 96.5 22 178.3 132.2 82 226.5 I68.o 43 34.5 25.6 03 82.7 61.4 63 130.9 97-I 23 179-1 i32.8 83 227.3 i68.6 44 35.3 26.2 o4 83.5 62.0 64 131.7 97.7 24 179.9 I33.4 84 228.1 I69.2 45 36 I 26.8 o5 84.3 62.5 65 I32.5 98.3 25 180.7 134.0 85 228.9 169.8 46136 9 27.4 o6 85.1 63.I 66 133.3 98.9 26 i8i.5 i34.6 86 229.7 17O.4 47 37.8 28.0 07 85.9 63.7 67 i34.I 99.5 27 182.3 135.2, 87 230.5 171.0 48 38.6 28.6 o8186.7 64.3 68 134.9 ioo.I 28 I83.1 35.8 88 231.3 171.6 49 39.4 29.2 09 87.5 64.9 69 135.7 100.7 29 183.9 i36.4 89 232.1 172.2 50 40.2 29.8 o 88.4 65.5 70 I36.5 ioi.3 30o 184.7 137.0 90 232.9 172.8 51 4I.0 30.4 III 89-2 66.I I71 137.3 10I.9 23 185.5 37.6 291 233.7 173.3 | 52 4I.8 31.0 2 90.0 66.7 72 I38.2 102.5 32 i86.3 138.2 92 234.5 173.9i 53 42.6 3i.6 13 90.8 67.3 73 139.0 io3.I 33 187.1 I38.8 93 235.3 174.5 54 43.4 32.2 i4 91.6 67-9 74 139.8 103.7 34 188.o 139.4 94 236.1 175.1 55 44.2 32.8 5 92.4 68.5 75.I4o.61 0o4.2 35 188.8 4o.o 95 236.9 175-7 56 45.o 33.4 16 93.2 69.1 76 I41.4 1o4.8 36 1 89.6 140.6 96 237.7 176.3 57 45.8 34.o 17 94.o 69-7 77 142.2 io5.4 37 19(.4 141.2 97 238.6 176.9 58 46.6 34.6 18 94.8 70.3 78 i43.0 Io6.o 38 191.2 141.8 98 239.4 177.5 59 47.4 35. 19 95.6 70o. 79 143.8 Io6.6 39 I92.0 142.4 99 240.2 1781 Dist.1 Dep. Lat.'Dist. Dep Ltat. Dist.t Dep. Lat. D)ist.l Dep. Lat. Dist. Dep. Lat. N.E.E. S.E./E. N.W.JW. S.W.W. [For 41 Points. Page 14] TABLE I. Difference of Latitude and Departure for 31 Points;:: -,.'N.E.N. N.W.N. S.E.S. S.W.4S. D)st. Lat. I)ep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I oo.8 oo.6 6i 47.2 38.7 I21 93.5 76.8 81 1139.9 Ii4.8 24i I86.3 152.9 2 o0.5 o0.3 62 47.9 39.3 22 94.3 77-4 82 I40.7 II5.5 42 187.1 153.5 3 02.3 OI.9 63 48.7 40.0 23 95.1 78.0 83 I4I.5 II6.I 43 187.8 154.2 4 o3. 02.5 64 49.5 40.6 24 95.9 78.7 84 142.2 116.7 44 1I88.6 154.8 5 03.9 03.2 65 50.2 41.2 25 96.6 79.3 85 I43.o II7.4 45 I89.4 155.4 6 o4.6 o3.8 66 51.o 41.9 26 97.4 79-9 86 I43.8 II8.0 46 190.2 156.I 7 05.4 0o4.4 67 51.8 42.5 27 98.2 80.6 87 I44.6 1 8.6 47 190.9 156.7 8 o6. 2 o5. 68 52.6 43.1 28 98.9 81.2 88 I45.3 II9.3 48 191.7 157.3 9 07.0 05.7 69 53.3 43.8 29 99.7 81.8 89 146.I 119.9 49 192.5 I58.o 10 07.7 06.3 70 54.I 44.4 30 1oo.5 82.5 go 146.9 20.5 50 193.3 58.6 ii o8.5 07.0 71 54.9 45.0 13i 10i.3 83.I 191 147.6 121.2 251 I94.0 159.2 12 o9.3 07.6 72 55.7 45.7 32 102.0 83.7 92 i48.4 I21.8 52 194.8 I59.9 13 io.o o8.2 73 56.4 46.3 33 102.8 84.4 93 149. 2 I22.4 53 195.6 i6o.5 i4 10.8 08.9 74 57.2 46.9 34 i03.6 85.o 94 i50.0 123.1 54 196.3 16I.I 5 11.6 o09.5 75 58.0 47.6 35 104.4 85.6 95 150.7 123.7 55 197.I 161.8 i6 12.4 10.2 76 58.7 48.2 36 105.1 86.3 96 I5I.5 124.3 56 I97.9 162.4 17 i3.1 10.8 77 59.5 48.8 37 105.9 86.9 97 152.3 125.0 57 198-7 163.0 i8 13.9 11.4 78 60.3 49.5 38 1o6.7 87.5 98 153. 125.6 58 199.4 163.7 19 I4-7 12.1 79 6i. 50o. 39 107.4 88.2 99 153.8 126.2 59 200.2 i64.3 20 15.5 12.7 80 61.8 50.8 4(0 108.2 88.8 200 54.6 126.9 60 201.0 164.9 21 16.2 I3.3 81 62.6 5i.4 141 109.0 89.4 o20 155.4 127.5 261 201.8 165.6 22 17.0 14.0 82 63.4 52.0 42 109.8 90.I 02 156.1 128.1 62 202.5 166.2 23 I7.8 I4.6 83 64.2 52.7 43 10o.5 90.7 03 1 56.9 I28.8 63 203.3 i66.8 24 i8.6 15.2 84 64.9 53.3 44 III.3 91.4 04 157-7 129.4 64 204-. 167.5 25 I9.3 15.9 85 65.7 53.9 45 I12. 92.0 05 I58.5 I3o.i 65 204.8 168.I 26 20.1 I6.5 86 66.5 54.6 46 112.9 92.6 o6 I59.2 130.7 66 205.6 168.7 27 20.9 17. 87 67.3 55.2 47 I 3.6 93.3 07 60o.o 131.3 67 206.4 I69.4 28 21.6 17.8 88 68.0 55.8 48 1I4.4 93.9 o8 16o.8 I32.0 68 207.2 170.0 29 22.4 i8.4 89 68.8 56.5 49 115.2 94.5 09 161.6 132.6 69 207.9 170.7 30 23.2 19.0 9o 69.6 57.1 50 116.0 95.2 o1 162.3 133.2 70 208.7 171.3 31 24.0 19.7 91 70.3 57-7 151 1I6.7 95.8 211 163.i 133.9 271 209.5 171.9 32 24.7 20.3 92 71.I 58.4 52 II7.5 96.4 I2 163.9 I34.5 72 2IO.3 172.6 33 25.5 20.9 93 71.9 59o. 53 118.3 97-I I3 164.7 i35.i 73 211.0 173.2 34 26.3 21.6 94 72.7 59.6 54 119.0 97.7 14 165.4 i35.8 74 211.8 173.8 35 27.I 22.2 95 73.4 60.3 55 119.8 98.3 i5 166.2 I36.4 75 212.6 I74.5 36 27.8 22.8 96 74.2 60. 56 120.6 99.0 i6 167.0 137.0 76 213.4 175.1 37 28.6 23.5 97 75.0 6i.5 57 12I.4 99.6 17 167-7137-7'77 214.I 175.7 18 29.4 24.1 98 75.8 62.2 58 122.1 100.2 i8 i68.5 i38.3 78 214.9 176.4 39 30.I 24.7 99 76.5 62.8 59 I22.9 100.9 19 169.3 I38.9 79 215.7 177-0 40 30.9 25.4 Ioo 77.3 63.4 60 123.7 io 1.5 20 170.1 139.6 80 216.4 177.6 4I 31.7 26.0 io1 78.1 64.I 161 124.5 102.1 221 170.8 140.2 281 2I7.2 178.3 42 32.5 26.6 02 78.8 64.7 62 125.2 o02.8 22 171.6 t40.8 82 2I8.0 178.9 43 33.2 27.3 03 79.6 65.3 63 I26.o o03.4 23 172.4 I41.5 83 218.8 179.5 44 34-0 27.9 o4 80.4 66.0 64 126.8 io4.0 24 I73.2 I42i. 84 2I9.5 180.2 45 34.8 28.5 05 81.2 66.6 65 127.5 I04.7 25 173.9 142.7 85 220.3 180.8 46 35.6 29.2 o6 81.9 67.2 66 128.3 105.3 26 174-7 I43.4 86 221.1 I81.4 47 36.3 29.8 07 82.7 67-9 67 129.1 I05.9 27 175.5 I44.o 87 221.9 182.1 48 37.1 30.5 o8 83.5 68.5 68 129.9 i o6.6 28 176.2 I44.6 88 222.6 182.7 49 37.9 31. 09o 84.3 69.I 69 130.6 107.2 29 177.0 145.3 89 223.4 183.3 50 38.7 31.7 io 85.0 69.8 70 13I.4 107.8 30 177.8 145.9 90 224.2 i84.o 51 39.4 32.4 1Ii 85.8 70.4 171 132.2 108.5 231 I78.6 146.5 291 224.9 i84.6 52 40.2 33.o 12 86.6 7I. - 72 I33.0 109.1 32 179.3 147.2 92 225.7 185.2 53 4I.0 33.6 i3 87.4 7.17 73 133.7 109.8 33 180.I 147.8 93 226.5 185.9 54 41.7 34.3 14 88.I 72.3 74 134.5 I I.4 34 180.9 148.4 94 227.3 i86.5 55 42.5 34.9 15 88.9 73.0 75 135.3 1ii.o 35 181 7 149. 95 228.0 187.1 56 43.3 35.5 16 89.7 73.6 76 136.o 111.7 36 182.4'49.7 96 228.8 187.8 57 44. 36.2 17 90.4 74.2 77 136.8 112.3 37 183.2 50o.4 97 229.6 I88.4 58 44.8 36.8 18 91.2 74.9 78 37.6 1 12.9 38 I84.o0 15:.o 98 230.4 189.o 59 45.6 37.4 19 92 0 75.5 79 i38.4 I 3.6 39 184.7 151.6 99 23 I1 189.7 60o 46.4 38.I 20 92.8 76.1 80 139.1 II4.2 40 185.5 152.3 300 231.9 190.3 Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. N.E.E. S.E.E. N.W.W. SW.W. [For, 4 Points. TABLE I. (Page 15 Difference of Latitude and Departure for 3- Points..- N.E.4N. N.W.~N. S.E.~S. S.W.~S. Dist. Lat. i Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat: Dep. Dist. Lat. Dep. 007oo 00.7 i 45.2 41.0 121 89-7 8.3 i81 I34.i 121.6 241 178.6 i61.8 2 01.51.3 62 45.9 46 22 90.4 81.9 82 134.9 122.2 49 179.3 162.5 3 02.2 o 2.0o 63 46.71 42.3!23 1.1 82.6 83 135.6 122. 5 43 180.1 i63.2 4 o3.o 02.7 64 47.4 43.0 24 91.9 83.3 84 136.3 123.6 44 80.8 163.9 5 o03.71 3.4 65 48.2 43. 7 25 92.6 83.9 85 1375. 124.2 45 189.5 1647 6 04.4 04.0 66 48.9 44.3 26 93.4 846 86 37.8 24.9 146 82.3 165.2 7 05.2 04.7 67 49.6 45.0 27 694. 85.3 87 i38.6 125.6 47 183.o 165.9 8 205.9 05.4 688 50.4 45.7 28 94.8 86.0 88 139.3 126.3 48 183.8 i66.5 9 16.7 16. o 69 51.1 46.3 29 95i. 86.6 89 0 140.0 126.9 49 184.5 i67.2 io 07.41 6.17 70 51.9 470o 30 96.3 87.3 9go 140.8 27.6 50 85. 1 i67.9 II 208.2o 07.4 6752.6 4 77 131 97 0. 88.0 211 i.5 128.3 251 2860. 168.6 12 08. 91 8. 1 72 53.3 48.4 32 97.8 88.6 92 142.3 128.9 52 1867 1869-2 13 09.6 108.7 73 154.1 49.o 33 98.5 89.3 93 93 43. 129.6 53 087.5 i69.9 14 Io.4 09.4 74 54.8 49.7 34 991 3 9o' 94 i 43.7 430.3 54 88.2 70.6 5 ii. ioI 75 55.6 504 5 ioo.o 90 7 95 i44.5 i34.4 55 188.9 171.2 36 6.79 102. 76 156.3 56. 36 ioo.8 91.3 96 145.2 i31.6 56 189.7 I71.9 17 12.6 11.4 77 157.1 5.7'37 101.5 92.0 97 i46.o 132.3 57 190.4 172.6 38 I3.3 12.1 78 57.8 52.4 38 102.3 92.7 98 i46.7 733.o 58 96 z. 173.3 39 24.1 12.8 79 58.5 53.81 319 1 99 47.4 33.6 59 73.9 17. 20 14.8 13.4 80 59.3 53.7 40 103.7 94.0 200 148.2 134.3 60 192.6 174.6 21 -15.6 -4. 8 60.5. - 54.4 141 45 94.7 201 648.9 435.0 26 8 93.4 175.3 22 36.3 14.8 82 6 0.8 55. 42 io5.2 95.4 02 149.7 135.7 I62 194. 175.9 23 17.o0 5.4 83 61.5 55.7 43 /16.5 960 03 150o.4 136.3 63 I494.9 176.6 24 17.8 o6.1 84 62.2 56.4 44 106.7 9-.7 04 151.2 1i37.o 64 195.6 177.3 45 18.5 16.8 85 63.5 571I 45 107.4 97.4 05 15.9 i 37-7 65 I96.4 1 78.o 26 19.3 17.5 86 63.7 57.8 46 1i8.2 98.0 06 152.6 138.3 66 197.1 1978.6 27 20.0 18.1 87 64.5 58.4 47 1i8.9 98-7 07 153.49 39.o 67 197.8 179.3 28 20.7 18.8 88 65.2 59. 48 109.7 99.4 o8 154.1 I39. 7 68 198.6 18o. 2 9 21.5 39.5 89 65.9 59.8 49 1.4 4oo.2 09 154.7 9 40.4 69 14993 180.6 30 223.2 20. 90 66.7 60.4 50 1.I Ii00. I 1155.6 14.0 70 200.6 181.3 3I 23.-o -o.8 -91 6. 3 -o 6 15.2 7I 1. i IoI. 3 21 156.3 141I 7 27I 2010.4 182.0 32 23.7 21 92 68.2 61.8 52 112.6 102..1 12 57.1 i42.4 72 201.5 182.7 33 24.5o 22.2 93 68.9 62.5 53 128.4 I02.7 3 11 57.8 143.0 73 202.3 183.3 34 25.2 22.8 94/ 69.6 63.2 54 714.1 I o3.4 / 4 58.6 1i437 74 203.0. 184.o 35 25.9 23.5 95 70.4 63.8 55 114.8 10io4.1 15 159.3 144.4 75 2o03.8 184.7 36 26.7 24.2 9671.1 64.5 56 115.6 1o4.8 16 i60.o 145.1 76 204.5 1i85.4 37 27.4 2-4.8 97 7.9 65. 57 171I6.3 Io5.4 87 1760.8 145.7 77 205.2 I86.o 38 28.2 25.5 98 72.6 65.8 58 117.1 106.1 18 161.5 146.4 78 2o6 o 86.7 39 28.9 26.2 99 73.4 66.5 59 17.8 106.8 9 J62.3 147.1 79 2067 187.4 40 29.6 26.9 19 8 74.1 67.2 6~1 i8.6 107.4 20 1 63.0 147.7 80 207.5 1o88.o 41 30.4 27.5 ioi 74.8 67.68 61 11.3 108.1 221 163.8'148.4 281 208.2 188.7 42 31.1 28.2 02 75.6 68.5 62 120.0 0o8.8 22 164.5 149.1 82 208.9 189.4 43 3.9 28.9 o3 76.3 69.2 63 120.8 / o9.5 23 165.2 149. 8 83 209.7 1.9044 32.6 29.b 04/ 77. 169.8 64 121.5 iio.i 24 166.0 150o.4 84 210.4 190o 7 45133.3 30.2 05 77.8 70.5 65 122.3 1o.8 25 166.7 151.1 85 211.2 191,.4 46 34. 130.9 o6 78.5 71.2 66 123.o 1p.5 26D 67.5 15.8 86 D2e.9 I92.t 7 i 34.8 31.6 07 79.3 7.9 67 1N23.7 112.2 27 W68.2 152.4 87 212.7 I9n -7 48 35.6 32.2 08 80.0 7a. 68 124.5 112.8 28 168.9 153.1 88 213.4 193.4 49 36.3 32.9 o09 80o.8 73.2 69 125.2 113.5 29-169.7 153.8 89 214.1 194.1 50o 37.0 33.6 io 81.5 73.9 70 126.o 114.2 30 170.4 154.5 go90 24.9 94.8 5137.8 34.2 ill 82.2 74.5 171 126.7I 114.8 231 171.2 155.1 291 215.6 195.4 52 38.5 34.9 2 83.0 75.2 72 I1927.4 /115.5 32 I7I1.9 155-8 92 216.4 196.1 53 39.3i35.6 13 83.7 75.9 73 I28.2 1 i6.2 33 172.6 156.5 93 /17.1 196.8 54140.0o 36.3 1 4 84.5 76.6 74 128.9 116.9 34 173.4 157.1 94 217.8 197.4 55 40.8136.9 5 85.2 77.2 75 I29.71717.5/ 35 174. 157.8 95 218.6 198.i 56 41.5 37. 16 86.0 77.9 76 130o.4 18.2 36 174.9 158.5 96 219.3 198.8 57 42.2 38.3 17 86.7 78.6 77 131.1 1 18.9 37 175.6 159.2 97 220.1 1995 58 43.0 39.o I 1 87.4 79-2 78 131.9 19g.5 38 176.3159.8 98 220.8 200. 59 43.7 39.6 9 88.2 79.9 79 32.6 20.2 39 I77.1 160.5 99 221.5 ~ 0.8 60 44.5 40.3 2-0 88.9 80o.6 80- 133.4 120o.9 40 I77. 8 161.2 300 5222.3 -.5 Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist.j Dep. Lat i. P _eL j N.EJE. S.E.pE. N.W.bW. S.W.1W. [For 41 Points, j 4~ ~ ~~~~~~~~~~~~~~~~~~~:: 4 4 4. Page 46] TABLE 1. DiiTerence of Latitude and Departure for 4 Points. --— > N.E. N.W. S.E. S.W. Distl.l Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist.1 Lat. Dep. 1io. 7 0o.7 61 43.I 43.I I2 85.6 85.6 181 128 8.0 128.0 24 170.4 70.4 2 01.4 01.4 62 43.8 43.8 22 86.3 86.3 82 128.7 128.7 42 171.1 171.1 3 02. I 02.1 63 44.5 44.5 23 87.0 87.0 83 29.4 129-4 43 171.8 171.8 4 02.8 (2.8 64 45.3 45.3 24 87.7 87-7 84 I30. 130o.1 44 172.5 172.5 5 o3.5 o3.5 65 46.0 46.o 25 88.4 88.4 85 i30.8 I30.8 45 173.2 173.2 6 o4.2 o4.2 66 46.7 46.7 26 89.1 89.1 86 131.5 131.5 46 173.9 173.9 7 o4.9 o4.9 67 647.4 47.4 27 89.8 89.8 87 132.2 132.2 47 174.7 I74.7 8 05.7 05.7 68 48.I 48.I 28 90.5 90.5 88 132.9 132.9 48 I75.4 175.4 9 o6.4 o06.4 69 48.8 48.8 29 91.2 9.2 89 i33.6 i33.6 49 176.1 176.1 Io 07.1 07. 70 49.5 49.5 30 91. 9 9.9 9 34.4 1 34.4 50 176.8 176.8 I 07.8 07.8 71 50.2 50.2 131 92.6 92.6 191 135.1 135.1 251 I77.5 177.5 I2 o8.5 o8.5 72 50.9 50.9 32 93.3 93.3 92 i35.8 I35.8 52 178.2 178.2 I3 09.2 09.2 73 5I.6 5.6 33 94.0 94.0 93 36.536.5 53 178. 178.9 14 o9.9 09.9 74 52.3 52.3 34 94.8 94.8 94 137.2 137.2 54 1796 1 796 i5 1o.6 Io.6 75 53.o 53.o 35 95.5 95.5 95 1379 37-9 55 i80.3 i80.3 16 11.3 11.3 76 53.7 53.7 36 96.2 96.2 96 38.6 i38.6 56 i8i.o i81.o 17 12.0 12.o0 77 54.4 54.4 37 96.9 96.9 97 139.3 I39.3 57 I181.7 181.7 18 12.7 12.7 78 55.2 55.2 38 97.6 97.6 98 140.0 40o.o 58 182.4 182.4 I9 I3.4 13.4 79 55.9 55.9 39 98.3 98.3 99 140.7 140.7 59 183.1 183.i 20 14.1 I4.1 80 56.6 5.6 40 99.0 99.0 200 I41.4 i4i.4 60 183.8 i83.8 21 14.8 14.8 8i 57.3 57.3 141 99-7 99-7 201 142.1 142.1 261 i84.6 i84.6 22 I5.6 15.6 82 58.o 58.0 42 0oo.4 00oo.4 02 142.8 I42.8 62 i85.3 I85.3 23 16.3 16.3 83 58.7 58.7 43 ioi.i ioi.I 03 I43.5 I43.5 63 I86.o i86.o 24 17.0 I7.0 84 59.4 59.4 44 ioi.8 ioi.8 04 144.2 144.2 64 186.7 186.7 25 17.7 17.7 85 60. 60.I 45 I02.5 102.5 o5 145.o0 45.0 65 187.4 187.4 26 8.4 18.4 86 60.8 60.8 46 103.2 o13.2 o6 145.7 145.7 66 188.1 i88. 2- I9.1 19.1I 87 61.5 61.5 47 103.9 103.9 07 464 46.44 67 188.8 188.8 28 I9.8 19.8 88 62.2 62.2 48 104.7 04-7 o8 I47 I 47-. 68 189.5 189.5 29 20.5 20.5 89 62.9 62.9 49 Io5.4 Io5.4 09 147.8 147.8 69 190.2 190.2 30 2I.2 21.2 90 63.66 6 50 o6.1 1o6.i Io i 48. 5 48.5 70 190.9 190.9 31 21.9 21.9 91 64.3 64.3 51 10o6.8 106.8 2I1 149.2 149.2 271 191.6 19I.6 32 22.6 22.6 92 65.1 65.I 52 107.5 107.5 I 149.9 149.9 72 192.3 192.3 33 23.3 23.3 93 65.8 65.8 53 1. 108. 08.2 13 50o.6 150.6 73 193.0 193.0 34 24.0 24.0 94 66.5 66.5 54 108.9 108.9 I4 151.3 I51.3 74 193.7 193.7 35 24.7 24.7 95 67.2 67.2 55 109.6 109.6 I5 152.0 152.0 75 194.5 194.5 36 25.5 25.5 96 67.9 67.9 56 IIo.3 IIO.3 16 152.7 I52.7 76 I95.2 19I5.2 37 26.2 26.2 97 68.6 68.6 57 I I I 1. o 7 I 53.4 I53.4 77 195.9 195.9 38 26.9 26.9 98 69.3 69.3 58 111.7 111.7 18 54.1 i54.1 78 I96.6 196.6 39 27.6 27.6 99 70.0 70.0 59 II2.4 12.4 i9 I54.9 154.9 79 I97.3 I97.3 40 28.3 28.3 Ioo 70.7'70.7 60 113.1 113.I 20 155.6 i55.6 80 198.o 198.0 41 29.0 29.0 IOI 7I1.4 71.4 161 13.8 113.8 221 i56.3 156.3 281 198.7 198.7 42 29.7 29.7 02 72.I 72.1 62 114.6 114.6 22 157.0 157.0 82 199.4 199.4 43 30.4 30.4 o3 72.8 72.8 63 115.3 115.3 23 I57.7 157.7 83 200.1 200.1 44 31.1 3.1 o04 73.5 73.5 64 116.o I 6.o 24 158.4 158.4 84 200.8 200.8 45 3I.8 3i.8 o5 74.2 74.2 65 116.7 116.7 25 159.1 159.1 85 201.5 201.5 46 32.5 32.5 o6 75.0 75.o 66 117.4 I 17.4 26 159.8 159.8 86 202.2 202.2 47 33.2 33.2 07 75.7 75.7 67 18.1 118.1 27 160.5 60o.5 87 202.9 202.9 48 33.9 33.9 o8 76.4 76.4 68 I 8.8 JI8.8 28 16I.2 16I.2 88 203.6 203.6'49 34.6 34.6 09 77.1 77.1 69 II9.5 1I19.5 29 I61.9 161.9 89 204.4 204.4 50 35.4 35.4 10 77.8 77.8 70 I20.2 120.2 30 162.6 162.6 90 205.1I 205.1 5i 36. 36.I I I I 78.5 785 17I I 20.9 120.9 231 I633.3 i63.3 291 205.8 205.8 52 36.8 36.8 12 79.2 79.2.72 121.6 121.6 32 i64.0 i64.0 92 206.5 206.5 53 37.5 37.5 13 79.9 79-9 73 122.3 122.3 33 i64.8 i64.8 93 207.2 207.2 54 38.2 38.2 14 80.6 80.6 74 123.0 123.0 34 i65.5 i65.5 94 207.9 207-9 55 38.9 38.9 I5 8I.3 8I.3 75 123.7 123.7 35 166.2 I 66.2. 95 208.6 208.6 56 39.6 39.6 I6 82.0 82.0 76 124.5 124.5 36 166 i66.66.9 96 209.3 209.3 57 40.3 40.3 17 82.7 82.7 77 I25.2 125.2 37 167.6 167.6 97 210.0 210.0 58 41.0 |i.o i8 83.4 83.4 78 125.9 125.9 38 i68.3 i68.3 98 210.7 210.7 59 41. 47 41.7 9 84. 84.I 79 126.6 126.6 39 169.0 169.0 99 211.4 211.4 60 42.4 42.4 20 84.9 84.9 8 127.3 127.3 40 169.7 169-7 300 212.1 2I2.I Dist.De. ep D ep. p. at.. Dist. Dep. Lat. Dist. Dep. at. N.E. N.W. S.E. S.W. [For 4 Points. Page 62],TAB LE III.' Meridional Parts.:,:2?2 M. 00 10 2 30 40 5' 60 70 8~90 10~ ll~ 120 130 m. -- o~ 16o i~o-8o...240: 3oo 36-J 421:: 482-5-2 603 664-725 787 o i 6i 121 18I 24i 3oi 362 422 483 543 6o4 665 726 788 I 2 2 62 122 i82 24J2 302 363 "423 484 544 605 666 727 789 9 3 3 63 i23 i83 243 303 364 4124 485 545 606 667 728 79~ 3 4 64 i24 i84 244 304 365 425'486 546 607 668 _729 791 4 5 - -i65 i2 -i85 -245 -305' 366 -426 487 -547'6'866 30 796'666 I26 186 2146 306 367 427 488 548 609 670 731 793 6 7 67 I27 I87 247,307 368 428 489 549 6i 671 732 794 7 8A 68 I28 i88 248 308 369 429 490 550 6 7i6-2 734 795 8 9 9 69 I29 i89 249 309 370 430 49i 55i 6?'6-3 735 796 9 10oIo 70: -I'- 10 --— 250 3 —io 371 -4 —i'492 55"-2'6i3' 674 736 -797 to I I II 7r i3i 191 25r 3rI 372 432 493 553 6f4 675 737 798 -, 12 2 72 i32 I92 252 ~3i2 373 433 494 554 6i5 676 738 799 I2 i 3 73 i33 i93 253 3i3 37'4 434 495 555 6i6 677 739 8od x3 14 1! 74 i34 194 254 3i4 375 435 496 556 617 678 740o 8oi t4'~-5 —-5"J"'75... 35 -i95 —255~ -3i5 4376' 4-97'557 6-'8 679 741 -802 i5 x 7 77 i37 197 257 317 378 438 499 559'62o 68I 743 804 17 x8 8 78 i38 i98 258 3i8 379 439 500 560 62i 682 744 8o5 18! Jr9 f9 79 139 199 259 319 38o 44o 5oi 56i 622 683 745 8o6 19j 20 -20 o" ioo 2- ~00 260o 3'-oo 38i 44i 502- 562- 623 68 4 872 21 2 81 i4l 20I 26i 32i 382 442 503 564 624 685 747 808 21 22 22 82 ~42 202 262 322 383 443 504 565 625 687 748 809 22 24 24 84 144 2o4 264 324 3-85 445 506 567 627 689 75o 8i2 2'5 2-5 85 ~4 205 -~65-..32'5 386 -'446 — 5o 56-8 686o7~82 26.26 86 i46 2o6 266 326 387/ 447 5o8 569 629 691 752.8 13 26 27 27 87 i47 207 267 327 388 448 5o9 570 63i 692 753 815 27 2828 88 i48 208 268 328 389 449 51o 57i 632 693 754 8i6 28 29 29 89 i49 209 269 330 390 450 5ii 572 633.694 755 817 99 3 0 9~ g5o A -o 21 7o 3 3 39i 45i 512 573 -634 695 756 8i8 30 3I 3I 9r i5r 211 27i 332 392 452 5i3 574 635 696 757 8I9 3I 3232 92 I52 2I2 272 333 393 453 5x4 575 636 697 788o3 3333 93 i53 2,3 273 334 394 454 515 576 637 698 759 82i 33 3434 94 i54 2i4 274 335 395 455 5i6 577 63;8 699 760 822 34 35'-3~5 -9~5- -55 -~i 5 — 75 —— 336:396 —456- -5I7 -578 —639 700 -76i 823 35! 3636 96 i56 2i6 276 337 397 457 518 579 640 7of 762 824 36: 3737 97 i57 217 277 338 398 458 5I9 58o 641 702 763 825 37 3838 98 i58 2i8 278 339 399 459 520 58i 642 703 764 826 38 39 39 99 -~q 29 279 34o 4oo 46o 521 582 643 704 765 827 139 TO- -'-ioo i6o 220 08 34i- 4oi 46I 522 583 644 7o5 766 828!4o 4i 4i oi i6l 221 28r 342 402 462 523 584 645 706 767 829 4i 4242 102 i62 222 282 343 403,463 524 585 646 707 768 830 42 4343 io3 I63 9- 3 283 34/4 404 464 525 586 647 708 769 83r 43 4444 IO4 i64 224 284 345 405 465 526 587 648 -709 770 832 44 45 -~o5 i65 -225 -285 -~36 ~ 4 5 7 710 -4 - 7o-771-334" 46 46 io6 i66 226 286 347 407 467 528 589 650 7II 772 834 46: 47 47 107 I67 227 287 348 408 468 529 590 651 72 773 835 47 48 48~ io8 i68 228 288 349 409 469 530 59i 65' 7i3 774 836 48 49 49 io9 I69 229 289 350 410 470 53i 592 653 714 775 837 49 50 I0~x ~7o 0 30 2.9o 35i' -4i 47i 532 593 654 7I5.777 -838 50 5i 51 III i7i 231 29i 352 412 472 533 594 655 7i6 778 8391. 51 52 52 i12 172 232 292 353 4i3 473 534 595 656 717 779 840 52 53 53 ii3 I73 9-3 3 293 354 414 474 535 596 657 718 780 84i 53 5 4 54 ii4 174 234, 294 355 415 476 536 597 658 7I9 78i 842 54i 5555 ~- 5 I75 235 295 356 4i6 477 537 598 659 720 782 843 55 56 56 u6 176 236 296 357 417 478 538 599 660 721 783 844 56 57 57 117 177 237, 297 358.4i8 479 539 600 66i 722 784 845:57 58 58 ii8 178 238 298 359 419 480 540 6oi 66a 723 785 846 58 F)959 ii9 I79 239 299 3o420 48I 54i 602 663 724 7868415 M.0~ 1~ 20 30 40 5~ 60 70 8~ 90'10~ 11~ 120 130 TABLE III.Pae 63'Meridional Parts. M. 140 15~ 160 7 170 180 _190 20 210 22~ 230 240 250 260 27 o 848 910 973 1035 1098 1i6i 1225 1289 i354 1419 i484 i55o 61i6. 1684 o I 850 911 974 36 99 63 26 90go 55 20 85 51 18 85 1 2 851 913 975 37 i1Ioo 64 27 91 56 21 86 52 19 86 3 852 914 976 38 oi 65 28 92 57 22 87 53 20 87 3 4 853 915 977 39 02 66 29 93 58 23 88 54 21 884 5 8.4 916 978 1o4i 1103 1167 1230 1295 1359 i 424 90o 556 1622 1689 5 6 855 917 979 42 05 68 32 96 60 25 91 57 23 906 7 856 918 98o 43 o6 69 33 97 6i 26 92 58 24 9 7 8 857 919 98 44 07 70 34 98 62 27 93 59 25 93 8 9 858 920 982 45 o8 71 35 99_63 28 94 60 26 94 9 io 859 921 -983 o46 1109 1172 1236 i3oo 1364 i43o 1495 i56i 1628 1695 io 11 860 922 984 47 10 73 37 o0 66 3i 96 62 29 96 11 12 861 923 985 48 II 74 38 02 67 32 97 63 30 9712 13 862 924 986 49 12 75 39 03 68 33 98 64 31i 9813 14 863 925 987 50 13 76 4o o4 69 34 99 665 32 99 14 15 864 926 988 i1r 1114 1177 1241 1305 1370 1435 1500 1567 1633 1700 15 16 865 927 989 52 15 78 42 o6 71 36 02 68 34 oi 16 17 866 928 99o 53 16 79 43 07 72 37 03 69 35 0317 18 867 929 991 54 17 8i 44 o8 73 38 0o4 70 37 04 18 ig 868 930 993 55 18 82 45 10_74 39 05 71 38 05 19 20 869 931 994 o1056 9 I3 i246 1311 1375 i44o i5o6 1572 1639 1706 20 21,870 932 995 57 20 84 48 12 76 41 07 73 40 0721 22 871 933 996 58 21 85 49 13 77 43 08 74 41 08 22.23 872 934 997 59 22 86 50 14 79 44 09 75 42 09 23 24 873 935 998 6o 23 87 51 15 80'45 10 77 43 11 24 25 874 936 999 1061 1125 ii88 1252 1316 i381 1446 151' 1578 i644 1712 25 26 875 937 00ooo 63 26 89 53 17 82 47 13 79 45 1326 27 876 938 oi 64 27 90 54 18 83 48 14 8 47 14 27 28 877 939 o02 65 28 91 55 19 84 49 15 8 48 15 28 29 878 941 o3 66 29 92 56 20 85 50 16 82 49 16 29 30 879 942 1004 1067 1130 1193 1257 1321 I386 i45i 1517 583 I650 1717 30 31 880 943 05 68 3i 94 58 22 87 52 18 84 51 18 3I 32 882 944 o6 69 32 95 59 24 88 53 19 85 52 2032 33 883 945 07 70 33 96 6o 25 89 55 20 86 53 2133 34 884 946 o8 71 34 98 61 26 go90 56 21 88 54 22 34 35 885 947 1009 1072 1135 1199 1262 1327 1392 1457 i522 1589 1656 1723 35 36 886 948 10 73 361200oo 64 28 93 58 24 90 57 2436 37 887 949 II 74 37 oi 65 29 94 59 91 58 2537 38 888 950 12 75 38 02 66 30 95 6o 26 92 59 26 38 39 889 951 i3' 76 39 o3 67 31 96 61 27 93 60 2739 40 890 952 1014 1077.1140 I204 1268 1332 13 97 1462 1528 1594 i66i 1729 40 41891 953 15 78 41 o05 69 33 98 63 29 95 62 30 41 42 892 954 16 79 42 o6 70 34 99 64 30o 96 63 31 42 A3 893 955 18 8o 44 07 71 35 1400 65 31 98 64 3243 44 894 956 I19 81 45 o8 72 36 oi 67 32 99166 3344 45 895 957 I020 1082 1146 1209 1273 1338 1402 1468 1533 1600 1667 1734 45 46 896 958 21 84 47 10 74 39 o3 69 35 oi 68 3546 47 897 959 22 85 48 ii 75 4o o05 70 36 02 69 36147 48898 960 23 86 49 12 76 4 06 71 37 03 70 38 48 49 899 961 24 87 50 13 77 42 07 72 38 o4 7I 39 49 50 900oo 962 1025 1088 1151 1215 1278 343 1408 1473 1539 1605 1672 1740 50 51 901 963 26 89 52 6 8o 44 09 74 40 06 73 41 5i 952 902 964 27 90 53 17 8i 45 io 75 41 08 75 42 52 53903965 28 91 54 18 82 46 ii1 76 42 09 76 43 53 154 904 966 29 92 55 19 83 47 12 77 43 10 77 4S454 55 9"o5 968 io3o 1093 1156 1220 1284- 1348I 413 -479 i544 16ii 1678 1746 55 56 906 969 31 94 57 21 85 49 14 8o 46 12 79 47 56 5- 907 970 32 95 58 22 86 5o0 15 8i 47 13 8o 48157 58 908 971 33 96 59 23 87 52 i6 82 48 14 8i 49158 59909 972 34 97 60 24 88 53 18 83 49 15 82 50 59 M.1 4 150 160 170 1.80 190 260 210 ~0 203 240 250 260 270 M Page 64] TABLE III Meridional Parts. M. 280 290 300 310 32~ 330 340 350 36 370 380 390 40 4 0 1751 1819 1888 1958 2028 2100 2171 224 2318 2393 2468 25.45 2623 2702 0 i 52 21 0o 59 3o 01-" 73 46 19 94 70 46 24 03 2 53 22 9jI 6o 31 02 74 47 20 95 71 48 25 o4 2 3 55 23 92 62 32 o3 75 48 22 96 72 49 27 o6 3 4 56 24 93 63 33 04 76 49 23 98 73 5 28 07 4 5 1757 1825 1894 1964 2034 2105 2178 2250 2324 2399 2475 2551 2629 2708 5 6 58 26 95 65 35 07 79 5 25 2400 76 53 31 10 6 7 59 27 96 66 37 o8 8o 53 27 01 77 54 32 11 7 8 6o 29 98 67 38 09 81 54 28 o3 78 55 33 12 8 9 6 3o 99 69 39 101 82 55 29 o4 8o 57 34 14 9 io 1762 1831 1900 1970 2040 21112184 2257 2330 2405 2481 2558 2636 2715 io I11 64 32 o01 71 4 13 85 58 32 o6 82 59 37 I611 12 65 33 02 72 43j 4 86 59 33 o8 84 6o 38 18 12 13 66 34 o3 73 44 15 87 6o 34 09 o 85 62 4/o 1913 14 67 35 o05 74 45 i6 88 6i 35 Io 86 63 4i 2. 114 15 1768 1837 i9o6 1976 2046 2117 2190 2263 2337 2411 2487 2564 2642 2722 15 i6 69 38 07 77 47 19 91 64 38 13j 89 66 44 23 16 17 70 39 o8 78 48 20 92 65 39 14 90go 67 45 2417 18 72 40.09 79 So 21 93 66 40 15 91 68 46 26 18 19 73 41 io 8o 51 22 94 68 42 16 92 69 48 2719 20 1774 1842 1912 1981 2052 2123 2196 2269 2343 2418 2494 2571 2649 2728 20 21 75 43 13 83 53 25 97 70 44 19 95 72 50 292 l1 229 76 45 I4 84 54 26 98 71 45 20 96 73 5i 3122 23 77 46 i5 85 56 27 99 72 46 22 98 75 53 32 23 24 78 47 i6 86 57 28 2200 74 48 23 99 76 54 33 24 25 1780 1i848 1917 1987 2058 2129 2202 2275 2349 2424 2500 2577 2655 2735 25 26 81 49 i8 88 59 31 o3 76 5o 25 oI 78 57 36l26 27 82 50 20 go90 60 32 04 77 51 27 o3 8o 58 37J27 28 83 52 21 91 6i 33 o5 79 53 28 o4 8 59 3928 29 84 53 22 92 63 34 07 8o 54 29 o05 82 6 40 29 30 1785 1854 1923 1993 2064 2135 2208 2281 2355 2430 250o6 2584 2662 2742 30 31 86 55 24 94 65 37 o09 82 56 32 o8 85 63 43 3i 32 87 56 25 95 66 38 io 83 58 33 og 86 65 44 32 33 89 57 27 97 67 39 5'185 59 34 10 88 66 4633 34 90 58 28 98 69 4o 13 86 6o 35 12 89 67 47 34 35 1791 i86o 1929 1999 2070 2141 2214 2287 2361 2437 2513 2590 2669 2748 35 36 92 61 30 2000 71 43 i5 88 63 38 14 91 70 50 36 37 93 62 3i oi 72 44 16 90 64 39 15 93 71 51 37 38 94 63 32 02 73 45 17 91 65 40 17 94 73 5238 39 95 64 34 04 75 46 19 92 66 42 18 95 74 54 39 40 1797 i865 1I935 2005 2076 2147 2220 2293 2368 2443 2519 2597 2675 2755 4o 4' 98 66' 36 o6 77 49 21 95 69 44 21 98 76 56 41 42 99 68 37 07 78 50 22 96 70 45 22 99 78 5842 43 i800oo 69 38 8 79 51 24 97 71 47 23:2601 79 59 43 44 oi 70 39 10 8o 52 25 98 73 48 24 o02 8 60 44 45 i8o02 1871 1941 2011 2082 2153 -2226 2299 2374 2449 2526 2603 2682 2762 45 46 o3 72 42 12 83 55 27 23o1 75 5i 27 o4 83 63 46 47 o05 73 43 13 84 56 28 02 76 52 28 -o6 84 64!47 48 06 75 44 i4 85 57 30 o3 78 53 3o 07 86 66 48 49 07 76 45 15 86 58 31 04L 79 54 3i o8 87 67149 50o i808 1877 1946 2017 2o88 i2159 2232 2306 2380 2456 2532 2610 *2688 2768 50 51 09 78 48 18 89 6i 33 07 8i 57 33 II 90 70 51 52 10'79 49 1990 62 35 o08 83 58 35 12 91 7152 53 II 8o 5 o g20 91 63 36 09 84 59 36 i4 92 7253 54 13 81 5 21 92 64 37 ii 85 61 37 15 94 74 54 55 i84 1883 1952 2022 2094 2165 2238 2312 2386 2462 2538 2616 2695 2775 55 56 5 84 53 24 95 67 39 i3 88 63 40 17 96 76 56 16 85 55 25 96 68 41 94 1 89 64 4 19 98 78 57 5 171 86 56 26 97 69 42 6 90go 66 42 20 99 79/58 59 18J 87 57 27 98 70 43 17 91 67 44 21 2700 8o 5j,.f 28- 290 30~ 310 "32 330 340 350 360 370 380 390 400 410 M. TABLE III. [Page 65 Meridional Parts. j. 420 43~ 440 450 46~ 47~ 48 490~ 500 510 52~ 530 54 550 I. o 2782 2863 2946 3030 316 3203 3292 3382 3474 3569 3665 3764 3865 3968 o 83 64 47 3 17 04 93 84 76 70 67 65 66 70 2 84 66 49 33 i8 o6 95 85 78 72 68 67 68 712 3 86 67 5o 34 20 07 96 87 79 74 70 69 70 73 3 4 87 69 5I 36 21 o9 98 88 81 75 72 70 71 75 4 5 2788 2870 2953 3037 3123 3210 3299 3390 3482 3577 3673 3772 3873 3977 5 6 90 71 54 38 24 12 330 91 84 78 75 74 75 78 6 7 91 73 56 4o 26 i3 02 93 85 80 77 75 77 807 8 92 74 57 41 27 i4 o3 94 87 82 78 77 78 82 8 9 94 75 58 43 29 i6 05 96 88 83 80 79 80 84 9 10 2795 2877 2960 3044 3i30 3217 3306 3397 3490 3585 368i 3780 3882 3985 io Ii 97 78 6i 46 31 I9 o8 99 92 86 83 82 83 87 11 12 98 8o 63 47 33 20 og 3400 93 88 85 84 85 89 2 13 99 81 64 48 34 2 2 11 02 95 90 86 85 87 911 3 14 280 82 65 50 36 23 12 o3 96 91 88 87 89 9214 15 2802 2884 2967 3051 3137 3225 3314 3405 3498 3593 3690 3789 3890 3994 15 i6 o3 85 68 53 39 26 16 07 99 94 91 90 92 96 16 17 05 86 70 54 4o 28 17 o8 350o 96 93 92 94 98 17 18 o6 88 71 55 42 29 19 10 o3 98 95 94 95 99 18 9 07 89 72 57 43 31 20 11 o4 99 96 95 97 4001 9i 20 2809g 2891 2974 3058 3r44 3232 3322 3413 3506 3601 3698 3797 3899 40o3 20 21 10 92 75 60 46 34 23 14 07 02 99 99 3901 05 21 12' II 93 76 6i 47 35 25 i6 og o4 3701 3800 02.o6 22 23 13 95 78 63 49 37 26 17 io o6 o3 02 o4 8 23 4 44 4 6 79 64 50 38 28 9 12 07 04 04 6 24 _ 2815 2897 2981 Jo 365 352 3240 3329 3420 35i4 360 3706 3806 3907 4012 25 i 26 17 99 2 67 53 4I 31 22 5 10 o8 07 09 4 26 7 1i8 2900 83 68 55 421 32 23 17 12 09 09 5 27 28 20 02 85 70 56 44 34 25 i8 14 II 11 13 17 28 9i 2 3 86 1 8 71 57 45 35 27 20 I5 I3 12 14 19 29 o0 2822 | 2904 2988| 3073 |3159 3247 3337 3428 352I 3617 3714 38i4 39I6 4021 30 31 24 o6 89 74 60 48 383 30 23 18 i6 16 18 22 3 23 25 07 91 75 62 50 40 3i 25 20 17 17 19 24 32 33 26 o8 92 77 63 51 41 33 26 22 19 19 21 26 33 3ij 28 10 93 78 65 53 43 34 28 23 21 21 23 28 34 35 2829 2911 2995 3080 3166 3254 3344 3436 3529 3625 3722 3822 3925 4029 35 36 30 I3 96 8i 68 56 46 37 31 26 24 24' 26 31 36 37 32 14 98 83 69 57 47 39 32 28 26 26 28 33 37 38 33 15 99 84 71 59 49 40 34 30 27 27 3o 35 38 39 34 17 3000 85 72 60 5o 42 36 3 29 291 32 37 39 40 2836 2918 3002 3087 3173 3262 3352 3443 3537 3633 373 13831 3933 4038 40 41 37 19 03 88 75 63 53 45 39 34 32 32 35 40 41 49. 39 2 05 go 76 65 55 47 40 36 34 34 37 42 42 43 40 | 22 06 91 78 66 56 48 42 38 36 36 38 44 43 44 41 24 07 93 79 68 58 50o 43 39 37 38 4 45 44 45 2843 2925 3009 3094 318 3269 3359 3345 3545 364i 3739 3839 3942 4047 45 46 44 26 o 95 82 71 6 1 53 47 43 4 41 44 49 46 47 45 28 12 97 84 72 62 54 48 44 42 43 45 51 47 48 47 29 13 98 85 74 64 56 50 46 44 44 47 52 48 49 48 31 14 3Ioo 87 75 65 57 51 47 46 46 49 54 49 50o 2849 2932 3016 3101ioi 388 3277 3367 3459 3553 3649 3747 3848 3951 4056 50 1 5I 33 17 o3 90 78 68 60 55 51 49 49 52 58 1 52 52 35 I9 o4 91 80 70 62 56 52 50 5 54 60 52 53 54 36 20 05 92 81 71 64 58 54 52 53 56 61 53 54 55 37 21 07 94 83 73 65 59 55 54 54 58 63 54 55 2856 2939 3023 3io8 3195 3284 3374 3467 356i 3657 3755 3856 3959 4065 55,56 58 40 24 10 97 86 76 68 62 59 57 58 6 67 56 57 59 42 26 IT 98 87 78 70 64 60 59 6o 63 69 57 58 60 i 43 27 3 3200 89 79 71 66 62 60 61 64 70 58 2 44 29 14 o| I 90 81 73 67 64 62 63 66 72 59. a 4~0 0 430 440 450 460 470 480 48 490 500 51~ 520 53 540 550 M. 9 -~~~~~~~~~~~l Page 66] TABLE III. Meridional Parts. AM. 560 570 58~ 590 600 610 620 630 640 650 660 670 68~ 69) M. o 4074 4183 4294 4409 4527 4649 4775 4905 75039 5179 534474 563 5795 I 76 84 96 II 29 51 77 07 42 8i 26 77 33 97 2 77 86 98 13 3 53 79 o 44 84 28 79 36 5800 2 3 79 88 4300 15 33 55 8i 12 46 86 31 82 39 o3 3 4 81 90 02 17 35 57 84 14 49 88 33 84 42 o6 4 5 4o83 4192 4304 4419 4537 4660 4786 4916 505i 5191 5336 5487 5644 5809 5 6 85 94 o6 21 39 62 88, 8 53 93 38 89 47 11 6 7 86 95 08 23 41 64 90 20 55 95 4i 92 5o 14 7 8 88 97 o09 25 43 66 92 23 58 98 43 95 52 17 8 9 90 99 II 27 45 68 94 25 60 5200 46 97 55 20 9 io 4092 4201 4313 4429 4547 4670 4796 4927 5062 5203 5348 5500oo 5658 5823 io II 94 o3 i5 3i 49 72 98 29 65 o05 5 o02 6o 25 11 12 95 05 17 33 51 74 4801 3i 67 07 53 o05 63 28 I2 13 97 07 19 34 53 76 o3 34 69 10 56 07 66 3113 14 99 08 21 36 55 78 05 36 71 12 58 io 68 34 14 15 4101 4210 4323 4438 4557 468o 4807 4938 5074 5214 536i 55313 5671 5837 15 i6 o3 12 25 40 59 82 09 40 76 17 63 15 74 39'16 17 o4 14 27 42 62 84 11 43 78 19 66 i8 76 42 17 18 06 i6 28 44 64 87 i4 45 8i 22 68 2o 79 45 i8 19 o8 i8 30 46 66 89 i6 47 83 24 71 23 82 48 19 20 4110 422o 4332 4448 4568 4691 48i8 4949 5085 5226 5373 5526 5685 5851 20 21 12 21 34 50 70 93 20 51 88 29 76 28 87 5421, 22 13 23 36 52 72 95 22 54 90 31 78 3 go90 5622 23 15 25 38 54 74 97 24 56 92 34 80 33 93 59 23 24 17 27 40 56 76 99 26 58 95 36 83 36 95 62 24 25 4119 4229 4342 4458 4578 4701 4829 4960 5097 5238 5385 5539 5698 5865 25 26 21 31 44 60 80 03 31 63 99 41 88 41 5701 68 26 27 22 32 46 62 82 05 33 65 5102 43 90 44 o4 71 27 28 24 34 47 64 84 07 35 67 04 46 93 46 06 7428 29 26 36 49 66 86 10 37 69 06 48 95 49 09 76 29 30o 4128 42384351 4468 4588 4712 4839 4972 5108 5250 5398 5552 5712 5879 30 3, 30 40 53 70 90 I4 42 74 53 54o01 54 15 82 3i 32 32 42 55 72 92 i6 44 76 i3 55 o3 57 17 85 32 33 33 44 57 74 94 i8 46 78 i5 58 06 59 20 88 33 34 35 46 59 76 96 20 48 81 18 60 08 62 23 91 34 35 4137 4247 4361 4478 4598 4722 485o 4983 5120 5263 54ii 5565 5725 5894 35 36 39 49 63 8o0 4600 24 52 85 22 65 13 67 28 96 36 37 41 51 65 82 02 26 55 87 25 67 i6 70 31 99 37 38 42 53 67 84 04 28 57 90 27 70 18 73 34 5902 38 39 44 55 69 86 06 31 59 92 29 72 21 75 36 05 39 40 4i46 4257 4370 4488 4608 4733 4861 4994 5i32 5275 5423 5578 5739 5908 40 41 48 59 72 90go 10 35 96 96 34 77 26 8o 42 141 42 5o 60 74 92 12 37 65 99 36 80 28 83 45 1442 43 52 62 76 94 i4 39 68 5001 39 82 3i 86 47 17 43 44 53 64 78 95 16 4r 70 03 41 84 33 88 50 1944 45 4155 4266-43 80 4497 46i8 4743 4872 500 5 5i43 5287 5436 -5591 5753 5922 45 46 57 68 82 99 20 45 74 08 46 89 38 94 56 25 46 47 59 70 8445oi 23 47 76 io 48 92 41 96 5-8 28 147 48 6i 72 86 03 25 50 79 12 51 94 43 99 6i 3i 48 49 62 74 88 05 27 52 81 14 53 97 46J5602 64 34 49 50 4164'4275 4390 4507 4629 4754 4883 5017 5155 5299 5448 5604 57675937 50 51 66 77 92 09 31 56 85 19 58 53oi0 51 07 70 4o 51 52 68 79 94 11 33 58 87 21 60 04 54 10 72 43 52 53 70 81 96 13 35 60 90 23 62 o6 56 12 75 46 53 54 72 83 98 15 37 62 92 26 65 09 59 i5 78 48 54 55 4173 4285 4399 4517 4639 4764 4894 5028 5167 53ii 546i 5617 5781 5951 55 56 75 874401 19 41 66 96 30 69 14 64 20 83 54/56 57 77 89 03 21 43 69 98 33 72 i6 66 23 86 5757 58 79 91 05 23 45 71 4901 35 ~ 74 19 69 25 89 6058 59 8i 92 07 25 47 73 o3 37 76 21 71 28 92 63 59 M. 560 570 580 590 600 610 620 630 640 650 660 670 680 69 M. TABLE III. [Page 67 Meridional Parts. im. 790 710 720 730 740 750 760 770 780 790 80~ 81~ 82~ 833 M. ol 566 6i46 6335 6534 6746 6970 7210 7467 7745 8o46 8375 8739 9145 9606 o 69 49 38 38 49 74 14 72 49 5I 81 45 53 14 2 72 52 41 41 53 78 18 76 54 56 87 52 6 22 2 3 75 55 45 45 57 82 22 81 59 6i 93 58 67 31 3 4 78 58 48 48 6o 86 27 85 64 67 98 65 74 39 4 5 5981 616i 6351 6552 6764 6990 7231 7490 7769 8072 84o4 8771 9182 9647 5 6 841 64 54 55 68 94: 35 94 74 77 10 78 89 556 7 86 67 58 58 71 97 39 98 78 83 6 84 96 64 7 8 89 70 61 62 75 7001 43 7503 83 88 22 9I1 9203 72 8 9 92 73 64 65 79 05 47 07 88 93 27 97 11 80 9 1o 5995 6177 6367 6569 6782 7009 7252 7512 7793 8099 8433 8804 921-8 9689 1io 11 98 8o 71 72 86 13 56 6 98 8o104 39 10 25 97 11 12 6ooi 83 74 76 go90 17 60 21 7803 09 45 17 33 9706 12 13 04 86 77 79 93 21 64 25 08 15 5i 23 40 14 13 14 07 89 8o 83 97 25 68 30 13 20 57 3o 48 23 14 151 6i00I 6192 6384 6586 *6801 7029 7273 7535 7817 8125 8463 8836 9731 i5 i6 13 95 87 90 04 33 77 39 22 3 69 43 62 40 i6 17 6 98 go90 93 8 37 8i 44 27 36 74 49l70 48 17 i8 19 6201 94 97 12 41 85 48 32 4i 8o 56 77 57 18 19 22 o5 97 6600 i5 45 89 53 37 47 86 63 85 65 19 20 6025 6208 6400o 6603 6819 7048 7294 7557 7842 8152 8492 8869 9292 9774 20 21 28 II o3 07 23 52 98 62 47 58 98 76 9300 83 21 22 31 14 07 10 26 56 7302 66 52 63 85o4 83 07 91 22 23 34 17 io 14 30 60 06 71 57 68 Jo 89 15 9800 23 24 37 20 13 I7 34 64 ii 76 62 74 16 96 22 0924 25 6040o 6223 6417 6621 6838 7068 7315 7580 7867 8179 8522 8903 9330o 9817 25 26 43 26 20 24 4i 72 19 85 72 85 28 09 37 2626 27 46 30 23 28 45 76 23 89 77 90g 34 i6 45 35 27 28 49 33 27 3i 49 8o 28 94 82 96 40 23 53 44 28 29 52 36 3o 35 53 84 32 99 87 8201 46 3o 6o 5229 30 6o055 6239 6433 6639 6856 7088 7336 7603 7892 8207 8552 8936 9368 9861 3o 31 58 42 37 42 60 92 41 08 97 12 58 43 76 70 31 32 61 45 4o 46 64 96 45 12 7902 18 65 50 83 79 32 33 64 49 43 49 68 7100 49 17 07 23 71 57 91 88 33 34 67 52 47 53 71 04 53 22 12 29 77 63 99 9734 35 6070 6255 6450 6656 6875 7108 7358 7626 7917 8234 8583 8970 9407 9906 35 36 73 58 53 60 79 12 62 31 22 4o 89 77 i4 15736 37 76 6i 57 63 83 16 66 36 27 45 95 84 22 24 37 38 79 64 6o 67 86 20 71 4o 32 51 8601 91 30 33 38 39 82 68 63 70 90 24 75 45 37 56 07 98 38 42 39 40 6085 6271 6467 6674 6894 7128 7379 7650 7942 8262 8614 9005 9445 9951 zo 41 88 74 70 77 98 32 84 54 48 67 20 12 53 6041 42 91 77 73 81 6901 36 88 59 53 73 26 i8 6i 69 42 43 94 80 77 85 05 40 92 64 58 79 32 25 69 7843 44 97 83 8o 88 09 45 97 68 63 84 38 32 77 87i44 45 61006287 6483 6692 6913 7149 7401 7673 7968 8290 8644 9039 9485 9996 45 46 03 90 87 95 17 53 06 78 73 95 5 46 93 10005 46 47 06 93 90 99 20 57 10 83 78 8301 57 53 9501 10015 47 48 o09 96 94 6702 24 61 14 87 83 07 63 6o 09 10024 48 49 12 99 97 06 28 65 g19 92 89 12 69 67 17 10033 49 50 6115 633 6500oo 6710 6932 7169 7423 7697 7994 83i8 8676 9074 9525 10043 50 51 18 o6 o4 13 36 73 27 7702 99 24 82 8i 33 10052 51 52 21 09 07 17 4 77 32 06 8oo004 29 88 88 4i oo6i 52 53 24 12 II 20 43 8i 36 11 09 35 95 96 4910071 53 54 27 15 14 24 47 85 41 i6 14 4i 8701 9103 57 10080 54 55 6i3o 6319 6517 6728 6951 7189 7445 7721 802o 8347 8707 9110 9565 10089 55 56 33 22 21 31 55 94 49 25 25 52 14 17 73 oog10099 56 57 36 25 34 35 59 98 54 3o 3o 58 20 24 81 0io08 57 58 40o 28 28 38 63 7202 58 35 35 64 26 3i 89 ioii8 58 59 43 32 31 42 66 o6 63 4o 4o 69 33 38 981012759 3r. 700 710 70 730 740 750 769 770 780 79 800 81~ 820 830 M. TABLE XXI. rPage 133 For turning Degrees and Minutes into Time, and the contrary. D. H.. D. H.M. D. H. M. D. H M. D H. H. D. H. M..M. S'.. M. M. S. M.. S. M. M. S. M. M. S. M. M. S. l o. 4 61 4- 4 121 8. 4 181 12. 4 241 i6. 4 3o0 20. 4 2 o. 8 62 4. 8 I22 8. 8 182 I2. 8 242 I6. 8 302 20. 8 3 0o.1 63 4.12 123 8.12 I83 12.12 243 16.1 303 20.12 4.. 0.16 64 4. 6 124 8.i6 I84 12.16 244 I6.16 304 20.16 5 0.20 65 4.20 I25 8.20 I85 12.20 245 16.20 305 20.20 6 0.24 66 4.24 I26 8.24 i86 12.24 246 16.24 306 20.24 7 0.22867 4.28 127 8.28 I87 12.28 247 16.28 307 20.28 8 0.32 68 4.32 128 8.32 88 12.32 248 16.32 308 20.32 9 0.36 69 4.36 I29 8.36 189 12.36 249 i6.36 309 20.36 o1 o.40 70 4.40 I30 8.40 190 I2.40 250 i6.40 3o1 20.40 ii 0.44 71 4.44 31i 8.44 191 12.44 251 I6.44 311 20.44 12 o.48 72 4.48 I32 8.48 192 12.48 252 i6.48 312 20.48 13 o.52 73 4.52 133 8.52 193 12.52 253 16.52 313 20.52 14 o.56 74 4.56 I34 8.56 194 12.56 254 16.56 3i4 20.56 15 I. 0 75 5. o 135 9. o 195 13. 0 255 17 o0 315 21. o i6 I. 4 76 5. 4 36 9. 4 I96 I3. 4 256 17- 4 316 21. 4 17 I. 8 77 5. 8 i37 9. 8 197 3. 8 257 17. 8 317 21. 8 18 I.12 78 5.12 I38 9.12 198 13.12 258 17.12 318 2I.I2 I9 i.I6 79 5.i6 139 9.16 199 3.1i6 259 17.16 319 21.16 20 1.20 80 5.20 i40 9.20 200 13.20 26o 17.20 320 21.20 21 1.24 81 5.24 14i 9.24 201 13.24 261 17.24 321 21.24 22 1.28 82 5.28 142 9.28 202 13.28 262 17.28 322 21.28 23 1.32 83 5.32 143 9.32 203 13.32 263 17.32 323 21.32 24.36 84 5.36 i44 9.36 204 i3.36 264 17.36 324 21.36 25 I.40 85 5.40 I45. 9.40 205 r3.40 265 17.40 325 21.40 26 1.44 86 5.44 i46 9.44 206 i3.44 266 I7.44 326 21.44 27 I.48 87 5.48 147 9.48 207 i3.48 267 17.48 327 21.48 28 1.52 88 5.52 I48 9.52 208 I3.52 268 17.52 328 21.52 29.56 89 5.56 I49 9.56 209 i3.56 269 17.56 329 21.56 30o 2. 90 6. 150 IO. o 210 I4. 0 270 i8. o 330 22. o 31 2. 4 9I 6. 4 151 Io. 4 211 14. 4 271 18. 4 331 22. 4 32 2. 8 92 6. 8 152 io. 8 212 I4. 8 272 i8. 8 332 22. 8 33 2.I2 93 6.12 I53 10.12 213 I4.I2 273 I8.12 333 22.12 34 2.16 94 6.i6 I54 1o.i6 214 14.I6 274 I8.i6 334 22.16 35 2.20 95 6.20 I55 10.20 2I5 I4.20 275 I8.20 335 22.20 36 2.24 96 6.24 i56 10.24 216 4.-24 276 I8.24 336 22.24 37 2.28 97 6.28 157 IO.28 217 14.28 277 18.28 337 22.28 38 2.32 98 6.32 I58 10.32 218 14.32 278 18.32 338 22.32 39 2.36 99 6.36 159 o1.36 219 14.36 279 18.36 339 22.36 40 2.40. 4o0. 6o 1o.40 220 I4.40 280 I8.40 340 22.40 41 2.44 IOI 6.44 161 Io.44 22 14.44 28I 84.44 34 22.44 42 2.48 I02 6.48 I62 Io.48 222 14.48 282 18.48 342 22.48 43 2.52 103 6.52 i63 10.52 223 14.52 283 1.8.52 343 22.52 44 2.56 io4 6.56 i64 io.56 224 I4.56 284 18.56 344 22.56 45 3. o o05 7. o 165 II. o 225 15. 0 285 19. 0 345 23. 0 46 3. 4 io6 7- 4 66 1. 4 226 15. 4 286 I9. 4 346 23. 4 47 3. 8 107 7- 8 167 ii. 8 227 15. 8 287 I9. 8 347 23. 8 48 3.12 108 7.12 i68 11.12 228 15.12 288 19.I2 348 23.1.2 49 3.i6 109 7.-6 169 11.I6 229 i5.I6 289 19.16 349 23.16 50 3.20 i1o 7.20 170 11.20 230 15.20 290 19.20!,o 23.20 51 3.24 III 7.24 171 11.24 231 15.24 291 19.24 351 23.24 52 3.28 112 7.28 172 11.28 232 I5.28 292 19.28 352 23.28 53 3.32 113 7.32 173 11.32 233 15.32 293. 19.32 353 23.32 54 3.36 114 7.36 174 11.36 234 15.36 294 19.36 354 23.36 55 3.40 15 7.40 I75 II.4o 235 15.40 295 19.40 355 23.40 56 3.44 116 7.44 176 11.44 236 15.44 296 19.44 356 23.44 57 3.48 117 7.48 177 11.48 237 I5.48 297 19.48 357 23.48 58 3.52 118 7.52 178 11.52 238 15.52 298 I9.52 358 23.52 59 3.56 II9 7.56 179 11.56 239 15.56 299 19.56 359 23.56 60 4. o 120 8. 0 i80o 12. 0 240 16. 0 300 20. 0 360 24. o Prnge 13-] TABLE XXIL. Proportional Logarithms. h rm h m h m h m Ih m h nn h m h m h m 00 0' Oo 1' 00 2' 00 3/ 00 4/ 00 5 00 6/ 00 71 00 s S. -, 12.2553 1.9542 1.7782 1.6532 i.5563 1.4771 1.4102 1.3522 0 i 4.0334 2481 9506 7757 6514 5549 4759 491 3513 2 -3.7324 2410 9471 7734 6496 5534 4747 4o81 35o4 2 3 5563 2341 9435 7710 6478 5520 4735 4071 3495 3 4 43i4 2272 9400 7686 6460 5506 4723 4o6I 3486 4 5 3.3345 2.2205 1.9365 1.7663 1.6443 1.549 1-.4711 I.4050.3477 5 6 2553 2139 9331 7639 6425 5477 4699 4040 3468 6 7 i883 2073 9296 7616 6407 5463 4688 4030o 3459 7 8 13o3 2009 9262 7593 6390 5449 4676 4020 3450 8 9 0792 1946 9228 7570 6372 5435 4664 4o0o 344i 9 o 3.0o334 2.1883.995 -1.7547 1.6355 1.5421.4652 -1.4000 -— 1.3 432 io' 2.9920 1822 9162 7524 6338 5407 4640 3989 3423 ir 12 9542 1761 9128 7501 6320 5393 4629 3979 3415 12 13 9195 1701 9096 7479 63o3 5379 4617 3969 3406 13 14 8873 1642 9063 7456 6286 5365 4606 3959 3397 14 15 2.8573 2.1584.9031.743 17 4.6269 i.5351 1.4594 1.3949 1.3388 15 i6 8293 1526 8999 7412 6252 5337 4582 3939 3379 J6 17 8030 1469 8967 7390 6235 5324 4571 3929 3371 17 18 7782 1413 8935 7368 6218 5310 4559 3919 3362 18 19 7547 i358 8904 7346 6201 5296 4548 3910 3353 19 20 2.7324 2.1303 1.8873 1.7324 1.6185 1.5283 1.4536 1.3900 1.3345 20 21 7112 1249 8842 7302 6i68 5269 4525 3890 3336 21 22 6910 1196 8811 7281 6i51 5256 4514 3880 3327 22 23 6717 1O43 8781 7259 6i35 5242 4502 3870 3319 23 24 6532 1091 8751 7238 6118 5229 4491 3860 33o10 24 25 2.6355 2.1040 1.8721 1.7217 1.6102 1.5215 i.448o0.3851 1.3301 25 26 6185 0989 8691 7196 6o85 5202 4468 384i 3293 26 27 602I 0939 866i 7175 6069 5089 4457 3831 3284 27 28 5863 0889 8632 7154 6053 5175 4446 3821 3276 28 29 5710 o840 8602 7133 6037 5162 4435 3812 3267 29 30 2.5563 2.0792 1.8573 1.7II2 1.6020 1.5149 1-4424 0.3802 0.3259 3o 31 542I 0744 8544 7091 6oo005 536 4412 3792 3250 31 32 5283 0696 85i6 7071 5989 5123 4401 3783 3242 32 33 5149 0649 8487 7050 5973 51o1 4390 3773 3233 33 34 5019 0603 8459 7030 5957 5097 4379 3764 3225 34 35 2.4894 2.0557 I.8431 1.70I0 1.5941 i.5089 1.4368 1.3754 1.3216 35 36 4771 0512 8403 6990 5925 5071 4357 3745 3208 36 37 4652 0467 8375 6970 5909 5058 4346 3735 3199 37 38 4536 0422 8348 6950 5894 5045 4335 3726 3190 38 39 4424 0378 8320 6930 5878 5032 4325 3716 3i83 39 40 2.4314 2.0334 1.8293 1.6910 1.5863 1.5019 i.4314 0.3707 1.3174 40 41 4206 0291 8266 6890 5847 5007 43o3 3697 3166 41 42 4102 0248 8239 6871 5832 4994 4292 3688 3158 42 43 4oo000 0206 8212 6851 5816 4981 4281 3678 3149 43 44 3900 oI64 8186 6832 5801o 4969 4270 3669 3141 44 45 2.3802 2.0122 1.8159 1.6812 1.5786 1.4956 1.4260 1.3660 i.333 45 46 3707 00oo8 833 6793 577I 4943 4249 365o 3024 46 47 3613 oo0040 8107 6774 5755 4931 4238 3641 3116 47 48 3522 0000 8o81 6755 5740 4918 4228 3632 31o8 48 49 3432 1.9960 8o55 6736 5725 4906 4217 3623 3100 49 50 2.3345 1.9920 1.8030 3 1.677 i.5710 1.4894 1.4206 1.3613 1.3091 50 51 3259 9881 8oo004 6698 5695 488i 4096'36o4 3083 51 52 3174 9842 7979 6679 568o 4869 4i85 3595 3075 52 53 3091 9803 7954 666i 5666 4856 4175 3586 3067 53 54 3o0o 9765 7929 6642 5651 4844 4164 3576 3059 54 55 2 293i 1.9727 1.7904 1.6624 1.5636 1.4832 1.4154 1.3567 1.3051 55 56 2852 9690 7879 66o5 5620 4820 4143 3558 3043 56 57 2775 9652 7855 6587 5607 48o8 4133 3549 3o34 57 58 2700 9615 7830 6568 5592 4795 4122 354o 3026 58 59 2626 9579 7806 655o 5578 4783 4112 353 3o018 59 S. 0~ 00 0 1 002~ 00 3' 40 4 00 5' 00 6( 0~ 7/' 00 / S. TABLE XXII. [Page 133 Proportional Logarithms. I km I h m h m h m hkm h m h km km 00 9' 0~ 10' 00 11 0~ 12' 0~ 13 00 14' 00 15' 0 1 00 171 1~ 0 1.3o010o 1.2553 1.2139 1.1761 1.1413.09io 1.0792 1.0512 1.0248 1 3002 2545 2132 1755 i4o8 1086 0787 0507 0244 1 2 2994 2538 2126 I749 1402 io81 0782 0502 0240 2 3 2986 2531 2119 I1743 1397 1076 0777 0498 0235 3 4 2978 2524 2113 1737 1391 1071 0773 0493 0231 4 5 1.2970 1.2517 1.2106 1.1731 1.1386 1.10o66 1.0768 1.0489 1.0227 5 6 2962 2510 2099 1725 1380 1061 0763 0484 0223 6 7 2954 2502 2093 1719 1374 io55 0758 048o 0219 7 8 2946 2495 2086 1713 1369 io5o 0753 0475 0214 8 o10 1.2931 1.2481 1.2073 1.1701 1.1358 1.1040 1.0744 1.0467 1.0206 o10 11 2923 2474 2067 1695 1352 o1035 0739 0/62 0202 I f 12 2915 2467 2061 1689 1347 o1030 0734 0458 0197 12 13 2907 2460 2054 1683 1342 1025 0730 o453 0193 13 14 2899 2453 2048 1677 i336 1020 0725 0449 0189 14 5 -1.2891 1.2445 1.2041 1.1671 1.1331 1.1015 1.0720 1.0444 1.0185 1 16 2883 2438 2035 1665 1325 1009 0715 o44o o0181 i6 17 2876 2431 2028 i66o 1320 i004 0711 0435 0176 17 18 2868 2424 2022 i654 1314 0999 0706 o43i 0172 i8 19 2860 2417 2016 i648 1309 0994 0701 0426 0168 19 20 1.2852 1.2410 1.2009 1.1642 1.13o3 1.0989 1.0696 1.0422 1.oi64 20 21 2845 2403 2oo003 1636 1298 0984 0692 0o48 oI60 21 22 2837 2396 1996 i63o 1292 0979 0687 0413 0156 22 23 2829 2389 1990 1624 1287 0974 0682 0409 o0151 23 24 J 2821 2382 1984 1619 1282 0969 0678 o040o4 0147 24 25 1.2814 1.2375 1.1977 1.1613 1.1276 1.0964 1.0673.o4o00o 1.043 25 26 2806 2368 197 160o7 1271 0959 o668 0395 o39 26 27 2798 2362 1965 160o 1266 0954 0663 0391 0135 27 28 2791 2355 1958 1595 126o 0949 0659 0387 o131 28 29 2783 2348 1952 1589 1255 0944 o654 0382 0126 29 30 1.2775 1.2341 1.1946 [.1584 1.1249 1.0939 1.0649 1.o378 1.0122 3o 31i 2768 2334 1939 1578 1244 0934 o645 0374 0118 31 32 2760 2327 1933 1572 1239 0929 0640 0369 0114 32 33 2753 2320 1927 i566 1233 0924 0635 0365 0110 33 34 2745 2313 1921 1561 1228 0919 0631 0360 010oio6 34 35 1.2738 1.2307 1.1914 1.1555 1.1223 1.0914 1.0626 i.0356 1.0102 o 35 36 2730 2300 1908 1549 1217 0909 0621 0352 oo0098 36 37 2722 2293 1902 1543 1212 0904 0617 0347 0093 37 38 2715 2286 1896 i538 1207 0899 0612 o343 0089 38 39 2707 2279 889 I532 120 0894 o060o8 0339 oo85 39 40 i.2T700 1.2272 1.1883 1.1526 1.1196 1.0889 1.0603 1.0334 1.oo0081 40 41 2692 2266 1877 1520 1191 o884 0598 0330 0077 41 42' 2685 2259 1871 i515 1186 o88o 0594 0326 0073 42 43 2678 2252 i865 1509 ii8o 0875 0589 032I 069 43 44 2670 2245 1859 1503 1175 0870 0585 0317 oo0065 44 45 7.2663 1.2239 i.1852 1.1498 1.1170 1.0865 t.o580 1.o3i3 i.oo6r 45 46' 2655 2232 i846 1492 ii64 0860 0575 0308 0057 46 47 2648 2225 84o0 1486 1159 0855 o571 o3o4 oo53 47 48 2640 2218 1834 i48i 1154 0850 o566 o3oo 0300 049 48 49 2633 2212 1828 1475 1149 0845 0562 0295 0044 49 50 1.2626 1.2205 1.1822 1.1469 i.i143 i.o84o i.o0557 1.0291 i.oo4o 50 51 2618 2198 1816 1464 1138 0835 0552 0287 0036 51 52 2611 2192 1809 1458 1133 0831 0548 0282 0032 52 53 2604 2185 1803 1452 1128 0826 0543 0278 0028 53 54 2596 2178 1797 447 1123 0821 o0539 0274 0024 54 55 1.2589 1.2172 1.1791 1.1441 1.1117 1.0816 1.0534 1.0270 1.0020 55 56 2582 2165 1785 1436 1112 o811 o53o 0265 0016 56 57 2574 2159 1779 143o 1107 o806 0525 0261 0012 57 58 2567 2152 1773 1424 1102 o8oi o0521 0257 oo0008 58 59 256o 2145 1767 1419 1097 0797 o516 022 0o004 59 S. 00 9/1 00 10 0 11'I 00 12' 00 13' 00 14' 00 15 00 16' 00 17I S. Page 134] TABLE XXII. Proportional Logarithms. Im km km h m Im km A: m km Il?n h m h m h mh 00 18' 0 19 00 20' 0~ 21' 0~ 22' 00 23' 00 24' 00 25' 0 261 0~ 27' 0~ 28' 00 29' S 0 10000 9765 9542 9331 9128 8935 8751 8573 84o3 8239 8o81 7929 0 1 9996 9761 9539 9327 9125 8932 8748 8570 8400 8236 8079 7926! 2 9992 9758 9535 9324 9122 8929 8745 8568 8397 8234 8076 7924 2 3 9988 9754 9532.9320 9119 8926 8742 8565 8395 8231 8073 7921 3 4 9984 9750 9528 9317 9115 8923 8739 8562 8392 8228 8071 7919 4 5 9980 9746 9524 9313 9II2 8920 8736 8559 8389 8226 8068 796 5 6/997 97426 9521 9310 6 9976 9742 9521 9310 9g9o 8917 8733 8556 8386 8223 8o66 7914 6 7 9972 9739 9517 993o6 906 8913 8730 8553 8384 8220 8063 7911 7 8 9968 9735 9514 9303 9102 8910 8727 855o 838i 8218 8o6I 7909 8 9 9964 9731 9510 9300 9099 8907 8724 8547 8378 8215 8058 7906 9 10 9960 9727 9506 9296 9096 8904 8721 8544 8375 8212 8055 7904 10 II 9956 9723 9503 9293 9092 8901 8718 8542 8372 8210 8o53 7901 II 12 9952 9720 9499 9289 9089 8898 8725 8539 8370 8207 8050 7899 12 13 9948 9716 9496 9286 9086 8895 8712 8536 8367 8204 8o48 7896 13 14 9944 9712 9492 9283 9083 8892 8709 8533 8364 8202 8o45 7894 14 15 9940 9708 9488 9279 9079 8888 8706 853o 836i 8199 8043 7891 15 i6 9936 9705 9485 9276 9076 8885, 8703 8527 8359 8196 8040o 7889 26 17 9932 9701 9481 9272 9073 8882 8700 8524 8356 8194 8037 7887 17 i8 9928 9697 9478 9269 9070 8879 8697 8522 8353 8191 8035 7884 18 19 9924 9693 9474 9266 9066 8876 8694 8519 8350 8i88 8032 7882 19 20 9920 9690 947' 9262 go63' 8873 869 185i6 83-48 8i86 80o3 7879 20 21 9916 9686 9467 9259 9060 8870 8688 85i3 8345 8i83 8027 7877 21 22 9912 9682 9464 9255 9057 8867 8685 851o 8342 8181 8025 7874 22 23 9908 9678 9460 9252 9053 8864 8682 8507 8339 8178 8022 7872 23 24 9905 9675 9456 9249 9050 8861 8679 8504 8337 8175 8020 7869 24 25 9901 9671 9453 9245 9047 8857 8676 8502 8334 8173 8017 7867 25 26 9897 9667 9449 9242 9044 8854 8673 8499 833i 8170 8014 7864 26 27 9893 9664 9446 9238 9041 885i 8670 8496 8328 8167 8012 7862 27 28 9889 966o 9442 9235 o9037 8848 8667 8493 8326 8i65 8009 7859 28 29 9885 9656 9439 9232 9034' 8845 8664 8490 8323 8162 8007 7857 29 30o 9881 9652 9435 9228 9031 8842 866i 8487 8320 8I59 804" 7855 30 31 9877 9649 9432 9225 9028 8839 8658 8484 8318 8157 8002 7852 31 32 9873 9645 9428 9222 9024 8836 8655 8482 83i5 8i54 7999 7850 32 33 9869 9641 9425 9218 9021 8833 8652 8479 8312 8152 7997 7847 33 34 986 9638 9421 9215 9018 8830 8649 8476 8309 8149 7994 7845 34 35 9861 9634 9418 9212 o9015 8827 864 8473 8307 8i46 7992 7842 35 36 9858 9630 9414 9208 9012 8824 8643 8470 8304 8144 7989 7840 36 37 9854 9626 9411 9205 9008 8821 8640 8467 83o01 8141 7987 7837 37 38 9850 9623 9407 9201 900oo5 8817 8637 8465 8298 8i38 7984 7835 38 39 9846 9619 9404 9198 9002 8814 8635 8462 8296 8i36 7981 7832 39 40 9842 9615 9400 9195 8999 88ii 8632 8459 8293 833 7979 830 4 41 9838 9612 9397 9191 8996 88o8 8629 8456 8290o 8131 7976 7828 41 42 9834 9608 9393 9188 8992 8805 8626 8453 8288 8128 7974 7825 42 43 9830 9604 9390 9i85 8989 8802 8623 8451 8285 8125 7971 7823 43 44 9827 9601 9386 9181 8986 8799 8620 8448 8282 8123 7969 7820 44 45 923 997 9383 9178 8983 8796 8617 8445 8279 8120 7966 7818 45 46 9819 9593 9379 9175 8980 8793 8614 8442 8277 8117 7964 7815 46 47 98i5 959o 9376 9171 8977 8790 8611 8439 8274 8115 7961 7813 47 48 9811,9586 9372 9168 8973 8787 8608 8437 8271 8112 7959 7811 48 49 9807 9582 9369 965 8970 8784 86o5 8434 8269 8io 0 7956 7808 49 50 9803 9579 9365 9162 8967 8781 8602 843i 8266 8107 7954 78(06 50 51 9800 9575 9362 9158 8964 8778 8599 8428 8263 81o4 7951 7803 51 52 9796 9571 9358 9155 8961 8775 8597 8425 8261 8102 7949 7801 52 53 9792 9568 9355 9152 8958 8772 8594 8423 8258 8099 7946 7798 53 54 9788 9564 9351 9148 8954 8769 8591 8420 8255 8097 7944 7796 54 55 9784 9561 9348 9i45 8951 8766 8588 8417 8253 8094 794I 7794 55 56 9780 9557 9344 9142 8948 8763 8585 84,4 8250 8091 7939 7791 56 57 9777 9553 9341 9138 8945 8760 8582 84ii 8247 8089 7936 7789 57 58 9773 9550 9337 9'35 8942 8757 8579 8409 8244 8086 7934 7786 58 59 9769 9546 9334 9132 8939 8754 8576 84o6 8242 8o84 7931 7784 59 S. 00 18' 0 d 19' 00 2/ 0' 0 201'00 22' 00 23' 00 24' 00 25' 0 26' 00 27' 00 28'o 29' S. TABLE XXII. [Page 135 Proportional Logarithms. h m hr m h m h m?/h m h m 1 m h m h m h m h nc S. 00 30' 0~ 311 00 32' 0 33 0~ 341/00 35' 00 36 100 371 0~ 38/ 00 39/ 00 40' 00 41'' 0 7782 7639 7501 7368 7238 7112 6990 6871 6755 6642 6532 6425 c I 7779 7637 7499 7365 7236 7110 6988 6869 6753 6640 6530 6423 2 7777 7634 7497 7363 7234 7108 6986 6867 6751 6638 6529 6421.2 3 7774 7632 7494 7361 7232 7106 6984 6865 6749 6637 6527 6420 3 4 7772 7630 7492 7359 7229 7104 6982 6863 6747 6635 6525 64i8 4 5 7769 7627 7490 7357 7227 7102 6980 }686i 6745 6633 6523 6416 5 6 7767 7625 7488 7354 7225 7100 6978 6859 6743 663i 6521 64i4 6 7 7765 7623 7485 7352 7223 7098 6976 6857 6742 6629 659 64i3 7 8 7762 7620 7483 7350 7221 7096 6974 6855 6740 6627 65i8 64ii 8 9 7760 7618 7481 7348 7219 7093 6972 6853 6738 6625 6516 6409 9 o1 7757 7616 7479 7346 7217 7091 6970 6851 6736 6624 6514 6407 10 II 7755 7613 7476 7344 7215 7089 6968 6849 6734 6622 6512 40o6 11 12 7753 7611 7474 7341 7212 7087 6966 6847 6732 6620 6510 6404 12 i3 7750 7609 7472 7339 7210 7085 6964 6845 6730 66i8 6509 6402 13 14 7748 7607 7470 7337 7208 7083 6962 6843 6728 6616 6507 64oo00 14 15 7745 7604 7467 7335 7206 7208-1 6960 684 6726 6614 6505 6398 15 i6 7743 7602 7465 7333 7204 7079 6958 6840 6725 6612 6503 6397 16 17 7741 7600 7463 7330 7202 7077 6956 6838 6723 6611 6501 6395 17 i8 7738 7597 7461 7328 7200 7075 6954 6836 6721 6609 6500 6393 18 19 7736 7595 7458 7326 7198 7073 6952 6834 6719 6607 6498 6391 9i 20 7734 7593 7456 7324 7196 7071 6950 6832 6717 66o5 6496 6390 30 21 7731 7590 7454 7322 7193 7069 6948 683o0 6715 66o3 6494 6388 21 22 7729 7588 7452 7320 7191 7067 6946 6828 6713 66o01 6492 6386 22 23 7726 7586 7450 7317 7189 7065 6944 6826 6711 66oo00 6491 6384 23 24 7724 7583 7447 7315 7187 7063 6942 6824 6709 6598 6489 6383 24 25 7722 7581 7445 73/3 7185 7061 6940 6822 6708 6596 6487 6381i 25 26 7719 7579 7443 7311 7183 7059 6938 6820 6706 6594 6485 6379 26 27 7717 7577 7441 7309 7181 7057 6936 68i8 6704 6592 6484 6377 27 28 7714 7574 7438 7307 7179 7055 6934 6816 6702 6590 6482 6376 28 29 7712 7572 7436 7304 7177 7052 6932 68i4 6700 6589. 648o0 6374 29 30 7710 7570 7434 7302 7175 7050 6930 6812 6698 6587 6478 6372 30 31 7707 7567 7432 7300 7172 7048 6928 6810o 6696 6585 6476 6371 31 32 7705 7565 7429 7298 7170 7046 6926 6809 6694 6583 6475 6369 32 33 7703 7563 7427 7296 7168 7044 6924 6807 6692 6581 6473 6367 33 34 7700 7560 7425 7294 7166 7042 6922 68o5 6691 6579 6471 6365 34 35 7698 7558 7423 7291 7164 7040 6920 6803 6689 6578 6469 6364 )-35 36 7696 7556 7421 7289 7162 7038 6918 6801 6687 6576 6467 6362 36 37 7693 7554 7418 7287 7160 7036 6916 6799 6685 6574 6466 6360 37 38 7691 7551 7416 7285 7158 7034 6914 6797 6683 6572 6464 6358 38 39 7688 7549 7414 7283 7156 7032 6912 6795 6681 6570 6462 6357 j'39'4o 7686 7547 7412 7281 7154 7030'6910 6793 6679 6568 646o 6355 4.41 7684 7544 7409 7279 7152 7028 6908 6791 6677 6567 6459 6353 41 42 7681 7542 7407 7276 7149 7026 6906 6789 6676 6565 6457 6351 i 43 7679 7540 7405 7274 7147 7024 6904 6787 6674 6563 6455 6350 43 44 7677 7538 7403 7272 7145 7022 6902 6785 6672 656i 6453 6348 l 44'45 7674 7535 74013 2707143 7020 6900 6784 6670 6559645 6346 45 46 7672 7533 7398'7268 7141 7018 6898 6782 6668 6558 6450 6344 46 47 7670 7531 7396 7266 7139 7016 6896 6780 6666 6556 6448 6343 47 48 7667 7528 7394 764 737 7014 6894 6778 6664 6554 6446 634i 48 49 7665 7526 7392 7261 7135 7012 6892 6776 6663 6552 6444'6339 49 50 7663 7524 7390 7259 7133 7010 6890 6774 666i 655o 6443 6338 5c0 51 7660 7522 7387 7257 7131 7008 6888 6772 6659 6548 6441 6336 51 52 7658 7519 7385 7255 7129 7006 6886 6770 6657 6547 6439 6334 52 53 7655 7517 7383 7253 7127 7004 6884 6768 6655 6545 6437 6332 53 54 7653 7515 7381 7251 7124 7002 6882 6766 6653 6543 6435 6331 54 55 7651 7513 7379 7249 7122 7000 688i 6764 665i 6541 6434 6329 55 56 7648 7510 7376 7246 7120 6998 6879. 6763 6650 6539 6432 6327 56 57 7646 7508 7374 7244 71/8 6996 6877 6761 6648 6538 6430 6325 57 58 7644 75o6 7372 7242 7116 6994 6875 6759 6646 6536 6428 6324 58 50 7641 7503 7370 7240 7114 6992 6873 6757 6644 6534 6427 6322 59. 00 30' 00 31' 00 30~ 00533/004' 0 356/10~ 37' 00 38' 0 39' 00 40/'00 41' S. Page 136] TABLE XXII. Proportional Logarithms. h m hI mlh m h t A m h m h m h m h m h m h m S. 00 42' 00 43 00 44 0~ 45' 00 46' 0~ 471 0 48' 00 49'10~ 50 00 51' 0~ 52' 00 53 S. 6320 6218 6118 6021 5925 5832 5740 5651 5563 5477 5393 53o 10 1 6319 6216 6117 6019 5924 5830 5739 5649 5562 5476 5391 5309 I 2 6317 6215 6115 6017 5922 5829 5737 5648 5560 5474 5390 5307 2 3 63i5 6213 6i13 6016 5920 5827 5736 5646 5559 5473 5389 53o6 3 4 6313 6211 6112 6014 5919 5826 5734 5645 5557 5471 5387 53o5 4 5 63I2 62310 6io 5917 5824 5733 5643 5556 5470 5386 5303 5 6 6.3o 6208 6108 6o011 5916 5823 5731 5642 5554 5469 5384 5302 6 7 6308 6206 6107 6009 5914 5821 5730 5640 5553 5467 5383 5300 7 8 6306 6205 6105 600o8 5913 5819 5728 5639 5551 5466 5382 5299 8 9 63o5 6203 6io3 6006 5911 58i8 5727 5637 5550 5464 538o 5298 9 io 63o3 6201 6102 6005 599 5816 5725 5636 5549 5463 5379 5296 io II 63oi 6200 6ioo 6003 59o8 5815 5724 5635 5547 546i 5377 5295 ii 12 63oo 6198 6099 6oor 5906 58i3 5722 5633 5546 5460 5376 5294 12 13 6298 6196 6097 6000ooo 5905 5812 5721 5632 5544 5459 5375 5292 13 i4 6296 6195 6095 5998 5903 58o10 5719 5630 5543 5457 5373 5291 14 15 6294 6193 6094 5997 5902 5809 7I8 5629 554i 5456 5372 5290 15 16 6293 6191 6092 5995 5900 5807 5716 5627 5540 5454 5370.5288 16 17 6291 6190 6090 5993 5898 5806 5715 5626 5538 5453 5369 5287 17 18 ~6289 6i88 6089 5992 5897 5804 5713 5624 5537 5452 5368 5285 18 g19 6288 6186 6087 5990 5895 58o3 5712 5623 5536 545o 5366 5284 19 20 6286 6i85 6o85 5989 5894 580o 5710 5621 5534 5449 5365 5283 20 21 6284 6183 6o84 5987 5892 58o00 5709 5620 5533 5447 5364 5281 21 22 6282 6i8i 6082 5985 5891 5798 5707 56i8 5531 5446 5362 5280 22 23 6281 6179 6o8i 5984 5889 5796 5706 5617 5530 5445 536i 5279 23 24 6279 6178 6079 5982 5888 5795 5704 5615 5528 5443 5359 5277 24 25 6277 6176 6077 5981 5886 5793 5703 56i4 5527 5442 5358 5276 25 26 6276 6174 6076 5979 5884 5792 5701 56i3 5526 5440 5357 5275 2.6 27 62746173 6074 5977 5883-.5790 5700 5611I 5524 5439 5355 5273 -7 28 6272 6171 6072 5976 5881 5789 5698 5610 5523 5437 5354 5272 2.8 29 6271 6169 6071 5974 5880 5787 5697 56o8 5521 5436 5353 5271 29 30 6269 668 6069 5973 5878 5786 5695 5607 5520 5435 5351 5269 3 31 6267 6166 6067 5971 5877 5784 5694 5605 55i8 5433 5350 5268 31 32 6265 6i65 6066 5969 5875 5783 5692 56o4 5517 5432 5348 5266 32 33 6264 6i63 6064 5968 5874 5781 5691 5602 5516 5430 5347 5265 33 34 6262 1661 6063 5966 5872 5780 5689 56o01 5514 5429 5346 5264 34 35 6260 6i6o 6061 5965 5870 5778 5688 5599 5513 5428 5344 5262 35 36 6259 6158 6059 5963 5869 5777 5686 5598 5511 5426 5343 5261 36 37 6257 6156 6o58 5961 5867 5775 5685 5596 5510 5425 534i 5260 37 38 6255 6155 6o56 5960 5866 5774 5683 5595 55o8 5423 5340o 5258 38 39 6254 6i53 6o55 5958,5864 5772 5682 5594 5507 5422 5339 5257 39 40 6252 615o 6o53 5957 5863 5771 568o 5592 55o6 5421 5337 5256 4o 41 625o 650o 6o5 5955 586i 5769 5679 5591 5504 5419 5336 5254 41 42 6248 6148 6050 5954 586o 5768 5677 5589 5503 5418 5335 5253 42 43 6247 6146 6o48 5952 5858 5766 5676 5588 55o01 5416 5333 5252 43 44 6245 6145 6o46 5950 5856 5765 5674 5586 5500oo 54i5 5332 525o 44 45 6243 6143 6045 5949 5855 5763 5673 5585 5498 5414 533i 5249 45 46 6242 6i41 6043 5947 5853 576i 5671 5583 5497 5412 5329 5248 46 47 6240o 6i4o 6042 5946 5852 5760 5670 5582 5496 5411 5328 5246 47 48 6238 6i38 6040 5944 585o 5758 5669 5580 5494 54o9 5326 5345 48 49 6237 6i36 60o38 5942 5849 5757 5667 5579 5493 54o8 5325 5244 49 50 6235 635 6037 594 5847 5755 5666 5578 5491 5407 5324 5242 50 51 6233 6133 6o35 5939 5846 5754 5664 5576 549o 54o5 5322 5241 5i 52 6232 6131 6o33 5938 5844 5752 5663 5575 5488 5404 5321 5240 52 53 6230 6i3o 6032 5936 5843 5751 5661 5573 5487 5402 532o 5238 53 54 6228 6128 6o3o 5935 5841 5749 5660 5572 5486 5401 53i8 5237 54 55 6226 6126 6029 5933 5839 5748 5658 5570 5484 5400o 5317 5235 55 56 6225- 6125 6027 593i 5838 5746 5657 5569 5483 5398 53i5 5234 56 57 6223 6123 6025 593o 5836 5745 5655 5567 548i 5397 5314 5233 57 58 622 6121 6o024 5928 5835 5743 5654 5566 5480 5395 5339 5 5 31 523 58 59 6220 620o 6022 5927 5833 5742 5652 5564 5478 5394 5311 523o 59 S. 00 42' 00 43' 00 44' 00 451 00 46 100 47 00 48'0 4 00 50 0~ 51' 00 52' 00 531 S. TABLE XXII. CP 1.37 Proportional Logarithms. kt m h m m h m hm kh t m h m tt m kl m h m h m 00 541 0~ 55 100 56/ 0~ 57' 0 58/ 00 59' 1 0', 10 " IC' 10 3/ 1 4' 105 ~ 0 5229 5149 5071 4994 49i8 4844 477I 4609 4629 4559 449 4424 5227 5I48 5070 4993 4917 4843 4770 4698 4628 4558 4490 4422 1 2 5226 5146 5068 4991 4916 4842 4769 4697 4626 4557 4489 442r 2 3 5225 5145 5067 4990 4915 484r 4768 4696 4625 4556 4488 4420 3 4 5223 5144 5o66 4989 4913 4839 4766 4695 4624 4555 4486 449 5 5222 543 5064 4988 4912 4838 4765 4693 4623 4554 4485 4418' 6 5221 5141 5o63 4986 4911 4837 4764 4692 4622 4552 4484 4417 6 7 5219 5i40 5062 4985[ 4910 4836 4763 4691 4u21 455I 4483 4416 7 8 5218 5139 5061 4984 4908 4834 4762 4690 4619 455o 4482 44i5 8 9 5217 5137 5059 4983 4907 4833 4760 4689 46i8 4549 4481 4414 9 Io 5215 5136 5o58 4981 4906 4832 4759 4688 4617 4548 448o 44i2 io ii, 5214 535' 5057 4980 4905 483i 4758 4686 46i6 4547 4479 4411 ii 12 5213 5133 5055 4979 4903 483o 4757 4685 46i5 4546 4477 44io 12 13 5211 5132 5o54 4977 4902 4828 4756 4684 46i4 4544 4476 4409 3 14 5210 5131 5o53 4976 4901 4827 4754 4683 4612 4543 4475 44o8 14 15 5209 5129 5o05 4975 4900 4826 4753 4682 46i 4542 4474' 4407 15 6 5207 5128 5o5o4974 4899 4825 4752 468o 46io 454i 4473 44o6 6 17 5206 5127 5049 4972 4897 4823 4751 4679 4609 454o 4472 44o5 17 i8 5205 5125 5048 4971 4896 4822 4750 4678 46o8 4539 4471 44o4 18 19 5203 5124 5o46 4970 4895 4821 4748 4677 4607 4538 4469 4402 19 20 5202o 5123 5o45 4969 4894 4820 4747 4676 46o6 5 4468 144o i 0 21 5201 5122 5o44 4967 4892 4819 4746 4675 46o4 4535 4467 44oo 22 5199 5120 5o43 4966 489 4817 4745 673 46o3 4534 4466 43 99 23 5198 5119 5041 4965 4890 4816 4744 4672 4602 4533 4465 o4398 3 24 5197 5118 5040 4964 4889 4815 4742 4671 46o 4532 4464 4397,4 25 5195 5116 5039 4962 4887 4814 4748 467o 46oo 4531 4463 4396 25 26 5194 5i15 5037 4961 4886 4812 4740 4669 4599 3o 4462 4395 26 27 5193 5114 5o36 4960 4885 48ii 4739 4668 4597 4528 4460 4394 27 28 5191 5112 5o35 4959 4884 48 4738 4666 4596 4527 4459 4393 28 29 5190o Sii 5o34 4957 4882 4809 4736 4665 4595 4526 4458 4391 29 30o 5.89 uio 5032 4956 4881 48o8 4735 4664 4594 4525 4457 4390 30 31 5187 5108 5031 4955 488o 48o6 4734 4663 4593 4524 4456 4389 31 32 5i86 5107 5030 4954 4879 48o5 4733 4662 4592 4523 4455 4388 32 33 5185 5106 5028 4952 4877 48o4 4732 466o 4590 4522 4454 4387 33 34 5i83 5105 5027 4951 4876 4803 4730 4659 4589 45 4453 4386 34 35 5i82 5103 5026 4950 4875 4801 4729 4658 4588 4519 4452 4385 35 36 5181 5102 5025 4949 4874 48oo00 4728 4657 4587 8 445o 4384 36 37 579 o 5023 4947 4873 4799 4727 4656 4586 4517 4449 4383 37 38 5178 5099 5022 4946 4871 4798 4726 4655 4585 45i6 4448 438i 38 39 5177 5098 5021 4945- 4870 4797 4724 4653 4584 4515 4447 438o 39 40 5175 5097 5019 4943 4869 4795 4723 4652 4582 4514 4446 4379 4o 41 5174 5095 5oi8 4942 4868 4794 4722 4651 458i 4512 4445 4378 4i 42 5173 504 5017 494/ 4866 4793 4721 465o 458o 45i 4444 43I- 42 43 5172 5093 5o016 4940 4865 4792 4720 4649 4579 4io 4443 4376 43 44 5170o 5092 5oi014 4938 4864 4791 4718 4648 4578 4509 444i 4375 44 45 5169 5090 5oi013 4937 4863 4789 4717 4646 4577 45o8 444o 4374 45 46 5168 5089 5012 4936 4861 4788 4716 4645 4575 4507 4439 4373 46 47 5166 5088 5011oii 4935 486o 4787 4715 4644 4574 45o6 4438 4372 47 48 5i65 5o86 5009 4933 4859 4786 4714 4643 4573 45o5 4437 4370 48 49 54 5085 5008 4932 4858 4785 4712 4642 4572 45o3 4436 4369 49 50 5162 5084 5007 4931 48564783 4711 46402 4435 4368 5i 5161 5082 5005 493o 4855 4782 4710 4639 4570 45oi 4434 4367 51 52 6 5o 5o8i 5004 4928 4854 4781 4709 4638 4569 oo 4433 4366 52 53 5158 5080 5003 4927 4853 4780 4708 4637 4567 4499 443 4365 5 3 54 5157 5079 5002 4926 4852 4778 470-7 4636 4566 4498 443o 4364 54 55 5i56 5077 5000 4925 4850 47774705 463 4565 4497 429 4363 55 56 5154 5076 4999 4923 4849 4776 4704 4633 4564 4495 4428 4362 56 57 5153 5075 4998 4922 4848 4775 4703 4632 4563 4494 4427 436i 57 58 5152 5073 4997 4921 4847 4774 4702 463i 4562 4493 4426 4359 58 59 550 5072 4995'4920 14845 4772 4701 463o 456 4492 4425 4358 59 S. 00 541 00 5~500~ 561 00 57 / 58/ 059/ 10/ 101 V 10 2 10 4' 1. S.J 18 Page 138] TABLE XXII. Proportional Logarithms. h k m hm h m h n h m h m h m km h m h m h k, h m 10 6' 10 7 J1 ~ 8 91 10 10 1 11 I 10 1' 1 10 14 10 15' 1016' 10 17 o 4357 4292 4228 4i64 4102 040o4o 3979 3919 386o 3802 3745 3688 o 4356 4291 4227 4163 4o101 4o39 3978 3919 3859 38oi 3744 3687 I 2. 4355 4290 4226 4162 400oo 4038 3977 3918 3858 3800 3743 3686 2 3 4354 4289 4224 416i 4099 4037. 3976 3917 3857 3799 3742 3685 3 4 4353 4288 4223 416o 4098 4o36 3975 3916 3856 3798 3741 3684 4 5 4352'4287 4222 459407 4035 3974 3915 3856 3797 3740 3683 5 6 435i 4285 4221 4i58 4096 4o34 3973 3914 3855 3796 3739 3682 6 7 4350 4284 4220 4157 4095 4033 3972 3913 3854 3795 3738 368i 7 8 4349 4283 4219 4i56 4093 4032 3971 3912 3853 3794 3737 368o 8 9 4347 i 4282 4218 4i55 4092 4031 3970 3911 3852 3793 3736 3679 9 10 - 4346 428s 4217 4i54 4091 4030 3969 3910 385i 3792 3735 3678 10 ii 4345 4280 4216 4i53 4090 4029 3968 3909 385o 3792 3734 3677 1, 12 4344 4279 4215 4152 4089 4028 3967 3908 3849 3791 3733 3677 12 13 4343 4278 4214 4i5r 4o88 4027 3966 3907 3848 3790 3732 3676 13 14 4342 4277 4213 4i50.o 4087 4026 3965 3906 3847 3789 3731 3675 14 15 4341 4276 4212 4149 4086 4025 3964 3905 3846 3788 3730 3674 15 16 434o 4275 4211 4147 4o85 4024 3963 39o4 3845 3787 3729 3673 16 17 4339 4274 4210 4i46 4084 4023 3962 3903 3844 3786 3728 3672 17 i8 4338 4273 4209 4145 4o83 4022 3961 3902 3843 3785 3727 3671 18 19 4336 4271 4207 4144 4082 4021 3960 3901 3842 3784 3727 3670 19 20 4335 4270 4206 4143 4o08 4020 3959 3900 384i 3783 3726 3669 20 21 4334 4269 4205 4142 4080 4019 3958 3899 384o 3782 3725 3668 2I 22 4333 4268 4204 414 4079 4o018 3957 3898 3839 3781 3724 3667 22 23 4332 4267 4203 4i4o 4078 4027 3956 3897 3838 3780 3723 3666 23 24 4331 4266 4202 4139 4077 40o6 3955 3896 3837 3779 3722 3665 24 25 4330 4265 4201 4138 4076 40o5 3954 3895 3836 3778 3721 3664 25 26 4329 4264 4200 4237 4075 4oi4 3953 3894 3835 3777 3720 3663 26 27 4328 4263 4199 4i36 4074 4013 3952 3893 3834 3776 3719 3663 27 28 4327 4262 4198 4135 4073 4012 3951 3892 3833 3775 3728 3662 28 29 4326 4261 4197 4i34 4072 402' 3950 3891 3832 3774 3717 3661 29 30 4325 4260 4196 4133 4071 4o01o 3949 3890 3.831 3773 3716 3660 30 31 4323 4259 4195 4132 4070 4009 3948 3889 3830 3772 3715 3659 31 32 4322 4258 4194 4i3i 4069 4oo8 3947 3888 3829 3771 3714 3658 32 33 4321 4256 4293 413o 4068 4007 3946 3887 3828 3770 3723 3657 33 34 4320 4255 4192 4129 4067 4006 3945 3886 3827 3769 3712 3656 34 -35 439 42544191 428 4o66 4oo005 3944 38858 826 3768 3711 3655 35 36 43i8 4253 4189 4127 4o65 4oo4 3943 3884 3825 3768 37o0 3654 36 37 43I7 4252 4i88 4126 4o64 4oo3 3942 3883 3824 3767 3709 3653 37 38 43I6 4251 4187 4I25 4o63 4002 3942 3882 3823 3766 3709 3652 38 39 43i5 4250 4i86 4124 4o62 4oo001 3940 3881 3822 3765 3708 365I 39 40 4314 4249 1 85 4122 4o6 4oo000 3939 388 3821 3764 3707 365 4 41 4313 4248 4184 4121 4o6o 3999 3938 3879 3820 3763 3706 3649 41 42 4311 4247 4i83 4120 4o59 3998 3937 3878 3820 3762 3705 3649 42 43 430o 4246 4182 4119 4o58 3997 3936 3877 3819 3761 3704 3648 43 44 430o9 4245 4181 4118 4o56 3996 3935 3876 38i8 3760 3703 3647 44 45 4308 4244 4-80 4117 4055 3995 3934 3875 1 387 3759372 3646 45 46 I4307 4243 4179 41i6 4054 3993 3933 3874 38i6 3758 3701 3645 46 47 43o06 424 4178 4115 4o53 3992 3932 3873 38i5 3757 3700 3644 47 48 43o5 424o 4177 4114 4o52 399I 3931 3872 3814 3756 3699 3643 48 49 43o4 4239 4176 4i3 4o051 3990 3930 3871 38i3 3755 3698 3642 49 50 4303 4238 475 4112 4050o 3989 3929 3870 38122 754 3697 364i 5o 5 4302' 4237 4174 4111 4049 3988 3928 3869 3811 3753 3696 3640 51 52 430i 4236 4173 4iio 4o48 3987 3927 3868 3810 3752 3695 3639 52 53 4300 4235 4172 4109 4047 3986 3926 3867 3809 375i 3694 3638 53 54 4/298 4234 4171 4io8 4o46 3985 3925 3866 38o8 3750 3693 3637 54 55 4297 4233 4269 4107 4o45 3984 3924 3865 3807 3749 3693 3636 55 56 4296 4232 4168 4io6 4o44 3983 3923 3864 38o6 3748 3692 3635 56 57 4295 4231 4167 4ro5 4o43 3982 3922 3863 3805 3747 3691 3635 57 58 4294 4230 4i66 4io4 4042 3982 3922 3862 3804 3746 3690 3634 58 59 4293 4229 4i65 4103 4o41 3980 392o 386i 38o3' 3746 3689 3633 59 S. 10 6' 10 10 8' 10 9' 10 10 11 IV 12' 1 13' 14' 1 15'11 16' 10 17' TABLE XXII. [Page 139 Proportional Logarithms. h m h mh m h m h m h m h m I m I 77i h m t M I S. 1~ 18/ 10 19' 1~ 20' 1 21' 10 22/ 1.~ 23 10 24' 1~ 25' 10 26' 10 27' 10 28's 1 2 9' o 3632 3576 3522 3468 34I5 3362 33Io 3259 3208 3158 3io8 3059 o I 3631 3576 3521 3467 34i4 336I 3309 3258 3207 3157 3I07 3058 2 3630 3575 3520 3466 3413 3360 3308 3257 3206 3156 3Io6 3057 2 3 3629 3574 3519 3465 3412 3359 3307 3256 3205 3 55 3o05 3056 3 4 3628 3573 35i8 3464 3411 3358 3306 3255 3204 3154 3io5 3056 4 5 3627 3572 3517 3463 34o1 3358 3306 3254 3204 3153 3Io4 3055 5 6 3626 3571 35i6 3463 3409 3357 3305 3253 3203 3I53 3i03 3054 6 7 3625 3570 35I5 3462 3438 3356 3304 3253 3202 3152 3I02 3053 8 3624 3569 35I5 346I 3438 3355 3303 3252 3201 3151 3o10 3052 8 9 3623 3568 35i4 3460 3407 3354 3302 325I 3200 3150 3o10 3052 9 1i 3623 3567 3513 3459 3406 3353 330o 3250 3 o99 3I49 3Ioo 3051 Io i I 3622 3566 3512 3458 34 3352 344300 3249 3I98 3I48 3099 3050 ii 12 3621 3565 3511 3457 3404 3351 3300 3248 3I98 3148 3098 3049 12 I3 3620 3565 3510 3456 3403 3351 3299 3247 3197 3147 3097 3048 13 14 36I9 3564 3509 3455 3402 3350 3298 3247 3196 3I46 3096 3047 14 15 36i8 3563 3508 3454 3401 3349 3297 3246 3295 3I45 3096 3047 15 16 3617 3562 3507 3454 3400 3348 3296 3245 3194 3144 3095 3046 16 17 36i6 356i 3506 3453 3400 3347 3295 3244 3193 3I43 3094 3045 17 i8 36I5 3560 3506 3452 3399 3346 3294 3243 3193 3I43 3093 3044 18 19 36I4 3559 3505 345I 3398 3345 3294 3242 3192 3142 3092 3043 19 20 361i3 3558 3504 3450 3397 3345 3293 3242 39I 34 I3091 3043 20 21 3612 3557 3503 3449 3396 3344 3292 324I 39go 3140 3091 3042 2I 22 36ii 3556 3502 3448 3395 3343 3291 3240 3I89 3I39 3090 304I 22 23 36io 3555 3350I 3447 3394 3342 3290 3239 3i88 3i38 3089 3o4 23 24 36o 3555 3500oo 3446 3393 3341 3289 3238 3i88 3i38 3088 3039 24 25 3609 3554 3499 3446 3393 3340 3288 3237 3187 3I37 3087 3039 25 26 3608 3553 3498 3445 3392 3339 3288 3236 3i86 3i36 3087 3038 26 27 3607 3552 3497 3444 3391 3338 3287 3236 3i85 3I35 3086 3037 27 28 3606 355I 3497 3443 3390 3338 3286 3235 3184 3I34 3085 3036 28 29 3605 3550 3496 3442 3389 3337 3285 3234 383 3i33 3084 3035 29 30 3604 3549 3495 3441 3388 3336 3284 3233 3183 3I33 3083 3034 30 31 3603 3548 3494 3440 3387 3335 3283 3232 3182 3132 3082 3034 3I 32 3602 3547 3493 3439 3386 3334 3282 3231 3181 313I 3082 3033 32 33 3601 3546 3492 3438 3386 3333 3282 323I 3i80 3i30 3081 3032'33 34 3600 3545 349I 3438 3385 3332 3281 3230 3179 3I29 3080 303I 34 35 3599 3545 3490 3437 3384 3332 3280 3229 3I78- 3129 3079 3030 35 36 3598 3544 3489 3436 3383 3331 3279 3228 3I78 3128 3078 3030 36 37 3598 3543 3488 3435 3382 3330 3278 3227 3177 3127 3078 3029 37 38 3597 3542 3488 3434 3381 3329 3277 3226 3176 3I26 3077 3028 38 39 3596 354i 3487 3433 3380 3328 3276 3225 3175 3I25 3076 3027 39 40 3595 3540 3486 3432 3379 3327 3276 3225 3174 3124 3075 3026 40 4I 3594 3539 3485 343I 3379 3326 3275 3224 3173 3I24 3074 3026 4I 42 3593 3538 3484 3431 33 33 3325 37 223 3173 323 3073 3025 42 43 3592 3537 3483 3430 3377 3325 3273 3222 3172 3122 3073 3024 43 44 3591 3536 3482 3429 3376 3324 3272 3221 317I 312I 3072 3023 44 45 3590 3535 3481 3428 3375 333 3323 327 3220 320 30 22 45 46 3589 3535 3480 34,27 3374 3322 3270 3220 3I69 3II9 3070 3022 46 47 3588 3534 3480 3426 3373 3321 3270 3219 3i68 31I9 3069 3021 47 48 3587 3533 3479 3425 3372 3320 3269 3218 3168 3ri8 3069 3020 48 49 3587 3532 3478 3424 3372 3319 3268 327 367 367 7 3068 3019 49 50 3586 353i 3477 3423 337I 339 3267 3216 3I66 3I16 3067 3018 50. 51 3585 3530 3476 3423 3370 33i8 3266 32I5 3I65 31I5 3066 30I8 51 52 3584 3529 3475 3422 3369 3317 3265 32I4 3r64 3I4 3065 3017 52. 53 3583 3528 3474 3421 3368 33i6 3265 3214 3.163 3Ii4 3065 3016 53 54 3582 3527 3473 3420 3367 33i5 3264 32I3 3163 3II3 3064 30i5 54 55 358I 3526 3472 3419 3366 33I4 3263 3212 3162 3II22 3063 3oi4 55 56 3580 3525 347I 348 3365 33 3 3262 32II 316 3111 3062 3o04 56 57 3579 3525 3471 3417 3365 3313 3261 3210 3 3i 3061 3oi3 |57 58 3578 3524 3470 34i6 3364. 3312 3260 32c 3159 31i1 3o6o 3012 58 59 3577 3523 3469 3415 3363 33ii 3259 3209 3 0i58 6 3011 59 3. 10~ IPi 119~ 3 01' 1 21' 1~ 22' 110 3'118 24'J11 25'11~ 26f 10 271 C 28'111 29' S. 36[/34 39./33 36J I 3 3 0,J33 Page 10] TABLE XXII. Proportional Logarithms. h n A ~ a n o h n im rn m h m h m A m h m h m h m IC, 1 0' 3 3 1 33 03 34/11 35/ 1 3611O 37 110 38' 1~ 39]10 40'11~ 41/ S. 0 3o01 2962 2915 2868 2821 2775 2730 2685 2640 2596 2553 2510 0 I 3009 2962 2914 2867 2821 2775 2729 2684 2640 2596 2552 2509 I 2 3009 2961 2-93 2866 2820 2774 2729 2684 2639 2595 2551 2508 2 3 3008 2960 2912 2866 2819 2773 2728 2683 2638 2594 2551 2507 3 4 3007 2959 2912 2865 2818 2772 2727 2682 2638 2593 2550 2507 4 5 3006 2958 2 11 2864 2818 2772 2726 2681 2637 2593 2549 2506 5 6- 3005 2958 2910 2863 2817 2771 2725 2681 2636 2592 2548 2505 6 7 3005 2957 2909 2862 2816 2770 2725 2680 2635 2591 2548 2504 7 8 3004 2956 2909 2862 2815 2769 2724 92679 2635 2591 2547 2504 8 9 3oo003 955 2908 2861 281 2769 2723 2678 2634 2590 2546 2503 9 io 3002 2954] 2907 2860 2814 2768 2722 2678 2633 2589 2545 2502 10 ii 3ooI 2954 2906 2859 2813.2767 2722 2677 2632 2588 2545 2502 II 12 3001 2953 2905 2859 28I2 2766 2721 2676 2632 2588 2544 2501 12 13 3000 2952 2905 2858 2811 2766 2720 2675 2631 2587 2543 2500 13 14 2999 2951 2904 2857 2811 2765 2719 2675 2630 2586 2543 2499 14 15 2998 2950 2903 2856 2810 2764 2719 2674 2629 2585 2542 2499 T5 16 2997 2950 2902 2855 2809 2763 2718 2673 2629 2585 2541 2498 16 17 2997 2949 2901 2855 2808 2763 2717 2672 2628 2584 2540 2497 17 i8 2996 2948 2901 2854 2808 2762 2716 2672 2627 2583 2540 2497 18 19 2995 2947 2900 2853 2807 2761 2716 2671 2626 2583 2539 2496 i9 20 2994 2946 2899 2852 2806 2760 2715 2670 2626 2582 2538 2495 20 21 2993 2946 2898 2852 2805 2760 2714 2669 2625 2581 2538 2494 21 22 2993 2945 2898 2851 2805 2759 2713 2669 2624 2580 2537 2494 22 23 2992 2944 2897 2850 92804 2758 2713 2668 2624 2580 2536 2493 23 24 2991 2943 2896 2849 2803 2757 2712 2667 2623 2579 2535 2492 24 25 2990 2942 2895 2848 2802 2756 2711 2666 2622 2578 2535 2492 25 26 2989 2942 2894 2848 2801 2756 2710 2666 2621 2577 2534 2491 26 27 2989 2941 2894 2847 2801 2755 2710 2665 2621 2577 2533 2490 27 28 2988 294o 2893 2846 2800 2754 2709 2664 2620 2576 2533 2489 8 29 2987 2939 2892 2845 2799 2753 2708 2663 2610 2575 2533 2489 29 30 2986 2939 2891 2845 2798 2753 2707 2663 2618 2574 25301 i248 3o 31 2985 2938 2891 2844 2798 2752 2707 2662 2618 2574 2530 2487 3i 32 2985 2937 2890 2843 2797 2751 2706 2661 2617 2573 2530 2487 32 33 2984 2936 2889 2842 2796 2750 2705 2660 2616 2572 2529 2486 33 34 2983 2935 2888 2842 2795 2750 2704 2660 2615 2572 2528 2485 34 35 2982 2935 2887 2841 2795 2749 2704 2659 2615 2571 2527 2485 35 36 2981 2934 2887 2840 2794 2748 2703 2658 2614 2570 2527 2484 36 37 2981 2933 2886 2839 2793 2747 2702 2657 2613 2569 2526 2483 37 38 2980 2932 2885 2838 2792 2747 2701 2657 2612 2569 2525 2482 38 39 2979 2931 2884 2838 2792 2746 2701 2656 2612 2568 2525 2482 39 40 2978 2931 2883 2837 2791 2745 2700 2655 2611 2567 2524 2480 40 41 2977 2930 2883 2836 2790 2744 2699 2655 2610 2566 2523 2480 41 42 2977 2929 2882 2835 2789 2744 2698 2654 2610 2566 2522 2480 42 43 2976 2928 2881 2835 2788 2743 2698 2653 2609 2565 2522 2479 43 44 2975 2927 2880 2834 2788 2742 2697 2652 2608 2564 2521 2478 44 45 2974 2927 2880 2833 2787 2740 2696 2652 2607 2564 2520 2477 45 46 2973 2926 2879 2832 2786 2741 2695 2651 2607 2563 2520 2477 46 4Z7 2973 2925 2878 2831 2785 2740 2695 2650 2606 2562 2519 2476 47 48 -972 2924 2877 2831 2785 2739 2694 2649 2605 2561 2518 2475 48 49 2971 2924 2876 2830 2784 2738 2693 2649 2604 256I 2517 2475 49 50 2970 2923 2876 2829 2783 2738 2692 2648 2604 2560 2507 2474 5o 51 2969 2922 2875 2828 2782 2737 2692 2647 2603 2559 2506 2473 51 52 2969 2921 2874 2828 2782 2736 2690 2646 2602 2559 2515 2472 52 53 2968 2920 2873 2827 2781 2735 2690 2646 2601 2558 2515 2472 53 54 2967 2920 2873 2826 2780 2735 2689 2645 2601 2557 2514 2471 54'55 2966 2919 2872 2825 2779 2734 2689 2644 2600 2556 2513 2470 55 56 2965 2918 2871 2825 2779 2733 2688 2643 2599 2556 2512 2470 56 57 2965 2917 2870' 2824 2778 2732 2687 2643 2599 2555 2512 2469 57 58 2964 2916 2869 2823 2777 2732 2687 2642 2598 2554 2511 2468 58 59 2963 2916 2869 2822 2776 2731 2686 264 2597 2553 2510 2467 59 S. 10 3010 31, 3X 10 33'i10 34' 10P35' 0 3"6' 1 37' 10 38' 10 39J11 40' 11 41 S. TABLE XXII. [Page 141 Proportional Logarithms. Ith m h m A h m h m h m h m l m h ml m I m S 1~ 42/11~ 431~ 44' 1~ 450 1~ 46/ 1~. 47 1~ 48' 1 491'1~ 50' 1~ 51 10 52' 1 53 S. o 2467 2424 2382 234I 23oo 2259 22I8 2178 2139 2099 2061 2022 0 I 2466 2424 2382 2340 2299 2258 2218' 2178 2138 2099 2060 2021 1 2 2465 2423 2381 2339 2298 2258 22I7 2I77 2I37 2098 2059 2021 2 3 2465 2422 2380 2339 2298 2257 2216 2176 2137 2098 2059 2020 3 4 2464 2422 2380 2338 2297 2256 2216 2176 2136 2097 2058 2019 4 5' 2463 2421 2379 2337 2296 2256 2215 2175 2136 2096 2057 2019 5 6 2462 2420 2378 2337 2296 2255 22I4 2174 2135 2096 2057 201.8 6 7 2462 2419 2378 2336 2295 2254 2214 2174 2134 2095 2056 2017 7 8 246I 2419 2377 2335 2294 2253 22I3 2173 2I34 2094 2055 2017 8 9 2460 2418 2376 2335 2294 2253 2212 2I72 2133 2094 2055 20I6 9 10 246o 2417 2375 2334 2293 2252 2212 2172 2132 2093 2054 2016 10 II 2459 2417 2375 2333 2292 2251 2211 2I71 2132 2092 2053 2015 II 12 2458 2416 2374 2333 2291 2251 2210 2170 2131 2092 2053 2014 12 I3 2458 2415 2373 2332 2291 2250 2210 2170 2130 2091 2052 2014 13 14 2457 2415 2373 233I 2290 2249 2209 2169 2130 2090 2052 2013 14 15 2456 2414 2372 2331 2289 2249 2208 21692129 2090 2051 2012 15 I6 2455 2413 2371 2330 2289 2248 2208 2168 2128 2089 2050 2012 16 17 2455 2412 2371 2329 2288 2247 2207 2167 2128 2088 2050 2011 17 18 2454 24I2 2370 2328 2287 2247 2206 2167 2127 2088 2049 2010 18 19 2453 2411 2369 2328 2287 2246 2206 2166 2126 2087 2048 2010 19 20 2453 2410 2368 2327 2286 2245 2205 2165 226 2o86 2048 2009 20 21 2452 2410 2368 2326 2285 2245 2204 2165 2125 2086 2047 2009 21 22 2451 2409 2367 2326 2285 2244 2204 2164 2124 2085 2046 2008 22 23 2450 2408 2366 2325 2284 2243 2203 2163 2I24 2085 2046 2007 23 24 2450 2408 2366 2324 2283 2243 2202 2163 2'23 2084 2045 2007 24 25 2449 2407 2365 2324 2283 2242 2202 2I62 2122 2083 2044 2006 25 26 2448 2406 2364 2323 2282 2241 2201 2161 2122 2083 2044 2005 26 27 2448 2405 2364 2322 2281 224I 2200 2I6I 2121 2082 2043 2005 27 28 2447 2405 2363 2322 2281 2240 2200 2160 2I20 2081 2042 2004 28 29 2446 2404 2362 2321 2280 2239 2I99 2159 2120 2081 2042 2003 29 30 2445 2403 2362 2320 2279 2239 2198 2159 2119 2080 2041 2003 30 31 2445 2403 2361 2320 2279 2238 2198 2158 2118 2079 2041 2002 31 32 2444 2402 2360 2319 2278 2237 2197 2157 2118 2079 2040 2001 32 33 2443 2401 2359 2318 2277 2237 2196 2157 2117 2078 2039 2001 33 34 2443 2401 2359 2317 2277 2236 2196 2156 2116 2077 2039 2000 34 35 2442 2400 2358 2317 2276 2235 2195 2155 2116 2077 2038 2000 35 36 2441 2399 2357 2316 2275 2235 21294 2155 2115 2076 2037 1999 36 37 2441 2398 2357 2315 2274 2234 2194 2154 2 115 2075 2037 1998 37 38 2440 2398 2356 2315 2274 2233 2193 2I53 2114 2075 2036 1998 38 39 2439 2397 2355 2314 2273 2233 2192 2153 2113 2074 2035 1997 39 40 2438 3 2396 55 2313 2272 2232 2192 2I52 2113 2073 2035 I996 40 41 2438 2396 2354 2313 2272. 2231 2191 2151 2112 2073 2034 1996 4I 42 2437 2395 2353 2312 2271 2231 2190 2151 2111 2072 2033 1995 42 43 2436 2394 2353 2311 2270 2230 2190 2150 2111 2072 2033 1994 43 44 2436 2394 2352 2311 2270 2229 2189 2149 2110 2071 2032 994 44 45 2435 2393 2351 2310 2269 2229 2188 2149' 219 2070 2032 1993 45 46 2434 2392 2350 2309 2268 2228'288 2148 2I09 2070 2031 1993 46 47 2433 2391 2350 2309 2268 2227;2187 2147 2108 2069 2030 1992 47 48 2433 2391 2349 2308 2267 2227 2I86 2147 2107 2068 2030 I991 48 49 2432 2390 2348 2307 2266 2226 2186 2I46 2107 2068 2029 1991 49 50 243I 2389 2348 2307 2266 2225;2185 2145 2106 2067 2028 1990 50 51 243I 2389 2347 2306 2265 2225 2184 2145 2105 2066 2028 1989 5I 52 2430 2388 2346 2305 2264 2224 2I84 2144 2105 2066 2027 1989 52 53 2429 2387 2346 2304 2264 2223 2183 2143 2104 2065 2026 1988 53 54 2429 2387 2345 2304 2263 2223 2182 2143 21o3 2064 2026 1987 54 55 2428 2386 2344 2303 2262 2222 2182 2142'2103 2064 2025 1987 55 56 2427 2385 2344 2302 2262 2221 2181 2141 2102 2063 2025 1986 56 57 2426 2384 2343' 2302 2261 2220 2180 2141 2101 2062 2024 1986 57 58 2426 2384 2342 2301 2260 2220 2180 2140 2101 2062 2023 1985 58 59 2425 2383 2342 2300 2260 2219 2179 2139 2100 2061 2023 1984 59 S. 10i 4 4 442' 1~1 43/ Il 4 -610 47 1~ 48 10 49^ 10 50' 16 5V 1~ 52 1~ 53/' S ~~~~~~10 421, 110 451'10 461 7/Z Page 142] TABLE XXII. Proportional Logarithms. Jim ihm h m tm n m m mh n m I in m k m S. 1054/ 10 55/ 1561 1057/ 158s 10 59l 20 0' 2" 1' 2~ 2' 2' 3 2 4/~. o 1984 1946 1908 1871 i834 1797 1761 1725 169 i1654 1619 o I 1983 1945 1908 1870 i833 1797 1760 1724 1689 I653 i6.8 i 2 1982 1944 1907 1870 1833 1796 i76oI 1724 1688 1652 1617 2 3 1982 1944 1906 1869 1832 1795 I759 1723 1687 1652 I617 3 4 1981 1943 o906 I868 i831 1795 1759 I722 1687 1651 1616 4 5 1981 1943 1905 i868 i831 794 1758 1722 i686 i651 1616 5 () 1980 1942 1904 1867 i830 1794 1757 1721 i686 165o0 165 6 7 1979 1941 1904 1867 183o 1793 1757 1721 i685 1650 1614 7 8 1979 17941 193 i866 1829 1792 1756 1720 1684 I649 i614 8 9 1978 1940 1903 i865 1828 1792 1755 1719 i684 i648 1613 9 10 1977 1939 1902 i865 1828 1791 1755 1719 i683 i648 1613 io 1'r 1977 1939'9oi i864 1827 1791 1754 1718 i683 1647 1612 II 12 1976 1938 190o1 1863 1827 1790 1754 1718 1682 1647 1612 12 13 1975 1938 1900 i863 1826 1789 1753 1717 i681 i646 1611 13 14 c19)75 1937 T1899 i862 1825 1789 1752 1717 i68i i645 1610o 14 15 1974 1936 1899 1862 1825 1788 1752 1716 1680 1645 1610o 15 i6 1974 1936 1898 1861 1824 1788 1751. 1715 68o0 1644 1609 16 17 1973 1935 1898 i86o 1823 1787 1751 1715 1679 i644 1609 17 i8 1972 1934 1897'I 860 1823 1786 1750 1714 1678 1643 i6o8 18 19 1972 1934 1896 1859 i822 1786'749 1714 1678 1643 1607 19 o20 1971 1933 1896 1859 1822 1785 1749 1713 1677 1642 1607 20 21 1970 1933 1895 1858 1821 1785 1748 1712 1677 I641 16o6 21 22 1970 1932 1894 1857 1820 1784 1748 1712 1676 i64 1i6o6 22 23 1969 1931 1894 1857 1820 1783 1747 1711 1676 1640 1605 23 24 1968 1931 1893 1856 1819 1783 1746 1711 1675 I64o 1605 24 25 1968 1930 1893 1855 1819 1782 1746 1710 1674 1639 1604 226 1967 1929 1892 1855 1818 1781 1745 1709 1674 i638 1603 26 27 1967 1929 1891 1854 1817 1781 1745 1709 1673 i638 1603 27 28 1966 1928 1891 1854 1817 1780 1744 1708 1673 1637 1602 28 29 1965 1928 1890 i853 i8i6 1780 1743 1708 1672 1637 1602 29 30 1965 1927 1889 1852 Iii6 779 1743 1707 1671 1636 1601 30 31 1964 1926 1889 i852 18i5 1778 1742 1706 1671 1635 i6oo 31 32 1963 1926 1888 1851 1814 1778 1742 1706 1670 1635 1600oo 32 33 1963 1925 i888 I85o 1814 1777 174I 1705 1670 i634 1599 33 34 1962 1924 1887 I850 1813 1777 1740 1705 1669 1634 1599 34 35 1962 1924 1886 1849 1812 1776 1740 1704 1668 1633 1598 35 36 1961 1923 i886 1849 1812 1775 1739 1703 i668 i633 1598 36 37 1960 1923 i885' 848 1811 1775 1739 1703 1667 1632 1597 37 38 1960 1922 1884 1847 1811 1774 1738 1702 1667 1631 1596 38 39 1959 1921 i884 1847 1810o 1774 1737 1702 1666 i63i 1596 39 ~40 1958 1921 1883 1846 1809o 1773 1737 1701 1665 163o 1595 40 41 1958 1920 i883 i846 1809 1772 1736 1700 i665 i630 1595 41 4') 957 1919 1882 i845 1808 1772 1736 1700 1664 1629 1594 42 43 1956 1919 i88i i844 i808 1771 1735 1699 i664 1628 1593 43 44 1956 1918 1881 i844 1807 1771 1734 1699 i663 1628 1593 44 45 55 1918 i88o0 843 80o6 1770 1734 1698 1663 1627 1592 45 46 1955 1917 i880 i843 i8o6 1769 1733 I697 1662 I627 1592 46 47 1954 1916 1879 1842 1 8o5 1769 1733 1697 i66i 1626 1591 47 48 1953 1916 1878 1841 18o5 1768 1732 1696 i661 1626 1591 48 49 1953 1915 1878 i841 18o4 1768 1731 1696. i66o 1625 1590o 49 50 1952 1914 1877 1840 1803 1767 1731 1695 1660 1624 1589 50 51 I951 1914 1876 2839 1803 1766 1730 I694 1659 1624 1589 51 52 1951 1913 1876 1839 1802 1766 1730 1694 i658 1623 1588 52 53 i95o 1913 1875 i838 18o2 1765 1729 1693 1658 1623 i588 53 54 i9go 1912 I175 i838 18o1 1765 1728 1693 i657 1622 1587 54 55 1949 1911 1874 1837 18oo 1764 1728 1692 1657 1621 1587 55 56 1948 1911 1873 1836 i8oo 1763 1727 1692 1656 1621 i586 56 57 1948 1910 1873 i836 1799 1763 1727 1691 i655 1620 i585 57 58 1947 [909 1872 1835 1798 1762 1726 169o0 655 1620 1585 58 59 1946 1909 1871 i835 1798 1762. 1725 1690go 654 1619 i584 59 S. 1054' 1 55' 1056' 1' 57' o58'1 / 0'' 2~ I3' 2 2 42.S. TABLE XXII. [Pae 143 Proportional Logarithms. h m h m m /I m k m h km m h m m h m I S. 5' 20 6' 20 71' f 2 8 20 9f 2010' 201/ 2012 2013/ 2014' 2015'. 0 i584 1549 8i5 I48i 1447 1413 I380 1347 1314 1282 1249 0 I i583 1548 1514 I480 1446 1413 1379 1346 1314 1281 1249 i 2 1582 i548 I 154 1479 446 1.42 1379 I46 12 1248 3 i582 1547 1513 1479 1445 1412 1378 I45 13i3 128o 1248 3 4 1.58i 1547 1512 1478 1445 1411 1378 1345 1312 280 1247 4 5 1581 546 1512. 1478 i444 1411 1377 i344 1311 1279 1247 5 6 158o 1546 15' 1477 1443 141i 1377 1344 1311 1278 1246 6 7 158o r545 1511 1477 1443 1409 1376 i343 i3 1278 1246 7 8 i578 i544 1 510 1476 1442 1409 1376 1343 1310 1277 1245 8 8 1578 1544 1510 1476 1442 1408 1375 1343 1309 1277 1245 9 io 1578 1543 1509 1475 1441 14o8 1374 1342 1309 1277 1244 9o ii 1577 i543 i5oS 147 1441 1407 1374 134i 1308 1276 124: 11i 12 1577 1542 I50S 1474 1440 1407 1373 i340o i38 1275 1243 12 i3 1576 I542 15o07 1473 1440 i4o6 1373 134o 1307 1275 1242 13 14 1576''1541 15o7 1473 1439 1406 1372 1339 1307 1274 1242 I4 15 1575 54o I506 1472 1438 1405 1372 339 1306 1274 1241 15 16 1574 i540 15oG 1472 i1438 1404 1371 i338 r306 1273 1241 16 17 1574 1539 1505 1471 1437 I404 1371 1338 1305 1273 1240 17 "I8 5-73 1539 15044 1470 1437 1403 1370 1337 3o4 1272 1240 18 19 1573 1538 150)4 1470 i436 1403 1370 1337 1304 1271 1239 19 20 1572 i538 1503o 1469 i436 1402 1369 1336 -303 1271 1239 20 21 1571 1537 50o3 1469 i435 1402 i368 1335 1303 1270 I238 21 22 1571 1536 1502 1468 435 1401 1368 1335 1302 1270 1238 22 23 1570 1536 1502 1468 i1434 1401oi 1367 1334 1302 1269 1237 23 24 1570 1535 1501oi 467 1433 1400 1367 1334 1301o 1269 1237 24 25 1569 535 1500oo 1467 1433 1399 i366 1333 1301 1268 1236 25 26 1569 1534 i5oo i466 1432 1399 1i366 1333 1300 1268 1235 26 27 i568 i534 1499 i465 1432 1398 1365 1332 1300 1267 1235 27 28 1567 1533 i499 i465 1431 1398 i365 1332 1299 1267 1234 28 29 1567 1532 1498 1464 i431 1397 1364 1331 1298 1266 1234 29 30 1566 1532 1498 1464 1430 1397 i363 1331 1298 1266 1233 30 I31 1566 1531 1497 1463 1429 1396 1363 1330 1297 1265 1233 31 32 i565 1531 1496 i463 1429 1396 1362 1329 1297 1264 1232 32 33 1565 1530 1496 1462 1428 1395 1362 1329 1296 1264 1232 33 34 i564 1530 1495 i461 1428 1394 i36r 1328 1296 1263 1231 34 35 1563 1529 1495 i461 1427 1394 1361 1328 1295 i263 1231 35 36 1563 1528 1494 1460 1427 1393 136o 1327 1295 1262 1230 36 37 1562 1528 1494 i46o 1426 1393 i36o 1327 1294 1262 1230 37 38 1562 1527 1493 1459 1426 1392 1359 1326 1294 1261 1229 38 39 1561 1527 1493 1459 1425 1392 1359 1326 1293 1261 1229 39 40 I561 1526 1492 i458 1424 1391 i358 1325 1292 1260 1228 40 41 156o 1526 1491 i458 1424 1391 1357 1325 1292 1260 1227 41 42 1559 1525 1491 1457 1423 1390 1357 1324 1291 1259 1227 42 43 1559 1524 1490 1456 1423 1389 1356 1323 1291 1259 1226 43 44 1558 1524 1490 1456 1422 1389 1356 1323 1290 1258 1226 44 45 1558 1523 1489 i455 1422 1388 135 1322 1290 1257 1225 45 46 1557 1523 1489 1455 1421 1388 1355 1322 1289 1257 1225 46 47 i556 i52/ 1488 i454 1421 1387 I354 1321 1289 1256 1224 47 48 i556 1522 1487 1454 1420 1387 i354 1321 1288 1256 1224 48 49 i555 1521 1487 i453 141 1oi386 135.3 320 1288 1255 1223 49 50 i555 1520 1486'6 1452 1419 i386 1352 1320 1287 1255 1223 5o8 51 1554 1520 I486 I142 1418 1385 1352 1319 1287 1254 1222 51 52 554 1513 1485 1451 1418 1384 1351 1319 1286 1254 1222 52 53 1553 1519 i485 1451 1417 i384 i351 I3i8 1285 1253 1221 53 54 -1i552 1518 i484 1i450 1417 i383 i35o 1317 1285 1253 1221.54 J5 I 55 151i8 i483 i450 1416 383 I35o 1317 1284 1252 1220 "55 56 1551 1517 i483 1449 1416 1382 1349 i 36 1284 1252 1219 56 -57 1551 1516 1482'449 1415 1382 1349 13i6 1283 I251 1219 57 58 1550 15i6 1482 i448 1414 1381 i348 1315 1283 1250 1218 58 59 1550 1515 148I 144o7 1414 1381 1348 i3i5 1282 1250 1218 59 2S 51 2I. o 5 20 _.i 20 8. 20 9 2 107 2011/ 20 1 2013 i 2 14T/ 201/ S. [50/56/,9L 48/i:4IT9 I,57tI2 22I,6 ~V/4 Page 144] TABLE XXII. Proportional Logarithms. h m kh m h m h Am mh m h h m h m m h m S. 2~ fl' 2 17' 2~ 18' 2~ 191/2 220 20211 2022 20~231 2241 225' 2026/ S 0 1217 1186 II54 1123 io 1o61 130 999 o o93 oog I 1217 Ii85 1153 1122 oI 9 10oo60 I2 9 09969 0939 0909 I 2 1216 I 84 1153 1122 o0o 1i060 1029 o 998 0968 0938 0908 2 3 1216 1184 1152 1121 I090 I059 1028 0998 0968 0938 0908 3 4 1215 183 1152 1120 1089 I058 I028 0997 0967 0937 0907 4 5 I215 1183 I151 1120 I 189 o158 1027 0997 0967 0937 0907 5 6 1214 1182 115I II9 io88 1057 1027 0996 0966 0936 0906 6 7 1214 1182 1150 1119 io88 1057 1026 0996 0966 0936 0906 7 8 1213 1181 150 1118 1087 10o56 1026 0995 0965 0935 0905 8 9:,213 1181 49 ii8 1087 Io56 1025 0995 0965 0935 0905 9 o1 12I2 1180 1149 1117 Io86 1o55 1025 0994 0964 0934 0904 IO II 121I ii8o 1148 1117 io86 io55 1024 0994 0964 0934 0904 I 12 121I 1179 ii48 I I6 Io85 io54 I024 0993 o963 0933 oo93 12 13 1210 1179 I 47 116 o185 Io54 I023 0993 0963.0933 0903 13 14 12IO 1178 1147 1115 1084 1053 1023 0992 0962 0932 0902 14 i5 1209 1178 1146 1115 o1084 I o1053 1022 0992 0962 0932 0902 15 I6 1209 1177 1146 1114 o183 1052 1022 0991 0961 0931 0901 16 I7 1208 1177 II45 I I4 o083 1052 1021 0991 0961 0931 0901 17 I8 I208 1176 1145 1113 o82 o5I o02 099go og60 0930 ogoo00 18 19 1207 1175 I I44 1113 1082 o1051 I020 0990 0960 og30 0900 Ig 20 1207 1175 1143 1112 1081 1050 1020 0989 0959 0929 o899 20 21 1206 II74 1143 1112 o8i Io050 1019 0989 0959 0929 0899 2I 22 1206 1174 1142 II 1o80 1049 1019 0988 o958 0928 0898 22 23 1205 1173 1142 1 11I Io80 1049 Ioi8 0988 0958 0928 0898 23 24 1205 1173 1141 IIo 1079 io48 ioi8 0987 0957 0927 0897 24 25 1204 II72 I 111 IIIO 1079 I048 IOI7 0987 0957 0927 0897 25 2 6 1204 1172 1140 1109 1078 1047 1017 0986 0956 0926 0896 26 27 1203 1171 140 1109 1078 1047 o016 0986 0956 0926 0896 27 28 I202 1171 1139 1108 I077 I046 ioi6 o985 0955 0925 0895 28 29 1202 II70 1139 Iio8 I076 o146 ioi5 0985 0955 0925 0895 29 30 1201 1170 1138 1107 1076 I045 o1015 0984 0954 0924 0894 30 31 1201 1169 r38 10o6 1075 145 1014 0984 0954 0924 0894 3i 32 1200 1169 1137 1106 -1075 I044 ioi4 0983 0953 0923 0893 32 33 I 200 168 1137 1105 1074 1044 10I3 0983 0953 0923 0893 33 34 I i99 II68 36 Ii5 I074 1043 o103 0982 0952 0922 0892 34 35 1199 II67 1136 1104 1073 1043 1012 0982 0952 0922 0892 35 36 198 1167 1135 I104 1073 1042 10o2 0981 0951 0921 0891 36 37 1198 i66 I135 iio3 I072 1042 IOIT 0981 0951 0921 0891 37 38 1197 1165 i34. 1103 1072 I04I IOi 09o80 0950 0920 0890 38 39 1197 II65 ii34 II02 1071 Io04I IOO o980 og50 o920 o890 39 40 1196 1164 1133 1102 1071 o140 I009 0979 0949 0919 0889 4o 4I 1196 1164 1132 IIo 1070 1040 I009 0979 0949 09g9 0889 4i 42 1195 1163 1132 IIOI 1070 1039 o008 0978 0948 o918 o888 42 43 195 ii63 1I31 IIoo o169 1039 oo8 0978 0948 o9g8 o888 43 44 1194 1162 1131 IIoo 1069 0o38 1007 0977 0947 0917 0887 44 45 2193 1162 1130 I099 io68 1037 1007 0977 0947 0917 0887 45 46 1193 ii6i ij3o 1099 1068 1037 Ioo6 0976 0946 0916 o886 46 47 1192 1161 1129 1098 1067 0o36 o006 0976 0946 0916 o886 47 48 II92 1160 1I29 0g98 I067 Io36 o005 0975 0945 0915 o885 48 49 lg19 ii6o II28 1097 1o66 1035 o005 0975 0945 o9g5 o885 49 50 1191 1159 1128 1097 o1066 1035 oo004 0974 0944 0914 o884 50 5I Ig9o 1I59 II27 1096 Io65 io34 0oo4 0974 0944 0914 o884 51 5,:I90 ii58 1127 1096 o065 1034 o003 0973 0943 0913 o883 52 53 xi89 1158 1126 1095 o164 i033 ioo3 0973 0943 0913 o883 53 54 1189 1157 1126 1095 1I64 o133 1002 0972 0942 0912 o883 54 55 i188 1157 1125 1094 o63 1032 1002 0972 0942 0912 0882 55 56 1188 1156 1125 o094 0o63 1032 Iooi 0971 0941 o09. I 0882 56 57 1187 1156 1124 1093 1062 o031 IooI 0971 094I 0911 088i 57 58 1187 1155 1I24 I092 1062 103i Iooo 0970 0940 0910 088I 58 59 |186 iI54 1123 1092 0o6i1 030 iooo o, 970 o940 09O1 o880 59 S. 20161 2T17/ 20 18 2 ~19 t 2020 21 222 023/ 20 241 2 2 512026/ S. TABLE XXII. [tge 145 Proportional Logarithms. h m A m h m h m h km h m h m h m h m hkm, I m S 2 7 2y 28' ~ 29' 20 30' 20 31/ 20 32' 2~ 33:' 2 34' 2035' 2~ 36' 20 37/ S. 0 o880 0850 o821 0792 0763 0734 0706 0678 o649 0621 0594 0 o879 o850 0820 0791 0762 0734 0705 0677 0649 0621 0593 I 2 0879 o849 0820 0791 0762 0733 0705 0677 0648 0621 0593 2 3 0878 0849 0819 0790 0762 0733 0704 0676 o648 0620 0592 3 4 0878 o848 0o81 0790 0761 0732 0704 0676 o648 0620 0592 4 5 0877 1848 o818 0789 0761 0732 0703 0675 0647 o619 o091 5 6 0877 0847 o818 0789 0760 0731 0703 0675 o647 0619 o59i 6 7 0876 0847 o817 0788 0760 0731 0703 0674 o646 o68 o5g91 7 8 0876 o846 o817 0788 0759 0730 0702 0674 o646 o6i8 o5go 8 9 0875 o846 o816 0787 0759 0730 0702 0673 o645 0617 o590 9 Io 0875 o845 o816 0787 0758 0730 0701 0673 0645 0617 0589 Io 1i 0874 o845 o8i6 0787 0758 0729 0701 0672 o644 o6I6 0589 ii 12 0874 o844 o8i5 0786 0757 0729 0700 o672 o644 o6I6 o588 12 13 0873 o844 o815 0786 0757 0728 0700 0671 o643 o665 0588 13 14 0873 o843 08i4 0785 0756 0728 o699 067I o643 o6I5 0587 I4 15 0872 o843 o8I4 0785 0756 0727 0699 0670 0642 o6I5 0587 5 I6 0872 0842 0813 0784 0755 0727 0698 0670 0642 06i4 o586 16 17 0871 0842 o813 0784 0755 0726 0698 0670 o641 0614 o586 17 i8 0871 o84 o812 0783 0754 0726 0697 0669 o64i o63 o585 8 19 0870 o84i o812 0783 0754 0725 o697 0669 o64,I o613 o585 19 20 o870 o84o o8ii 0782 0753 0725 0696 o668 o640 0612 o585 20 21 0869 o840 o8i- 0782 0753 0724 0696 o668 o640 o612 o584 21 22 0869 0839 o8io 0781 0752 0724 0695 0667 0639 o6ii 0584 22 23 o868 0839 o8io 078I 0752 0723 0695 0667 0639 o6 1 o583 23 24 o868 o838 o809 0780 075i 0723 0694 o666 o638 06o0 o583 24 25 0867 o838 0809 0780 o751 0722 0694 o666 o638 o6io 0582 25 26 0867 0837 o808 0779 0751 0722 0694 o665 0637 0609 0582 26 27 o866 0837 o808 0779 0750 072I 0693 o665 0637 0609 058I 27 28 o866 o836 0807 0778 0750 072I 0693g o664 o636 o609 0581 28 29 0865 o836 0807 0778 0749 072I 0692 o664 o636 0608 o580 29 30 o865 o835 o806 0777 0749 0720 0692 o663 o635 o608 o580 30 3i o864 o835 o8o6 0777 0748 0720 0691. o663 o635 0607 0579 3i 32 o864 o834 o805 0776 0748 0719 0691 0663 o634 0607 0579 32 33 o863 o834 o805 0776 0747 0719 o6go o662 o634 0606 0579 33 34 o863 o834 o804 0775 0747 0718 o690 o662 o634 o606 0578 34 35 0862 o833 o804 0775 0746 0718 o689 o66 0o633 o6o0 0578 35 36 o862 o833 o803 0774 0746 07I7 0689 066 o633 o6o5 0577 36 37 086i 0832 o803 0774 0745 0717 o688 o660 0632 o604 0577 37 38 086i 0832 o802 0774 0745 07I6 o688 o660 0632 o6o4 0576 38 39 o86o o83i 0802 0773 0744 07I6 0687 0659 063i o603 0576 39 40 o860 o83I o8oi 0773 0744 o7I5 0687 0659 o63I o603 0575 4o 41 0859 o830 o80I 0772 0743 0715 o686 o658 o630 o602 0575 4 42 0859 o830 o80o 0772 0743 07I4 o686 o658 o630 0602 0574 42 43 o858 0829 o8oo 0771 0742 0714 o686 o657 0629 0602 o574 43 44 o858 o829 o800 0771 0742 0713 o685 o657 o629 060i o573 44 45 o857 0828 o799 0770 0741 07I3 0685 o656 0 628 060o1 573 45 46 0857 o828 0799 0770 074I 07I2 o684 o656 0628 o600 o573 46 47 o856 0827 0798 0769 0740 07I2 o684 0655 0628 o600 o572 47 48 o856 0827 0798 0769 0740 0711 o683 o655 0627 0599 o572 48 49' o855 0 826 0797 0768 0740 0711 0683 655 0627 0599 0571 49 50 o855 0826 0797 0768 0739 0711 0o682 o654 0626 o598 057 50 5 o855 o825 0796 0767 0739 0710 o682 0654 0626 o598 0570 5 52 o854 o825 0796 0767 0738 07I0 o68I o653 0625 o597 057o 52 53 o854 0824 o795 0766 0738 0709 o68I o653 o625 o597 o569 53 54 o853 0 o824 o795 0766 0737 0709 680 o652 0624 0596 o569.1 54[ 55 o853 o823 0794 0765 0737 0708 o680 o652 0624 o596 o568 55 56 0852 0823 0794 0765 0736 0708 0679 0651 0623 0596 0568 56 57 0852 0822 0793 0764 0736 0707 0679 o65i 0623 o595 0568 57 58 o85i 0822 0793 0764 0735 0707 0678 o650 0622 o595 0567 58 59 o85 0821 0792 0763 0735 0706 0678 o650 0622 o594 0567 59 s-20- ~ 20 28'"20 291 20 30'20 311 fL 3' I - S. 227 202S 229' 20~30 2~31V 23o 2033'1 234/ 20 35' 2~ 36' 237' 1S. 19 page 1461 Tage 146] TABLE XXII. Proportional Logaritbhms. j m i, m i n hm m km Jm h m m Itn m Ih m' 2038 239 40' 2 4 42' 20 43 2044/ 2045/ 2046' 20471 248 o o566 0539 0512 o484 o458 o43i 0404 0378 o352 0326 o0300 I 0566 o538 o51 0484 0457 0430 o4o4 0377 o35i 0325 0299 1 2 0565 o538 o511 0484' 0457 0430 0403 0377 o35i 0325 0299 2 3 o565 0537 05io o483 o456 o430.o403 0377 o350 0324 0298 3 4 o564 0537 0o51 0483 o456 0429 04o3 0376 o350 o324 0298 4 5 0564 0536 0509 0482 0455 0429 0402 0376 0349 0323 0297 5 6 o563 o536 050 0482 o0455 0428 0402 0375 0349 o323 0297 6 7 0563 0536 0508 o48 0454 0428 04o01 0375 0349 0323 0297 7 8 0562 o535 0508 o048 0454 0427 o4oi 0374 o348 0322 0296 8 9 o562 0535 0507 0480 0454 0427 040o 0374 o348 0322 0296 9 Io 0562 0534 0507 o48o o453 0426 0400 0374 0347 0321 0295 10 ii o56i o534 0507 048o0 0o453 0426 0399 0373 0347 o32 0295 II 12 o56r 0533 o5o6 0479 0452 0426 0399 0373 0346 0320 0294 12 13 0560 0533 o0506 0479 0452 0425 0399 0372 o346 0320 0294 13 14 o56o 0532 0505 0478 o45i 0425 0308 0372 0346 03/9 0294 14 15 o559 0532 0505 0478 o451 0424 0398 0371 o345 0319 0293 15 16 0559 o531 o504 0477 0450 0424 0397 0371 0345 0319 0293 i6 17 0558 o53i 0504 0477 0450 0423 0397 0370 0344 o318 0292 17 18 o558 o531 0503 0476 o450o'0423 0396 0370 o344 o318 0292, 18 19 0557 o530 0503 0476 0449 04322 0396 0370 0343 17 03291 19 20 0557 0530 0502 0475 0449 0422 0395 0369 0343 0317 0291 20 231 0557 0529 0502 0475 0448 0422 0395 0369 0342 o3i6 0291 21 22 o0556 0529 0502.0475 o448 0421 0395 0368 0342 o036 0290 22 23 0556 -0528 o5oi 0474 0447 0421 0394 o368 0342 0316 0290 23 24 o555 0528 o50o 0474 0447 0420 0394 0367' o34 I o3i5 0289 24 25 0555 0527 0500 0473 0446 0420 0393 0367 o34i o3i5 0289 25 26 0554 0527 0500 0473 o446 o419 0393 0366 0340 i o34 0288 26 27 o554 0526 0499 0472 0446 0419 0392 o366 o34o0 034 0288 27 28 0553 0526 0499 0472 445 04418 0392 0366 0339 o3i3 0288 28 029 0553 0526 0498 0471 o445 04i8 0392 o365 0339 o033 0287 29 30 0552 0525 0498 0471 o444 0418. 0391 0365 0339 0313 0287 30 31 0552 0525 0498 0471 I444 0417 0391 0364 0338 0312 0286 31 32, 0552 0524 0497 0470 0443 0417 0390 o364 o338 0312 0286 32 33 o551 o524 0497 0470 0443 o4i6 0390 o363 0337 o3ii 0285 33 34 0551 0523 o4Z96 0469 0442 o4i6 0389 o363 0337 0o311 /0285 34 35 0550 0523 0496" 0469 04 2 5 0o 89 0363 0336 0310 0285' 35 36 0550 0522 0495 0468 0442 o4i5 o388 0362 o336 o3io 0284 36 37 0549 0522 0495 o468 0441 o4i4 0388 0362 0336 0310 0284 O 37 38 0549 0521 0494 0467 0441 0414 0388 o36i -o335 0309 0283 38 39 o548 0521 0494 0467 0440 0414 0387 o361 0o335 030o 0283 39 40 0548 0521 0493 0467 0440 o413 0387 0360 0334 0308 0282 40 41 0547 0520 0493 0466 0439 0413 0386 036o o334 o3o8 0282 41 42 0547 0520 0493 0466 0439 0412 0386 0359 0333 0307 0282 42 43 o546 0519 0492 0465 0438 0412 0385 0359 o333 0307 0281 43 44 o546 0519 0492 0465 0438 041i o385 0359 o333 0307 0281 44 45 0546 0518 0491 0464 0438 o041"I o384 0358 0332 0306 0280 45 46 0545 o0518 o49 0464 0437 o410 0384 0358 o332 0306 028o 46 47 0545 0517 0490 0463 0437 o410 o384 0357 o33i 0305 0279 47 48 0544 0517 o49o 0463 0436 o04i o383 0357 o33 03o5 0279 48 49 o0544 0517 0489 0462 0436 0409 0383 0o56 o33o o304 0279 49 50 0543 0516 0489 0462 0435 0409 0382 0356 0330 0304 0278 50 5i 0543 o056 0489 0462 435 o0408 0382 o356 0329 o3o4 0278 51 52 o542 o055 0488 o46 0o434 0408 o38 0o355 0329 o3o3 0277 52 53 0542 0o55 0o488 0461 0434 0407 o381 0355 0329 0303 0277 53 54 o54i o5i4 0487 0460 0434 0407 o381 0354 0328 0302 0276 54 55 0541 0514 0487 0460 o433 o4o6 0380 0354 0328 0302 0276 55 56,0541 0o53 0486 0459 o0433 046 380 0353 0327 0301o 0276 56 57 o54o0 053 0486 0459 0432 0406 0379 o353 0327 030t 0275 57 58 o540 i o0512 o485 0458 0432 04o05 0379 0353 0326 o3oo 0275 58 59 0539 o0512 0485 o458 o43 040o5 378 o352 o326 o0300oo 0274 59 I "' 00 39' 20i40 241' 20 42' 20 43 20 44 20 45 20 46 20 47/ 2048 S. TABLE XXII. LPag 147 Proportional Logarithms. h in hm k m h m km n kmh m \?n km. kAIm m Akm s. o49' 250 & 251 20 52" 2 53 20 5 51 2556 2057 2058 20 59/ S. 0 0274 0248 0223 0197 0172 0147 0122 0098 0073 0049 0024 0 1 0273 0248 0222 0197 0172 0147 0122 0097 0073 0048 0024 1 3 0273 0247 02 o21 o0196 0171 oI 46 0121 o0096 0072 0047 0023 3 4 0272 0247 0221 0196 0171 oi46 021 oo0096 0071 0047 0023 4 5 0272 0246 0221 0I95 0170 i45 0120 0096 007 00oo46 0022 5 6 0271 0246 0220 0195 0170 o45 0120 0095 0071 oo0046 0022 6 7 02.7 0245 0220 0194 0169 oi44 O119 0095 0070 0046 0021 7 8 0270 0245 0219 o194 0o69 oi44 or119 0094 0070 0045 0021 8 9 0270 0244 02i9 0194 0169 oi43 0119 oo0094 oo0069 0045 002 9 10 0270 0244 0219 o0193 o68 o43 o8 oo0093 oo0069 0044 0020 10 II 0269 0244 0218 o0193 oJ68 oi43 oi18 0093 0068 0044 0020 II 12 0269 0243 0218 0192 0167 0142 0117 0093 oo68 0044 I0019 12 3 0268 0243 0217 0192 0167 0142 0117 0092 oo0068 oo43 oo019 3 14 0268 0242 0217 0192 oi66 o4 0117 0092 6067 oo0043 0019 4 15 0267 0242 0 o216 0191 o66 o4i o16 oo0091 0067 0042 oo i8 15 i6 0267 0241 0216 0191 oi0166 0141 oi6 0091 oo66 0042 00oo8 16 17 0267 0241 0216 0190 o0165 oi4o oi5 0091 0066 0042 0017 17 18 0266 o241 0215 o0190o o0165 oi40 oiI5 009o 0066 0oo4 0017 8 1i 0266 0240 025 0189 o64 0139 o4 ooo0090 oo65 oo0041 0017 9 20 0265 0240 024 o0189 o64 o0139 o4 oo0089 oo0065 0040 ooi6 20 21 0265 0239 0214 0189 oi63 0139 0114 0089 0064 0oo4o oo6 21 22 0264 0239 023 o0188 oi63 oi38 0113 0089 0064 0040 0oo5 22 23 0264 0238 0213 oi0188 o0163 oi38- oii3 oo88 0064 0039 ooI5 23 24 0264 0238 0213 0187 0162 o0137 0oI12 oo0088 oo63 oo0039 oo0015 24 25 0263 0238 0212 0187 0162 0137 0112 0087 oo63 oo38 0014 25 26 0263 0237 0212 0187 oi6i oi36 0112 0087 0062 oo0038 o004 26 27 0262 0237 0211 o086 o0161 o36 0111 0087 o0062 oo0038 oo003 27 28 0262 0236 0211 o86 0161 o36 o011 oo0086 0062 0037 0013 28 29 0261 0236 0211 o85 o0So oi35 oiio 0086 oo6i 0037 001oo2 29 30 0261 0235 0210 o85 o0160 0135 o011o 0085 oo0061 00oo36 001 3 31 026 0235 0210 o0184 o059 oi34 o011o 0085 0060 oo36 oo001 3 32 0260 0235 0209 oi84 0159 oi34 0109 0084 0060 oo36 00oo11 32 33 0260 0234 0209 oi84 Oi58 o34 0109 0084 oo6o 00oo35 ooii 33 34 0259 0234 0208 oi83 oi58 oi33 oio8 0084 oo0059 oo3 ooo 34 35 025-9 0233 0208 oi83 5oi8 oi33 1 oio8 oo83 o59 oo034 oio 35 36 0258 0233 02o8 0182 0157 o0132 0107 0083 oo0058 oo0034 ooio 36 37 0258 0233 0207 o82 o0157 o0132 0107 0082 0058 0034 ooo0009 37 38 0258 0232 0207 oi8i 0156 oi31 0107 0082 0057 oo0033 ooo0009 38 39 0257 0232 0206 o0181 o56 31 010oio6 oo0082 0057 oo0033 00oo8 39 40 0257 0231 0206 oi8i oI56 oi3i oi6 oo008 0057 00 oo32 00oo08 4e 4I 0256 0231 0205 oi80 oi55 o30o oio5 oo8 0056 0032 000oo8 4 42 0256 023o 0205S oi8o0 o55 oi3o oio5 oo8o 0056 oo3i 0007 42 43 0255 0230 0205 0179 0o54 0129 oo5 0080 00ooSS oo3 0007 43 44 255 0230 0204 0179 o0154 0129 oo4 oo0080 oo0055 00oo3 ooo0006 44 4,5 0255 0229 0204 0179 oi53 0129 0104 0079 0055 oo0030 oo0006 45 46 0254 0229 0203 0178 oi53 0128 o0o3 0079 oo54 0030 ooo6 1'46 47 0254 0228 0203 0178 0153 0128 oo0103 0078 0054 0029 oo0005 47 48 0253 0228 0202 0177 0152 0127 oo3 00oo78 0053.0029 oo0005 48 49 0253 0227 0202 0177 0152 0127 0102 0077 oo0053 0029 0004 49 50o 0252 0227 0202 0176 15I 0126 0102 0077 oo0053 0028 0oo4 50 51 0252 0227 0201 0176 o51 0126 0101 0077 0052 0028 oo0004 51 52 0252 1 0226 0201 0176 OI51 012.6 0101 0076 0052 0027 0003 52 53 0251 0226 0200 OI75 OI50 0123 0100 0076 0051 0027 0003 53 54 0251 0225 0200 oi75 o15o 0125 0100oIoo 0075.oo5i 0027 0002 54 55 0250 0225 0200 0174 0149 0124 01 0075 0051 0026 0002 55 56 0250 0224 0199 0174 0149 0124 0099 0075 0050 0026 0002 56 57 0250 0224 0199 0174 o48 0124 0099 0074 oo0050 0025 ooo 57 58 0249 0224 0198 0173 oi48 0123 oo98 0074 0049 0025 oooi 58 59 0249 0223 o0198 0173 oi48 0123 oo0098 0073 oo0049 0025 oooo 59 S. 2049' 20150/ 20~51' 2052 2`53 2054' 2055' 20561 f20571 2058' ~259' S. TABLES Page 1601 TABLE XXIV. Of Natural Sines. Prop.. 10 0 40 Prop. PIr^ ___ _ZL;iLr!sjO- ____________________ _____0_______ _____ _______30p_________ arts 29 N N.siie.N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 2 O o 0oo0 io00000 01745- 99985 034o0o 99939 05234 99863 06976 99756 60 2, oo02 oo o 100000 01774 9994 03519 99938 05263 99861 07005 99754 59 2 2 00058 0ooooo0 o08o3 99984 03548 99937 05292 99860 07034 99752 58 2 i 3 00087 1000looooo0 01oi832 99983 03577 99936 05321 99858 07063 99750 57 2 2 4 oo016 00ooooo 01862 99983 03606 99935 o535o 99857 07092 99748 56 2.2 i o)145 looooo 01891 99982 03635 99934 05379 99855 07121 99746 55 2 3 6 00175 0000ooooo 01920 99982 03664 99933 o54o8 99854 07150 99744 54 2 3 7o00204 soooool 01o49 99983 03693 99932 05437 99852 07179 99742 53 2 4 8 00233 100oooo 0178 99980 03723 99931 o55466 9985I 07208 99740 52 2 4 9 )00262 oooo100000 02007 99980 03752 99930 o5495 99849 07237 99738 51 5 1o 002 10ooooo00 02036 9979 03781 99929 05524 99847 07266 99736 0 2 5 i 00320o 99999 02065 99979 o38o10 9927 o5553 99846 07295 99734 49 2 6 12 00349 99999 02094 99978 03839 99926 05582 99844 07324 99731 48 2 6 13 0o378 99999 02123 99977 03868 99925 o561i 99842 07353 99729 47 2 714 00407 99999 02152 77 03897 99924 05640 99841 07382 99727 46 2 -7 5 oo436 99999 02181 99976 03926 99923 05669 99839 07411 99725 45 2 8 6 00oo465 9999 02211 99976 03955 99922 05698 99838 07440 99723 44 i 8 17 00495 99999 02240 99975 03984 99921 05727 99836 07469 99721 433 i 9 18 00524 99999 02269 99974 o0403 99919 05756 99834 07498 99719 42 1 9 19 o00553 99998 02298 99974 04042 99918 05785 99833 07527 99716 41 i io 20 00582 99998 02327'99973 o4071 199917 o5814 9983r 07556 99714 40 1 io 21 oo0611 99998 02356 99972 041oo 99916 05844 99829 07585 99712 39 1 11 22 0064o 99998 02385 99972 04129 999I5 05873 99827 07614 99710 38 i II 23 oo669 99998 02414 99971 04,59 9993 05902o 99826 07643 99708 37 12 24 00698 99998 02443 99970 04188 99912 05931 99824 07672 99705 36 i 12 25 00727 99997 02472 99969 04217 999I1 05960 99822 07701 99703 35 1 13 26 00756 99997 02501 99969 o4246 9991 05989 9982I 07730 99701 34 i 13 27'00785 99997 02530 99968 04275 99909 o6o018 99819 07759 99699 33 i 14. 28 oo0081 99997 02560 99967 o4304 99907 06047 99817 07788 99696 32 1 r4 29 00844 99996 025.89 99966 04333 999o0 06076 99815 07817 99694 31 i 15 30 00873 99996 02618 99966 04362 99905 o61o5 99813 07846 99692 30 o iS 31 00902 99996 02647 99965 0439i 999o4 0o634 99812 07875 99689 29 1 15 32 o00o93 99996 o02676 99964 64420 99902 o0663 99810 07904 99687 28 i i6 33 oo00960 99995 02705 99963 04449 9990o 06192 99808 07933 99685 27 1 6 34 00oo989 999951 02734 99963 04478 9990o 06221 99806 07962 99683 26 i 17 35 oio18 99995 02763 99962 04507 99898 06250 99804 07991 99680 25 i 17 36 0o047 99995 02792 99961 04536 99897 o6279 99803 08020 99678 24 1 18 37 01076 99994 02821 99960 o4565 99896 o63o8 99801 o38o49 99l76 3 i 18 }38 oio5 99994 02850 99959 04594 99894 06337 99799 o8078 996'-3 22 i 19 39 ois34 99994 02879 99959 04623 99893 o6366 99797 08107 99671 21 1 9 4o o01164 999931 02908 99958 04653 99892 06395 99795 o8i36 99668 20 i 20 41 01193 99993 02938 99957 04682 99890 06424 99793 o8165- 99666 19 i 20 42 01222 99993 02967 99956 04711 99889 o6453 99792 08194 99664 18 i 21 43 01251 99992 02996 99955 04740 99888 06482 99790 o8223 99661 17 I 21 44 01280 99992 03025 99954 04769 99886 0651i 99788 o8252 99659 i6 I 22 45 01309 99991 o3o54 99953 04798 99885 o654o 99786 o8281 99657 i5 i 22 46 o0338 9999i o3o83 99952 04827 99883 06569 99784 o83io 99654 14 0 23 47 01367 99991 03112 99952 04856 99882 o6598 99782 08339 99652 13 o 23 48 01396 999901 o341 9995I o4885 9988I 06627 99780 08368 99649 i2 i 24 49 01425 99990 03170 99950 04914 99879 o6656 99778 08397 99647 II J 24 50 01o454 99989 0o319999949 04943 99878 o6685 99776 o8426 99644 io o 25 51 oi483 99989 3228 99948 04972 99876 06714 99774 o08455 99642 9 0 25 52 o0153 99989 03257 99947 o5oo00 99875 06743 99772 o8484 99639 8 0 26 53 o01542 99988 03286 99946 o5o3o 99873 06773 99770 o85i3 99637 7 0 26 54 01571 99988 o33s6 99945 1 0559 99872 06802 99768 08542 99635 6 o 27 55 or600oo 99987 o335 99944 o5o88 99870 o683o 99766 08571 99632 5 f 27 56 01629 99987 03374 99943 o5117 99869 o686o 99764 o86oo 99630 4 o 28 57 oi658 99986 o34o03 99942 0o546 99867 6889 99762 08629 99627 3 o 28 58 o01687 99986 03432 9994 05i75 99866 06918 99760 o08658 99625 2 29 59 01716 99985 o3461 99940 05205 99864 06947 99758 08687 99622 i o 29 6 01745 99985 03490 99939 05234 99863 06976 99756 08716 99619 o 0 N. cos. N. sine. N. cos.N. sine. N. cos. N. sine. N. cosIN. i n1. N.ncos. -N. sine. M- __ 890 880 870 860 850 TABLE XXIV. [Page 161 Of Natural Sines. Prop. 50 6i 70 0 I Prop' plrLS 60* 70 v pairtg 29 A I N. sine. N..os. N. sine. N. cos. N. sine. N. cos. N. sine. N, cos. N. sine. N. cos. 4 c 08716 99619 10453 99452 12187 9255 13917 99027 15643 98769 60 4 0 T 08745 99617 10482 99449 12216 99251 13946 99023 15672 98764 59 4 1 2 08774 99614 10511 99446 12245 99248 13975 99019 15701 98760 58 4 [ 3 o88o3 99612 10540( 99443 12274 99244 14004 99015 15730 98755 57 4 2 4 o8831i 996o 10569 99440 12302 99240 14o33 9oi0i 15758 98751 56 4 2 5 o886o 99607 10597 99437 12331 99237 14o6i 99006 15787 98746 55 4 3 6 o8889 996o04 10626 99434 12360 99233 14090 99002 i5816 98741 54 __4 7 7 (08918 99602 10655 99431 12389 99230 14119 98998 T5845 98737 53 4 4 8 o8947 9y599 1o684 99428 12418 99226 14148 98994 15873 98732 52 3 4 9 08/76 99596 1(713 99424 12447 99222 14177 98990 15902 98728 5i 3 5 10 09005 99594 10742 99421 12476 99219 14205 98986 15931 98723 5o 3 5 i1 09034 99591 10771 99418 12504 99215 14234 98982 15959 98718 49 3 6 12 09063 99588 10800 99415 12533 99211 14263 98978 15988 98714 48 3 6 13 09002 99586 io899 99412 12562 99208 14292 98973 16017 98709 47 3 7 14 09121 99583 1o858 99409 12591 99204 14320 98969 16o46 98704 46 3 7 135 (9130 99580 10887 99406 12620 99200 14349 98965 16074 98700 45 3 8 i6 09179 99578 10916 99402 12649 99197 14378 98961 i6io3 98695 44 3 8 17 09208 99575 io945 99399 12678 99193 14407 98957 16132 98690 43 3.9 18 09237 99572 10973 99396 12706 99189 i4436 98953 i6i6o 98686 42 3 9 19 09266 99570 11002 99393 12735 99186 14464 98948' 16189 98681 41 31 10 20( 00295 99567 11031 99390 12764 99182 14493 98944 16218 98676 4o 3 10 21 09324 99564 iio6o 99386 12793 99178 14522 98940 16246 98671 39 3 Ii 22 09353 99562 11089 99383 12822 99175 14551 98936 16275 98667 38 3 ii 23 09382 99559 1iii8 99380 12851 99171 I4580 98931 163o4 98662 37 2 12 24 09411 99556 11147 99377 12880 99167 146o8 98927 i6333 98657 36 2 12 25 0944o 99553 11 176 99374 12908 99163 14637 98923 1636i 98652 35 2 13 26 09460 99551 11205 99370 12937 99160 i4606 98919 -16390 98648 34 13 2.7 09498 99548 11234 99367 12966 99156 14695 98914 16419 98643 33 2 14 28 09527 99545 11263 99364 12995 99152 14723 98910 16447 98638 32 2; 14 29 o9556 99542 11291 99360 13024 99148 14752 98906 16476 98633 31 2 i5 3o o9585 99540 11320.99357 i3o53 99144 14781 98902 16505 98629 30 2 i5 31 09614 99537 11349 99354 i3o8i 9914i i48io 98897 i6533 98624" 29 15 32 09642 99534 11378 99354 13139 99137 14838 98893 16562 98614 27 ~ 163 3 09671 99534 11407 993457 1313 99137 74838 98889 16569 98614 27 2 6' 34 09700 99528 1i436 99344 i3i68 99129 14896 98884 16620 98609 26 2 17 35 09729 99526 ii465 99341 13197 99125 14925 98880 16648 98604 25 2 17 36 09758 99523 11494 99337 13226 99122 1 -4954 98876 16677 98600 24 2 18 37 09787 99520 11523 993 4" 13254 99118 14982 98871 16706 98595 23 2 i8 38 09816 99517 11552 99331 13283 99114 1i5oi 98867 16734 98590 22 1 19 39 09845 99514 ii58o 99327 13312 99110 15o4o 98863 16763 98585 21 1 19 40 09874 99511 11609 99324 13341 99106 15069 98858 16792 98580 20 1 20 41 09903 99508 1i638 99320 13370 99102. 15097 98854 16820 98575 19 1 20 42 09932 99506 11667 99317 13399 99098 15126 98849 168409 98570 i8 1 21. 43 09961 995o3 11696 99314 13427 99094 15i55 98845 16878 98565 17 i 21 44 09990 99500 11725 99310 i3456 99091 i5i84 98841 169(06 98561 i6 I 22 45 10019 99497 11754 99307 i3485 99087 15212 98836 16935 98556 I5 i 22 46 ioo48 99494 11783 99303 i3514 99083 15241 98832 16964 98551 14 I 23 47 10077 99491 11812 99300 i3543 99079 15270 98827 16992 98546 13 i 23 48 ioio6 9488 1184o 99297 13572 99075 15299 98823 17021 98541 12 I 24 49 ioi35 99485 51869 99293 i36oo 99071 15327 98818 17050 98536 II 1 24 50 ioi64 99482 11898 99290 13629 99067 15356 98814 17078 98531 1 1o 25 Si 10192 99479 11927 99286 13658 99063 i5385 98809 17107 98526 9 i 25 52 10221 99476 11956 99283 13687 99059 i5414 98805- 17136 98521 8 1 26 53 10250 99473 11985 99279 13716 99055 15442 98800 17164 98516 7 0 26 54 10279 99470 12014 99276 13744 99051 15471 98796 17193 98511 6 o 27 55 10308 99467 12043 99272 13773 99047 155oo 98791 17222 98506 5 o 27 56 10337 99146 12071 99269 i13802 99043 15529 98787 1725o 985oi 4 o 28 57 io366 99461 12100 99265 i383i 99039 15557 98782 17279 98496 3 o 28 58 io395 99458 12i2? 99262 1386o 99o35 i5586 98778 17308 98491 2 0 29 59 10424 99455 12158 99258 13889 99031 i56i5 98773 17336 98486 i o 29 60 104I53 994f52 12187 99255 13917 99027 i5643 98769 17365 98481 o o N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. M 84 830 820 81~ 80" III Page 16] TABLE XXIV. Of Natural Sines. op 100 110 120 130 140 Prop. parts I__ _1 _: - parts 28 li N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 6 0 o 17365 98481 19081 98163 20791 978I5 22495 97437 24192 97030 60 6 0 T 17393 98476 9109 98I57 20820 97809 22523 97430 24220 97023 59 6 I 2 17422 98471 19138 98152^ 20848 97803 22552 97424 24249 97015 58 6.1 3 17451 98466 1967 98146 20877 97797 22580 97417 24277 97008 57 6 2 4 I7479 98461 I 995 98140 209o5 97791 22608 97411 24305 97001 56 6 2 5 17508 98455 19224 98135 20933 97784 22637 97404 24333 96994 55 6 3 6 17537 98450 19252 98129 20962 97778 22665 97398 24362 96987 54 5 3 7 17565 98445 19281 98124 20990 97772 22693 9739I 24390 96980 53 5 4 8 I7594 98440 I9309 98118 21019 97766 22722 97384 24418 96973 52 5 4 9 17623 98435 19338 98112 21047 97760 22750 97378 24446 96966 5r 5 5 10 17651 98430 19366 98107 21076 97754 22778 97371 24474 96959 50 5 5 1 17680 98425 I9395 98o10 21104 97748 22807 97365 24503 96952 49 5 6 12 17708 98420 19423 98096 21132 97742 22835 97358 24531 96945 48 5 16 13 17737 98414 19452. 98090 21161 97735 22863 97351 24559 96937 47 5 7 14 17766 98409 I9481 98084 21189 97729 22892 97345 24587 96930 46 5 7 5 17794 98404 19509 98079 21218 97723 22920 97338 246I5 96923 45 5 7 16 17823 98399 19538 98073 21246 97717 22948 97331 24644 969I6 44 4 8 I7 17852 98394 19566 98067 21275 97711 22977 97325 24672 96909 43 4 8 I8 17880 98389 19595 98061 21303 97705 23005 97318 24700 96902 4 4 9 19 17909 98383 19623 98056 21331 97698 23033 97311 24728 96894 41 4 9 20 179383 78 962 98378 965 9805 2360 97692 23062 97304 24756 96887 40 4 I 21 17966 98373 19680 98044 21388 97686 23090 97298 24784 96880 39 4 o 22 17995 98368 19709 98039 21417 97680 23118 97291 24813 96873 38 4 Ir 23 I8023 98362 19737 98033 21445 97673 23146 97284 24841 96866 37 4 ii 24 18052 98357 19766 98027 2I474 97667 23175 97278 24869 96858 36 4 12 25 I808i 98352 I9794 98021 21502 97661 23203 97271 24897 96851 35 4 12 26 18109 98347 183983 986 21530 97655 23231 97264 24925 96844 34 3 13 27 i8i38 98341 19851 98o010 2559 97648 23260 97257 24954 96837 33 3 13 28 18166 98336 I9880 98004 21587 97642 23288 97251 24982 96829 32 3 14 29 18195 98331 19908 97998 21616 97636 23316 97244 25010 96822 31 3 14 30 18224 98325 I9937 97992 21644 97630 23345 97237 25038 96815 30 3 14 31 18252 98320 19965 97987 21672 97623 23373 97230 25066 96807 29 3 I5 32 18281 98315 I9994 97981 21701 976I7 23401 97223 25094 96800 28 3 15 33 I8309 98310 20022 97975 2I729 97611 23429 97217 25122 96793 27 3 6 34 i8338 98304 20051 97969 21758 97604 23458 97210 25151 96786 26 3 16 35 18367 98299 20079 97963 21786 97598 23486 97203 25179 96778 25 3 17 36 18395 98294 20108 97958 21814 97592 23514 97196 25207 96771 24 2 I7 37 18424 98288 20136 97952 21843 97585 23542 97189 25235 96764 23 2 18 38 I8452 98283 20165 97946 21871 97579 23571 97182 25263 96756 22 2 i8 39 i848I 98277 20193 97940 21899 97573 23599 97I76 2529I 96749 21 29 19 40 18509 98272 20222 97934 21928 97566 23627 97169 25320 96742 20 2 19 41 18538 98267 20250 97928 21956 97560 23656 97162 25348 96734 g19 2 20 42 18567 98261 20279 97922, 21985 97553 23684 97155 25376 96727 i8 2 20 43 18595 98256 20307 97916 220I3 97547 23712 9748 25404 96719 17 2 21 44 18624 98250 20336 97910 22041 97541 23740 9714I 25432 96712 16 2 21 45 18652 98245 20364 97905 22070 97534 23769 97134 25460 96705 15 2 21 46 868I1 98240 20393 97899 22098 97528 23797 9727 2548 96697 14 22 47 18710 98234 20421 97893 22I26 97521 23825 97120 25516 96690 13 I 22 48 18738 98229 20450 97887 22155 97515 23853 97113 25545 96682 12 I 23 49 18767 98223 20478 9788I 22I83 97508 23882 97106 25573 96675 11 I 23 50 18795 98218 20507 97875 22212 97502 23910 97100 25601 96667 10 I 24i 51 18824 98212 20535 97869 22240 97496 23938 97093 25629 96660 9 I t J 52 18852 98207 20563 97863 22268 97489 23966 97086 25657 96653 8 I 25 53 I888i 98201 20592 97857 22297 97483 23995 97079 25685 96645 7 1 25. 54 18910 98196 20620 97851 22325 97476 24023 97072 25713 96638 6 I | 26 55 18938 98I90 20649 97845 22353 97470 24051 97065 25741 96630 5 I 26 56 18967 98I85 20677 97839 2238297463 24079 97058 25769 96623 4 27 57 18995 98179 20706 97833 22410 97457 24108 97051 25798 966i5 3 o 27 58 19024 98I74 20734 97827 22438 97450 24I36 97044 25826 96608 2 0o 28 59 19052 98168 20763 97821 22467 97444 24I64 97037 25854 966)o I o 28 60 19081 98163 20791 978i5 22495 97437 24I92 97030 25881 96593 o o N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. l 790 780 770 76~ 75~ TABLE XXIV. [Page 163 Of Natural Sines. Prop. 15o 16~ 170 180o 19~ rro. parts alp 27 M N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 9 25882 96593 27564 96126 29237 95630 30902 95106 3255794552 60 9 i 259IO 96585 27592 96118 29265 95622 3092995097 32584 94542 5 9 1 2 2 9538 96578 27620 96110 29293 95613 30957 95088 32612 94533 58 9 3 25966 96570 27648 96102 29321 95605 30985 95079 32639 94523 57 9 2 4 25994 96562 27676 96094 29348 95596 31012 95070 32667 94514 56 8 2 5 26022 96555 27704 96086 29376 95588 31o04 95061 32694 94504 55 8 3 6 26050 96547 27731 96078 29404 95579 3io68 95052 32722 94495 54 8 3 7 26079 96540 27759 96070 29432 95571 31095 95043 32749 94485 53 & 4 8 26107' 96532 27787 96062 29460 95562 31123 95033 32777 94476 52 8 4 9 26135 96524 27815 96054 29487 95554 3I51i 95024 32804 94466 5i 8,5 20 26163 96517 27843 96046 29515 95545 31178 95oi5 32832 94457 50 8 5 1J 26191 96509 27871 96037 29543 95536 31206 95006 32859 94447 49 7 5 12 26219 96502 27899 96029 29571 95528 31233 94997 32887 94438 48 7 -6 J3 26247 96494 27927 96021 29599 95519 3126 94988 32914 9428 47 7 6 14 26275 96486 27955 96013 29626 95511 31289 94979 32942 94418 46 7 7 15 26303 96479 27983 96005 29654 95502 3i3i6 94970 32969 94409 45 7 7 16 26331 96471 28011 95997 29682 95493 3i344 94961 32997 94399 44 7 8 17 26359 96463 28039 95989 29710 95485 31372 94952 33024 94390 43 6 8 18 26387 96456 28067 95981 29737 95476 31399 94943 33o51 94380 42 9 19 26415 96448 28095 95972 29765 95467 31427 94933 33079 94370 4o 6 9 20 26443 96440 28123 95964 29793 95459 31454 94924 331o6 94361 4o 6 9 21 26471 96433 28150 95956 29821 9545o 31482 94915 33i34 9435 39 6 10 22 26500oo96425 28178 95948 29849 95441 31510 94906 33161 94342 38 6 1o 23 26528 96417 28206 95940 29876 95433 31537 94897 33189 94332 37 6 11 24 26556 96410 28234 95931 29904 95424 3i565 94888 33216 94322 36 5 II 25 26584 96402 28262 95923 29932 95415 31593 94878 33244 94313 35 5 12 26 26612 96394 28290 95915 29960 95407 31620 94869 33271 94303 34 5 12 27 26640' 96386 28318 95907 29987 95398 31648 94860 33298 94293 33 5 13 28 26668 96379 28346 95898 300oo5 95389 31675 94851 33326 94284 32 5 1.3 29 26696 96371 28374 95890 30043 95380 31703 94842 33353 9427 3 5 14 30 26724 96363 28402 95882 30071 95372 31730 94832 33381 94264 30 5 — 14 - 3 2-6752 96355 28429 95874 30098 95363 31758 94823 334o8 94254 29" 4 14 32 26780 96347 28457 95865 30126 95354 31786 94814 33436 94245 28 4 i5 33 26808 96340 28485 95857 3o'54 95345 31813 94805 33463 94235 27 4 15 34 26836 96332 28513 95849 30o82 95337 31841 94795 33490 94225 26 4 i6 35 26864 96324 28541 95841 30209 95328 3i868 94786 335i8 94215 25 4 i6 36 26892 96316 28569 95832 30237 95319 31896 94777 33545 94206 24 4 17 37 26920 96308 28597 95824 30265 95310 31923 94768 33573 94196 23 3 17 38 26948 96301 28625 95816 30292 95301 31951 94758 336oo 94186 22 3 i8 39 26976 96293 28652 95807 3o32o 95293 31979 94749 33627 94176 21 3 i8 40 27004 96285 28680 95799 30348 95284 32006 94740 33655 94167 20 3 I8 41 27032 96277 2(8708 95791 30376 95275 32034 94730 33682 94157 19 3 19 42 27060 96269 28736 95782 30403 95266 32061 94721 33710 94147 i8 3 19 43 27088 96261 28764 95774 3o431 95257 32089 94712 33737 94137 17 3 20 44 27116 96253 28792 95766 30459 95248 32116 94I02 33764 94127 16 2 20 45 27144 96246 28820 95757 30486 95240 32144 94693 33792 94118 15 2 21 46 27172 96238 28847 95749 305r4 95231 32171 94684 33819 94108 14 2 21 47 27200 9623o 28875 95740 3o542 195222 32199 94674 33.846 94o98 13 2 22 48 27228 96222 28903 95732 30570 95213 32227 94665 33874 94088 12 2 22 49 27256 96214 28931 9572, 4 30597 95204 32254 94656 33901 94078 II 2.3 50- 27284 96206 28959 957i5 30625 95195 32282 94646 33929 94068 10 2 23 51 27312 96198 28987 95707 3o653 95186 32309 94637 33956 94058 9 1 23 52 27340 96:9o 29o15 95698 3068o 95177 32337 94627 33983 94049 8 1 24 53 27368 96182 29042 95690 30708 95168 32364 94618 34oI 94039 7 1 24 54 27396 96174 29070 9568i 30736 95159 32392 94609 34o38 94029 6 i 25 55 27424 96166 29098 95673 30763 95150 32419 94599 34o65 94019 5 25 56 27452 96158 29126 95664 30791 95142 32447 94590 34093 94009 4 1 26 57 27480 96150 29154 95656 30819 95133 32474 94580 34120 93999 3 o 26 58 27508 96142 29182 95647 30846 95124 32502 94571 34147 939891. o 27 59 27536 96134 29209 95639 30874 95115 32529 94561 34175 93979 i 0 27 60 27564 96126 29237 95630 30902 95106 32557 94552 34202 93969 o o N. ens. N. 1i1e. N. cos. N. sine. N. Nco.iN. sine. N. co.. sie. N. cos.,N. sine.- 740 73 72~ 71~ 700 Ptge 164]1 TABLE XXIV. Of Natural Sines. Prop. _ 20" 2 ____ __ Prop. 27 mli N. sine. N. cos. N. sine. N. cos. N. sine. N. c os. N. sine. N. cos. N. si cos. 11 0 o 34202 93969 35837 93358 37461 92718 39073 92050 40674 19355 6o ii T 34229 93959 35864 93348 37488 92707 3100oo 9203 40700 91343 59 Ii i 2 34257 93949 35891 93337 375r5192697 39127 92028 40727 91331 58 ii 1 3 34284 93939 35918 93327 37542 92686 39153 92016 40753 91319 57 10 2 4 3431i 93929 35945 93316 37569 92675 39080 92005 40780 91307 56 io 2 5 34339 93919 35973 93306 37595 92664 39207 91994 40806 91295 55 io 3 6 34366 93909 36oo000 93295 37622 92653 39234 91982 4o833 91283 54 jo 3 7 34393 93899 36027 93285 37649 92642 39260 91971 40860 91272 53 i 4 8 34421 936 89 36o54 93274 37676 92631 39287 91959 40886 91260 52 i0 44 9 34448 93879 36o08o 93264 37703 92620 39314 91948 40913 91248 5i 9 5 io 34475 93869 36io8 93253 37730 92609 39341 91936 4o0939 91236 5S 9 5 ii 34503 93859 36i35 93243 37757 92598 39367 91925 40966 91224 49 9 5 12 34530 93849 36162 93232 37784 92587 39394 91914 40992 91212 48 9 6 13 34557 93839 36190 93222 37811 92576 39421 91902 41019 91200 47 9 6 14 34584 93829 36217 93211 37838 92565 39448 91891 41045 91188 46 8 7 i5 346T2 93819 36244 93201 37865 92554 39474 91879 41072 91176 45 8 7 i6 34639 93809 36271 9.3190 37892 92543 39501 91868 41098 91164 44 8 8 17 34666 93799 36298 93180 37919 92532 39528 91856 41125 91152 43 8 8 i8 34694 93789 36325 93169 37946 92521 39555 91845 4i15i 91140 42. 8 9 19 34721 93779 36352 93159 37973 92510 39581 91833 41178 91128 41 8 9 20 34748 93769 36379 93148 37999 92499 39608 91822 41204 91116 40 7 9 2i 34775 93759 364o6 93137 38026 92488 39635 91810 41231 91104 39 7 io 22 348o3 93748 36434 93127 38053 92477 39661 91799 41257 91092 38 7 10 23 3483o 93738 3646i1 93116 38080o 92466 39688 91787 41284 91080 37 7 11 24 34857 93728' 36488 93106 38107 92455 39715 91775 4i3Jo 91068 36 7 ii 25 34884 93718 36 519 3095 38i34 92444 39741 91764 41337 91056 35 6 12 26 34912 93708 36542 93084 38i6i 92432 39768 91752 41363 91044 34 6 12 27 34939 93698 36569 93074 38188 92421 39795 91741 41390 91032 33 6 13 28 34966 93688 36596 93063 38215 92410 39822 91729 41416 91020 32 6 13 29 34993 93677 36623 93052 38241 92399 39848 91718 4i443 91008 31 6 14 30 35021 93667 36650o 93042 38268 92388 39875 91706 41469 90996 30 6 14 3i 35o48 93657 36677 93031 38295 92377 39902 91694 41496 90984 29 5 14 32 35075 93647 36704 93020 38322 92366 39928 91683 41522 90972 28 5 15 33 35bO2 93637 367311 9301 38349 92355 39955 91671 4i549 90960 27 5 15 34 3530o 93626 36758 92999 38376 92343 39982 91660 41575 90948 26 5 i6 35 35157 93616 36785 92988 384o3 92332 4ooo8 648 41602 90936 25 5 16 36 35184 93606 36812 92978 38430o 92321 4oo0035 91636 41628 90924 24 4 17 37 35211 93596 36839 92967 38456 92310 40062 91625 4i655 90911 23 4 17 38 35239 93585 36867 92956 38483 92299 400oo88 91613 41681 90899 22 4 i8. 39 35266 93575 36894 92945 385io 92287 4oii5 91601 41707 90887 21 4 18 4o 35293 93565 36921 92935 38537 92276 40oi4 91590 41734 90875 20 4 i8 41 35320 93555 36948 92924 38564 92265 40o68 915.78 4760 90863 19 3 19 42 35347 93544 36975 92913 38591 92254 40195 91566 41787 90851 r8 3 19 43 35375 93534 37002 92902 38617 92243 40221 97555 41813 19039 17 3 20 44 35402 93524 37029 92892 38644 92231 40248 91543 4184o 90826 i6 3 20 45 35429 93514 37056 92881 38671 92220 40275 91531 4i866 90814 15 3 21 46 35456 93503 37083 92870 38698 92209 4030oi 91519 41892 90802 4 3 21 47 35484 1 93493 37110 928S9 38725 92198 40328 9i508 41919 90790 13 2 22 48 35511 93483 37137 92849 38752 92186 40355 91496 4194 90778 12 2 I22 49 35538 93472 37164 92838 38778 92175 40381 91484 41972 90766 11 2 23 50 3556.5 93462 37191 92827 38805 92164 4o4o8 91472 41998 90753 10 2 23 51 35592 93452 37218 92816 38832 92152 4o434 91461 42024 90741 9 2 23 52 35619 93441 37245 92805 38859 92141 4046i 91449 42051 90729 8 i 24 53 35647 93431 37272 92794 38886 92130 40488 91437 42077 90717 7 1 ~2.4 54 35674 93420 37299 92784 38912 92119 405i4 91425 42104 90704 6 25 7 OT 94o 37326 92773 383 939 92107 4o054i 91414 4213o 90692 5 i 25 56 35728 93400 37353 92762 38966 92096 40567 91402 42156 90680 4 I 26 57 35755 93389 37380 92751 38993 92o85 40594 91390 42183 90668 3 i 26 58 35782 93379 37407 92740 39020 92073 40621 91378 42209 90655 2 o 27 59 3581o 93368 37434 92729 39046 92062 40647 1966 42235 90o643 i o 27 60 35837 93358 37461 92718 39073 92050 40674 91355 42262 90631 0o' N. cos. N. sine. N. Ncos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. M 690 680 670 66~ 650 TABLE XXIV. [Page 165 Of Natural Sines. parts 250 260 270 230 290 plp 26 M N. sine. N. s os. N. sine. N. cos. N. sine. N. cos. N. sine. cos. N. sie. N. cos. 4 o o 42262 90631 43837 89879 45399 89101 46947 88295 48481 87462 60 14 o i 42288 90618 43863 89867 45425 89087 46973 88281 48506 87448 59 14 1 2 42315 90606 43889 89854 4545 89074 46999 88267 48532 87434 58 14 3 42341 90594 43916 89841 45i77 89061 47024 88254 48557 87420 57 2 4 42367 90582 43942 89828 45503 89048 47050 88240 48583 87406 56 13 2 5 42394 90569 43968 89816 45520 89035 47076 88226 48608 87391 55 i3 3 6 42420 90557 43994 89803 45554 89021 47101 88213 48634 87377 54 13 3 7 42446 90545 44020 89790 4558o 89008 47127 88199 48659 87363 53 12 3 8 42473 90532 44046 89777 45606 88995 47153 88185 48684 87349 5 12 4 9 42499 90520 44072 89764 45632 88981 47178 88172 48710 87335 51 12 4:o 42525 90507 44098 89752 45658 88968 47204 88158 48735 87321 50 12 5 ii 42552 90495 44124 89739 45684 88955 47229 88144 48761 87306 49 Ii 5 12 42578 90483 44151 89726 45710 88942 47255 8813o 48786 87292 48 ii 6 13 42604 90470 44177 89713 45736 88928 47281 88117 488 1187278 47 11 6 I4 42631 90458 44203 89700 45762 88915 47306 880o3 48837 87264 46 i 7 15 42657 90446 44229 89687 45787 88902 47332 88o89 48862 87250 45 ii 7 i6 42683 90433 44255 89674 45813 88888 47358 88075 48888 87235 44 i1 7 17 42709 90421 44281 89662 45839 88875 47383 88062 48913 87221 43 io 8 I8 42736 90408 44307 89649 45865 88862 47409 88o48 48938 87207 42 10 8 i9 42762 90396 44333 89636 45891 88848 4743Z4 88o34/ 48964 87193 41 1o 9 20 42788 90383 44359 89623 45917 88835 47460 88020 48989 87178 40 9 9 21 42815 90371 44385 89610 45942 88822 47486 88oo6 4904, 87164 39 9 10 22 42841 90358 444r 189597 45968 888o8 47511 87993 4904o 87150 38 9 i0 23 42867 90346 44437 89584 45994 88795 47537 87979 49065 87136 37 9 io 24 42894 90334 44464 89571 46020 88782 47562 87965 49090 87121 36 8 11 25 42920 90321 44490 89558 46046 88768 47588 8795 49116 87107 35 8 11 26 42946 90309 44516 89545 46072 88755 47614 87937 49141 87093 34 18 12 27 42972 90296 44542 89532 46097 88741 47639 87923 4966 87079 33 8 12 28 42999 90284 44568 89519 46123 88728 47665 87909 49192 87064 32 7 13 29 43025 90271 44594 89506 46149 88715 47690 87896 49217 870o5 31 7 13 30 43o51 90259 44620 89493 46175 88701 47716 87882 49242 87036 3 7 13 31 43077 90246 44646 89480 46201 88688' 47741 87868 49268 87021 2 7 14 32 43104 90233 44672 89467 46226 88674 47767 87854 49293 87007 28 7 14 33 4313o 90221 44698 89454 46252 88661 47793 87840 49318 86993 277 6 15 34 43156 90208 44724 89441 46278 88647 478I8 87826 49344 86978 26 6i 15 35 43182 90196 44750 89428 46304 88634 47844 87812 49369 86964 25 C i6 36 43209 90o83 44776 89415.4633o 88620 47869 87798 49394 86949 24 6 16 37 43235190171 44802 89402 46355 88607 47895 87784 49419 86935 23 5 i6 38 43261 90158 44828 89389 46381 88593 47920 87770 49445 86921 22 5 17 39 43287 90146 44854 89376 46407 8858o 47946 87756 4947 86906 21r 5 I7 40 43313 90133 4488o 89363 46433 88566 47971 87743 49495 86892 20 5 18 41 43340 90120 44906 89350 46458 88553 47997 87729 49521 86878 ig 4 18 42 43366 90108 44932 89337 46484 88539 48022 87715 49546 86863 i 1 4 19 43 43392 90095 44958 89324 465io1 88526 48048 87701 49571 86849 7 ^ 19 44 4348 90oo82 44984 89311 46536 88512 48073 87687 49596 86834 i6 4 20 45 43445 90070 45010o 89298 46561 88499 48099 87673 49622 86820 i5 4 20 46 43471 90057 45o36 89285 46587 88485 48124 87659 49647 86805 14 3 20 47 43497 90045 45062 89272 46613 88472 481.5o 87645 49672 86791 3[ 3 21 48 43523 90032 45o88 89259 46639 88458 48175 87631 49697 86777 12 3 22 51 43602 89994 45166 89219 46716 88417 48252 87589 49773 86733 / 2 23 52 43628 89981 45192 89206 46742 88404 48277 87575 49798 86759 8 2 23 53 43654 89968 45218 89193 46767 88390 48303 87561 49824 86704 7 2 23 54 4368o 89956 45243 89180 46793 88377 48328 87546 49849 86690 6 2455 43706 89943 45269 8967 468 88363 48354 875 32 49874 86675 5 24 56 43733 89930 4529 89153 46844 88349 48379 87518 49899 8666i 4 25 57 43759 89918 45321 89140 46870 88336 48405 87504 49924 86646 3 i 25 58 43785 89905 45347 89127 46896 88322 4843o 87490 49950 86632 2 o 26 59 438ii 89892 45373 89114 46921 88308 48456 87476 49975 86617 1 o 26 60 43837 89879 45399 89o101 46947 88295 4848 87462 500oooo00 8660o3 o N N. cos. N. sine. N. cos.. sine. N. cos. N. sine. N. cos. N. sine. N. cos.N. sine. AM 64~ 630 662 61~ 600 Pa^e 166] TABLE XXIV. Of Natural Sines. rop. 300 310 32o 330 340 Prop. p_ r parts 2 MI N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 16 o- 0 o 5oooo 866o3 51504 857I7 52992 848o5 54464 83867 55949 82904 6o 1 6 o 502.5 86588 51529 85702 53017 84789 54488 8385i 55943 82887 59 i i 2 5oo5o 86573 51554 85687 53o41 84774 54513 83835 55968 82871 58 I 5 i 3 50076 86559 51579 85672 53o66 84759 54537 83819 55992 82855 57,5 2 4 e5010 86544 5i6o4 85657 53091 84743 54561 838o4 56oi6 82839 56 i5 5 50126 8653o 51628 85642 53rI5 84728 54586 83788 56040 82822 55 i5 3 6 50o51 86515 5'653 85627 5314o 84712 5460o 83772 56o64 828o6 54 3 7 50176 865oi 51678 85612 53164 84697 54635 83756 56o88 82790 53 I4 3 8 5020I 86486 517o3 85597 53189 84681 54659 83740 56112 82773 52 14 4 9 50227 8647 51728 85582 53214 84666 54683 83724 56136 82757 51 14,4 50252 86457 51753 85567 53238 84650 54708 83708 56160 82741 50 13 5 i 50277 86442 51778 85551 53263 84635 54732 83692 56i84 82724 49 13 12 50302 86427 5i8o3 85536 53288 84619 54756 83676 56208 82708 48 13 "5"13 i50327 "86413 5 1828 85521 53312 84604 547^183660 56232 82692 47 13 i 14 50352 86398 5i852 85506 53337 84588 54805 83645 56256 82675 46 12 o615 50377 86384 51877 8549r 53361 84573 54829 83629 56280 82659 45 12 2i6 50oo3 86369 51902 85476 53386 84557 54854 83613 56305 82643 44 12 7 17 5o428 86354 5I927 8546i 5341i 84542 54878 83597 56329 82626 43 II 18 50453 8634o 5r952 85446 53435 84526 54902 8358i 56353 8260o 42 If 6 19 50478 86325 51977 85431 53460 845ii 54927 83565 56377 82593 41 ii 8 20 505o3 863io 52002 854i6 53484 84495 54951 83549 56401 82577 40 ii 9 21 50528 86295 52026 854o 535o09 8448o 54975 83533 56425 82565 39 io 9 22 50553 86281 52051 85385 53534 84464 54999 83517 56449 82544 38 io 10 23 50578 86266 52076 85370 53558 84448 55024 83501 56473 82528 37 To 10 24 5o060o3 86252 52101 85355 53583 84433 55048 83485 56497 82511 36 IO io 25 50628 86237 52126 185340 53607 84417 55072 83469 56521 82495 35 9 ii 26 5o654 86222 52151 85325 53632 84402 55097 83453 56545 82478 34 9 if 27 50679 86207 521,75 8531o 53656 84386 55I21 83437 56569 82462 33 9 12 28 50704 86292 52200 85294 5368i 84370 55145 83422 56593 82446 32 9 12 29 50729 86178 52225 85279 53705 84355 55269 834o5 56617 82429 3i 13 3o 5o754 86i63 52250 85264 5373o 84339 55294 83389 5664i 82413 30 8 3 3i 50779 86i48 52275 85249 53754 84324 55218 83373 56665 82396 29 8 21.3 32 508o4 86 33 52299 85234 53779 84308 55242 83356 56689 82380 28 7 i4 33 50829 86129 52324 85228 538o4 84292 55266 8334o 56713 82363 27 7 14 34 5o854 860o4 52349 85203 53828 84277 55292 83324 56736 82347 26 7 5 35 50o879 86089 52374 85i88 53853 84261 55315 833o8 5676o 82330 25 7 5 36 5090o4 86074 52399 85173 53877 84245 55339'83292 56784 82314 24 6 i5 37 50929 8659 52423 85157 53902 84230 55363 83276 568o8 82297 23 6 i6 38 50954 86045 52448 85142 53926 84254 55388 83260 56832 82281 22 6 16 39 50979 86030o 52473 85127 53952 84198 55412 83244 56856 82264 21 6 17 4o 5ioo04 86oi5 52498 85iI2 53975 84282 55436 83228 5688o 82248 20 5 17 4i 51029 86oo000 52522 850o96 54o00o 8467 5546o 83222 56904 82231 19 5 i8 42 5105i4 85985 52547 85o81 54024 84I51 55484 83295 56928 82214 i8 5 i8 43" 51079 85970 52572 8S566 54049 84135 55509 83179 56952 82198 17 5 18 44 5io04 85956 52597 85o05i 54073 8420o 55533 83i63 56976 82181 1 6 4 i9 45 51129 85942 52621 85o35 54097 841o4 55557 83147 57000 82165 15 4 i' 46 5ii54 85926 52646 85020 54122 84o88 55581 83131 57024 82148 14 4 0 2( 47 52179 859ii 52671 85oo5 54146 84072 556o5 83.ii5 57047 82232 13 3 20 48 SIo04 85896 52696 84989 5417i 84o57 5563o 83098 5707r 821 5 2, 3 2(1 49 5229 8588i 52720 84974 54i9518404i 55654 183082 57095 82098 3 2 5o 5s1254 85866 52745 84959 5422o 84025 55678 83066 57119 82082 io 3 1 51i 51279 8585I 5277o 84943 54244 84009 55702 83050 5743 820o65 9 2 22 52 5i3o4 85836 52794 84928 54269 83994 55726 83o34 57167 82048 8 2 2-2. 53 51329 85825 52819 84923 54293 83978 55750 83017 5719i 82032 7 2 23 54 5i354 858o6 52844 84897 54317 83962 55775 830oo 572i5 82o05 6 2 23 55 51379 85792 52869 84882 54342 83946 55799 82985 57238 81999 5 23 56 5140o4 857777 52893 84866 54366 83930 55823 82969 57262 82982 4 i 24 57 51429 85762 52918 84851 54391 839i5 55847 82953 57286 8i965 3 i 24 58 58 454 85747 53943 84836 544i5 83899 55871 82936 573o1 81949 2 i 25 59 51479 85732 52967 84820 5444o 83883 55895 82920 57334 81932 i 0 25 60 550o4 85717 52992 848o5 54464 83867 559i9 82904 57358 819i5 o 0 N. cos. N. sine. N. cos.N. sine. N. cos.9 N. sine. N. cos. N. sine. N. cos. N. sine. M 590 580 570 56~ 550 TABLE XXIV. [Page 167 Of Natural Sines. Prop. 350 360 370 30 390 Prop. parts 35 36 37 38 39 I 23 N. sine. N. c os. N. si.N. cos. N. sine.nN. cos. N. sine.N, cos. N. se. N. cos. 8 o o 57358 81915 58779 80902 60182 79864 6i566 78801 62932 77715 6o 1 c T 57381 81899 58802 80885 60205 79846 61589 78783 62955 77696 59 I8 i1 2 574o5 81882 58826 80867 60228 79829 61612 78765 62977 77678 58 17 1 T 3 57429 81865 58849 8o85o 60251 79811 6i635 78747 63000 77660 57 17 2 4 57453 8i848 588731 8833 60274 79793 6i658 78729 63022 77641 56 17 2 1 5 57477 81832 58896 8o8i6 60298 79776 6i681 78711 63o45 77623 55 17 2 6 57501 81815 58920' 80799 60321 79758 61704 78694 63o68 77605 54 i6 3 7 57524 81798 58943 80782 60344 79741 61726 78676 63090 77586 53 16 3 8 57548 81782 58967 80765 60367 79723 61749 78658 63113 77568 52. 16 3 9 5757a 81765 58990 80748 60390 79706 6I772 78640 63i35 77550 5 I5 4 10 57596 81748 59014 80730 6o044[ 79688 61795 78622 63i58 77531 50 i5 4 1 57619 81731 59037 80713 60437 7967r 61818 78604 63180 77513 49 15 5 12 57643 81714 59o6i 80696 60460o 79653 61841 78586 63203 77494 48 I4 5 13 57667 8168 59084 80679 60483 79635 6i864 78568 63225 77476 47 14 5 14 57691 8i68I 59108 80662 6o050o6 79618 61887 78550 63248 77458 46 14 6 i5 57715 81664 59131 80644 60529 79600 61909 78532 63271 77439 45 I4 6 i6 57738 81647 59154 80627 60553 79583 61932 78514 63293 77421 44 13 7 17 57762 8163i 59178 80610 60576 79565 61955 78496 633i6 77402 43 13 7 18 57786 816i4 59201 80593 60599 79547 61978 78478 63338 77384 42 13 19 578o0 81597 59225 80576 60622 79530 62001 78460 63361 77366 4i 12 8 20 57833 8i580 59248 8o558 60645 79512 62024 78442 63383 77347 40 12 8 21 57857 8i563 59272 8o541 60668 79494 62046 78424 63406 77329 39 12 8 22 57881 8i546 59295 80524 60691 79477 62069 784o05 63428 77310 38 II 9 23 57904 8153o 593i8 80507 60714 79459 62092 78387 63451 77292 37 II 9 24 57928 {8i53 59342 80489 60738 7944' 62115 78369 63473 77273 36 Ii 10 25 57952 81496 59365 80472 60761 79424 62138 78351 63496 77255 35 II 10 26 57976 81479 59389 80455 60784 79406 62160 78333 63518 77236 34 io 10 27 57999 81462 59412 80438 60807 79388 62183 78315 63540 77218 33 io ii 28 58023 8i445 59436 80420 60830 79371 62206 78297'63563 77199 32 1o 11 29 58047 81428 59459 8o4o3 60853 79353 62229 78279 63585 77181 31 9 12 3o 58070 81412 59482 8o386 60876 79335 62251 78261 636o8 77162 3o0 12 31 58094 813.95 59506 8o368 60899 79318 62274 78243 63630 77144 299 9 12 32 58118 81378 59529 80o35 60922 79300 62297 78225 63653 77125 28 8 13 33 58i4i 8r36r 59552 8o334 60945 79282 62320 78206 63675 77107 27 8 13 34 58i65 8i344 59576 803i6 60968 79264 62342 78188 63698 77088 26 8 13 35 58189 8132.7 59599 80299 60991 79247 62365 78170 63720 77070 25 8 i4 36 58212 81310o 59622 80282 6o1015 79229 62388 78152 63742. 77051 24 7 14 37 58236 81293 59646 80264 6o1038 79211 62411 78134 63765 77033 23 7 i5 38 5826o 81276 59669 80247 610o6 79193 62433 78116 63787 77014 22 7 15 39 58283 81259 59693 80230 60o84 79176 62456 78098 63810 76996 21 6 15 4o 58307 81242 597i6 86212 61107 79158 62479 78079 63832 76977 20 6 i6 4i 5833o 81225 59739 8oi95 61130 79140 62502 78061 63854 76959 19 6 i6 42 58354 81208 59763 80178 61153 79122 62524 78043 63877 76940 i8 5 16 43 58378 81191 59786 80160 61176 79105 62547 78025 63899 76921 17 5 17 44 584o01 8174 598o9 8o0143 61199 79087 62570 78007 63922 76903 i6 5 17 45 58425 81157 59832 8o125 61222 79069 62592 77988 63944 76884 15 5 18 46 58449 8ii4o 59856 8oio8 61245 79051 62615 77970 63966 76866 14 4 i8 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 13 4 i8 48 58496 8110o6 59902 80073 61291 79016 62660 77934 64o011'76828 12 /4 19 49 58519 81089 59926 8oo0056 61314 78998 62683 77916 64033 76810 3 19 So 58543 81072 59949 8oo38 61337 78980 62706 77897 64056 76791 Io 3' 20o 5 58567 8io55 59972 80021 63601 78962 62728 77879 64078 76772 9 3, 20o 52 5859o 8o1038 59995 8ooo3 61383 78944 62751 77861 64100 76754! 8 220 53 58634 8ioo2 60049 79986 61406 78926 62774 77823 64i23 35 7 7.1 2. 21 54 58637 81004 60042 79968 61429 7896 627746 77824 6445 7673517 6 i1 55 5866i 80987 6oo65 79951 6i451 78891 62819 77806 64167 76698 5 2 21 56 58684 80970 60089 79934 61474 78873 62842 77788 64190 76679 4 1 22 57 58708 8o953 60112 79916 61497,78855 62864 77769 64T12 76661 3 1 22 58 58731 80936 6oi35 79899 61520 78837 62887 7775i 64234 76642 2 i 23 59 58755 80919 60o58 79881 6i543 78819 62909 77733 64256 76623; I o 23 60 58779 80902 60182 79864 6i566 78801 62932 77715 64279 76' 64. 0 0! N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sins. M 540 530 520 51~ 50~ ^::gse 1&s] TABLE XXIV. Of Natural Sines. 400 1 0 Prop. p j | 40A 410 ___ 42____ 430 4 44~ pars ^I A.I'N.- sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 19 o 64 279 700o4 65606 7547I 669I3 74314 68200 73I35 69466 7I934 6o0 19 o 1 6430r 76586 65628 75452 66935 74295 68221 73I16 69487 71914 59 19 1 2 64323 76567 65650 75433 66956 74276 68242 73096 69508 71894 58 i8 I 3 64346 76548 65672 754I4 66978 74256 68264 73076 69529 71873 57 18 I 4 64368 76530 65694 75395 66999 74237 68285 73056 69549 71853 56 18 2 5 64390 7651I 65716 75375 67021 74217 68306 73036 69570 7I833 55 17 2 6 64412 76492 65738 75356 67043 74198 68327 73016 69591 71813 54 I7 3 7 64435 76473 65759 75337 67064 74178 68349 72996 69612 71792 53 17 3 8 64457 76455 65781 75318 67086 74159 68370 72976 69633 71772 52 i6 3 9 64479 764.36 65803 75299 67107 74139 68391 72957 69654 71752 51 16 4 o 6450o 764I7 65825 75280 67I29 74120 684I2 72937 69675 71732 50 16 4 ii 64524 76398 65847 75261, 67151 74100 68434 72917 69696 71711 49 I6 4 12 64546 76380 65869 75241 67172 74080 68455 72897 69717 71691 48 IS 5 i3 64568 7636I 65891 75222 67I94 74(061 68476 72877 69737 7I671 47 15 5 4 64590 76342 65913 75203 67215 74041 -68497 72857 69758 71650 46 15 6 I5 64612 763 3 65935 75I84 67237 74022 685i8 72837 69779 71630 45 4 6 i6 64635 76304 65956 75165 67258 74002 68539 728I7 69800 71610 44 14 6 17 64657 76286 65978 75146 67280 73983 6856I 72797 69821 7I590 43 14 7 i8 64679 76267 66000 75I26 67301 73963 68582 72777 69842 71569 42 13 7 19 64701 76248 66022 75107 67323 73944 686o3 72757 69862 71549 4 1i3 7 20 64723 76229 66044 75088 67344 73924 68624 72737 69883 71529 40 13 8 21 64746 762o0 66o606 75069 67366 73904 68645 72717 69904 7I508 39 I2 8 22 64768 76192 66088 75050 67387 73885 68666 72697 69925 71488 38 12 8 23 64790 76173'66109 75030 67409 73865 68688 72677 69946 71468 37 12 9 24 64812 76154 6613I 75011 67430 73846 68709 72657 69966 71447 36 ii 9 25 64834 76135 66i53 74992 67452 73826 68730 72637 69987 71427 35 ii o1 26 64856 76116 66175 74973 67473 73806 68751 72617 70008 71407 34 ii o0 27 64878 76097 66197 74953 67495 73787 68772 72597 70029 71386 33 io 10 28 64901 76078 66218 74934 675I6 73767 68793 72577 70049 7I366 32 io II 29 64923 76059 66240 74915 67538 73747 68814 72557 70070 71345 3I io i 30 64945 76041 66262 74896 67559 73728 68835 72537 70091 7I325 30 io ii 3I 64967 76022 66284 74876 67580 73708 68857 72517 70112 71305 29 9 12 32 64989 76003 66306 74857 67602 73688 68878 72497 70132 71284 28 9. 12 33 650Io 75984 66327 74838 67623 73669 68899 72477 70153 71264 27 9 12 34 65033 75965 66349 74818 67645 73649 68920 72457 70174 71243 26 8 13 35 65055 75946 66371'74799 67666 73629 68941 72437 70195 71223 25 8 13 36 65077 75927 66393 74780 67688 73610 68962 72417 70251 71203 24 8 14 37 65Ioo 75908 664I4 74760 67709 73590 68983 72397 70236 71182 23 7 ~ I4 38 65122 75889 66436 74741 67730 73570 69004 72377 70257 71162 22 7. 4 39 65I44 75870 66458 74722 67752 73555 69025 72357 70277 71141 21 7 5 4o 65166 75851 66480 74703 67773 73531 69046 72337 70298 71121 20. 6 i5 41 65i88 75832 66501 74683 67795 73511 69067 72317 70319 71100 19 6 15 42 65210 75813 66523 74664 678I6 7349I 69088 72297 70339 71080 18 6 16 43 65232 75794 66545 74644 67837 73472 69I09 72277 70360 71059 17 5 I6 i44 65254 75775 66566 74625 67859 73452 69130 72257 70381 71039 6 5 1 17 45 65276 75756 66588 74606 67880 73432 69151 72236 70401 71019 5 5 17 46 65298 75738 66610 74586 6790I 734I3 69172 722I6 70422 70998 I4 4 7 47 65320 757I9 66632 74567 67923 73393 69193 72196 70443 70978 13 4 i8 48 65342 75700 66653 74548 67944 73373 692I4 72176 70463 70957 I2 4 18 49 65364 75680 66675 74528 67965 73353 69235 72156 70484 70937 II 3 18 50o 65386 75661 66697 74509 67987 73333 69256 72I36 70505 70916 io 3 i9'51 65408 75642 66718 74489 68008 73314 69277 72116 70525 70896 9 3 19 52 65430 75623 66740 74470 68029 73294 69298 72095 70546 70875 8 3 9 53 65452 75604 66762 74451 6805i 73274 69319 72075 70567 70855 7 2,2 i54 65474 75585 66783 7443 68072 73254 69340 72055 70587 70834 6 2 20 55 |65496 75566 66805 74412 68093 73234 69361 72035 70608 70813 5 2 i S1 56 655I8 75547 66827 74392 68II5, 73215 69382 720I5 70628 70793 4 I 21 57 65540 75528 66848 74373 68I36 73I95 69403 71995 70649 70772 3 I 21 58 65562 75509 66870 74353 68157 73175 69424 71974 70670 70752 2 I 22 59 65584 75490 66891 74334 68179 73i55 69445 71954 70690 70731 I o 22 60 65606 7547I 66913 743I4 68200 73i35 69466 71934 70711 70711 C 0 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos.[N. sine. N. cos. N. sine. M - 490 480 470 460 450 Page 170] TABLE XXVI. Logarithms of Numbers. No. 100 ~ 1600. _Log. 00000 ~20412. No. 0 1 2 3 4 5 6 7 8 9 100 00000 ooo43 oo o87 ooi3o 0 73 00217 0oo0260 oo3o3 oo346 00389 43 42 TOI 00432 00475 00oo58 oo56 00604 00647 00689 00732 00775 oo8 I - - 102 oo860oo00903 00945 00988 010o3o 01072 oi15 01157 01199 01243; 9 8 ro3 01284 01326 o0368 o4o o01452 01o494 0o536 01578 01620 o01662 2, 104 01703 01745 01787 01828 01870 1OII2 o01953 01o995 02036 02078 4 105 0211C 02160 02202 02243 02284 02325 02366 02407 02449 02490 5 22 21 io6 02531 02572 02612 02653 02694 02735 02776 02816 02857 02898 6 26 25 107 02938 02979 030oi 03060 o30oo o3i4i o318 03222 0o3262 03302 7 30 29 io8 03342 o3383 03423 o3463 0o35o3 o3543 0o3583 03623 o3663 03703 8 34 34 io9 03743 03782 03822 03862 03902 03941 03981 04021 o4o060o o4o10o 9 39 38 I1o o4139 04179 04218 04258 04297 o4336 04376 o44i5 04454 04493 41 40 Iii 04532 lo457i o46oo4650 04689 04727 047661o48o5' o4844 o4883 112 04922046 0496 99 05038 05077 05II5 05154 05192 05231 05269 I 4 4 113 o53o8 5346 5385 05423 05461 o55oo00 05538 05576 o5614 05652 2 8 8 114 05690 05729 05767 o58o5 o5843 05881 05918 05956 05994 06032 3 12 12 6070 6__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _6_ _ _ _ _ _6__ _3_ _ _ _ _ 3__ _ _ _ _ _ _ _ _ _ _ 8 4 i6 i6 115 06070 o0610o8 06145 o06183 06221 06258 06296 o6333 0637 o6408 5 2 2 116 o6446 06483 06521 o6558 06595 o6633 06670 06707 06744 06781 6 25 24 117 o6819 0o6856 06893 0o6930 06967 07004 07041 07078 07115 07151 7 118 07188 07225 07262 07298 07335 07372 07408 07445 07482 07518 8 33 119 07555 07591 07628 07664 07700 07737 07773 07809 07846 07882 3736 120 07918 07954 07990 08027 080o63 08099 o835 08171 08207 08243 121 08279 o8314 o835o o8386 08422 o8458 08493 08529 08565 o860o 39 38 122 o8636 08672 08707 08743 08778 o8814 o8849 o8884 08920 08955 l 4, 123 o8991 09026 09061 09096 09132 09167 09202 09237 09272 09307 2 8 8 I24 09342 09377 09412 09447 09482 09517 09552 09587 09621 09656 3 12 11 125 09691 09726 0976 0979o 5 o09830 09864 09899 09934 o9968 iooo3 4 16 I5 126 10037 10072 1o0o6 o014o 10175 10209 10243 10278 10312 o346 5 20 19 127 1o38o 1o415 10449 io483 10517 10551 1o585 10619 o653 10o687 6 23 23 128 10721 10755 10789 10823 10857 10890 10924 10958 10992 11025 7 27 27 129 11059 110o93 11126 11160 11193 11227 11261 11294 1132-7 11361 8 3i 30 13o 11394 11428 1146 1T1494 11528 11561 11594 11628 11661 11694 9 34 131 11727 11760 11793 11826 1186o 11893 11926 11959 11992 12024 37 36' 132 12057 12090 12123 12156 12189 12222 12254 12287 12320 12352 ~ i33 12385 12418 12450 12483 12516 12548 12581 12613 12646 12678 4 4/ 134 12710 12743 12775 12808 12840 12872 12905 12937 12969 13001 2 7 7 135 13033 13066 13098 13130 13162 13194 13226 13258 13290 13322 4 15 14 i36 13354 13386 134i8 i345o 13481 13513 13545 13577 i3609 1 364 5 19 18 137 13672 13704 13735 13767 13799 i383o 13862 13893 13925 13956 6 22 22 i38 13988 14019 14o51 14082 14114 i4145 14176 14208 14239 14270 7,26 25 139 43o01 14333 4364 14395 14426 14457 14489 14520 14551 14582 8 30 29 140 14613 -4644 14675 14706 14737 14768 14799 14829 486o 1489I 9 33 32 141 14922 14953 14983 i5o04 15o45 15076 i5io6 15137 i5i68 15198 35 34 142 15229 15259 15290 15320 15351 i538i 15412 15442 15473 155o3 - 143 15534 15564 15594 15625 15655 15685 15715 15746 15776 158o6 I 4 3 144 15836 15866 15897 15927 15957 15987 16017 16047 16077 16107 2 7 7 145 16i37 16167 16197 16227 16256 16286 163i6 16346 16376 16406 3 10 18 4 7 i46 16435 16465 16495 16524 16554 16584 16613 16643 16673 16702 s i S 1 7 16732 1676I 16791 16820 i685o 16879 169o9 16938 16967 16997 6 2I 20 148 17026 17056 17085 17114 17143 17173 17202 17231 17260 17289 / 149 17319 17348 17377 17406 17435 17464 17493 17522 17551 17580.8.27 150 176o09 17638 17667 17696 17725 17754 17782 17811 17840 17869 9 32 3 151. 17898 17926 17955 17984 i8oi3 18o41 18070 18099 18127 18156 - 152 i8i84 18213 18241 18270 18298 18327 18355 18384 18412 18441 33132 153 18469 18498 18526 i8554 18583 18611 18639 18667 18696 18724 3 3 154 18752 18780 i88o8 18837 18865 18893 18921 18949 18977 19005 2 7 6 155 19033 19061 18 1908919117 19145 19173 19201 19229 19257 19285 3 io io i56 19312 i9340 19368 19396 19424 19451 19479 19507 19535 19562 4 13 13 157 i9590 19618 19645 19673 19700 19728 19756 19783 19811 19838 5 17 i6 i58 19866 19893 i9924 19948 19976 2ooo3 20030 20058 20085 20112 6 20 19 159 20140 201 201 194 20222 202149 20276 20303 20330 20358 20385 7 23 22 No. 0 1 2 3 4 5 6 7 8 9 93029 TABLE XXVI. [Page 171 Logarithms of Numbers. No. 1600o 2200. Log. 20412 34242. No. 0 2 3 4 5 6 7 8 9:6o 20412 20439 20466 20493 20520 20548.20575 20602 2o629 20656 31 30 161 20683 20710 20737'20763 20790 20817 20844 20871 20898 20925 i 3 3 162 20952 20978 2100oo5 21032 21059 21085 21112 21139 21165 21192 2 6 6 163 21219 21245 21272 21299 21325 21352 21378 21405 21431 21458 3 9 9 164 21484 21511 21537 21564 921590 21617 21643 21669 21696 21722 4 12 12 -65 21748 21775 21801 21827 21854 21880 21906 21932 21958 21985 5 16 15 i66 22011 22037 22063 22089 22115 22,41 22167 22194 22220 22246 6 19 i8 167 22272 22298 22324 22350 22376 22401 22427 22453 22479 22505 7 22 21 168 2253i 22557 22583 22608 22634 22660 22686 22712 22737 22763 8 25 24 169 22789 22814 22840 22866 22891 22917 22943 22968 22994 23019 9 28 27 170 23045 23070 23o96 23121 23147 23172 23198 23223 23249 23274!J 9 28 171 23300 23325 23350 23376 23401 23426 23452 23477 23502 23528 172 23553 23578 23603 23629 23654 23679 23704 23729 3754 2375 779 3 1 3 173 23805 23830 23855 23880 23905 23930 23955 23980 24005 24030 2 6 6 174 240.55 24080 241o 5 24130 24155 24180 24204 24229 2425 24279 8 175 24304 24329 24353 24378 24403 24428 24452 24477 24502 24527 I5154 176 24551 24576 2460, 24625 24650 24674 24699 24724 24748 24773 6 17 17 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 250o8 7 20 20 178 25042 25066 25091 25115 25139 25164 25188 25212 25237 25261 8 23 22 179 25285 25310 25334 25358 25382 25406 25431 25455 25479 25503 9 26 25 i80 25527 2551 25575 25600 25624 25648 25672 25696 25720 25744 27 2 181 25768 25792 25816 25840 25864 25888 25912 25935 25959 25983 i -- 82 26oo007 26o031 26055 26079 26102 26126 26150 26174 26198 26221 3 3 183 26245 26269 26293 26316 26340 26364 26387 26411 26435 26458 2 5 5 i84 26482 26505 26529 26553 26576 26600oo 26623 26647 26670 26694 3 8 8 185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 4 " 3 i86 26951 I26975 26998 27021 27045 27068 27091 27114 27138 27161 5 "413 187 27184 27207 27231 27254 27277 27300 27323 27346 27370 27393 6 J 6 6 i88 27416 27439 27462 27485 27508 27531 27554 27577 27600 27623 7 I9 8 189 27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 8 22 21 190 27875 27898 27921 27944 27967 27989 28012 28035 28058 2808 9124 ~23 191 28703 28126 28,49 28171 28194 28217 28240 28262 28285 28307 25 24 192 28330 28353 28 375 28398 28421 28443 28466 28488 28511 28533 3 2 193 28556 28578 28601 28623 28646 28668 28691 28713 28735 28758 2 5 i'94 287802o 8803 28825 28847 28870 28892 28914 28937 28959 28981 3 8 - 195 2900oo3 29026 29048 29070 29092 29115 29137 29259 29181 29203 4 io TO 196 29226 29248 29270 29292 29314 29336 29358 29380 29403 29425 5 i3 I3 197 29447 29469 29491 295i3 29535 29557 29579 29601 29623 29645 6 15 14 198 29667 29688 29710 29732 29754 29776 29798' 29820 29842 29863 7 i8 17 199 29885 29907 29929 29951 29973 29994 3oo16 3oo38' 30060 30081 8 20 19 200 3oio3 30125 3oi46 30o68 30190 30211 30233 30255 30276 30298 9 23122 201 j30320 3o34 3o363 3o384 3o4o6 30428 30449 30471 30492 3o514 23 22) 202 3o535 30557 30578 3o6oo00 3062 30643 30664 30685 30707 30728. 203 30750 30771 30792 3o8i4 30835 3o856 30878 30899 30920 30942 I 2 2 204 30963 30984 31oo6 3ro27 3io48 3o1069 3o091 31112 31133 3154 2 5 4 -205 31175 31197 31218 31239 312-60 128i 3o302 - 2 31323 4 3r351366 206 31387 3i4o8 31429 3i450 31471 31492 31513 31534 31555 31576 5 12 II 207 397 36i8 31639 3i66o 3i68i 31702 31723 31744 31765 31785 6 14 13 208 3i8o6 31827 3i848 31869 31890 31911 3193r 31952 31973 32994 7 i6'iS 209 32015 32035 32056 32077 32098 32118 32139 32160 32181 32201 8 18 18 210 o 32222 32243 32263 32284 32305 32325 32346 32366 32387132408 9 21 20 211 32428 32449 32469 32490 32510 32531 32552 32572 32593 32613 212 32634 32654 32675 32695 32715 32736 32756 32777 32797 32818 21 20 213 32838 32858 32879 32899 32919 32940 32960 32980 3300oo 33021 1 2 214 33o41 33062 33082 33102 33122 33143 33163 33i83 33203 33224 2 4 4 215 133244 33264 33284 33304 33325 33345 33365 33385 33405 33425 3 6 6 216 33445 33465 33486 33506 33526 33546 33566 33586 33606 33626 4 8 8 217 33646 33666 33686 33706 33726 33746 33766 33786 33806 33826 5 i 218 33846 33866 33885 33905 33925 33945 33965 33985 34o005 34025 6 13 12 219 34044 340o64 34084 34io04' 34124 34143 34i63 34183 34203 34223 7 15 14 N8 17 i6 No. 0 1 2 3 4 5 6 7 8 9 9,1918 Page 172] TABLE XXVI. Logarithms of Numbers. No. 2200 —2800. Log. 34242 —-— 44716. No. 0 1 2 3 4 5 6 7 8 9 220 34242 34262 34282 343oi 34321 3434i 3436i 3438o 34400o 34420 0 221 34439 34459 34479 34498 345i8 34537 34557 34577 34596 346i6 1 2 222 34635 34655 34674 34694 34713 34733 34753 34772 34792 348ii 2 4 223 3483o 3485o 34869 34889 34908 34928 34947 34967 34986 35005 3 6 224 35025 35o44 35o64 35083 35102 35122 3541 35160o 35i8o 35199 4 8 225 35218 -35238 35257 35276 35295 35315 35334 35353 35372 35392 5 ic 226 354ri 3543o 35449 35468 35488 35507 35526 35526 355645 35583 6 12 227 356o3 35622 3564 356660 35679 35698 35717 35736 35755 35774 7'4 228 35793 35813 35832 35851i 35870 35889 35908 35927 35946 35965 8 i6 229 35984 36oo003 6021 36o4o 36o59 36078 36097 36116 36i35 36i54 9 I8 230 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 19 231 3636i 36380 36399 364i8 36436 36455 36474 36493 365ii 36530 232 36549 36568 36586 36605 36624 36642 3666 3668o 36698 36717 2 233 36736 36754 36773 36791 368io 36829 36847 36866 36884 36903 3 6 234 36922 36940 36959 36977 36996 3714 377033 37051 37070 37088 4 8 235 37107 37125 37144 37162 37181 37199 37218 37236 37254 37273 5 io 236 37291 37310 37328 37346 37365 37383 37401 37420 37438 37457 6 ii 237 37475 37493 37511 37530 37548 37566 37585 37603 37621 37639 7 I 238 37658 37676 37694 37712 37731 37749 37767 37785 37803 37822 8 I5 239 37840 37858 37876 37894 37912 37931 37949 37967 37985 38003 240 38021 38039 38057 38075 38093 38112 3830o 38 48" 38 166- 38184 241 38202 38220 38238 38256 38274 38292 383io 38328 38346 38364 _ 242 38382 38399 38417 38435 38453 38471 38489 38507 38525 38543 1 2 243 3856i 38578 38596 386i4 38632 3865o 38668 38686 38703 38721 2 4 244 38739 38757 38775 38792 388io 38328 38846 38863 3888 38899 3 5 245 38917 38934 38952 38970 38987 39005 39023 39041 39058 39076 7 246 39094 39111 39129 39146 39164 39182 39199 39217 39235 30252 6 9 247 39270 39287 39305 39322 39340 39358 39375 39393 39410 39428 6 2 1 248 39445 39463 39480 39498 39515 39533 39550 39568 39585 39602 7 13 249 3962.o 39637 39655 39672 39690 39707 39724 39742 39759 39777 8 250 39794 39811 39829 39846 39863 39881 39898 39915 39933 3995o - 251 39967 39985 40002 400oo19 40037 4oo54 40071 4o0088 4oo6 40123 17 252 4o40o 40157 40175 40192 40209 40226 40243 40261 40278 40295 253 4o0312 40329 4o346 4o364 4o38i 40398 4o4I5 40432 40449 4o466 3 254 4o483 4050oo 40518 4535 40552 40569 4o586 40603 40620 40637 3 5 255 40654 40671 4o688 40705 40672240739 40756 40773 40790 40807 4 7 256 40824 4o84i 4o858140875 40892 40909 4o0926 40943 40960 40976 5 9 257 40993 41010 41027 41044 4io61 41078 41095 41111 41128 41145 6 io 258 41162 41179 41196 41212 41229 41246 41263 41280 41296 413i3 7 12 259 4i33o 41347 4r363 41380 41397 41414 4i43o 41447 41464 4i48i 8 i4 260,41497 41514 4i53i 41547 4564 4i58i 45971 464 4i163i 41647 2__7 261 4i664 4i681 41697 41714 41731 41747 41764 4178o 41797 41814 16 262 4i83o 41847 41863 4i88o 41896 41913 41929 41946 41963 41979 263 41996 42012 42029 4'2oo45 42062 42078 42095 42111 42127 42144 I 2 ~264 42160 42177 42193 42210 42226 42243 42259 42275 42292 42308 2 3 265 42325 42341 42357 42374 42390,424o06 42423 42439 42455 42472 4 6 266 42488 42504 42521 42537 42553 42570 42586 42602 42619 "2635 5 8 267 42651 42667 42684 42700 42716 142732 42749 42765 42781 42797 6 268 428I3 4283o 42846 42862 42878 42894 42911 42927 42943 42959 7 II 269 42975 42991 43008 43024 43o04o 43o56 43072 43088 431o4 43120 8 i3 270 43r36 43152 43169 43i85 43201 43217 43233 43249 43265 43281 9 14 271 43297 43313 43329 43345 4336i 43377 43393 43409 43425 4344i 272 43457 43473 43489 435o5 43521 43537 43553 43569 43584 436oo 15 273 43616 43632 43648 43664 43680 43696 43712 43727 43743 43759 i 2 27 43775 43791 43807 43823 43838143854 43870 43886 43902 43917 2 275 43933 43949 43965 43981 43996 44012 44028 44o44 44059 44075 3 5 276 44091 44107 44122 44i38 44154 447o0 44i85 44201 44217 44232 4 6 i 277 44248 44264 44279 44295 443ii 44326 44342 44358 44373 44389 5 8 /078 44404 44420 44436 4445i 44467 4483 I44498 445i4 44529 4454 6 9 279 44560 44576 44592 446o7 44623 44638 44654 44669 44685 144700 7 i1 No. 0 -1 3 4 i5 6 7 8) 9 I 9 4 TABLE XXVI. [Page 173 Logarithms of Numbers. No. 2800 3400. Log. 44716 —53148. No. 0 1 2 3 4 5 6 7 8 9 280 44716 44731 44747144762 44778 44793 4480og 44824 4484o 44855 16 281 44871 44886 44902 4497 44932 44948 44963 44979 44994 45o010 282 45025 45o4o 45o56 45071 45o86 45102 45117 45i33 45i48 45i63 2 3 283 45179 45194 45209 45225 45240 45255 45271 45286 453o01 45317 3 5 284 45332 45347 45362 45378 45393 45408 45423 45439 45454 45469 4 6 285 45484 45500 455i5 45530 45545 45561 45576 45591 45606 45621 5 8 286 45637 45652 45667 45682 45697 45712. 45728 45743 45758 45773 6 io 287 45788 45803 458i8 45834 45849 45864 45879 45894 45909 45924 7 Ii 288 45939 45954 45969 45984 4600ooo 46o5 46030 4645 46060 46075 8 13 289 46090 46o105 46120 4635 4650o 46i65 46i8o 46195 462o0 46225 9 14 290 46240 46255 46270 46285 46300 46315 4633o 46345 46359 46374 291 46389 464o4 46419 46434 46449 46464 46479 46494 46509 46523 292 46538 46553 46568 46583 46598 46613 46627 46642 46657 46672 15 293 46687 46702 46716 46731 46746 46761 46776 46790 46805 46820 294 46835 4685o 46864 46879 46894 46909 46923 46938 46953 46967 I 295 46982 4 6997 47012]47026 47041 466927100 47114 9 296 47 47129147144 47159 47173 47188 47202 47217 47232 47246 47261 / 6 297 47276 47290 47305 47319 47334 47349 47363 47378 477392 7 5 298 47422 47436 47451 47465 47480 47494 47509 47524 47538 47553 6 9 299 47567 47582 47596 47611 47625 47640 47654 47669 47683 47698 7 3o0 47712 47727 4774I 47756 47770 47784 47799 47813 47828 47842 8 12 301o 47857 47871 47885 47900oo 4794 47929 47943 47958 47972 47986 9 14 302 48ooi 48oi5 48029 48o44 48o58 48073 48087 48101 48116 4813o 303 48144 48159 48173 48187 48202 48216 48-30 48244 48259 48273 3o4 48287 48302 48316 48330 48344 48359 48373 48387 484o0 484i6 305 4843o 48444 48458 48473 48487 485oii 485i5 4853o 48544 48558 14 306 48572 148586 486oi 486i5 48629 48643 48657 48671 48686 48700 1 1 307 48714 48728 48742 48756 48770 48785 48799 488i3 48827 48841 2 3 308 48855 48869 48883 48897 48911 48926 48940 48954 48968 48982 3 4 309 48996 49010 49024 49038 49052 49066 49080 49094 49108 49122 4 6 31o 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 8 7 311 492i76 49290 49304 49318 49332 493,46 49360 49374 49388 49402 o 8 312 49415 49429 49443 49457 49471 49485 49499 49513 49527 4954I 7 10 313 49554 49568 49582 49596 49610 49624 49638 49651 49665 49679 8 ii 314 49693 49707 49721 49734 49748 49762 49776 49790 49803 49817 9.3 315 4983i 49845 49859 49872 49886 499oo00 49914 49927 49941 49955 3i6 49969 49982 49996 5ooio 50024 50037 50o5 5oo65 50079 50092 317 5010oo6 50120 50133 50147 50161 50174 50188 50202 50215 50229 3T8 5o0243 50256 502o70 50284 50297 50311 50325 50338 50352 5o365 __s _ 319 50379 50393 504o6 50420 5o0433 50447 5o46I 50474 50o488 5050l oi 320 "505 1550529 50542 50556 50569 50583 50596 506io 5o623 5063 7 321 5o651 5o664 50678 50691 50705 50718 50732 50745 50759 50772 5 322 50786 50799 5o8i3 50826 50840 50853 5o866 50880 50893 50907 4 323 50920 150934 50947 50961 50974 50987 51001o 51014 51028 51041 /z i 3324 51055 5o1068 51081 51095 51108 51121 51ii35 5I148 51162 51175 6 7 9 325 51i88 51202 51215 51228 51242 51255 51268 51282 51295 51308 8 io 326 51322 51335 51348 51362 51375 5i388 514o2 5i4r5 51428 5i44r 1,2 327 5r455 5i468 5148i 51495 5150o8 51521 51534 51548 51561 55574 328 515871 56oi 51614 51627 5i64o 51654 51667 5168o 51693 51706 329 51720 51733 51746 51759 51772 51786 51799 15812 51825 51838 330 5i85i 51865 51878 51891 5go1904 51917 51930 51943 51957 51970 12 331 5i983 51996 52009 52022 52035 52048 52061 52075 52088 52101..i. 332 52114 52127 52140 52153 52166 521-79 52192 52205 52218 52231 333 52244 52257 52270 52284 52297 52310 52323 52336 52349 52362 3: 334 52375 52388 52401 524i4 52-427 52440 52453 52466 52479 52492 4 5 335 52504 52517 52530 52543 52556 52569 52582 52595 52608 52621 5 6 336 52634 52647 52660 52673 52686 52699 52714 52724 52737 52750 6 7 337 52763 52776 52789 52802 52815 52827 52840 52853 52866 52879 7 8 338 52892 529o0 5 52917 52930 5943 52956 5a969 52982 52994 53007 8 io 339 53020 53033 53046 53058 5307i 530o84 53097 53110 53122 53135 9 I No. 0 1 2 3 4 5 6 17 8 9 Page 1761 TABLE XXV1. Logarithms of Numbers. No. 4600 - 5200. Log. 66276 71600. No. 0 1 2 3 4 5 6 8 9 460 66276 66285 66295 66304 663i4 66323 66332 66342 6635I 66361 10 46i 66370 66380 66389 66398 66408 66417 66427 66436 66445 66455 -_ 462 66464 66474 6648.3 66492 66502 66511 66521 66530 66539 66549 2 2 463 66558 66567 66577 66586 66596 66605 666i4 66624 66633 66642. 3 464 66652 6666i 66671 66680 66689 66699 66708 66717 66727 66736 4 4 465 66745 66755 66764 66773 66783 66755 792 668o1 668ii 66820 66829 5 5 466 66839 66848 66857 66867 66876 66885 66894 66904 66913 66922 6 6 467 66932 66941 66950 66960 66969 66978 66987 66997 67006 67015 7 7 468 67025 67034 67043 67052 67062 67071 67080 67089 67099 67108 8 8 469 67117 67127 67136 67145 67154 67164 67173 67182 67191 67201 9 9 470 67210 672I9 67228 67237 67247 67256 67265 67274 67284 67293 47r 67302 6731I 67321 67330 67339 67348 67357 67367 67376 67385 472 67394 67403 67413 67422 67431 67440 67449 67459 67468 67477 473 67486 67495 67504 67514 67523 67532 67541 67550 67560 67569 474 67578 67587 67596 67605 67614 67624 67633 67642 67651 67660 475 67669 67679 67688 67697 67706 67715 67724 67733 67742 67752 476 67761 67770 67779 67788 67797 67806 67815 67825 67834 67843 477 67852 67861 67870 67879 67888 67897 67906 67916 67925 67934 478 67943 67952 67961 67970 67979 67988 67997 68006 68oi5 68024 479 68034 68043 68052 68o61 68070 68079 68088 68097 680o6 68115 480 68124 68I33 68142 68I15 6816o68869 68178 68187 68196 68205 481 68215 68224 68233 68242 68251 68260 68269 68278 68287 68296 482 68305 68314 68323 68332 68341 6835o 68359 68368 68377 68386 483 68395 68404 68413 68422 6843i 68440 68449 68458 68467 68476 484 68485 68494 68502 68511 68520 68529 68538 68547 68556 68565 485 68574 68583 68592 6860o 686o 6866869 8628 68637 68646 68655 9 486 68664 68673 68681 68690 68699 68708 68717 68726 68735 68744 i 487 68753 68762 68771 68780 68789 68797 688o6 688I5 68824 68833 2 2 488 68842 68851 68860 68869 68878 68886 68895 68904 68913 68922 3 3 489 68931 68940 68949 68958 68966 68975 68984 68993 69002 690oII 4 490 69020 69028 69037 69046 69055 69064 69073 69082 6909o 69099 5 5 49I 69108 69117 69126 69I35 69144 69152 69161 69I70 69179 6918 66 5 492 69197 69205 69214 69223 69232 69241 69249 69258 69267 69276 7 6 493 69285 69294 69302 69311 69320 69329 69338 69346 69355 69364 8 7 494 69373 69381 69390 69399 69408 69417 69425 69434 69443 69452 9 8 495 69461 69469 69478 69487 69496 69504 69513 69522 69531 69539 496 69548 69557 69566 69574 69583 69592 696o0 69609 69618 09627 497 69636 69644 69653 69662 69671 69679 69688 69697 69705 69714 498 69723 69732 69740 69749 69758 69767 69775 69784 69793 69801 499 69801 69819 69827 69836 69845 69854 69862 69871 69880 69888 500oo 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 5o0 69984 69992 70001 70010 70018 70027 70036 70044 70053 70062 502 70070 70079 7o088 70096 7b005 70114 70122 70o 31 7040 70I48 503 70157 70165 70174 70183 70191 70200 70209 70217 70226 70234 504 70243 70252 70260 70269 70278 70286 70295 70303 70312 70321 505 70329 70338 70346 70355 70364 70372 70381 70389 70398 70406 506 70415 70424 70432 70441 70449 70458 70467 70475 70484 70492 507 70501 70509 70518 70526 70535 70544 70552 70561 70569 70578 50o8 70586 70595 70603 70612 7062I 70629 70638 70646 70655 70663 509 70672 70680 70689 70697 70706 70714 70723 70731 70740 70749 5i0 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 8 51T 70842 70851 70859 70868 70876 70885 70893 70902 70910 70919 512 70927 70935 70944 70952 70961 70969 70978 70986 70995 71003 2 513 71012 71020 71029 71037 71046 71054 71063 71071 71079 71088 3 2 514 71096 71105 71113 71122 7130 73 739 7147 71155 71164 71172 41 3 515 71181 71189 7119i8 7206 71214 71223 71231 71240 71248 71257 5 4 5I6 71265 71273 71282 71290 71299 71307 71315 71324 71332 71341 6/j5. 517 71349 7I357 7I366 71374 71383 71391 71399 714 7148 71416 71425 6 5I8 7I433 71441 71450 71458 71466 71475 71483 71492 71500 71508 86. 519 1715I7 7525 71533 7I542 71550 71559 71567 7I575 71584 71592 9i7 No. 0 1 3 4 5 6 7 8 9 TABLE XXVI. [Page 177 Logarithms of Numbers. No. 5200 —5800. Log. 71600 -. 76343. No. 0 1 2 3 4 5 6 7 8 9 520 7i600 7I609 7I6I7 7i625 7I634 71642 7I650 7I659 7I667 7675 9 521 71684 71692 71700 71709 71717 71725 71734 71742 71750 71759 522'71767 71775 7I784 71792 7800 7I809 7I817 7825 71834 71842 I 523 7185) 71858 71867 71875 71883 7892 7900 7 7 908 71917 7195 3 524 7193' 71941 71950 71958 71966 71975 71983 71991 71999 72008 4 4 525 72016 72024 72032 7204I 72049 72057 72o66 72074 72082 72090 5 5 526 72099 72107 72115 72123 72I32 72I40 72I48 72I56 72I65 72173 6 5 527 72181 72189 72I98 72206 722I4 72222 72230 72239 72247 72255 7 6 528 72963 72272 7 7228 72288 72296 2304 723I3 72321 72329 72337 8 7 529 72346 72354 72362 72370 72378 72387 72395 72403 724I1 72419 9 8 53o 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 53I 72509 72518 72526 72534 72542 72550 72558 72567 72575 72583 532 72591 72599 72607 726I6 72624 72632 -2640 72648 72656 72665 533 72673 72681 72689 72697 72697 2705 727 72705 7273 72722 72730 72738 72746 534 72754 72762 72770 72779 72787 72795 72803 72811 72819 72827 535 7835 /2843 72852 72860 72868 72876 72884 72892 72900 72908 536 72916 72925 72933 72941 72949 72957 72965 72973 72981 72989 537 72997 73006 73014 73022 7303 73022 73 73038 73046 73054 73062 73070 538 73078 73086 73094 73102 73111 73119 73127 73I35 73I43 73151 539 7_73159 73167 73175 73183 73191 73199 73207 73215 73223 73231 64o 73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 541 73320 73328 73336 7733368344 73352 7 773368 73376 73384 73392 542 73400 73408 73416 73424 73432 73440 73448 73456 73464i 73472 543 73480 73488 73496 73504 73512 73520 73528 73536 73544 73552 544 73560 73568 73576 73560 73568 73576 73584 73592 73600 73608 7366 73624 73632 545 73640 73648 73656 73664 73672 73679 73687 73695 73703 73711 8 546 73719 73727 73735 73743 73751 73759 73767 73775 73783 73791 I 547 73799 73807 73815 73823 73830 73838 73846 73854 73862 73870 2 2 548 738 73878 73886 73894 73902 73910 73918 73926 73933 73941 73949 3 2 549 73957 73965 73973 73981 73989 73997 74005 74013 74020 74028 4 3 550 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 5 4 551 74115 74123 74131 74139 74147 74I55 74162 7417 74178 74186 6 5 552 74I94 74202 74210 742I8 74225 74233 74241 74249 74257 74265 7 6 553 74273 74273 74280 74288 74296 74304 74312 74320 74327 74335 74343 8 6 554 74351 74359 74367 74374 74382 74390 74398 74406 74414 74421 9_ 7 555 74429 7 4437 744454429 744374453 7 64445 74453 7446 74468 74476 74484 74492 74500 556 74507 74515 74523 74531 74539 74547 74554 74562 74570 74578 557 74586 74593 74593 74601 74609 746 774624 74632 74640 74648 74656 558 74663 74671 74679 74687 74695 74702 74710 74718 74726 74733 559 74741 74749 74757 74764 74772 74780 74788 74796 74803 74811 56o 74819 74827 74834 74842 74850 74858 74865 74873 7488I 74889 561 74896 74904 74904 912 7 4920 74927 74935 74943 74950 74958 74966 562 7497 74981 7498 9 74997 75005 75012 75020 75028 75035 75043 563 75051 75059 75066 75074 75082 75089 75097 75105 75113 75120 564 75128 75136 75143 75151 75159 75166 75174 75182 75189 75197 565 75205 7523 75220 7522 775236 75243 75251 75259 75266 75274 566 75282 75289 575297 75305 75312 75320 75328 75335 75343 75351 567 75358 75366 75374 75381 75389 75397 75404 75412 75420 75427 568 75435 75442 75450 75458 75465 75473 75481 75488 7 5496 75504 569 75511 75519 75526 75534 75542 75549 75557 75565 75572 75580 570 75587 75595 75603 7561I0 75618 75626 75633 75641 75648 75656 7 571 7566 75671 75679 75686 75694 75702 75709 75717 75724 75732 572 75740 75747 75755 75762 75770 75778 75785 75793 75800 75808 2 I 573 75815 75823 75831 75838 75846 75853 75861 75868 75876 75884 3 2 574 75891 75899 75906 759I4 75921 75929 75937 75944 75952 75959 4 3 575 75967 75974 75982 75989 75997 76005 76012 76020 76027 76035 5 4 576 76042 76050 76057 76065 76072 76080 76087 76095 76103 76110 6 4 577 76118 76125 76133 0 4 7648 76155 76I63 76170 76178 76185 7 5 578 76193 76200 7628 76215 I 76223 7630 76238 76245 76253 76260 8 6 579 76268 76275 76283 76290 76298 76305 76313 76320 76328 76335 9 6 No. |O 1 2 3 4 5 6 7 8 9 23 Page 178] TABLE XXVI. Logarithms of Numbers. No. 5800 —— 6400. Log. 76343 — 80618. No. 0.1 2 3' 4 5 6 7 8 9 580 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 8 58I 76418 76425 76433 76440 76448 76455 76462 76470 76477 76485 582 76492 76500 76507 76515 76522 76530 76537 76545 76552 76559 2 2 583 76567 76574 76582 76589 76597 76604 76612 76619 76626 76634 3 2 584 76641 76049 76656 76664 76671 76678 76686 76693 7670I 76708 4 3 585 76716 76/23 76730 76738 76745 76753 76760 76768 76775 76782 5 4 586 76790 76797 76805 768I2 76819 76827 76834 76842 76849 76856 6 5 587 76864 76871 76879 76886 76893 76901.76908 76916 76923 76930 7 6 588 76938 76945 76953 76960 76967 76975 76982 76989 76997 77004 8 6 589 77012 779 7726 77034 7704I 77048 77056 63 77706 77078 9.7 590 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 59I 77159 77166 77173 71 77I 7788 77195 77203 77210 77217 77225 592 77232 77240 77247 77254 77262 77269 77276 77283 7729I 77298 593 77305 77313 77320 77327 77335 77342 77349 77357 77364 7737I 594 77379 77386 7739377 774 0 8 7745 77422 77430 77437 77444 595 77452 77459 77466 77474 77481 77488 77495 77503 77510 77517 596 77525 77532 77539 77546 77554 77561 77568 77576 77583 77590 597 77597 77605 77612 77619 77627 77634 77641 77648 77656 77663 598 77670 77677 77685 77692 77699 77706 77714 77721 77728 77735 599 77743 77750 77757 77764 77772 77779 77786 77793 77801 77808 600 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 601 77887 77895 77902 77909 779I6 77924 77931 77938 77945 77952 602 77960 77967 77974 7798I 77988 77996 78003 78010 78017 78025 603 78032 78o39 780 46 78053 78061 78068 78075 78082 78089 78097 604 78104 78111 78i 18 78125 78132 78140 78147 78 78654 78 78168 605 78176 78183 78190 78197 78204 78211 78219 78226 78233 78240 7 606 78247 78254 78262 78269 78276 78283 78290 78297 78305 78312 607 78319 78326 78333 78340 78347 78355 78362 78369 78376 78383 608 78390 78398 78405 78412 78419 78426 78433 78440 78447 78455 3 609 78462 78469 78476 78483 78490 78497 78504 78512 78519 78526 4 3 6i0 78533 78540 78547 78554 78561 78569 78576 78583 78590 78597 5 4 611 78604 78611 786I8 78625 78633 78640 78647 78654 78661 78668 6 4 612 78675 78682 78689 78696 78704 78711 78718 78725 78732 78739. 7 5 613, 78746 78753 78760 78767 78774 78781 78789 78796 78803 78810 8 6 614 78817 78824 78831 78838 78845 78852 78859 78866 78873 78880 9 6 615 78888 78895 78902 78909 78916 78923 78930 78937 78944 7895I 616 78958 78965 78972 78979 78986 78993 79000 79007 79014 79021 617 79029 79036 79O43 79050 79057 79064 79071 79078 79085 79092 618 79099 79106 79113 79120 79127 79134 79141 79148 79155 79162 619 79169 79176 79183 79190 79r97 79204 79211 79218 79225 79232 620 79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 621 79309 79316 79323 79330 79337 79344 79351 79358 79365 79372 622 79379 79386 79393 79400 79407 79414 79421 79428 79435 79442 623 79449 79456 79463 79470 79477 79484 79491 79498 79505 79511 624 79518 79525 79532 79539 79546 79553 79560 79567 79574 79581 625 79588 79595 79602 79609 79616 79623 79630 79637 79644 79650 626 79657 79664 79671 79678 79685 79692 79699 79706 79713 79720 627 79727 79734 79741 79748 79754 79761 79768 79775 79782 79789 628 79796 79803 79810 79817 79824 7983I 79837 79844 79851 79858 629 79865 79872 79879 79886 79893 79900 79906 799I3 79920 79927 630 79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 6 63i 80003 80010 80017 80024 8o030 80037 80044 8005i 80058 80065 -~ 632 80072 80079 80085 80092 80099 80106 8o0I3 80120 80I27 80o34 I I 633 8o040 80147 8oi54 8oI6i 80168 80175 80182 80I88 80195 80202 634 80209 80216 80223 80229 80236 80243 80250 80257 80264 80271 42 635 80277 80284 80291 80298 80305 80312 80318 80325 80332 80339 5 3 636 80346 80353 80359 80366 80373 80380 80387 80393 80400oo 80407 6 4 637 80414 80421 80428 8o434 8044I 80448 80455 80462 80468 80475 7 4 638 80482 80489 80496 80502 80509 805I6 80523 80530 80536 80543 8 5 639 8o55o 80557 80564 80570 80577 80584 8059I 80598 80604 806I 9 5 No. 0 1 2 3 4 5 6 7 8 9 TABLE XXVI. [Page 179 Logarithms of Numbers. No. 6400 -7000. Log. 80618 84510. No. 0 1 2 3 4 5 6 7 8 9 640 806I 80625 80632 80638 8o645 80 52 80659 8o665 80672 80679 7 641 80686 80693 80699 80706 80713 80720 800726 8733 80740 80747 77 642 80754 80760 80767 80774 80781 80787 80794 8o80o 808o8 80814 2 643 80821 808.28 8o835 8o84i 8o848 8o855 80862 8 o868 80875 80882 3 2 644 80889 80895 80902 809o9 809 16 80922 80929 80936 80943 80949 4 3 645 80956 80963 8C969 80976 80983 809o0 80996 80oo3 8ioi0 81017 5 4 646 81023 81o3o 81037 8jo43 to5 05 810o7 8I064 81070 8I077 8io84 6 4 647 8o109 81097 81104 81111 1117 81124 81131 8II37 81144 8r115 7 5 648 8ii 8 8II64 81171 81178 8 8i849 8 8i98 81204 8121I 812I8 8 6 649 81224 8123I 81238 81245 81251 81258 8I265 81271 81278 81285 9 6 650 8291 818 8i325 8i3ii 83 83325 8333i 8i338 8I345 8I35i 651 81358 8I365 81371 81378 8i385 81391 8i398 81405 8141 8i418 652 81425 8I43 8i438 8i445 8145i 81458 81465 8I47I 81478 8485 653 81491 81498 81505 8I5 815i8 81525 8i53I 8I538 81544 8i55i 654 8558 81564 81571 81578 8I584 8I591 81598 81604 81611 81617 655 81624 8163 81637 8i644 8165I 81657 8I664 87 8 1 8I677 8684 656 8I690 81697 81704 81710 81717 81723 8I730 81737 81743 81750 657 81757 81763 8I770 81776 81783 81790 81796 8i803 81809 81816 658 81823 81829 81836 81842 8i849 8I856 8I862 8I869 8I875 81882 659 81889 8I895 81902 8 998 819I5 81921 81928 8I935 8194I 81948 66c 81954 81961 8I968 81974 81981 81987 81994 82000 82007 82014 66 8200 82027 8233 8040 82046 82053 82060 82066 82073 82079 662 82086 82092 82099 82105 82112 821 9 82125 8232 82 238 82145 663 82151 82158 82164 8217I 82178 82184 82191 82197 82204 82210 664 82217 82223 82230 82236 82243 82249 82256 82263 82269 82276 665 82282 82289 82295 82302 82308 82315 82321 82328 82334 82341 666 82347 82354 82360 82367 82373 82380 82387 82393 82400 82406 667 824I3 82419 82426 82432 82439 82445 82452 82458 82465 82471 668 82478 82484 82491 82497 82504 82510 82517 82523 82530 82536 669 82543 82549 82556 82562 82569 82575 82582 82588 82595 82601 670 82607 826I4 82620 82627 82633 82640 82646 82653 82659 82666 671 82672 82679 82685 82692 82698 82705 82711 82718 82724 82730 672 82737 82743 82750 82756 82763 82769 82776 82782 82789 82795 673 82802 82808 82814 82821 82827 82834 82840 82847 82853 82860 674 82866 82872 82879 82885 89 8289 82905 82911 829I8 82924 675 82930 82937 82943 82950 82956 82963 82969 82975 82982 82988 676 82995 83001 83008 83o04 83020 83027 83033 -3040 83046 83052 677 83059 83065 83072 83078 83085 83091 83097 830o4 8310o 83117 678 83123 83129 83i36 83142 83149 83155 83161 83168 83174 83181 679 83187 83193 83200 83206 8323 83219 83225 83232 83238 83245 680 83251 83257 83264 8327 83276 83283 83289 83296 83302 83308 68i 83315 83321 83327 83334 83340 83347 833 833583359. 83366 83372 682 83378 83385 83391 83398 83404 834o0 83417 83423 83429 83436 683 83442 83448 83455 8346i 83467 83474 83480 83487 83493 83499 684 83506 83512 83518 83525 8353 183537 83544 83550 83556 83563 685 83569 83575 83582 83588 83594 8360I 83607 836i3 83620 83626 6 686 83632 83639 83645 8365i 83658 83664 83670 83677 83683 83689 T 687 83696 83702 83708 83715 83721 83727 83734 83740 83746 83753 2 688 03759 83765 83771 83778 83784 83790 83797 83803 83809 838i6 3 2 689 83822 83828 83835 8384i 83847 83853 8386o 83866 83872 83879 4 2 690 83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 5 3 691 83948 83954 83960 83967 83973 83979 83985 83992 83998 84004 6 4 692 84o01 84017 84023 84029 84036 84042 84048 84055 8406i 84067 7 4 693 84073 84o8o 84086 84092 84098 84o05 841 184117 84123 84I30 8 5 694 84136 84142 84r48 84155 8416i 84167 84173 8418o 84i86 84192 9 5 695 84198 84205 8421 84217 84223 84230 84236 84242 84248 84255 696 84261I 84267 84273 84280 84286 84292 84298 84305 843 84317 697 84393 84330 84336 84342 84348 84354 84361 84367 84373 84379 698 84386 84392 84398 844o4 844io 84417 84423 84429 84435 84442 699 84448_ 84454 84460 84466 84473 84479 84485 8449i 84497 845o04 No. 0 1 2 3 4 5 6 7 8 9. No-Z. j. — 8 t 8 446 1 84 84479 6 7 8 9 Page 180] TABLE XXVI. Logarithms of Numbers. No. 7000 — 600. Log. 84510 — 88081. No. 0 1 2 3 4 5 6 7 8 9 700 845I0 845I6 84522 84528 84535 8454I 84547 53 844559 84566 7 701 84572 84578 84584 84590 84597 84603 84609 846i5 84621 84628 I 702 84634 84640 84646 84652 84658 84665 84671 84677 84683 84689 2 I 703 184696 84702 84708 84714 84720 84726 84733 84739 84745 8475 3 2 704 84757 84763 84770 84776 84782 84788 84794 848oo 84807 84813 4 3 705 84819 84825 8483I 84837 84844 84850 84856 84862 84868 84874 5 4 706 84880 84887 84893 84899 84905 849 I 84917 84924 84930 84936 6 4 707 84942 84948 84954 84960 84967 84973 84979 84985 8499i 84997 7 5 708 85003 85009 85oi6 85022 85028 85o34 85040 85046 85052 85058 8 6 709 85o65 85071 85077 85083 85089 85095 85foi 85107 8514 851I20 9 6 710 85I26 85132 85i38 85I44 85150 85156 85i63 85169 85175 8518I 7II 85187 85193 85199 852o5 85211 85217 85224 85230 85236 85242 712 85248 85254 85260 85266 85272 85278 85285 85291 85297 853o3 713 85309 853i5 85321 85327 85333 85339 85345 85352 85358 85364 7I4 85370 85376 85382 85388 85394 854oo 85406 85412 854I8 85425 715 8543 85437 85443 85449 85455 85461 85467 85473 85479 85485 716 85491 85497 85503 85509 855i6 85522 85528 85534 85540 85546 717 85552 85558 85564 85570 85576 85582 85588 85594 85600 85606 718 85612 85618 85625 85631 85637 85643 85649 85655 8566 185667 719 85673 85679 5685 8 85691 85697 85703 85709 85715 85721 85727 720 85733 85739 85745 8575 85757 85763 85769 85775 85781 85788 721 85794 858oo 858o6 85812 85818 85824 85830 85836 85842 85848 722 85854 85860 85866 85872 85878 85884 85890 85896 85902 85908 723 85914 85920 85926 85932 85938 85944 85950 85956 85962 85968 724 85974 85980 85986 85992 859986oo4 86oo0 86o06 86022 86028 725 86034 86040 8646 86052 86058 86052 88 8664 86070 86076 86082 86088 6 726 86094 86Ioo 86io6 86112 86 8 86124 86i30 86136 86141 86147 i 727 86i53 86159 86165 86I7 86177 86I83 86I89 86195 86201 86207 2 I 728 86213 86219 86225 86231 86237 86243 86249 86255 86 6267 3 2 729 86273 86279 86285 8629I 86297 86303 86308 86314 86320 86326 4 2 730 86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 5 3 73I 86392 86398 86404 864io 864I5 86421 86427 86433 86439 86445 6 4 732 8645i 86457 86463 86469 86475 8648r 86487 86493 86499 86504 7 4 733 86510 86516 86522 86528 86534 86540 86546 86552 86558 86564 8 5 734 86570 86576 8658i 86587 86593 86599 86605 86611 86617 86623 9 5 735 86629 86635 8664 86646 86652 86658 86664 86670 86676 86682 736 86688 86694 86700 86705 86711 86717 86723 86729 86735 86741 737 86747 86753 86759 86764 86770 86776 86782 86788 86794 86800 738 86806 86812 86817 86823 86829 86835 8684 186847 86853 86859 739 68684 86870 86876 86882 86888 86894 86goo 86906 869 1 86917 740 86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 74I 86982 86988 86994 86999 87005 87011 87017 87023 87029 87035 742 87040 87046 87052 87058 87064 87070 87075 87081 87087 87093 743 87099 871o5 87111 87116 87122 87128 87134 874o0 87I46 8715I 744 87 857 87163 87I69 87175 87181 87186 87192 87198 87204 87210 745 872I6 87221 87227 87233 87239 87245 87251 87256 87262 87268 746 87274 87280 87286 87291 87297 87303 87309 87315 87320 87326 747 87332 87338 87344 87349 87355 87361 87367 87373 87379 87384 748 87390 87396 87402 87408 87413 874I9 87425 8743I 87437 87442 749 87448 87454 87460 87466 8747T 87477 87483 87489 87495 87500 750 87506 87512 87518 87523 8752987 87535 8754 87547 755287558 5 751 87564 87570 87576 87581 87587 87593 87599 87604 876i0 876I6 T- I 752 87622 87628 87633 87639 87645 87651 87656 87662 87668 87674 2 J 753 87679 87685 87691 87697 87703 87708 87714 87720 87726 87731 3 2 754 87737 87743 87749 87754 87760 87766 87772 87777 87783 87789 4 2 755 877 878 87795 8780 87806782 [87818 87823 87829 87835 8784/ 87846 5 3 756 87852 87858 87864 87869 87875 87881 87887 87892 87898 87904 6 3 757 87910 87915 87921 87927 87933 87938 87944 87950 87955 887961 7 4 758 87967 87973 87978 87984 87990 87996 88o00 88007 88o03 88oi8 8 4 759 88024 88o03 88036 88041 88047 88053 88058 88064 88070 88076 9 5 No. 0 1 2 3 4 5 6 7 8 9 TABLE XXVI. [Page 181 Logarithms of Numbers. No. 7600 8200. Log. 88081 —91381. No. 0 1 2 3 4 5 6 7 8 9 760 880o8 88087 88093 88098 88o14 88110 88116 88121 88127 88i33 6 76I 88I38 88I44 88I50 88I56 88i6i 88I67 88173 88178 88i84 88I90 ~ 762. 88I95 8820o 207 2 882 7 882 24 88 230 88235 8824 88247 2 763 88252 88258 88264 88270 88275 88281 88287 88292 88298 88304 3 2 764 88309 88315 88321 88326 88332 88338 88343 88349 88355 88360 4 2 765 88366 88372 88377 88383 83988389 8395 88400 88406 88412 8847 5 766 88423 88429 88434 88440 88446 8845 88457 88463 88468 88474 6 4 767 88480 88485 88491 88497 88502 88508 88513 88519 88525 88530 7 4 768 88536 88542 88547 88553 88559 88564 88570 88576 88581 88587 8 5 769 88593 88598 88604 886io 88615 88621 88627 88632 88638 88643 9 5 770 88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 771 88705 88711 88717 88722 88728 88734 88739 88745 88750 88756 772 88762 88767 88773 88779 "88784 88790 8879588o 88807 8882 888 773 88 8824 888 88835 8884 8846 8852 88829 88835 888468 774 88874 88880 88885 8889I 88897 902 8890 08 88913 88919 88925 775 883 8893 8936 8894i 88947 88953 8895889 964 88969 8895 8898 776 88986 88992 88997 89003 89009 89 89020 89025 89031 89037 777 89042 89048 89053 89059 89064 89070 89076 9 898 8987 89092 778 89098 89104 89109 89I5 89120 89126 89131 89137 89143 89148 779 89I54 89159 89165 89170 89176 89182 89187 89193 89198 89204 780 89209 89215 89221 89226 89232 89237 89243 89248 89254 89260 781 89265 89271 89276 89282 89287 89293 89298 89304 89310 89315 782 89321 89326 89332 89337 89343 89348 89354 89360 89365 8937I 783 8937689382 89382 7 89393 89398 89404 89409 89415 8042 89426 784 89432 89437 89443 89448 89454 89459 89465 89470 89476 8948I 785 89487 89492 89498 89504 89509 89515 89520 89526 89531 89537 786 89542 89548 89553 89559 89564 89570 89575 89581 89586 89592 787 89597 89603 896o9 89614 89620 89625 89631 89636 89642 89647 788 89653 89658 89664 89669 89675 89680 89686 89691 89697 89702 789 89708 89713 89719 89724 89730 89735 89741 89746 89752 89757 790 89763 89768 89789 7749779 89785 89790 89796 89801 89807 89812 791 89818 89823 89829 89834 89840 89845 89851 89856 89862 89867 792 89873 89878 89883 89889 89894 899)00 89905 899 1 89916 89922 793 89927 89933 89938 89944 89949 89955 89960 89966 89971 89977 794 89982 S9988 89993 89998 90004 90009 90015 90020 90026 90031 795 9oo37 goo42 go90048 90053 g00oo59 9064 90069 90075 90oo080 90086 796 9c091 90097 90102 90108 90113 90119 90124 90129 90135 90140 797 o9046 90151 901 57 90162 90168 90173 90179 90184 90189 90195 798 90201 90)206 90211 90217 90222 90227 90233 902.38 90244 90249 799 90255 902o6 90o266 90271 90276 90282 90287 90293 90298 90304 -80 90309 90314 9(0320 -90325 90331 90336 90342 90347 90352 90358 8oI 90363 90369 90374 90380 90385 90390 90396 90401 90407 90412 802 90417 90423 90428 90434 90439 90445 90450 90455 90461 90466 803 90472 90477 90482 90488 90493 90499 90504 90509 g9055 90520 8o4 90526 9053 90536 90542 90547 90553 go558 90~563 90569 90574 8o5 90580 9 o5 9 585 9o9 0596 9060I 90607 90612 90617 90623 o90628 5 806 90634 o90639 90644 90650 90655 90660 90666 90671 90677 90682 J 807 90687 90693 90698 90703 90709 907I4 90720 90725 90730 90736 2 1 80o 90741 90747 90752 90757 90763 90768 90773 90779 90784 90789 3 2 809 90795 90800 90806 90811 90816 90822 90827 9832 90838 90843 4 810 90849 90854 90859 90865 90870 90875 9088i 90886 o90891 9097 5 3 81r 90902 90907 90913 19098 90924 90929 90934 90940 90945 9o950 6 3 812 90956 90961 90966 9972 90977 90982 90988 90993 90998 91004 7 4 8I3 9oo009 g914 91020 91025 9030 91036 91041 91046 91052 91057 8,4 8I4 91062) gi68 973 901078 91084 91089 91094 91100 9105 91110 915 815 91I6 9121 91126 9I132 91I37 9142 9 148 91153 91158 91I64 816 91169 9II74 |9180 91 85 9190 91196 91201 91206 912 92129217 817 91222 91228 91233 91238 91243 91249 91254 91259 91265 91270 818 91275 9I281 91286 91291 91297 91302 91307 91312 91318 91323 819 91938 91334 91339 9I344 91350 9I355 9I360 9365 9I371 9376 No. 0 l 2 3 5 6 7 8 Page 182] TABLE XXVI. Logarithms of Numbers. No. 8200 —-8800. Log. 91381- 94448. No. 0 1 2 3 4 5 7 8 9 820 91381 91387 91392 9397 91403 91408 9143 191418 91424 91429 6 82I 91434 91440 9I445 91450 91455 91461 91466 91471 91477 91482 I'822 91487 91492 9I498 91503 91508 91514 91519 91524 91529 91535 2 823 91540 91545 91551 91556 91561 91566 91572 91577 91582 91587 3 824 9I593 9I598 91603 91609 91614 91619 91624 91630 91635 91640 4 825 164516 9I65 656 91661 91666 91672 9I67 1677 91689 91687 9693 5 3 826 91698 91703 9I709 91714 91719 9I724 9T730 91735 91740 9I745 6 4 827 91751 91756 91761 91766 91772 9I777 91782 91787 9I793 9I798 7 4 828 9I803 91808 91814 918I9 9I824 9I829 91834 91840 91845 91850 8 5 829 91855 91861 91866 91871 91876 91882 91887 91892 91897 91903 9 830 91908 9I913 91918 91924 91929 9 91934 9 1939 9944 915 919955 831 91960 91965 9197I 9I976 19 998 1986 91991 91997 92002 92007 832 92012 92018 92023 92028 92033 92038 92044 92049 92054 92059 833 92065 92070 92075 92080 92085 92091 92096 92101 92106 92111 834 921I73 92122 92I27 921 92137 92143 92148 92153 92158 92I63 835 92169 92174 92179 92184 -92189 92195 92200 92205 92210 92215 836 92221 92226 92231 92236 9224I 92247 92252 92257 92262 92267 837 92273 92278 92283 92288 92293 92298 92304 92309 92314 92319 838 92324 92330 92335 92340 92345 92350 92355 9236I 92366 9237T 839 92376 92381 92387 92392 92397 92402 92407 924I12 924I8 92423 840 92428 92433 92438 92443 92449 92454 92459 9 2464 92469 92474 841 92480 92485 92490 92495 92500 992505 925II 92516 92521 92526 842 92531 92536 92542 92547 92552 92557 92562 92567 92572 92578 843 92583 92588 92593 92598 92603 92609 9 2614 926 92624 92629 844 92634 92639 92645 92650 92655 92660 92665 92670 92675 92681 845 92686 92691 92696 92701 92706 92711 92716 92722 92727 92732 5 846 92737 92742 92747 92752 92758 92763 92768 92773 92778 92783 847 92788 92793 92799 92804 92809 928I4 92819 92824 92829 92834 I 848 92840 92845 92850 92855 92860 92865 92870 92875 92881 92886 2 849 9289 6 92 929 92909296 92911 92916 92921 92927 92932 92937 3 2 850 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 5 85i 92993 92998 93003 93008 930I3 93018 93024 93029 93034 93039 6 3 852 93044 93049 93054 9 93059 93o64 93069 93075 93080 93085 93090 7 4 853 93095 93100 9305931 93o 93115 93120 93125 93131 93136 9341I 8 4 854 93146 93I51 93I56 93I6I 93166 93171 93176 93I8I 93I86 93192 9 5 855 93I97 93202 93207 93212 932I7 93222 93227 9332 93 3237 93242856 93247 93252 93258 93263 93268 93273 93278 93283 93288 93293 857 93298 93303 93308 93313 93318 93323 93328 93334 93339 93344 858 93349 93354 93359 93364 93369 93374 93379 93384 93389 93394 859 93399 93404 93409 93414 93420 93425 93430 93435 93440 93445 86 930 93459345 93460 93465 93470 93475 9348 93485 9490 93495 861 93500 93505 93510 93515 93520 93526 93531 93536 93541 93546 862 9355I 93556 93561 93566 93571 93576 9358I 93586 9359I 93596 863 93601 93606 93611 93616 93621 93626 93631 93636 93641 93646 864 93651 93656 93661 93666 93671 93676 93682 93687 93692 93697 865 93702 93707 93712 93717 93722 93727 93732 93737 93742 93747 866 93752 93757 93762 93767 93772 93777 93782 93787 93792 93797 867 93802 93807 938I2 93817 93822 93827 93832 93837 93842 93847 868 93852 93857 93862 93867 93872 93877 93882 93887 93892 93897 869 93902 93907 93912 93917 93922 93927 93932 93937 93942 93947 870 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 4 871 94002 94007 94012 940I7 94022 94027 94032 94037 94042 94047 _ 872 94052 94057 94062 94067 94072 94077 94082 94086 94091 94096 I' 873 9IO 94ro6 94111 94116 94121 94126 94I3I 94136 94141 94146 2 874 94151 94156 9416I 94166 94171 94176 9418I 94186 94191 94T96 3 4 2 875 94201 94206 94211 94216 9422194226 94231 9236 4 9240 94245 5 876 94250 94255 94260 94265 94270 94275 94280 94285 94290 94295 6 877 94300 94305 943io 943I5 94320 94325 94330 94335 94340 94345 / 878 94349 94354 94359 94364 94369 94374 94379 94384 94389 94394 8 3 879 94399 94404 94409 94414 94419 94424 94429 94433 94438 94443 9 4 No. 0 1 2 3 4 5 6 7 8 9 TABLE XXVI. [Page 183 Logarithms of Numbers. No. 8800 9400. Log. 94448 97313. No. 0 1 2 3 4 5 6 7 8 9 88o 94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 5 88i 94498 94503 94507 94512 94517 94522 94527 532 94537 94542 __. 882 94547 p4552 94557 94562 94567 9457I 94576 9458 94586 94591 I 883 94596 9460o 946o6 94611 9 9466 9462I 94626 94630 94635 94640 2 884 94645 94650 94655 9466o 94665 94670 94675 94680 94685 94689 3 2 885 94694 94699 94704 94709 94714 94719 94724 94729 94734 94738 5 3 886 94743 94748 953 75 9475 3 94 768 94773 94778 94783 94787 6 3 887 94792 94797 94802 94807 948T2 94817 94822 94827 94832 94836 7 4 888 94841 94846 94851 94856 94861 94866 94871 94876 94880 94885 8 4 889 94890 94895 9490o 94905 9491o 94915 94919 94924 94929 94934 9 5 890 94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 891 94988 94993 94998 95002 95007 95012 95)17 95022 95027. 95032 892 95036 95.04I 95046 95051 95056 9506 95066 9507I 95(75 95080 893 95085 95090 95095 95100 95I05 95I09 95114 95II9 95124 95129 894 95134 95139 9543 95148 951 95395158 95163 95168 95173 95177 895 95182 95187 95192 95197 95202 95207 95211 95216 95221 95226 896 95231 95236 95240 95245 95250 95255 95260 95265 95270 95274 897 95279 95284 95289 95294 95299 95303 95308 95313 95318 95323. 898 95328 95332 95337 95342 95347 95352 95357 9536I 95366 95371 899 95376 95381 95386 95390 95395 95400 95405 954o1 95415 95419 900 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 901 95472 95477 95482 95487 95492 95497 95501 95506 95511 95516 902 95521 95525 95530 95535 95540 95545 95550 95554 95559 95564 903 95569 95574 95578 95583 95588 95593 95598 956o2 95607 95612 904 956I7 95622 95626 95631 95636 95641 95646 95650 95655 95660 905 95665 95670 95674 95679 95684 95689 95694 95698 95703 95708 906 957I3 95718 95722 95727 95732 95737 95742 95746 9575I 95756 907 9576I 95766 95770 95775 95780 95785 95789 95794 95799 95804 908 95809 95813 95818 95823 95828 95832 95837 95842 95847 95852 909 95856 95861 95866 95871 95875 9588o 95885 95890 95895 95899 9I0 95904 95909 95914 95918 95923 95928 95933 95938 95942 95947 911 95952 95957 9596I 95966 9597I 95976 95980 95985 95990 95995 912 95999 96004 96009 960I4 96019 96023 96028 96033 96038 96042 913 96047 96052 96057 9606I 96066 96071 96076 96080 96085 960o 9 914 96095 96099 96104 96I09 96 14 96 18 96123 96128 96I33 96137 915 96142 96I47 96152 96I56 9616i 96I66 96171 96175 968o1 96I85 9I6 96190 96194 96I99 96204 96209 962I3 96218 96223 96227 96232 917 96237 96242 96246 9625I 96256 96261 96265 96270 96275 96280 9g8 96284 96289 96294 96298 96303 96308 96313 963I7 96322 96327 9I9 96332 96336 96341 96346 96350 96355 96360 96365 96369 96374 920 9637 969384 96388 96393 96398 96402 96407 96412 96417 9642I 92I 96426 9643I 06435 96440 96445 96450 96454 96459 96464 96468 922 96473 96478 96483 96487 96492 96497. 96501 96506 965II 96515 923 96520 96525 96530 96534 96539 96544 96548 96553 96558 96562 924 96567 96572 96577 96581 96586 9659I 96595 96600 96605 96609 925 966I4 96619 96624 96628 96633 96638 96642 96647 96652 96656 4 926 96661 96666 96670 96675 96680 96685 96689 96694 96699 96703 -— 0 927 96708 96713 96717 96722 96727 96731 96736 96741 96745 96750 2 928 96755 96759 96764 96769 96774 96778 96783 96788 96792 96797 3 929 96802 96806 96811 968I6 96820 96825 96830 96834 96839 96844 4 93o 96848 96853 96858 96862 96867 96872 96876 9688I 96886 96890 5 2 931 96895 96900 96904 96909 96914' 96918 96923 96928 96932 96937 5 2 932 96942 96946 9695I 96956 96960 96965 96970 96974 96979 96984 7 3 933 96988 96993 96997 97002 97007 97011 970I6 97021 97025 97030 8 3 934 97035 97039 97044 97049 97053 97058 97063 97067 97072 97077 9 4 935 97o8 976 97890 795 97100 97104 7I0 974 1 4 9718 97123 936 97I28 97132 97137 971 974 46 9756 1 97155 97I5 97165 976 69 937 97174 97179 97183 97188 97192 97197 97202 97206 97211 97216 938 97220 97225 97230 97234 97239 97243 97248 97253 97257 97262 939 97267 97271 97276 97280 97285 97290 97294 97299 97304 97308 No. I 0 1 2 3 4 5 6 7 8 9 Page 184] TABLE XXVI. Logarithms of Numbers. No. 9400 ~- 10000. Log. 97313 —- 99996. No. 0 1 2 3 4 5 6 7 8 9 940 97313 97317 9732297327 97331 97336 9734 97345 97350 97354 941 c7359 97364 97368 97373 97377 97382 97387 973 9397396 97400. 942 97405 97410 97414 97419 97424 97428 97433 97437 97442 97447 2 943 97451 97456 97460 97465 97470 97474 97479 97483 97488 7493 3 2 944 97497 97502 97506 975II 9751 6 975 20 97525 97529 97534 97539 4 2 94-5-5 97543 97548 97552 97557 97562 97566 97571 97575 97580 97585 5 3 946 97589 97594Z 97598 97603 97607 97612 97617 97621 97626 97630 6 3 947 97635 97640 97644 97649 97653 97658 97663 97667 97672 97676 7 4 948 97681 97685 97690 97695 97699 97704 97708 97713 97717 97722 8 4 949 97727 9773i 97736 97740 97745 97749 97754 97759 97763 97768 9 5 950 977,2 97777 97782 97786 97791 97795 97800 97804 97809 97813 951 97818 97823 97827 97832 97836 97841 97845 97850 97855 97859 952 97864 97868 97873 97877 97882 97886 97891 97896 97900 97905 953 97909 97914 97918 97923 97928 97932 97937 97941 97946 97950 954 97955 97959 97 9798 97968 97973 97978 97982 97987 97991 97996 955 980oo 98o05 98o0 98014 98')1 98023 98028 98032 98037 98041 956 98046 98050 98055 98059 98064 98068 98073 98078 98082 98087 957 98091 98096 98oo100 98o105 9810998II4 98118 98123 98127 98132 958 98137 98141 98146 98150 98155 98159 98164 98168 98173 98177 959 98182 98186 98191 98195 98200 98204 98209 98214 98218 98223 960 98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 961 98272 98277 98281 98286 98290 98295 98299 98304 98308 98313 962 98318 98322 98327 98331 98336 98340 98345 98349 98354 98358 963 98363 98367 98376 98381 98385 98390 98394 98399 98403 964 98408 984-12 98417 98421 98426 9843o, 98435 98439 98444 98448 965 98453' 98457 98462 98466 98471 98475 98480 98484 98489 98493 906 98498 98502 98507 98511 98516 98520 98525 98529 98534 98538 967 98543 98547 98552 98556 98561 98565 98570 98574 98579 98583 968 98588 98592 98597 98601 986o5 98610 98614 98619 98623 98628 96 98632 98637 986 98 696 9865 98655 98659 98664 98668 98673 9709 98677 82 9868641 986 98650 98655700 98704 98096 9868 98717 971 98722 98726 98731 98735 98740 98744 98749 98753 98758 98762 972 98767 98771 98776 98780 98784 98789 98793 98798 98802 98807 973 98811 98816 98820 98825 98829 98834 98838 98843 98847 98851 974 98856 9886o 98865 98869 98874 98878 98883 98887 98892 98896 975 989oo 98905 98909 98914 98918 98923 98927 98932 98936 98941 976 98945 98949 98954 98958 98963 98967 98972 98976 98981 98985 977 98989 98994 98998 99003 99007 99012 99016 99021 99025 99029 978 99034 99038 99043 99047 99052 99056 99061 99065 99069 99074 979 99078 99083 99087 99092 99096 99100 99105 99109 99114 99118 98o 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 98 99 9967 99171 99176 99180 99185 99189 99193 99198 99202 99207 982 99211 99216 99220 99224 99229 99233 99238 99242 99247 99251 983 99255 99260 99264 99269 99273 99277 99282 99286 99291 99295 984 993oo300 99304 99308 99313 99317 99322 99326 99330 99335 99339 985 99344 99348 99352 99357 99361 99366 99370 99374 99379 99383 4 986 99388 99392 99396 99401 99405 99410o 99414 99419 99423 99427 -' 987 99432 99436 99441 99445 99449 99454 99458 99463 99467 9947I oI 988 99476 99480 99484 99489 99493 99498 99502 99506 99511 99515 2' 989 99520 99524 99528 99533 99537 99542 99546 99550 99555 99559 3 1 990 99564 99568 99572 99577 9958I 997 999585 590 9594 99599 99603 4 2 991 99607 99612 99616 99621 99625 99629 99634 99638 99642 99647 5 2 992 99651 99656 99660 99664 99669 99673 99677 99682 99686 99691 6 2 993 99695 99699 99704 99708 99712 99717 99721 99726 99730 99734 7 3 994 99739 99743 99747 99752 99756 9976o 99765 99769 99774 99778 3 995 99782 99787 99791 9979 9 99800 99804 99808 99813 99817 99822 996 99826 99830 99835 99839 99843 99848 99852 99856 99861 99865 997 99870 99874 99878 99883 99887 99891 99896 99900 99904 99909 998 99913 99917 99922 99926 99930 99935 999399944 99948 99952 999 99957 99961 99965 99970 99974 99978 99983 99987 99991 99996 No. 0 1 2 3 415 6 7 8 9 TABLE XXVII. [ge l85 Log. Sines, Tangents, and Secants. 00 179~ M I our A.M. Hour P.r. Sine. Difi'.l/Cosecant. Tangent. Diff. 1'lCotangent Secant. Cosine. N 012 0 0 0 0 Inf. Neg. Infinite. Inf. Neg. Infinite. 0.00000 10.00000 60o 1 ii 5952 o 86.463731301031i3.53627 6.46373/30oT313.53627 oo0oo 00000 59 2 5944 i6 76476 17609 23524 7647617609 23524 ooooo 00000 58 3 59 36j 0 24 94085 12494 o5915 94085 12494 05915 00000 00000 57 4 59 28 0 32 7.06579 9691 12.93421 7.6579 969 12.93421 oooo ooooo00000 56 51 159 20 0 0 40 7.16270 7918 12.83730 7-16270 7918 12.8373o0 i0 ooooo o. ooooo 55 6 59 12 0 48 241881 6694 75812 24188 6694 75812 00000 00000 54 7 59 41 0 56 30882 58oo00 69118 30882 5800 69118 00000 ooooo 53 8 58 56 4 36682 5115 63318 36682 5115 633i8 00000 00000 52 9 58 48 I 12 41797 4576 58203 41797 4576 58203 00000 00000oooo 5 io I 58 40o0 1 20 7.46373 41391i2.53627 7.46373 413912.53627 1o.looooo o.ooooo 5o II 58 32 I 28 50512 3779 49488 50512 3779 49488 00000 00000 49 12 58 24 1 36 54291 3476 45709 54291 3476 45709 00000 00000 48 13 58 16 1 44 57767 3218 42233 57767 3219 42233 00000 0oooo 47 14 58 8 I 52 60985 2997 39015 6o0986 2996 39014 00000 00000 46 15 i1 580 0-02 07.63982 2802-12.36018 7.63982 2803 12.36018 I0.000000.00000 45 16 57 52 2 8 66784J 2633 33216 66785 2633 33215 ooooo ooooo 44 17 57 44 2 16 69417 2483 3o583 69418 2482 30582 0000oooo 999999 43 i8 57 36 2 24 71900 2348 281oo 71900 2348 28oo00 00001 99999 42 19 57 28 2 32 74248 2227 25752 74248 2228 25752 00001 99999 4[ 20 11 57 20-0 2 40 7.76475 2119 12.23525 776476 211912.23524 Io.oooo00001 9.99999 4 21 57 12 2 48 78594 2021 21406 78595 2020 21405 0000I 99999 39 22 57 4 2 56 8o6i5 1930 19385 8o6i5 1931 19385 00001 99999 38 23 56 56 3 4 82545 1848 17455 82546 1848 17454 0000oooo 99999 37 24 56 48 3 12 84393 1773 15607 84394 1773 156o06 00001 99999 36 2511i 56 4o 0 3 20 7.86166 1704 12.13834 7.86167 170412.3833 i. o.ooo ol 9.99999 35 26 56 32 3 28 87870 1639 -12130 87871 1639 12129 oooo I 99999 34 27 56 24 3 36 89509 1579 10491 89510o 1579 10490 00001 99999 33 28 56 16 3 44 9ro88 1524 o8912 91089 1524 o8911 0000oooo 99999 32 29 56 8 3 52 92612 1472 07388 92613 1473 07387 00002 99998 31 3011o ii 56 o 0 4 0 7.94084 1424 12.05916 7.94086 142412.05914 10.00002 9.99998 -3o 31 55 52 4 8 95508 1379 04492 95510 1379 04490 00002 99998 29 32 55 44 4 16 96887 1336 0311ii3 96889 i336 031ri 00002 99998 28 33 55 36 4 24 98223 1297 01777 98225 1297 01775 00002 99998 27 34 55 28 4 32 99520 1259 oo00480 99522 1259 00478 00002 99998 26 35 11 55 20 0 4 4o 8.00779 122311.99221 8.00781 1223 11.99219 10.00002 9.99998 25 36 5512 4 48 02002 1190 97998 02004 1190 97996 00002 99998 2 37 55 4 4 56 03192 1158 96808 03194 115 96806 oooo00003 99997 23 38 54 56 5 4 04350 1128 95650 04353 1128 95647 oooo00003 99997 22 39 54 48 5 12 05478 1100oo 94522 o548i 11oo 94519 oooo3 99997 21 40o 11 54 40o 5 28.06578 1072 11.93422 8.o658i 107211.93419 io.oooo0 3 9.99997 20 41 54 32 5 28 0765 1046 92350 07653 1047 92347 0000oooo3 99997 19 42 54 24 5 36 08696 1022 91304 8700 1022 91300 oooo00003 99997 18 43 54 16 5 44 09718 999 90282 09722 998 90278 oooo3 99997 17 44 54 8 5 52 10717 976 89283 10720 976 89280 oo000o4 99996 i6 45 11 54. 0 o6 o 8.11693 954 11.88307 8.11696 95511.88304 10.00004 9.99996 iS 46 53 52 6 8 12647 934 87353 12651 934 87349 00004 99996 14 47 53 44 6 i6 i3581 914 86419 13585 915 86415 00004 99996 13 48 53 36 6 24 14495 896 855o5 145oo00 895 85500oo 00004 99996 12 49 53 28 6 32 15391 877 84609 15395 878 84605 00004 99996 ii 501o ii 53 20 0 6 4o8.16268 860 11.83732 8.16273 86011i.83727 10.00005 9.99995 10 51 53 12 6 48 17128 843 82872 17133 843 82867 00005 99995 9 52 534 4 6 56 17971 827 82029 17976 828 82024 oooo00005 99995 8 53 52 56 7 4 18798 812 81202 884 812 81196 o000oo5 99995 7 54 5:2 48 7 12 19610 797 803go 19616 797 8384 oooo00005 99991 6 5511i 52 4 00 7 20 8.20407 78211.79593 8.20413 782 11.795871 o.oooo6 9.99994 5 56 52 32 7 28 21189 769 78811 21195 769 78805 oooo00006 99994 4 57 52 24 7 36 21958 755 78042 21964 756 78036 oo00006f 99994 3 58 52 16 7 44 22713 743 77287 22720 742 77280 00006 99994 2 59 52 8 7 52 23456 730 76544 23462 730 76538 000oooo06 99994 1 60 52 0 8 0 24186 717 75814 24192 718 758o8 00007 99993 0 M Hourp.1.1HourA.M. Cosine. Diff. 1 Secant. CotangentjDiff. 1' Tangent. Cosecant. Sine. M 900 890 24 Page 186] TABLE XXVII. Log. Sines, Tangents, and Secants. 10 ____ _______ 178 Hour-A.M.lHoUr p.M. Sine. _Diffl Cosecant. Tangent. Diff. 1 Cotangent Secant. Cosine. o 52 o 8 o 8.24186 717 11.75814 8.24192 718 11.75808 10. 00007 999993 6o 5I 52 8 8 24903 706 75097 24910 706 75090 00007 99993 59 2 5 44 8 16 25609 695 74391 25616 696 74384 00007 99993 58 3 51 36 8 24 263041 684 73696 26312 684 73688 00007 99993 57 4 51 28 8 3)2 26988 673I 730r2 26996 673' 73004 0ooo8 999921 56 51 I 51 20-0 8 4o 8.27661 663 41.72339 8.27669 663 11.72331 1o.oooo89.99992 55 6 5r 12 8 48 28324 653 71676 28332 654 71668 00008 99992 54 7 51 4 8 56 28977 644 71023 28986 643 71014 00008 99992 53 8 5o 56 9 4 29621 634 70379 29629 634 70371 00008 99992 52 9 50 48 9 12 30255 624 69745 30263 625 69737 00009 99991 51 10 oI 50 4o 0 9 20 8.30879 6i6 ii.69[2 8.30888617 11.69112 10.00009 9.9999 50o ii 5o 32 9 28 31495 6o8 685o5 3i505 607 68495 0000o 99991 491 12 50o 24 9 36 32103 599 67897 32112 599 67888 00010 99990 48 i3 5o0 6 9 44 32702 590 67298 32711I 591 67289 00010 99990 47,14 50 8 9 52 33292 583 66708 33302 584 66698 000ooo 9999 46 1511 5o 0 0 0 o 8.33875575 11.66125 8.33886 575 11.66114 10.00010 9.9990 45 6 49 52 10o 8 344501 568 6555o 3446i 568 65539 00011 99989 441 17 49 44 0o 16 35o018 560 64982 35029 561 64971 oooii 99989 43 I8 49 36 io 24 35578 553 64422 35590 553 644I ooo ii 9998942 19 49 28 10 32 36r3i 547 63869 36143 546 63857 00011ii 99989 41 20 II 49 20 10 4o 8.36678 539 11.63322 8.36689 540 11i.63311 10.000129.99988 40 21 49 12 i 48 37217 533 62783 37229 533 62771 00012 99988 39 22 49 4 10 56 37750 526 6225o 37762 5247 62238 00012 9998838 23 48 56 1Ii 4 38276 520 61724 38289 520 61711 00013 99987 37 24 48 48 ii 12 38796 514 61204 38809 54 61191 ooo00013 99987 36 25 48 4o0 i11 20 8.39310 5o8 i1.669 8.39323 509 11.6o677 10.00013 9.99987 35 26 48 32 1I 28 39818 502 60182 39832 502 6o0168 ooo4 99986 34 27 48 24 ii 36 40320 496 59680 40334 496 59666 oooi4 99986 33 28 48 i6 ii 44 4o8i6 491 59184 4o83o 499 59170 oooi4 99986 32 29 48 8 11 52 41307 45 58693 41321 486 58679 ooo00015 99985 31 30o 1 48 00 12 o 8.41792 480 11.58208 8.41807 480 11.58193 Io.oo000159.99985 3o0 31 4 52 12 8 42272 474 57728 42287 475 5773 ooo0005 99985 29 32 47 44 1 2 16 42746 470 57254 42762 470 57238 o000oo6 99984 28 33 47 36 12 24 43216 464 56784 43232 464 56768 ooo00016 9998427 34 47 28 12 32 436801 459 56320 43696 460 563o4 ooo6 99984 26 35- I 47 20 0 12 4o 8.44139 455 11.5586i 8.44i56 455 1i.55844 1i0.000I79.99983 25 36 47 2 12 48 44594 45o 55406 446r 450 55389 00017 99983 24 37 q 4 12 56 450o44 445 54956 45061 446 54939 00017 99983 23 38 46 56 13 4 45489 44 54511i 45507 441 54493 00018 99982 221 39 46 48 13 12 45930 436 54070 45948 437 54052 00018 99982 211 40114640 013o 208.46366 433 11.53634 8.46385 432 11.53615 10.00018ooo 9.99982 20 4o 1 46 32 13 28 46799 427 53201 46817 428 53i83 00019 99981 19 42 46 24 13 36 47226 424 52774 47245 424 52755 00019 99981 181 43 46 I6 I3 44 47650 419 52350 47669 420 52331 00019 99981 17 44 46 8 13 52 48069 4i6 51i93i 48089 4i6 51911ii 00020 99980 i6 -45i1146 0 014 0 8.48485 411 11.51515 8.48505 412 11i.51495 10.000209.99980 i5 46 45 52 1i4 8 48896 408 511io4 48917 408 5o1083 00021 99979 i4 47 45 44 14 16 4 49304 4o4 50696 4932 404 50675 00021 99979 3 48 45 36 14 24 49708 4oo 50292 49729 401 50271 00021 99979 12 49 45 28 14 32 50108 396 49892 50130 397 49870 00022 99978 1i 50 1 45 20 o 14 40 8.50oo4 393 11.49496 8.50527 393 1.49473 10.00022-9.99978 io 51 45 12 14 48 50897 390 49103 50o920o 390 49080 00023 99977 9 52 45 4 14 56 51287 386 48713 51310 386 48690 00023 99977 8 53 44 56 15 4 51673 382 48327 51696 383 48304 00023 99977 7 54 44 48 15 12 52055 379 47945 52079 380 47921 00024 99976 6 55 1iI 44 40o 015 20 8.52434 376 11.47566 8.52459 376 1ii.4754i 10.000249.99976 5 56 44 32 15 28 52810o 373 47190 52835 373 47i65 00025 99975 4 57 44 24 15 36 53183 369 46817 53208 370 46792 00025 99975 3 58 44 i6 15 44 53552 367 46448 53578 367 46422 00026 99974 2 59 44 8 15 52 53919 363 4608r 53945 363 46055 00026 99974 60 44 0 i6 0 54282 36o 4578 543o8 36i 45692 00026 99974 o M Hourp.M.HoUrA.M. Cosine. Diffi. Secant. Cotangent Diff. 1 Tangent. Cosecant. -Sine. M 910 880 TABLE XXVII. [Pge 187, Log. Sines, Tangents, and Secants. 20 1770 Am HourA.M.Hourp.m. Sine.'Diff.l Cosecant. Tangent. DifflCotang ent Secant. Cosine. M - 1144 o 6 i6 08.54282' 36o 11.45718 8.543o8 361 11.45692 10.0002619.99974 6o 43 52 i6 8 54642 357 45358 54669 358 4533I 00027 99973 59 2 43 44 1 6 i6 54999 355 45oo001 55027 355 44973 00027 99973 58 3 43 36 16 24 55354 351 44646 55382 352 446i8 00028 99972 57 4 43 28 I6 32'55705 349 44295 55734 349 44266 00028 99972 56 51II 43 20 0 16 4o 8.56054 346 11.43946 8.56083 346 11.43917 I0.000299.99971 55 6 43 12 i6 48 56400 343 4360oo 56429 344 43571 00029 99971 54 7 43 4 16 56 56743 34i 43257 56773 341 43227 00030 99970 53 8 42 56 17 4 57084 337 42916 57114 338 42886 00030 99970 52 9 42 48 17 12 57421 336 42579 57452 336 42548 ooo3, 99969 1i io ii 42 4o 0 17 20 8.57757 332 11.42243 8.57788 333 11.42212 1o.0ooo31 9.9969 50 11 42 32 17 28 58089 33o 41911 58[21 33o 41879 00032 99968 49 12 4, 24 17 36 58419 328 4i58i 58451 328 41549 00032 99968 48 13 42 I6 17 44 58747 325 41253 58779 326 41221 00033 99967 47 14 42 8 17 52 59072 323 40928 59io5 323 40895 00033 99967 46 15 1142 01 0 i8 o 8.59395 320 11.40605 8.59428 321 11.40572 Io.o000339.99967 45 i6 41 52 18 8 59715 318 40285 59749 319' 40251 ooo34 99966 44 17 41 44 18 i6 6oo33 316 39967 6oo68 3i6 39932 00034 99966 43 18 41 36 i8 24 60349 313 39651 6o384 314 39616 00035 99965 42 19 41 28 i8 32 60662 311 39338 60698 311 39302 oo00036 99964 41 20 11 41 20 08 40 8.60973 309 11.39027 8.61009 0o 1.38991 I. o00036999964 40 21 41.12 i8 48 61282 307 38718 61319 307 3868i 00037 99963 39 22 4r 4 18 56 61589 3o5 38411 61626 3o5 38374 00037 99963 38 23 4o 56 19 4 61894 302 381o6 61931 3o3 38069 00038 99962 37 24 40 48 19 12 62196 301 37804 62234 3o01 37766 00038 99962 36 2511 40o 4o 0 019 2018.62497 298 11.37503 8.62535 299 11.37465 10.000399.99961 35 26 4o 32 19 28 62795 296 37205 62834 297 37166 00039 99961 34 27 40o 24 19 36 63091 294 36909 63131 295 36869 00040 99960 33 28 40 16 19 44 63385 293 36615 63426 292 36574 00040 99960 32 29 40 8 19 52 63678 290 36322 63718 291 36282 00041 99959 31 3o 4o 0 0 20 0 8.63968 288 11.36032 8.64009 289 11.35991 o.ooo4 9.99959 3o 31 39 52 20 8 64256! 287 35744 64298 287 35702 00042 99958 29 32 39 44 20 16 64543 284 35457 64585 285 354i5 00042 99958 28 33 39 36 20 24 64827 283 35173 64870 284 353o ooo43 99957 27 34 39 28 20 32 65110 281 34890 65154 281 34846 00044 99956 26 35 II 39 20 0 20 4o 8.65391 279 II.34609 8.65435 280 11.34565 1o.ooo449.99956 25 36 39 12 20o 48 65670 277 3433o 65715 278 34285 00045 99955 24 37 39 4 20 56 65947 276 34o53 65993 276 34007 00045 99955 23 38 38 56. 21 4 66223 274 33777 66269 274 33731 000oo46 99954 22 39 38 48 21 12 66497 272 33503 66543 273 33457 00046 99954 21 401i 38 40 021 208.66769 270 11.33231 8.668i6 271 i.33184 10.ooo47999953 20 41 38 32 21 28 67039 269 32961 67087 269 32913 ooo00048 99952 19 42 38 24 21i 36' 67308 267 32692 67356 268 32644 oo00048 99952 18 43 38 i6 21 44 67575 266 32425 67624 266 32376 00049 9995i 17 44 38 8 21 52 67841 263 32159 67890 264 32110 ooo49 9995i i6 4511 i38 0 022 08.681io4 263 11.31896 8.68154 263 11.31846 1o.ooo5o9.9995o 15 46 37 52 22 8 68367 260 31633 68417 261 31583 ooo51 99949 14 47 37 44 22 i6 68627 259 31373 68678 260 31322 ooo0005i 99949 13 48 37 36 22 24 68886 258 31114 68938 258 31062 00052 99948 12 49 37 28 22 32 69144 256 30856 69196 257 3080o4 00052 99948 II 5o ii 37 20 0 22 4o 8.694 24 ii.3o6oo 8.69453 255 11.30547 o10.000539.99947 10 51 37 12 22 48 69654 253 30346 69708 254 30292 00054 99946 9 52 37 4 22 56 69907 252 30093 69962 252 3oo0038 00ooo054 99946 8 53 36 56 23 4 70159 250 29841 70214 251 29786 oo00055 99945 7 54 36 48 23 12 70409 249 29591 70465 249 29535 q000oo56 99944 6 5511 i 36 40 0 23 20 8.70658 247 11.29342 8.70714 248 11.29286 1o.oo00o569.99944 5 56 36 32 23 28 70go5 246 29095 70962 246 29038 ooo00057 99943 4 57 36 24 23 36 71151i 244 28849 71208 245 28792 ooo00058 99942 3 53 36 i6 23 44 7i395 243 28605 71453 244 28547 00058 99942 2 59 36 8 23 52 71638 242 28362 71697 243 28303 00059 99941 i 6o 36 24 0 71880 240 28120 71940 24 28060 ooo00060 99940 0 M Hour P.M. Hour A.M. Cosine. Diff.i' Secant. Cotangent Diff. 1' Tangent. Cosecant. Sine. M 920 870 Page 188' TABLE XXVII. Log. Sines, Tangents, and Secants. 30_____________________________________ _________________ 176. HounrA,. Hour pim. Sine. Diff.' Cosecant. Tangent. Diff. lCotangent Secant. Cosine. M 0 iI 36 ol o0 24 o0 8.71880 240 11.28120o 8.71940 241 11.28060 10.000609.99940 i 35 52 24 8 72120 239 27880 72181 239 27819 ooo6o 99940 59 9 35 44 24 16 72359 238 27641 72420 239 27580 ooo00061 99939 58 3 35 36 24 24 72597 237 27403 72659 237 27341 00062 99938 57 4 35 28'24 32 72834 235 27166 72896 236 27104 ooo62 99938 56 511 35 200 024 4o 8.7306 234 11.26931 8.73132 234 11.26868 10.000639.99937 55 6 35 12 24 48 73303 232 26697 73366 234 26634 ooo64 99936 54 7 35 4 24 56 73535 232 26465 73600 232 26400 00064 99936 53 8 34 56 25 4 73767 230 26233 73832 231 26168 ooo65 99935 52 9 34 48 25 12 73997 229 26003 74063 229 25937 00066 99934 51 i0 134 40o 25 20 8.74226 228 11.25774 8.74292 229 11.25708 10.000669.99934 50 ii 3432 25 28 74454 226 25546 74521 227 25479 00067 99933 49 12 34 24 25 36 74680 226 25320 74748 226 25252 oo00068 99932 48 13 34 16 25 44 74906 224 25094 74974 225 25026 ooo68 99932 47 14 34 8 25 52 75130 223 24870 75199 224 24801 oo069 99931 46 1511 34 0 0 26 0 8.75353 222 11.24647 8.75423 222 11.24577 10.000709.99930 45 16 33 52 26 8 75575 220 24425 75645 222 24355 00071 99929 44 17 33 44 26 16 75795 220 24205 75867 220 24133 00071 99929 43 i8 33 36 26 24 76015 219 23985 76087 219 23913 00072 99928 42 19 33 28 26 32 76234 217 23766 76306 219 23694 00073 99927 41 20'I 33 20 26 40o 8.76451 216 11.2349 8.76525217 11.23475 10.000749.99926 4o 21 33 12 26 48 76667 216 23333 76742 216 23258 00074 99926 39 22 33 4 26 56 76883 214 23117 76958 215 23042 00075 99925 38 23 32 56 27 4 77097 213 22903 77173 214 22827 00076 99924 37 24 32 48 27 12 77310 212 22690 77387 213 22613 00077 99923 36 25 II 32 4o 0 27 20 8.7752' 1 211.22478 8.77600 211 11.22400 10.000779.99923 35 26 32 32 7 7 28 77733 210 22267 77811 211 22189 00078 99922 34 27 32 24 27 36 77943 209 22057 78022 210 21978 00079 99921 33 28 32 16 27 44 78152 208 21848 78232 209 21768 ooo8o 99920 32 29 32 8 27 52 78360 208 21640 7844 208 21559 ooo00080 9920 31 301I1i32 0 0 28 0 8.78568206 11.21432 8.78649 206 11.2351 10.000819.99919 3o 31 3i 52 28 8 78774 205 21226 78855 26 21145 00082 99918 29 32 31 44 28 16 78979 204 21021 79061 205 20939 ooo83 99917 28 33 31 36 28 24 79183 203 20817 79266 204 20734 00083 99917 27 34 31 28 28 32 79386 202 20614 79470 203 20530 ooo00084 999 1626 35 i 31 20 0 28 4o 8.79588 201 11.20412 8.79673 202 11.20327 10.000859.99915 25 36 3 12 28 48 79789 201 20211 79875 201 20125 oo00086 9994 24 37 3i 4 28 56 79990 199 20010 80076 201 19924 00087 99993 23 38 3o 56 29 4 80189 199 19811 80277 199 19723 00087 99913 22 39 30o 48 29 12 8o388 197 19612 80476 198 1952o4 00088 99912 21 40o i11 3 40 29 20 8.80585 197 11.19415 8.80674 198 11.19326 10.000ooo89.999911 20 41 3o 32 29 28 80782 196 19218 80872 196 19128 00090 99910 19 42 30 24 29 36 80978 195 19022 81o68 196 18932 00091 999og9 8 43 30 16 29 44 81173'94 18827 81264 195 18736 00091 99909 1.7 44 3o 8 29 52 81367 193 8633 81459 194 854 000 ooo2 99o 08 16 45i11 3o 0 0 3o o 8.856o 192 11.18440 8.81653 193 11.18347 10.000939.99907 15 46 29 52 3o 8 81752 192 18248 8i846 192 i854 00094 9990 14 47 29 44 3o 16 81944 190 18056 82038 192 17962 00095 99905 13 48 29 36 30 24 82134 190 17866 82230 g190 17770 00096 99904 12 49 29 28 30 32 82324 189 17676 82420 190 17580 00096 99904 ii 50 11 29 20 o 30 4o 8.82513 188 11.17487 8.82610 189 11.17390 10.00097l9.99993 1i 51 29 12 3o 48 82701 187 17299 82799 188 17201 00098 99902 9 3 29 4 3o 56 82888 187 17112 82987 188 17013 00099 99901 8 53 28 56 31i4 83075 186 16925 83175 186 16825 ooioo 99900 7 54 28 48 31 12 83261 185 16739 83361 i86 I6639 000oo 99899 6 55 11 28 40 031 20 8.83446 184 ii.i6554 8.83547 185 ii.i6453 10.001029.99898 5 56 28 32 31 28 8363o 183 16370 83732 184 16268 00102 99898 4 57 28 24 3i 36 838i3 183 16187 83916 184 J6084 oo1o3 99897 3 58 28 i6 3i 44 83996 i81 16004 84ioo 182 5goo 000ooio4 99896 1 59 28 8 3i 52 84177 181 15823 84282 182 15718 00105 99895 i 60 28 0 32 0 84358 181 15642 84464 182 5536 00oio6 99894 o M Ilour P.M. Hour A.M. Cosine. Diff. l' Secant. Cotangent Dif'. I' Tangent. Cosecant. Sine. M 930 86" TABLE XXVII. [Page 189 Log. Sines, Tangents, and Secants. 4o 1750 I Hour A.-r. Hour p.M. Sine. Diff.l' Cosecant. Tangent. Diff.l' Cotangent Secant. Cosine. M 0 I 128 0 0 32 0 8.84358 i8i- 11.15642 8.84464 182 11.5536 o1.0oo06 9.99894 60 27 52 32 8 84539 179 I546i 84646 i80 I5354 00107 99893 59 2 27 44 32 16 84718 I79 15282 84826 80o 15174 o0008 99892 58 3 27 36 32 24 84897 178 I5103 85006 179 14994 o009 99891 57 4 27 28 32 32 85075 177 14925 85185 178 I48I5 00109 99891 56 5 11 27 20 0 32 40 8.85252 177 11.4748 8.85363 177 11.14637 io.ooi 0o 9.99890 55 6 97 12 32 48 85429 176 14571 85540 177 14460 OOit 99889 54 7 27 4 32 56 85605 175 14395 85717 176 14283 00112 99888 53 8 2656 33 4 85780 175 14220 85893 176 14107 001131 99887 52 9 26 48 33 12 85955 173 14045 8606, 174 1393I ooII4 99886 51 10 1 26 40o 033 20 8.86128 173 11.13872 8.86243J 174 11.3757 o1.ooii5 9.99885 50 I 26 32 33 28 863o0 173 13699 8641'7 174 I3583 00116 99884 49 12 26 24 33 36 86474 171 I3526 86591 172 3409 00117 99883 48 13 26 61 33 44 86645 171 13355 86763 I72 13237 ooii8 99882 47 4 26 8 33 52 86816 171 i3184 86935 171 13065 ooII9 99881 46 511 26 o 0 34 o 8.86987 i169 11.30o3 8.87106 171 I 1.2894 1o.001o20 9.99880 45 16 25 52 34 8 87156 169 12844 87277 170 12723 00121 99879 44 17 25 44 34 i6 87325 169 12675 87447 169 I2553 00121 99879 43 I8 25 36 34 24 87494 67 12506 87616 169 12384 00122 99878 42 19 25 28 34 32 87661 i68 12339 87785 168 12215 00123 99877 4' 20 1i 25 20 0 34 40 8.87829 i66 11.1217i 8.87953 167 11.I2047 10.00124 9.99876 40 21 25 12 34 48 87995 I66 12005 88120 167 11880 00125 99875 39 22 25 4 34 56 88161 165 11839 88287 166 11713 00126 99874 38 23 24 56 35 4 88326 i64 11674 88453 i65 11547 00I27 99873 37 24 24 48 35 88490 164 11510 886i8 i65 II382 00I28 99872 36 25 I 24 40 o 35 20 8.88654 I63 11.11346 8.88783 I65 I.1127 10.00129 9.9987I 35 26 24 32 35 28 7 63 83 88948 i63 948 63 11052 ooi30 99870 34 27 24 24 35 36 88980 162 I1020 89111 163 10889 ooi03 99869 33 28 24 i6 35 44 89142 162 o1858 89274 163 10726 00132 99868 32 29 24 8 35 52 89304 160 10696 89437 161 I0563 00133 99867 31 30o 11 24 0 o 36 o 8.89464 161 1 i.o1536 8.89598 162 11.1402 I0. o I34 9.99866 30 31 23 52 36 8 89625 159 10375 89760 i60 10240 o0035 99865 29 32 23 44 36 16 89784 159 10216 89920 160 o0080 ooI36 99864 28 33 23 36 36 24 89943 159 10057 90080 i60 09920 00137 99863 27 34x 23 28 36 32 90o02 158 09898 90240 159 09760 ooi38 99862 26 35 11 23 20 o 3640 8.90260 157 11.09740 8.90399 I58 II.0960I1 10.0039 9.99861 25 36 23 I2 36 48 90417 157 09583 90557 158 09443 001oo40 99860 24 37 23 4 36 56 90574 156 09426 90715 157 09285 ooI4I 99859 23 38 22 56 37 4 9073o 155 09270 90872 157 09128 00142 99858 22 39 ^2 48 37 12 90885 155 o9115 91029 156 08971 ooI43 99857 21 4 11 22 40 0 37 20 8.91040 155 11.08960 8.91185 I55 i.o88i5 io.00144 9.99856 20 41 22 32 37 28 9.95 154 08805 91340 155 08660 o0045 99855 19 42 22 24 37 36 91349 I53 0865i 91495 I55 08505 oo046 99854 18 43 22 16 37 44 91502 153 08498 91650 153 08350,00147 99853 17 44.22 8 37 52 91655 152 08345 91803 i54 08197 ooI48 99852 16 45 11 22 0 0 38 o 8.91807 152 1I.08193 8.91957 i53 II.08043 10.00149 9.99851 15 46 21 52 38 8 91959 151 o804I 92110 152 07890 ooi50 99850 14 47 2I 44 38 16 92110 i 51 07890 92262 152 07738 00152 99848 13 48 21 36 38 24 92261 I50 07739 92414 I5I 07586 00153 99847 12 49 1 228 38 32 92411 150 07589 92565 151 07435 o0054 99846 ii 5o 121t 20 38 40 8.92561 149 11.o7439 8.92716 50o 11.07284 Io.ooi55 9.99845 io 51 21 12 38 48 92710 149 07290 92866 150 07134 ooi56 99844 9 52 21 4 38 56 92859 148 07141 93016 149 06984 o0057 99843 8 53 20 56 39 4 93007 I47 06993 93165 148 06835 ooi58 99842 7 54 20 48 39 12 93i54 I47 06846 933r3 I49 06687 00o59 9984r 6 55 11 20 4 0 39 20 8.93301 147 11.06699 8.93462 147 II.o6538 o.ooI60o 9.99840 5 56 20 32 39 28 93448 146 o6552 93609 147 06391 o0161 99839 4 57 20 24 39 36 93594 146 06406 93756 147 06244 00162 99838 3 58 20 6 39 44 93740 145 06260 93903 146 o6o97 ooi63 99837 2 59 20 8 39 52 93885 145 o6115 94049 146 o5951 00164 99836 1 60 20 0 4o 0 9403o 144 05970 94i95 145 o5805 ooI66 99834 o MI Hourp.M. HourA.M. Cosine. Dif.l' Secant. Cotangent Diff. 1 Tangent. Cosecant. Sine. M 94P 850 Paue 1901 Pag 19j ~TABLE XXVII. Si' Log. Sines, Tangents, and Secants. 50 A A B B C C 1740 lM ourA.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Difl. Cotangent Secant. Diff. Cosine. M I 20 00 o 8. 0 40 o 8.94o30 11.0597o 8.94195 o.0o58o5 1.ooi66 o 9.99834 60 i 9 52 40 8 94174 2 05826 94340 2 o566o 0o0167 99833 59 2 19 44 40o 6 94317 4 05683 94485 4 o555 001o68 0 99832 58 3 9g 36 40 24 94461 7 05539 94630 7 05370 00169 o 99831 57 4 19 28 4o 32 94603 9 05397 94773 9 05227 00170 0 99830 56 511 1920 0 4o 4o 8.94746 Ii 11.05254 8.94917 I 11.05083 10.00171 0 9.9982955 6 19 12 40 48 94887 13 05113 95060 13 04940 oo00172 0 99828 54 7 19 4 4o 56 95029 15 04971 95202 15 04798 00173 o 99827 53 8 18 56 41 4 95170 18 o483o 95344 i8 o4656 00175 o 99825 52 9 8 48 41 12 95310 20 04690 95486 20 04514'00176 0 99824 5S 10 I 18 40 o4r 20 8.95450 2211.04550 8.95627 22 11.04373 10.00177 0 9.99823 50 ii 18 32 41 28 95589 24 044iI 95767 24 04233 00178 o 99822 49 12 i8 24 41 36 95728 26 04272 95908 27 04092 00179 0 99821 48 i3 18 16 41 44 95867 29 04i33 96047 29 03953 ooi8o o 99820 47 14 18 8 41 52 96005 31 03995 96187 31 03813 00181 o 99819 46 15 111i i8 0 042 o 8.96143 33 11.03857 8.96325 33 ii.o3675 1I1oooi83 o 9.99817 45 16 17 52 42 8 96280 35 03720 96464 35 03536 00184 0 99816 44 17 17 44 42 6 96417 37 03583 96602 38 03398 oo00185 o 99815 43 18 17 36 42 24 96553 39 03447 96739 4o 03261 ooi86 0 99814 42 19 17 28 42 32 96689 42 o33ii 96877 42 03123 00187 o 99813 4, 20 11 17 20 042 40 8.96825 44 i.o3175 8.97013 4411.02987 1o.ooi88 0 9.99812 40 21 17 12 42 48 96960 46 03040 97150 46 02850 00190 0 99810 39 22 17 4 42 56 97095 48 02905 97285 49 02715 00191 0 99809 38 23 16 56 43 4 97229 S 02771 9742I 51 02579 00192 0 99808 37 24 i6 48 43 12 97363 53 02637 97556 53 02444 00193 o 99807 36 25 ii 16 4o 0 43 20 8.97496 5511.02504 8.97691 55 11.02309 10.00194.i 9.99806 35 26 16 32 43 28 97629 57 02371 97825 58 02175 00196 I 99804 34 27 16 24 43 36 97762 59 02238 97959 6o 02041 00197 I 99803 33 28 1i6 16 43 44 97894 61 02106 98092 62 o01908 00198 I 99802 32 29 16 8 43 52 98026 64 01974 98225 64 01775 00199 I 99801 3i 3011 ii 6 0 044 o8.98157 661I.01oi843 8.98358 6611.01i642 10.00200 I 9.99800 30 3 15 52 44 8 98288 68 01712 98490 69 oi5io 00202 I 99798 29 32 15 44 44 16 98419 70 0158i 98622 71 01378 00203 I 99797 28 33 15 36 44 24 98549 72 oi45i 98753 73 01247 00204 1 99796 27 34 15 28 44 32 98679 75 o01321 98884 75 oiii6 00205 I 99795 26 35 i 15 200 44 40 8.98808 7711.01192 8.99015 77 11.009oo 85 10.00207 I 9.99793 25 36 15 12 44 48 98937 79 oio63 99145 8o oo855 00208 I 99792 24 37 15 4 44 56 99066 8 00934 9925 82 00725 00209 99791 23 38 14 56 45 4 99194 83 oo806 99405 84 00595 00210 1 99790 22 39 i4 48 45 12 99322 86 00678 99534 86 oo466 00212 I 99788 21 40o I 14 4 0 45 20 8.99450 88 I.oo55o 8.99662 89 11.00338 10.0023 I 9.-99787 20 41 14 32 45 28 99577 90 00423 99791 91 00209 00214 I 99786 19 42 14 24 45 36 99704 92 00296 99919 93 ooo0008 00215 I 99785 8 43 i4 i6 45 44 99830 94 00170 9.00046. 95 10.99954 00217 I 99783 17 44 i4 8 45 52 99956 96 5 oO44 00174 97 99826 00218 I 99782 i6 451 14 0 o46 0 9.00082 99 o.99918 9.00301 100 10.99699 10.00219 I 9-99781 15 46 13 52 46 8 00207 101 99793 00427 102 99573 00220 I 99780 14 47 13 44 46 i6 00332 io3 99668 oo553 io4 99447 00222 I 99778 13 48 13 36 46 24 00oo456 io5 99544 00679 10o6 99321 00223 I 99777 12 49 13 28 46 32 0058i 107 99419 oo85 o108 99195 00224 I 99776 ii o5011 13 20 0 46 40 9.00704 110 10.99296 9.009301 III 10.99070 10.00225 I 9.99775 o10 51 13 12 46 48 00828 112 99172 o01055 113 98945 00227 1 99773 9 52 13 4 46 56 00951 II4 99049 01179 115 98821 00228 I 99772 8 53 12 56 47 4 01074 116 98926 o01303 1179 98697 00229 I 99771 7 54 12 48 47 12 o01196 r118 98804 01427 120 98573 00231 I 99769 6 55 ii 12 4o0 47 20 9.01318 121 10.98682 9.01550 122 -0.98450 10.00232 I 9.99768 5 56 12 32 47 28 01440 123 98560 01673 124 98327 00233 I 99767 4 57 12 24 47 36 oi561 125 98439 01796 126 98204 00235 I 99765 3 58 12 i6 47 44 01682 127 98318 01918 128 98082 00236 i 99764 2 59 12 8 4752 oi803 I 129 98197 02040 131 97960 00237 I 99763 1 60 120 48 0 01923132 98077 02162 133 97838 00239 I 99761 o II Hour P.M.1 Hour A.M. Cosine.'Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine.'M 950 A A B B C C 84" Seconds of time...... I 2s 3, 4 1 58 6" 7" A 6 33 49 66 82 99 5I Prop. parts ofols. B 17 33 50 66 83 ioo 116 0C o 0o I I I I TABLE XXVII. [Page 191 S', GI. Log. Sines, Tangents, and Secants. G.' _6 3A A B B C C 1730 M Hour A.m. Hour P.M. Sinr. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 01112 00 48 0o9.01-23 010.98077 9.02162 010.97838 10.00239 o 9.9976 60 I II 52 48 8 o03.43 2 97957 02283 2 97717 002401 99760 59 2 1i 44 48 16 o' 163 4 97837 02404 4 97596 00241 o 99759 58 3 11 36 48 24 02283 6 97717 02525 6 97475 00243 o 99757 57 4 1128 48 32 02402 7 97598 02645 8 97355 00244 o 99756 56 5 11 II 20 0 48 40 9.02520 9 10.97480 9.02766 9 10.97234 10.00245 0 9.99755 55 6 11 12 48 48 02639 ii 97361 02885 ii 9715 00247 0 99753 54 7 11 4 48 56 02757 13 97243 03005 13 96995 00248 o 99752 53 8 1o 56 49 4 02874 15 97126 0o324 15 96876 00249 0 99751 52 9 Io 48 49 12 02992 17 97008 03242 17 96758 00251 o 99749 51 io i i 4o 40 o 49 209g.o03109 19 10io.96891 9.03361 1910.96639 10.00252 0 9.99748 50 ii 10 32 49 28 03226 20 96774 03479 2! 96521 00253 o 99747 49 12 o10 24 49 36.03342 22 96658 03597 23 96403 00255 o 99745 48 1 03 io 16 49 44 o3458 24 96542 03714 24 96286 00256 o 99744 47 14 Io 8 49 52 03574 26 96426 03832 26 96168 00258 o 99742 46 1511 i 0 0o 50, o 9.03690 2810.96310 9.03948 2810.96052 10.00259 0 9-99741 45 16 9 5 50 8 03805 30 96195 04065 3o 95935 00260 o 99740 44 17 9 44 50o 16 03920 31 96080 04i81 32 95819 00262 o 99738 43 18 9 36 50 24 04034 33 95966 04297 34 95703 00263 o 99737 42 19 9 28 50o 32 04149 35 9585i o44i3 36 95587 00264 o 99736 41 20 II 9 20 o 50 4o 9.04262 37 10.95738 9.o4528 3810.95472 10.00266 o 9.99734 40 21 9 12 50o 48 04376 39 95624 o04643 39 95357 00267 I 99733 39 22 9 4 50 56 04490 41 95510 04758 41 95242 00269 I 99731 38 23 8 56 51 4 04603 43 95397 04873 43 95127 00270 I 99730 37 24 8 48 5i 12 04715 44 95285 04987 45 95013 00272 i 99728 36 25 ii 8 40 0 5120 9.04828 4610.95172 9.05101 47o1094899 10.00273 1 9.99727 35 26 8 32 5i 28 04940 48 95060 05214 49 94786 00274 i 99726 34 27 8 24 5i 36 05052 50o 94948 5328 Si 94672 00276 I 99724 33 28 8 6 51i 44 o0564 52 94836 o5441 53 94559 00277 I 9972.3 32 29 8. 8 Si 52 05275 54 94725 o555o3 54 94447 00279 I 99721 31 3o0 II 8 o 0 52 0 9.05386 5610.94614 9.05666 56 10.94334 10.00280 i 9.99720 30 31 7 52 52 8 05497 57 94503 05778 58 94222 00282 I 99718 29 32 7 44 52 16 05607 59 94393 05890 60 94110 I 00283 I 99717 28 33 7 36 52 24 05717 6i 94283 06002 62 93998 00284 i 99716 27 34 7 28 52 32 05827 63 94173 06113 64 93887 00286 i 99714 26 3511 720 o 52 409g.o5937 65 10.94063 9.06224 6610.93776 10.00287 I 9.99713 25 36 7 12 52 48 06046 67 93954 06335 68 93665 00289 I 99711 24 37 7, 4 52 56 06155 69 93845 o6445 69 93555 00290 I 99710 23 38 6 56 53 4 o6264 70 93736 o6556 71 93444 00292 I 99708 22 39 6 48 53 12 06372 72 93628 06666 73 93334 00293 I 99707 21 401o i 6 40 o 53 20 9.06481 74 10.93519 9.06775 7510.93225 10.00295 I 9.99705 20 41 6 32 53 28 06589 76 93411 06885 77 93115 00296 I 99704 19 42 6 24 53 36 06696 78 93304 06994 79 93006 00298 I 99702 i8 43 6 i6 53 44 o6804 8o 93196 07103 8i 92897 00299 I 99701 17 44 6 8 53 52 06911 8i 93089 07211 83 92789 oo3oi i 99699 i6 451 i 6 0 0 54 o 9.07018 83 10.92982 9.07320 84 10.92'580 10.00302 I 9.99698 15 46 5 52 54 8 07124 85 92876 07428 86 92572 oo00304 i 99696 14 47 5 44 54 16 07231 87 92769 07536 88 92464 oo0030o5 i 99695 13 48 5 36 54 24 07337 89 92663 07643 go90 92357 00307 I 99693 12 49 5 28 54 32 07442 91 92558 07751 92 92249 oo00308 99692 II 5011i 5 20 54 40 9.07548 93 10.92452 9.07858 9410.9242 o10.00310o I 9.99690 0 51 5 12 54 48 07653 94 92347 07964 96 9236 oo00311ii I 99689 9 52 5 4 54 56 07758 96 92242 08071 98 91929 oo00313 I 99687 8 53 4 56 55 4 07863 98 92137 08177 99 91823 oo3r4 I 99686 7 54 4 48 55 12 07968 ioo 3 92032 08283 i1 -91717 oo36 99684 7 6 55i 440o 55 20 9.08072 102 10.91928 9.o8389 10310.91611 10.0037 I 9.99683 5 56 4 32 55 28 08176 104 91824 o08495 15 9505 0031oo9 99681 4 57 4 24 55 36 08280 10io6 91720 08600oo 107 91400 00320 I 99680 3 58s 4 16 55 44 o8383 107 91617 08705 109o 91295 00322 99678 2 59 4 8 55 52 o8486 109 91514 08810 iII 91190 00323 I 99677 i 60o 4 0 56 0 08589 II1 91411 08914 113 91086 00325 1 99675 0 M Hori..-i. Hour A.M.I Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine M 960 A A B B C C 83" Seconds of time...... P s 32 4s 5s 6s I7 (A i4 28 42 56 69 83 97 Pro. arts of cols. B 14 28 42 56 70 84 98 0C 0o I I i I i ~6 5[ 44 o5~64 52 4836 0544[ 5 00277 z 99723 32 Page 192] TABLE XXVII.'~. Log. Sines, Tangents, and Secants. G'. 7~0 A A B B C C 172~ M Ilcur AM. Hour Pi. Sine. Diff.Cosecant. T o an-ent. Diff. Cotangent Secant. Diff. Cosine. M o ii 4 0 o 56 o 9.08589 0 10.91411I g.o8g4 0 10.g1086 10.00325 o 9.99675 6o i 352 56 8 08692 2 1308 09019 2 90981 00326 o 99674 59 2 3 44 61 6 08795 3 91205 0o123 3 90877 00328 o 99672 58 3 3 36 56 24 08897 5 91103 09227 5 90773 oo330 o 99670 57 4 3 28 56 32 08999 6 91001oo 09330 7 90670 00oo33i o 99669 56 51 i 3 2 0 56 40 9.09101ogo i 8 10.90899 909434 8 10.0go566 1o.oo333 o 9.99667 55 6 3 12 56 48 09202 TO 90798 09537 10 90463 00334 o 99666 54 7 3 4 56 56 09304 i 90696 09640 11 9036 00oo336 o 99664 53 8 2 56 57 4 09405 13 90595 09742 13 90258 00337 o 99663 52 9 248 57 12 09506 14 90494 09845 15 90155 00339 o 99661 51 10 11 2 40o 0 57 20 9g.og6o 61 i6 10.9go394 9.09947 i6 o10.900oo53 io.oo341 0 9.99659 50 i 2 32 57 28 09707 i8 90293 10049 18 89951 00342 0 99658 49 12 2 24 57 36 09807 19 90193 ii5 20 89850 00344 0 99656 48 13 2 6 57 44 09907 21 9003 10252 21 89748 oo00345 o 99655 47 14 2 8 57 52 ooo0006 22 89994 10353 23 89647 00347 0 99653 46 1511 2 0 0 58 0 9.o1010o6 24 o10.89894 9.-10454 24 10.89546 10.00349 0 9.99651 45 i6 52 58 8 10205 26 89795 io555 26 89445 0035o 99650 44 17 T144 58 i6 io304 27 89696 10o656 28 89344 oo00352 0 99648 43 18 T36 58 24 10402 29 89598 10756 29 89244 oo00353 99647 42 119 1 28 58 32 io5o 3o 8949 io856 31 8944 oo00355 T 99645 41 20 11 I 20 0 58 40 9.10599 32 10.89401 9.10956 33 10.89044 1o.oo357 I 9.99643 40 21 1 12 58 48 0697 34 893o3 iio56 34 88944 00358. 99642 39 22 1 4 58 56 10795 35 89205 1155 36 88845 oo0036o 9964 38 __3 0 56 59 4 10893 37 89107 19254 37 88746 00362 i 99638 37 24 0 48 59 12 0990 38 89010 353 39 88647 oo00363 I 99637 36 25 11 o 4o 0 59 20 9.11087 4o 10.88913 9.11452 4I 1o.88548 io.oo365 I 9.99635 35 26 0 32 59 28 1ii84 42 88816 11551 42 88449 00367 I 99633 34 27 0 24 59 36 11281 43 88719 11649 44 8835r oo368 99632 33 28 0 i6 59 44 11377 45 88623 I1747 46 88253 00370 i 99630 32 29 o 8 59 52 11474 46 88526 11845 47 88 55 00371 I 99629 3i 3oII 0 o o 0.11570 48 1o.88430 9.11943 49 10.88057 10.00373 I 9.99627 3o 3110o 59 52 0 8 i666 50o 88334 1204 51i 87960 00375 i 99625 29 32 59 44 o i6 11761 51 88239 12138 52 87862 00376 i 99624 28 33 59 36 0 24 11857 53 884o3 12235 54 87765 00378 1 99622 27 34 59 28 o 32 11958 54 88048 12332 55 87668 oo38o 99620 26 35 10 59 201 o 40 9-12047 56 10io.87953 9.12428 57 o10.87572 1o.oo382 i 9.99618 25 36 59 12 0 48 12142 58 87858 12525 59 87475 oo00383 I 99617 24 37 59 4 o 56 12236 59 87764 12621 60 87379 oo00385 1 99615 23 38 58 56 T 4 12331 6i 87669 12717 62 87283 00387 i 99613 22 139 58 48 12 12425 62 87575 12813 64 87187 oo388 99612 21 4o0 i-58-40 l 20 9.12519 6 10o.87481 9.129o09 65 10o.87091 10.00oo390 9.99610 20 41 58 32 28 1262 66 87388 300oo4 67 86996 00392 I 99608 19 642 58 24 36 12706 67 87294 3099 68 86901 00393 I 99607 18 43 581 6 44 12799 69 87201 13194 70 86806 00395 1 99605 17 44 58 8 1 52 12892 70 87108 13289 72 86711 00397 I 99603 i6 45 10o 58 0 2 0 9.12985 72 10.87015 9.13384 73 o10.86616 10.00399 9.99601 i5 46 57 52 2 8 13078 74 86922 13478 75 86522 oo00400oo 99600 14 47 57 44 2 16 13171 75 86829 13573 77 86427 00402 i 99598 13 48 5736.224 13263 77 86737 3667 78 86333 00o4o4 I 99596 12 49 57 28 2 32 i3355 78 86645 13761 80 86239 00405 1 99595 ii 50 10 57 20 1 2 4o0 9.13447 801Io.86553 9.13854 81 io.86i46 10.00407 i 9 99593 io 51 57 12 2 48 13539 82 86461 13948 83 86052 00409 i 99591 9 52 57 4 2 56 1363o 83 86370 14o41 85 85959 oo004i I 99589 8 53 56 56 3 4 i3722 85 86278 14134 86 85866 00412 i 99588 7 54 56 48 3 12 i3813 87 86187 14227 88 85773 oo00414 2 99586 6 551 i 56 401 3 20 9.13go904 88 10.86096 9.14320 go90 io.8568o io.oo416 2 9.99584 5 56 56 32 3 28 13994 go90 86no6 4412 91 85588 00418 2 99582 4 5 — 56 24 3 36 T4o85 91T 85915 145o4 93 8^496 oo419 2 99581 3 58 56 16 3 44 14175 93 85825 14597 95 85403 00421 2 99579 2 59 56 8 3 52 14266 95 85734 14688 96 85312 00423 2 99577 I 6o 56 o 4 o i4356 96 85644 14780 98 85220 00425 2 99575 0 H-lourp?.Mi.H ourA.Al. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M lJ7o A A B B C C 82" Seconds of time..... 2" 3s 4, 5 69 9sT Prop. parts of cols.A 12 24 36 48 60 72 84 Prop. parts of colsB 12 24 37 49 73 86 TABLE XXVII. [Page 193 S.' Log. Sines, Tangents, and Secants. GI. 8~ A A B B C C 1710 M HourA.lM. Hourp.Ar. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M o 1O 56 oi 4 o 9.14356 o io.85644 9.14780 o 10.85220 10.60425 0 9.99575 6o i 55 52 4 8 14445 I 85555 14872 I 85128 00426 o 99574 59 2 55 44 4 i6 14535 3 85465 14963 3 85037 00428 0 99572 58 3 55 36 4 24 I4624 4 85376 15o54 4 84946 00430 o 99570 57 4 55 28 4 32 14714 6 85286 i5145 6 84855 00432 o 99568 56 51io 55 201 4 40o 9.4803 7 o.85197 9.15236 7 Io.84764 10.oo 00434 o 9.99566 55 6 55 12 4 48 14891 8 85109 15327 9 84673 00435 o 99565 54 7 55 4 4 56 14980 Io 85020 15417 io 84583 00437 o 99563 53 8 54 56 5 4 150o69 ii 84931 i5508 12 84492 00439 o 99561 52 9 54 48 5 12 15157 i3 84843 15598 13 84402 00441 o 99559 51 o 10 54 401 5 20 9.15245 i4 10.84755 9.i5688 14 10.84312 Io.oo443 o 9.99557 50 ii 54 32 5 28 i5333 16 84667 i5777 i6 84223 004441 99556 49 12 54 24 5 36 15421 17 84579 15867 17 84133 004461 o 99554 48 13 54 I6 5 44 15508 18 84492 15956 19 84044 o6448 o 99552 47 14 54 8 5 52 15596 20 84404 16046 20 83954 00450 o 99550 46 5 10o 54 I 6 0 9.15683 21 10.84317 9.16135 22 io.83865 10.00452 0 9.99548 45 16 53 52 6 8 15770 23 84230 16224 23 83776 00454 I 99546 44 17 53 44 6 16 I5857 24 84143 i6312 25 83688 00455 I 99545 43 18 53 36 6 24 15944 25 84056 164o0 26 83599 00457 I 99543 42 19 53 28 6 32 i6o30 27 83970 16489 27 83511 00459 I 99541 41 2io0 i53 o 6 4o 9.16116 28 10o.83884 9.16577 29 io.83423 1o.oo46i I 9.99539 40 21 53 12 6 48 16203 30 83797 16665 3o 83335 00463 i 99537 39 22 53 4 6 56 16289 31 83711 16753 32 83247 00465 I 99535 38 23 52 56 7 4 16374 32 83626 16841 33 83159 00467 I 99533 37 24 52'48 7 12 I6460 34 83540 16928 35 83072 00468 I 99532 36 2510 52 401 I 7 20 9.16545 35o10.83455 9.17016 36 10.82984 10.00470 I 9.99530 35 26 52 32 7 28 16631 37 83369 17103 37 82897 00472 I 99528 34 27 52 24 7 36 16716 38 83284 17190 39 82810 00474 I 99526 33 28 52 i6 7 44 i68oi 39 83199 17277 40 82723 00476 1 99524 32 29 52 8 7 52 16886 4i 83114 17363 42 82637 00478 I 99522 3i 3o01o 52 o 8 0 9.16970 42 10.83030 9.17450 43 10.82550 10.00480 I 9.99520 3o 31 5i 52 8 8 17055 44 82945 17536 45 82464 00482 i 99518 29 32 5i 44 8 i6 17139 45 82861 17622 46 82378 00483 I 99517 28 33 5r 36 8 24 17223 47 82777 17708 48 82292 00485 I 99515 27 34 51 28 8 32 17307 48 82693 17794 49 828o6 00487'I 99513 26 3510 51 20 8 40 9.17391 49 10.82609 9.17880 50 10.82120 10.00489 I 9.99511 25 36 51 12 8 48 17474 51 82526 17965 52 82035 00491 I 99509 24 37 51 4 8 56 17558 52 82442 8o051 53 81949 00493 I 99507 23 38 50 56 94 17641 54 82359 18136 55 81864 00495 i 99505 22 39 50 48 9 12 17724 55 82276 18221 56 81779 00497 I 99503 21 40 o 50 4o01 9 20 9.17807 56 10.82193 9.18306 58 10.81694 10.00o.oo 499 I 9.99501 20 41 5o 32 9 28 17890 58 82110 I8391 59 81609 oo0050oi I 99499 19 42 50 24 9 36 17973 59 82027 18475 61 81525 00503 i 99497 8 43 50 16 9 44 18055 6i 81945 i8560 62 8i440 00505 i 99495 17 44 50 8 9 52 18137 62 8r863 i8644 63 81356 o0506 I 99494 16 45 Io 50 o I Io o 9.18220 63 10.81780 9.18728 65 10.81272 io.oo5o8 I 9.99492 15 46 49 52 io 8 18302 65 81698 18812 66 81188 -oo51o I 99490 14 7 49 44 1o 16 18383 66 81617 18896 68 81104 00512 I 99488 13 48 49 36 10 24 18465 68 81535 18979 69 81021 oo514 2 99486 12 49'49 28 1o 32 18547 69 8i453 19063 71 80937 oo5i6 2 99484 ii 50o 10o 49 20, 1 IO 4o 9.18628 71 10.81372 9.19146 72 1o.8o854 io.oo5i8 2 9.99482 10 51 49 2 o10 48 18709 72 81291 19229 74 80771 00520 2 99480 9 52 49 4 10 56 18790 73 81210 19312 75 80688 00522 2 99478 8 53 48 56 1f 4 18871 75 81129 19395 76 80605 00524 2 99476 7 54 48 48 ir 12 18952 76 81048 19478 78 80522 00526 2 99474 6 5510o 48 40 1I 20 9.19033 78.10.80967 9.19561 79 10.80439 10.00528 2 9.99472 5 56 48 32 11 28 19113 79 80887 19643 8i 80357 00530 2 99470 4 57 48 24 11 36 1919i 8o 80807 19725 82 80275 00532 2 99468 3 58 48 i6 1I 44 19273 82 80727 19807 84 80193 00534 2 99466 2 59 48 8 11 52 19353 83 80647 19889 85 80111 00536 2 99464 I 6o 48 0 12 0 19433 85 8o0567 19971 87 80029 00538 2 99462 0 M HourP.M. HourA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff.I Sine. MI g~8~ A A B B C C 810 Seconds of time...... 1 2" 3' 4" 5 6' 7" A 11 21 32 42 53 63 74 Prop parts of cols. B IT 22 32 43 54 65 76 C 0 0 I I ii 2 25 Page 194] TABLE XXVII. Log. Sines, Tangents, and Seca-nts. GW. 90 A A B B C C 1700 Al Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff Cosine. M 010o 48 0 I 12 0 9.19433 o 10.80567 9.I9971 0 10.80029 io.oo538 o 99j462 60 I 47 52 12 8 19513 I 80487 20053 I 79947 oo0540 o 99460 59 2 47 44 12 i6 19592 3 80408 20134 3 79866 00542 0 99458 58 3 47 36 12 24 19672 4 80328 20216 4 79784 00544 o 99456 57 4 47 28 12 32 19751 5 80249 20297 5 79703 oo546 0 99454 56 51io 4720 12 409.19830 610.80170 9.20378 6 10.79622 10.00548 0 9.99452 55 6 47 12 12 48 19909 8 80091 20459 8 79541 00550 0 99450 54 7 47 4 12 56 19988 9 80012 20540 9 79460 00552 0 99448 53 8 46 56 13 4 20067 Io 79933 20621 IO 79379 00554 o 99446 52 9 46 48 13 12 20145 II 79855 20701 12 79299 00556 o 99444 51 io io 46 40 1 i3 20 9.20223 13 10.79777 9.20782 13 10.79218 1o.oo558 0 9.99442 50o ii 46 32 13 28 20302 14 79698 20862 14 79138 00560 o 99440 49 12 46 24 13 36 20380 15 79620 20942 16 79058 00562 0 99438 48 13 46 i6 13 44 20458 i6 79542 21022 17 78978 00564 0 99436 47 14 46 8 13 52 20535 1 79465 21102 i8 78898 oo566 0 99434 46 1510o 46 0 I 14 o0 9.20o613 19 10.79387 9.21182 19 10.78818 o0.00568 I 9.99432 45 16 45 52 14 8 20691 20 79309 21261 21 78739 00571 I 99429 44 17 45 44 14 i6 20768 21 79232 21341 22 78659 00573 I 99427 43 i8 45 36 14 24 20845 23 79155 21420 23 78580 00575 I 99425 42 19 45 28 i4 32 20922 4 79078 2499 25 78501 00577 I 99423 41 20 10 45 20 14 40 9.20999 25 10.7900 9.21578 26 10.78422 10.00579 I 9.99421 4 21 45 12 14 48 21076 26 78924 21657 27 78343 00581 I 99419 39 22 45 4 I4 56 21153 28 78847 21736 28 78264 00583 I 99417 38 23 4456 15 4 21229 29?877I 21814 30 78186 00585 1 9941537 24 44 48 15 12 21306 3o 78694 21893 31 78107 00587 I 99413 36 25 10 44 40 15 20 9.21382 31 10.78618 9.21971 32 10.78029 10.00589 I 9.99411 35 26 44 32 15 28 21458 33 78542 22049 34 77951 00591 I 99409 34 27 44 24 15 36 21534 34 78466 22127 35 77873 00593 1 99407 33 28 44 16 15 44 21610 35 78390 22205 36 77795 00596 I 99404 32 29 44 8 15 52 21685 37 78315 22283 38 77717 00598 I 99402 31 30o 10 44 0 16 o 9.21761 38 10.78239 9.22361 39 10.77639 io0.00600oo I 9.994oo00 3o0 31 43 52 i6 8 21836 39 78164 22438 40 77562 00602 I 99398 29 32 43 44 16 i6 21912 40 78088 22516 41 77484 oo00604 1 99396 28 33 43 36 i6 24 21987 42 78013 22593 43 77407 oo6o6 i 99394 27 34 43 28 16 32 22062 43 77938 22670 44 7733 00oo608 I 99392 26 35 io 43 20 1 i6 40 9.22137 44 10.77863 9.22747 45 10.77253 io.oo6io I 9.99390 25 36 43 12 16 48 22211 45 77789 22824 47 77176 00612 I 99388 24 37 43 4 i6 56 22286 47 77714 22901 48 77099 oo00615 99385 23 38 42 56 17 4 22361 48 77639 22977 49 77023 00617 I 99383 22 39 42 48 17 12 22435 49 77565 23054 50 76946 00619 I 9938.1 21 40 10 42 40 1 17 20 9.22509 50 10.77491 9.23130 52 10.76870 10.00621 1 9.99379 20 41 42 32 17 28 22583 52 77417 23206 53 76794 60623 I 99377 19 42 42 24 17 36 22657 53 77343 23283 54 76717 00625 I 99375 18 43 42 I6 17 44 22731 54 77269 23359 56 76641 00628 2 99372 17 44 42 8 17 52 22805 55 77195 23435 57 76565 oo00630 2 99370 16 45 10 42 0 1 i8 0 9.22878 57 10.77122 9.23510 58 10.76490 10.00632 2 9.99368 15 46 4r 52 18 8 22952 58 77048 23586 6o 76414 oo00634 2 99366 14 47 4i 44 i8 16 23025 59 76975 23661 6i 76339 oo636 2 99364 13 48 4i 36 8 24 23098 6 76902 23737 62 76263 oo00638 2 99362 12 49 41 28 18 32 23171 62 76829 23812 63 76188 oo00641 2 99359 II 50 10 o 41 20 I 18 40 9.23244 63 10.76756 9.23887 65 10.76113 io.oo643 2 9.99357 o10 51 41 12 8 48 23317 64 76683 23962 66 76038 oo00645 2 99355 9 52 4r 4 i8 56 23390 65 76610 24037 67 75963 00647 2 99353 8 53 4o 56 19 4 23462 67 76538 24112 69 75888 oo00649 2 99351 7 54 40 48 19 12 23535 68 76465 24186 70 75814 00652 2 99348 6 55 10 4 440 1 1920 9.23607 69 10.76393 9.24261 71 10.75739 io.oo654 2 9.99346 5 56 4o 32 19 28 23679 71 76321 24335 73 75665 00656 2 99344 4 57 40 24 19 36 23752 72 76248 24410 74 75590 oo00658 2 99342 3 58 40 16 Ig 44 23823 73 76177 24484 75 75516 oo00660 2 99340 2 59 40 8 52 23895 4 76105 24558 76 75442 oo00663 2 99337 1 60 4 0 o 20 0 239671 76 76033 24632 78 75368 oo665 2 99335 o MIH HourP.i. Hour A.l. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 990 A A B B C C 800o Seconds of time...... Is" 2 3" 4" 5* 6' 7" (A o 19 29 39 47 57 68 Prop. parts of cols. B 0 19129 39 8 9 5 66'0 r2I 47 / 7' 76 o0C i 2 2 TABLE XXVII. [Page 195 CI. kLog. Sines, Tangents, and Secants. 0' 10~ A A B B C C 169~ M H$our A.M. H-our P.M. Sine. Diff. Cosecant: Tangent. Diff. Cotaient Secant. Diff. Cosine. M 01 i04 0 oI 20 09.23967 o010o.6033 9.24632 0 10.75368 I0o.0 665 0 9.99335 6o i 39 52 20 8 24039 i -75961 24706 1 75294 00667 0 99/333 50 2 39 44 20 16 241r10 2 75890 24779 2 75221 00669 o 99331 58 3 39 36 20 24 24181 3 75819 24853 4 75147 00672 0 99328 571 4 39 28 20 32 24253 5 75747 24926 5 75074 00674 o 99326 56 510 3920 2040o 9.24324 6 10.75676 9.25000 6 10.75000 10.00676 0 9. 9934 55 6 39 121 2'0 48{ 24395 7 75605 25073 7 74927 00678 o 99322 541 7 39 4 20 56 24466 8 75534 25146 8 74854 oo68i o 99319 53 8 38 56 21 4 24536 9 75464 25219 9 74781 oo683 o 99317 52 9 38 48 21 12 24607 io 75393 25292 II 74708 00685 o 993,5 51 o10 10o 38 4 1 21 20 9.246771 i' 10 75323 9.25365 12 10.74635 10.00687 0 9.99313 5o i1 38 32 21 28 24748 13 75252 25437 13 74563 0o69o 0 99310 49 12 38 24 21 36 2481.8 14 75182 25510 14 74490 00692 0 993o8 481 13 38 16 21 44 24888 15 75r12 25582 15 74418 00694 1 99306 471 14 38 8 21 52 24958s 16 7504- 25655 16 74345 oo00696 I 99304 46 i 5o 38 01 22 09.25028 17 10.74972 9.25727 i8 10.74273 10.00699 1 9.99301 45 16 37 52 22 8 25098 18 74902 257991 19 74201 00701 I 99299 44: 17 37 44{ 22 16 25168 19j 74832 25871 20 74129 007o03 99297 43 i8 37 36, 22 24} 25237 20 74763 25943 21 74057 00706 1 99294 42 19t 37 28 22 32 25307 22 74693 26015 22 73985 00708 1 99292 41: 20 10 37 20 1 22 40 9.25376 23 100.74624 9.26086 24,10.7391410.00710 2 9.99290 40 21 37 12 22 48 55445 24 74555 26158 25 73842 00712 1 99288 39 22 37 4 22 56 25514 25 74486 26229 26 73771 00715 I 99285 38 23 36 56 23 4 25583 26 74417 26301 27 73699 00717 I 99283 37 240 36 48 23 12 25652 27 74348 2637 28 73628 00719 I 99281 36, 25 io 36 40 I 23 20 9.25721 28 10.74279 9-26443 29 10.73557 10.00729 i 9.99278 35 26 36 321 23 28 25790 3o 74210 26513 31 73486 007241 99276 34 27 36 24 23 36 25858 3J 74142 26585 32'734 15 00726 i 99274 33 28 36 16 23 44 25927 32 74073 26655 33 73345 00729 1 99271 32 29 36 8! 23 52 25995 33 74005 26726 34 73274 00731 I 99269 31 3o 10 36 01 2-4 -0 926063- 34 10.73937 9-26797 35 10o.320310o.00733 9.99267 3o 31 35 521 24 81 26131 35 73869 26867 36 73133 00736 1 99264 29 32 35 44 24 i6 26199 36 73801 2 6937 38 73063 00738 1 99262 28 33 35 36 24 24 26267 38 73733 27008 39 72992 00740 1 99260 27 34 35 28 24 32 26335 39 73165 27078 4o 72922 00743 I 99257 26 35 io 35 20 1 24 4o 9-26403 40 10.73597 9.27148 41 10.72852 10.00745 i 9.-99255 25 136 35 12 24 48 26470 41 73530 27218 49 72782 00748 5 90252 24 37,35 4I 24 561 26538 41 73462 2'7288 44 72712 00750 i 9925o 23 3 34 56 25 41 26605 43 73395 27357 45 72643 00752 I 99248 22^ 39 34 48 25 12 26672 44 73328 27427 46 72573 0o5 2 9924 21 401o0 34 40 1 25 2019.26739 4) 10.73261 9-27496 47 10 7250i 10 00757 2 9.99243 21 41 34 32 25 28 26806 47 73194 27566 48 72434 00759 2 09241 191 42 34 24 25 36 26873. 48 73127 27635 49'72365 00762 2 99238 18 43 34 i6 25 44 26940 49 73060o 27704 51 72296 00764 2 99236 17 44 34 8 25 521 27007 50 72993 27773 5,2 72227 00767 2 09233 i6 45 10 34 0 126 9.27073 5.1 0.72927 9.2742 53 10.2158 0. 00769 2 9- 9233 15 46 33 521 26 8 27140 52 72860o 27911 54 72089, 00771 2 99229 14 47 33 44 26 16 27206 53 72794 27980 55 72020 00774 2 99226 13 48 33 36 26 24 27273 55 72727 28049 56 71951 007,76 2 99224 i12 49 33 28 26 321 27339 56 726611 221r 50 7 71883 00779 2 99221 I1 50o 10 33 20 26 40 9.27405 57 10.72595 9.28186 59 10.718:4 1 10,o 781 2 9.92-19 10 51 33 12 26 48 27471 58 72529 28-354 60 7746 00783 2 99217 91 52 33 4 26 56 27537 59 72463 28323 61 I 71677 00786 2 99214 8 53 32 56 27 4 2760') 6o 72398 28301 (6 716(0 00788 2 99212 7 54 32 481 27 12 27668 61 72332 984509 63 7:1541I 007911 2 99209 61 55 io 32 40 i 27 20 19.277.3 63 10.72266 9 28127 6'10 i 473 10.0(793 2 9992-07 5 56 32 32 27 28 27799 64/ 722o(1 28595 66 71405 00796 2 99204 57 32 24 27 361 27864J 65 721363 286('.) 6.7 71338 00798 2 99202 3' 58 32 16 27 44 2793(0 66 72070 2873o0 68j 7137o o00800o 2 99200 21 6o 32 0 28 O 28(60 68.7194( 28865 71 71135 008(05 2 99195 o I fourp.. HourA.M. I Cosine. Diff. Secant. Cota nn(tlif' i an t'. e. ang Cosecant. Dif. Sine. M 100~ A A B C C -79 Seconds of time...... P 1 3, 4_S 5 6 70 A 9 17 26 34 43 5160 Prop. parts of cols. B 9 1 26J35 44 53 62 C. lo i i i? a TABLE XXVII. Log. Sines, Tangents, and Secants. 11 A A B B C C 168s 3I Hour A.M. HourPr.i. Sine. Diff. Cosecant. Tangent. Diff.Cotangent Secant. Diff. Cosine. M 1032 0 28o 3.28060 o 2 28 0.7940 9.28865 o 10.7135 1.oo00805 o 9.99195 6 1 31 52 28 8 28125 i 71875 28933 z 710671 00808 99192 59 2 31 44 28 ~6 28190 2 71810 29000 2 71000 008oo8o 0 9990 58 3 31 36 28 24 28254 3 71746 29067 70933 00813 99187 57 4 3 2s8 28 32 28319 4 768 29134 4 70866 oo00815 o 99185 56 510 3 20 1 28 4o09.2838 5 10.71616 9.29201 5 10.70799 1000 818 o 99982 55 j6 3 I2 28 48 28448 6 71552 29268 6 70732 00820 o 99180 54 31 4 28 5o 6 28512 7 79488 2335 8 70665 00823 o 99177 53 8 30o56 209 28577 8 71423 29402 9 70598 0o8 99175' 30 48 29 012 28641 9 71359 29468 io_ 70532 00828 0 99172 5s 10o 3o0 o r 29 20 9.28705 10 10.71295 9.29535 ii 10.70465 ilo.oo830 o 9 o99170 1 ii 30 32 29 28 287691 II 71231 9601 12 70399 83 997 49 ~12 30 24' 2-9 366 28833 12 71167 29668 i3 70332 00835 I 99165 48 13 30 16 29 44 28896 13 711o04 29734 14 70266 oo838 I 99162 t47 14 30 8 20 52 28960 t4 71040 29800 So 70200 oo008o i 99160 46 5 soo o 0 -1 30 o 09-29024 16 10.70976 9.29866 16 10.70134 io.oo8431 I 9.99157 45 1i6 29 52 30 8 29087 17 70913 29932 17'70068 6o845 1 99155 44 17 29 44 3o i6 29150 z8 7085o0' 2999981 70002 oo848 1 99152 43 19 29 28 30 3j 29277 20o 70723 30130 20 69870 00853 I 99147 41 8 ~ 29 36 3 23 291377 19 70786 0oo64 19 ~ 69936 oo85oI 99]50I4 20 10 29 20.o i 30 40 9.29340 21 10.70660 9.30195 22 10.69805 io.0o855 I 9.99I5 21I 29 12 30 48 29403 22 70597 30261 23 69739 00858 1 99421 39 22 29 4 30 56 294S66 23 70534 30326 24 69674 oo86o0 1 9914o 38 23 28 56 3, 4 29529 24 70471 30391 25 69609 oo863 I 9971 373.24 28 48 321 E2 29591 25 70409 30457 276 69543 oo865 I 9935 136 25 10 28 0o 131 20 9.29054 26 10.70346 9.30522 27 10.69478 io.oo868 I 9.99132 35 26 - 28 32 31 2.8 29716 27 70284 30587 -28 69413 00870. 99130 34 27 2'8 24 31 36 29779 28 70221 30652 29 69348 00873 I 99127 33 128 28,61 31 44 29841 9 70159 30717 3o 69283 o0876 i 9924 32 29 28 8 31 52 29903 3o 70097 30782 31 69218 00878 I 99122 3i 301o 10 o0 32 o9.299 31 106 7oo0034 9.3084I6 3:'o.69154 to.oo88i - g9.991191 3 13 27 52 32 8 30028 32 69972 30911 33 69089 oo883 1 99t71 29 32 27 44 3 2 i6 300901 33 69910 30975 35 69025 oo00886 994 28 33 27 36 32 24 30oi5 34 69849 3o1040 36 68960 o00888 i 99112 27 34 _27 2 8 32 32 3021_3 35 69787 311o4 37 68896 oo8gI r 909 I 26 35 1o 27 20 i 32 40 9.30275 36 10.69725 9.31168 38 io.68832 10.00894. 2' 9.9916 25 36 27 32 48 3o336 37 69664 31233 39 68767 00896 2 99104 24 37 4' 32 56 30398 38 69602 31297 40 68703 00899 991 23 139 2 48 33 12 30521 40 69479 31425 42 68575 00904 2 99096 21 4. 10 26 40 i 33 20 9.3058241 i10.69418 9.31489 43 Io.685i1 10.009072 9.99093 20 41 26 32 33 28 30643 42 69357, 31552 44 68448 00909 2 99091 19 42 26 24 33 36 30704 43 6-9296 31616 45 68384 0091 2 99088 18 43 26 16 33 44 30765 45 69235 31679 46 68321 00914 2 99086 17 44 26 8 33 52 30826 46 69174 31743 47 68257 00917 99083 16 4510 26 01 i34 o 9.30887 47 10.69113 9.31806 49 10.68194 10.00920 2 9-99(80 i 46! 25 52 34 8 30947 48 69053 31870 50 68i3o 009422 2 99078 14 147 25 44 34 6 310oo8 49 68992 31933 51 68067 00925 2 - 99075 i3 48 25 36 34 24 3o168 50 68932 31996 52 68oo4 009298 2 99072 12 49 25 28 34 321 3129 5 68871 32059 53 67941 oo00930 2 9907(0 ot 50 10 25 20 134 40 9.31189 52 10.68811 9.32122 54 10.67878 10.00933 2 9.99067 10 51 2512 34 48 31250 53 68750 32185 55 67815 00936 2 99(64 9 52 25 4 334 56 31310o 54 68690 32248 56 67752 00938 2 99)62 8 53 24 56 35 4 3i370 55 6863 32311 57 67689 oo0941 2 99059 7 54 24 481 35 12 3i43o 56 68570 32373, 58 67627 00944 2 993056 6 55 io 24 4 35 20.31490 57 10.6850 9.32436 59 10.67564 10.00946 2 9.9954.5'56 a4 32 35 28 31549 58 68451 32498 6o 67502 00949 2 99o5 4:57 24 a44 35 36 3160o 59 68391 32561 6i 67439 00952 2 99048 3 58 2416N 35 44 31669 60 68331 32623 63 67377 0954 2 99046 2 59 24 8 3552 31728 6i 68272 32685 64 67315 00957 3 99043 i 160 244 o 36 o 31788 63 68212 32747 65 67253 00960 3 99040 OJ Mf iu. floorp..oA. Cosine. Diff. Secant. Cotain[get Diff. Tan t. Cosecant. Diff. Si7e.11 iGo0 A A B " B C 0 780 Seconds of time...... 56 2s 3s 4 5s 6 71 Prop. parts of c.ols. B 8 6 243 2 39 49 5 f54C o2 8 1 2 4 2 TABLE XXVII. [Page 197 s'. Log. Sines, Tangents, and Secants. G. l20 A A B B C C 16'70 M I- our A.M. fHour P.Mi. Sine. Diffi Cosecant. Tannt. Diff. Cotangent Secant. Diff. Cosine. A 0 10 24 0 136 o 9.31788 o 10.682,12 9.32747 o 10. 67253 o0.00 oo60 o 9.99040 60 I 23 52 36 8 31847 I 68153 32810 i 67190 0062 0 99038 59 2 23 44 36 16 31907 2 68093 32872 2 67128 00965 o 99035 58 3 23 36 36 24 31966 3 68o34 32933 3 67067 00968 o 99032 57 4 23 28 36 32 32025 4 67975 32995 4 67005 00970 0 99030 56 5 10 23 20 I 3640o 9.32084 5 10.6796 9.33057 5 10o.66943 1000973 o 9.99027 55 6 23 12 36 48 32143 6 67857 33119 6 6688I 00976 o 99024 54 7 23 4 36 56 32202 7 67798 33180 7 66820 00978 0 99022 53 8 22 56 37 4 32261 8 67739 33242 8 66758 00981 o 99019 52 9 2248 37 12 32319 9 67681 33303 9 66697 00984 o 9901651 Io 10 2240 1 37 20 9.32378 io 10.67622 9.33365 Jo 1o.66635 10.00987 o 9.99013 5o IJ 22 32 37 28 32437 Io 67563 33426 ii 66574 00989 I 99011 49 12 22 24 37 36 32495, I 67505 33487 12 66513 00992 1 99008 48 13 22 16, 37 44 32553 12 67447 33548 13 66452 009951 99005 47'4 22 8! 37 52 32612 13 67388 33609 1,4 66391 0oo998 I 99002 46 15 10 22 o0 1 38 0 9.32670 14 10.67330 9.33670 15 Io.6633o 1o.oi ooo a 9.99000 45 I'6 21 52 38 8 32728 15 67272 33731 I6 66269 oioo3 i 98997 44 17 21 44 38 I6 327861 16 67214 33792 1 7 66208 ooo0 6 i 98994 43 i8 21 36 38 24 32844 17 67156 33853 I8 66,47 01009 I 98991 42 19 21 28 38 32 32902 18 67098 33913 I19 66087 01011 i 98989 4i o2010 21 20 38 40 9.32960 19 10.67040 9.33974 20 10.66026 1o.oi 4 9.98986 4 21 21 12 38 48 33oi8 20 66982 34034 21 65966 01017 1 98983 39 22 21 4 38 56 33075 21 66925 34095 22 65905 01020 1 98980 38 23 20 56 39 4 33133 22 66867 34155 23 65845 01022 1 98978 37 24 20 48 39 12 33190 23 668o0 34215 24 65785 01025 1 98975 36 25 10 20 40 I 39 20 9.33248 24 I0.66752 9.34276 25 10.65724 10.01028 I 9-98972 35 26 20 32 39 28 333o5 25 66695 34336 26 65664 oio3i 98969 34 27 20 24 39 36 33362 26 66638 34396 27 65604 oo033 1 98967 33 28 20 i6 39 44 33420 27 6658o 34456 28 65544 oio36 98964 32 29 20 8 39 52 33477 28 66523 34516 29 65484 0oo39 I 98961 31 301Io 20 0o 4 o 09.33534 29 0.66466 9.34576 30 io.65424 10.01042 I 9.98958 3o 31 19 52 40 8 33591 29 66409 34635 31 65365 o01045 i 98955 29 32 19 44 40 r6 336471 30 66353 34695 32 653o5 01047 i 98953 28 33 19 36 40 24 33704 31 66296 34755 33 65245 oio5o 2 98950 27 34 19 28 4o 32 33761 32 66239 34814 34 65i86 oio53 2 98947 26 35 io 19 20 i4 40 4o 9.33818 33 10.66182 9.34874 35 10.65126 io.oio56 2 9.98944 25 36 19 12 40 48 33874 34 66126 34933 36 65067 01059 2 98941 24 37 19 4 4o 56 33931 35 66069 34992 37 65008 01062 2 98938 23 38 i8 56 41 4 33987 36 660o3 35051 38 64949 o64 2 98936 22 39 i8 48 41 12 34043 37 65957 35.i1 39 64889 01067 2 98933 21 4o 1 80 48 4o 4 20 9.34100 38 10.65o900 9.35170 40 o.6483o 10.01070 2 9.989301 2 4I i8 32 41 28 34i56 39 65844 35229 41 64771 01073 2 98927 19 42 IS 24 41 36 34212 40 65788 35288 42 64712 01076 2 98924 i8 43 i8 i6 41 44 34268 41 65732 35347 43 64653 01079 2 98921 17 44 i8 8 4r 52 34324 42 65676 35405 44 64595 oio8i 2 989191 6 451i8 0 142 9. 4 o9.3438o 430.65620 9.35464 45 o0.64536 io.oio84 2 9.98916 15 46 17 52 42 8 34436 44 65564 35523 46 64477 01087 2 98913 14 47 17 44 42 16 34491 45 65509 3558i 47 64419 01090 2 98910 13 48 I7 36 42 24 345471 46 65453 3564o 48 64360 01oo93 2 98907 12 49 17 28 42 32 34602 47 65398 35698 49 64302 01096 2 9890o4 i 5o 17 20 142 40 9.34658 48 10.65342 9.35757 o 10.64243 10.01099 2 9.98901 io 51 17 12 42 48 34713 48 65287 35815 5i 64i85 01102 2 98898 9 53 i6 56 43 4 34824 5o 65176 35931 53 64069 01107 2 98893 7 54 i6 48 43 12 34879 5 I 65121 35989 54 640o 1 onio 3 98890 1551o 6 40 I 43 20 9.34934 52 1o.65o66 9.36047 55 10.63953 io.o0ii3 3 9.98887 5 56 i6 32 43 28 34989 53 65oi1 36io5 56 63895 oiii6 3 98884 4 57 I6 24 43 36 35044 54 64956 36i63 57 63837 01119 3 98881 3 58 i6 16 43 44 35099 55 64901 36221 58 63779 01122 3 98878 2 59 i6 8 43 52 35154 56 64846 36279 59 63721 01125 3 98875 i 6o i6 0 44 0 35209o 57 64791 36336 6o 63664 01128 3 98872 0 1 Hour P.IM. our A.M. Cosie. Dff. Secant. CotanentDi. Tannt. Cosecant. nDiffi Sine. ll 102~ A A B B C CC 77 Seconds of time...... I'2 3S 4g 58 6S 7" (A 7 14 21 29 36 43 50 Prop. parts of cols. B 7 i5 22 3o 37 45 52. C o I.. T 2I 2i Page 1981 TABLE XXVII. S'. Log. Sines, Tangents, and Secants. G 130 A A B _ B C C 166~ M Hour A.M. Hour P.I. Sine. Diff.Cosecant. Tangent. Diff. Cotangen., Secant. Diff. Cosine. M 0 Io I6 o I 44 o 9.35209 o 10.6479I 9.36336 o Io.63864 10.0128 o 9.98872 60 5 52 44 8 35263 i 64737 36394 636o6 oi013 o 98869 59 2 15 44 44 i6 3538 2 64682 36452 2 63548 o0133 o 98867 58 3 i5 36 44 24 35373 3 64627 36509 3 63491 oii36 o 98864 57 4 I5 28 44 32 35427 4 64573 36566 4 63434 0I139 o 98861 56 5 10 5 20 1 44 40 9.35481 4 io.6459 9.36624 5 I0.63376 o.o1142 o 9.98858 55 6 i5 I2 44 48 35536 5 64464 3668 6 63319 0o145 o 98855 54 7 15 4 44 56 35590 6 644 ( 36738 6 63262 OII48 o 98852 53 8 4 56 45 4 35644 7 64356 36795 7 63205 O151 0 98849 52 9 i4 48 45 12 35698 8 64302 36852 8 63 48 oii54 o 98846 51 0 1o I4 4o I 45 20 9.35752 9 10.64248 9.3690 9 1o.63oi 10.01157 T 9.98843 50 ii 14 32 45 28 35806 io 64194 36966 io 63034 01i6oI 98840 49 12 14 24 45 36 35860 ii 64i40 37023 ir 62977 Oi63 I 98837 48 13 14 I6 45 44 35914 ii 64086 37080 12 62920 OII66 I 98834 47 14 14 8 45 52 35968 I2 64032 37137 13 62863 01169 I 98831 46 I5 10 14 o 1 46 o 9.36022 13 10.63978 9-37193 14 10.62807 10.0172 I 9.98828 45 i6 13 52 46 8 36075 I4 63925 37250 15 62750 01175 I 98825 44 17 i3 44 46 16 36I29 15 63871 37306 16 62694 01178 I 98822 43 i8 13 36 46 24 36182 i6 63818 37363 17 62637 oii8i I 988I9 42 19 I3 28 46 32 36236 17 63764 37419 I8 6258I o I84 I 98816 4I 20 10 13 20 I 46 40 9.36289 8 10.63711 9.37476 19 10.62524 10.01187 I 9.98813 40 21 13 12 46 48 36342 18 63658 37532 19 62468 OI90 I 98810 39 22 13 4 46 56 36395 19 63605 37588 20 62412 01193 I 98807 38 23 12 56 47 4 36449 20 6355r 37644 21 62356 o0196 I 98804 37 24 12 48 47 12 36502 21 63498 37700 22 62300 01199 I 98801 36 2510 12 40 1 47 20 9.36555 22 10.63445 937756 23 10.62244 10.01202 9.98798 35 26 I2 32 47 28 36608 23 63392 37812 24 62188 oI205 I 98795 34 27 12 24 47 36 36660 24 63340 37868 25 62132 o01208 I 98792 33 28 12 16 47 44 36713 25 63287 37924 26 62076 01211 I 98789 32 29 12 8 47 52 36766 25 63234 3798 62020 7 6 01214 98786 3i 30o 1012 0 1 48 0 9.36819 26 o10.63181 9.38035 28 10.61965 10.01217 2 9.98783 30 31 II 52 48 8 36871 27 63129 38091 29 6I909 01220 2 98780 29 32 II 44 48 i6 36924 28 63076 38147 30 6i853 01223 2 98777 28 33 II 36 48 24 36976 29 63024 38202 31 61798 01226 2 98774 27 34 If 28 48 32 37028 30 62972 38257 32 61743 0I229 2 98771 26 36 ii 12 48 48 37133 32 62867 38368 33 6I632 01235 2 98765 24 37 Ii 4 48 56 37185 32 62815 38423 34 61577 OI238 2 98762 23 38 10 56 49 4 37237 33 62763 38479 35 61521 01241 2 98759 22 39 io 48 49 12 37289 34 62711 38534 36 6i466 01244 2 98756 21 40 10 10 401 I49 20 9.37341 35 10.62659 9.38589 37 io.6I411 10.01247 2 9.98753 20 41 10.32 49 28 37393 36 62607 38644 38 6i356 01250 2 98750 19 42 10 24 49 36 37445 37 62555 38699 39 6I301 01254 2 98746 I8 43 io i6 49 44 37497 38 62503 38754 40 61246 01257 2 98743 17 44 io 8 49 52 37549 39 6245I 38808 4I 61192 01260 2 98740 16 4510 I 0 I 50 9.37600 39 o.62400 9.38863 42 10.61137 10.01263 2 9.98737 15 46 9 52 50 8 37652 40; 62348 38918 43 61082 01266 2,98734 14 47 9 44 50o 6 37703 41 62297 38972 44 61028 01269 2 98731 13 48 9 36 50 24 37755 42 62245 - 39027 45 60973 01272 2 98728 12 49 9 28 50 32 37806 43 62194| 39082 45 60918 01275 2 98725 ii 50 10 9 201 50 40 9.37858 44 10.62142 9.39136 46 io.60864/ 10.01278 3 9.98722 Io 51 9 12 50 48 37909 45 62091 39190 47 608o10 01281 3 98719 9 52 9 4 50 56 37960 46 62040 39245 48 60755 01285 3 987i5 8 53 8 56 51 4 3801o 47 6i989 39299 49 60701 01288| 3 98712 7 54 8 48' 5 12 38062 47 61938 39353 50 60647 0129I 3 987(9 6j 55 o10 8 401 51 20 9.38113 48 10.61887 9-39407 5I 10.60593 10.01294 3 9.98706 5 56 8 32 5r 28 38 64 49 6I836 3946I 52 60539 01297 3 98703 4 57 8 24 5S 36 382I5 50 61785 39515 53 60485 oI3oo 3 98700 3 58 8 i6 51 44 38266 51 61734 39569 54 60431 oi303 3 98697 2 59 8 8 5I 52 38317 52 61683 39623 55 60377 o1306 3 98694 I 60 8 o 52 o 38368 53 61632 39677 56 60323 01310 3 98690 0 MI Iour p.M.i HourAl.M. Cosine. Diff. Secant. CotangentDiff. Tangent. Cosecant.Diff. Sie. 103~ A A B B C C 760 Seconds of time...... 1" 2s 3" 4' 5s 6| 7' Prp prA t7 13 20 26 33 39 46 Prop. parts of cols. B 7 I4 2I 28 35 42 49 C 1 1 2 292 3 f Pamge 199 TABLE XXVIL Log. Sines, Tangents, and Secants, 14~ A _ A B B C C165I Mll Hour A.M. HourPM. ine. DiffCosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. l 0 10 8 0 52 0 9.38368 o io.6r632 9.39677 10o.60323 o.oi31 o 0 0.9860o 6o 1 7 52 52 8 388 i 61582 39731 i 60269 o01313 o 98687 59 2 7 44 52 i6 38469 2 6i53i 39785 2 60215 o01316 o 98684 58 3 7 36 Sa 24 38519 2 6148i 39838 3 60162 01319 o 98681 57 4 7 28 52 32 38570 3 6143o 39892 3 6010o8 o322 o 98678 56 510 7 20 I 5240 9.38620 4 10.61380 9.39945 4 o10.6005oo5 10.01325 o 9-98675 55 6 7 12 52 48 38670 5 6i330 39999 5 60001 01329 0 98671 541 7 7 4 52 56 38721 6 61279 40052 6 59948 o0332 0 9866853 l8 656 534 38771 7 61229 4oo6 7 59894 o0335 0 98665 52 9 6 48 53 12 38821 7 61179 4059 8 598, 01338 o 98662 511 i o10 6 4o 1 5320 9.38871 8 10.61129 9 40i 9 10.59788 10.o34 1 i 9-98659 0 ii 6- 32 53 28 38921 9 61079 40266 io ( 59734 o01344 2 98656 49 12 6 24 53 36 38971 io 61029 40319 io 59681 o0348 i 98652 48 13 6 16 53 44 39021 ii 60979 40372 11 59628 01351 i 98649 47 14 6 8 53 52 39071 60929 40425 12 59575 0o354 i o86/4646 151o 6 o 0154 09.3912 12 10.60879 9-40 8 13.59522 10.01357 1 9 98643 45 16 5 52 54 8 39170 i3 6083o 40531 i4 59469 oi36o i 9864 i 44 17 5 44 54 6 392201 4 60780 40584 15 59416 oi364 i 98636 43i I8 5 36 54 24 39270 15 60730 4o636 16 59364 01367/ 1 98633 42 19 5 28 54 32t 39319 15 6o681 4o689 17 59311 o01370 1 9863 0 4I 20 10 5 20 i 54 40 9.39369 16 10io.6o631 9.40742 17 10.59258 1(.o01373 9.98627 A40 21 5 12 54 48 39418 17 6o582 40795 8 59205 oI377 1 986)3 39 22 5 4 54 56 39467 s8 60533 40847 19 59153 01380 3 98630 38 23 4 56 55 4 39517 19 60483 40900 2 59100o o383 1 98617 371 24 4 48 5512 3956620 60434 4o0952 21 59048 0o3861 I 986141 36 25 1o 4 40 I 55 209o.39615 20 10.60385 9.4100o 22 10,58995 o10.01390 1 9 o86io 35 26 4 32 55 28 39664 21 6o336 41057 23 58943 01393 2 98607 134 27 4 24 55 36 39713 22 60287 41109 23 5889 01396 I 986,,.4 33 28 4 16 5544 39762 23 60238 41161 24 58839 01399 2 986)1 32 29 4 8 55 52 39811 24 60189 41214 25 58788 oi403 2 98597 3i 3oIo 4 0 56 o 9.39860 24 1 o.6o 14 9.41266 26 10.58734 10.01406 2 9 98594 30 31 3 52 56 8 39909 25 60091 41318 27 58682 01409 9 98591 291 32 3 44 56 i6 39958 26 60042 41370 28 58630 01412 2 98588 28 33 3 36 56 24 4ooo6 27 59994 41/22 29 58578 01416 2 9850841 27 34 3 28 56 32 4no55 28 5994 441474 30 58526 01419 98581 26 35 1o 3 20 56 40 9.40o03 29 10.59897 9,41526 3.0 o.58474 10.01422 2 9.98578 925 36 3 2 56 48 4052 29 59848 4578 3i 58422 0o426 2 985574 24 37 3 4 56 56 40200 30 598oo0 41629 3 5837 o01429 2 98571 23 38 2 56 57 4 40249 31 59751 4i68'i1 33 583'19' o01432 2 98568 22 39 2 48 57 12 40297 32 59703 41733 34 58267 oi435 a 9856521 401 I 0 4 57 20 9.40346 33 10o.59654 9.41784 35 10.5816 io.o01439 9 9.985612o 4 1: 2 3 57 28 40394 33 596061 41836 36 58164 o01442 2 -98558 1,9 42 2 24 57 36 404424 34 59558 41887 36 58ii13 oi445 2 98555 181 43 2 16 57 44 40o490 35 59510 41939 3.7 58061 01449 2 98551 17 44 2 8 57 52 40o538 36 59462 4199 38 58010 0o145 22 @9854z8 16 45102 1 58o 9.40586 37 o.59414 9.42041 39 10.57959 10.01455 2 9 98545 15 46 52 58 8 40634 37 59366 42o93 40 57907 01459 3 98540 1 41 47 1 44 58 16 40682 38 59318 42144 41 57856 01462 3 98538 i13 48 I 36 58 24 40730 39.59270 42195 42 57805 o0165 3 98535 i 49 s 28 58 32 40778 40 59222 42246 43 57754 01469 3 98531 1 50o io 20 158 40 9.4o0825 4 10io.59175 9.42297 43 10,57703 10.01472 3 9.98528 io 51 I 12 58.48 4o873 42 59127 42348 44 57652 oi475 3 98525 9 52 1 4 58 56 4o0921 42 590709 42399 45 57601 01479 3 98521I 8 53 0 56.59 4 40968 43 59032 42450 46 57550 o1482 3 98518 71 54 o 48 59 12 4i0i6 44 58984 42501 47 57499 o01485 3 98515 6 55 io o 4( 1 59 20 9.41063 45 io.8937 9.42552 48 10.57448 io.o1489 3 9.9851 5 56 0 32 59 28 41i1i 46 58889 42603 49 57397 01492 3 985o8 41 57 0o 24 59 36 41I58J 46 58842 42653 5o 57347 o01495 3 98505 3 58 o 16 59 44 41205 427 58795 42704 5o 57296 01499 3 98501 2 59 0 8 59 52 41252 48 58748 42755 51 57245 01502 3 984981 1 6o o 2 0 0 41300 49 58700 42805 52 57195 o5o6 3 984941 o Flour psi. -.lHourA.M. Cosine. Diff. Secant. Cotang-entiDiff. Tangent. Cosecant. Diff. Sine M 10~4 4 A B B C C 75, Seconds of time...... P 2 3' 4s 5S6' 7 A 6 18 24 31 37 43 Prop. parts of cols. B 7 13 20 26 33 39 4 ^0 0 1 1 3 2 2 3~~~~~~~~~~~~_ Page 200] ~R TABLE XXVII. S'. Log. Sines, Tangents, and Secants. G 150 A A B B C C 1640 M Hour A.M. Hour P.a. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. lDiff: Cosine. M 0 10 0 0 2 0 o 9.4I300 o 10.58700 9.42805 o I0.57195 I0.0506 0 9.98494 60 9 59 52 0 8 41347 I 58653 42856 I 57144 01509 0 98491 59 2 59 44 0o 6 41394 2 58606 42906 2 57094 OI512 o 98488 58 3 59 36 0 24 41441 2 58559 42957 2 57043 o0156 o 98484 57 4 59 28 o 32 4i488 3 58512 43007 3 56993 01519 o 98481 56 5 9 59 201 2 0 40 9.415-35 4 1.58465 9.43057 4 1o.56943 10.01523 o 9.98477 55 6 59 121 o 48 4582 5 584 54 43o08 5 56892 01526 98474 54 7 59 4 o 56 41628 5 58372 43i58 6 56842 01529 0 98471 53 8 58 561 I 4 4I675 6 58325 43208 7 56792 OI533 o 98467 52 9 58 48 1 12 41722 7 58278 43258 7 56742 01536 I 98464 51 1O 9 58 40 2 1 20 9.41768 8 10.58232 9.43308 8 10.56692 Io.oI54o I 9.98460 50 Ii 58 32 I 28 41815 8 58i85 43358 9 56642 oi543 I 98457 49 12 58 241 36 4I86I 9 58139 43408 10 56592 01547 I 98453 48 13 58 i6 I 44 41908 io 58092 43458 ii 56542 oi55o I 98450 47 4 58 8 I 52 41954 ii 58046 43508 I 56492 oo553 I 98447 46 5 9 58 0 2 2 0 9.42001 II 10.57999 943558 12 10.56442 10.01557 I 9.98443 45 16 57 52 2 8 42047 12 57953 43607 13 56393 to0560 1 98440 44 17 57 44 2 i6 42093 13 57907 43657 14 56343 o0564 I 98436 43 i8 57 36 2 24 42140 14 57860 43707 15 56293 01567 I 98433 42 19 57 28 2 32 42I86 14 57814 43756 i6 56244 01571 1 98429 41 20 9 57 20 2 2 40 9.42232 15 10.57768 9.43806 16 10.56194 10.01574 I 9.98426 40 21 57 12 2 48 42278 i6' 57722 43855 17 56i45 01578 I 98422 39 22 574 256 42324 17 57676 43905 18 56095 01581 I 98419 38 23 56 56 3 4 2370 17 57630 43954 19 56046 oi585 98415 37 24 56 48 3 I2 42416 18 57584 44004 20 55996 oi588 I 98412 36 25 9 56 40 2 3 20 9.42461 19 10.57539 9.44053 20 10.55947 10.01591 I 9.98409 35 26 56 32 3 28 42507 20 57493 44102 21 55898 01595 2 98405 34 27 56 24 3 36 42553 21 57447 44551 22 55849 01598 2 98402 33 28 56 16 3 44 42599 21 5740I 44201 23 55799 I602 2 98398 32 29 56 8 3 52 42644 22 57356 44250 24 55750 oi605 2 98395 31 30 9 56 0 2 4 0 9.42690 23 10.57310 9.44299 25 o.55701 10.01609 2 9.98391 30 31 55 52 4 8 42735 24 57265 44348 25 55652 01612 2 98388 29 32 55 44 4 16 42781 24 57219 44397 26 55603 o01616 2 98384 28 33 55 36. 4 24 42826 25 57174 44446 27 55554 01619 2 98381 27 34 55 28 4 32 42872 26 57128 44495 28 55505 01623 2 98377 26 35 9 55 20 2 4 40 9.42917 27 10.57083 9.44544 29 IO.55456 1o.01627 2 9.98373 25 36 55 12 4 48 42962 27 57038 44592 29 55408 oI630 2 98370 24 37 55 4 4 56 43008 28 56992 4464i 30 55359 oi634 2 98366 23 38 54 56 5 4 43053 29 56947 44690 3i 553io 01637 2 98363 22 39 54 48 5 I2 43098 30 56902 44738 32 55262 o0641 2 98359 21 40 9 54 40 2 5 20 9.43143 30 10.56857 9.44787 33 10.552I3 IO.OI644 2 9.98356 20 41 54 32 5 28 43i88 31 56812 44836 34 55164 oi648 2 98352 19 42 54 24 5 36 43233 32 56767 44884 34 55116 oi651 2 98349 i8 43 54 16 5.44 43278 33 56722 44933 35 55067 oi655 3 98345 17 44 54 8 5 52 43323 33 56677 44981 36 55019 oi658 3 98342 16 45 9 54 0 2 6 o 9.43367 34 I0.56633 9.45029 37 10.54971 10.OI662 3 9.98338 i5 46 53 52 6 8 43412 35 -56588 45078 38 54922 oi666 3 98334 14 47 5,3 44 6 16 43457 36 56543 45126 38 54874 01669 3 98331 13 48 53 36 6 24, 43502 36 56498 45I74 39 54826 01673 3 98327 12 49 53 28 6 32 43546 37 56454 45222 4o 54778 01676 3 98324 Ii 50 9 53 20 2 6 40 9.4359I 38 10.56409 9.45271 41 10.54729 IO.OI680 3 9.98320 io 51 53 12 6 48 43635 39 56365 453r9 42 5468 01683 3 98317 9 52 53 4 6 56 4368o0 39 56320 45367 43 54633 01687 3 98313 8 53 52 56 7 4 43724 4o 56276 454i5 43 54585 01691 3 98309 7 54 52 48 7 12 43769 41 56231 45463 44 54537 01694 3 98306 6 55 9 52 40 7 2 9.43813 42 10.56187 9.45511 45 10.54489 10.01698 3 9.98302 5 56 52 32 7 28 43857 43 56143 45559 46 5444i 01701 3 98299 4 57 52 24 7 36 43901 43 56099 45606 47 54394 01705 3 98295 3 58 52 16 7 44 43946 44 56054 45654 47 54346 01709 3 98291 2 59 52 8 7 52 43990 45 56010oo 45702 48 54298 01712 3 98288 i 60 52 0 8 o 44034 46 55966 45750 49 54250 01716 4 98284 M Hour P.M1. Hour A.l. Cosine. Diff. Secant. Cotangent Diff. Tan-ent. Csecant. Diff. Sine. M 105~ A A B B C C 74 Seconds of time...... 1" 2' 3' 4" 5 6 7s A 6 11 17 23 28 34 4o Prop. parts of cols. B 6 I 8 25 3I 37 43 "0 0 1 2 2 3 TABLE XXV II. [Page 201 Log. Sines, Tangents, and Secants. GI. 16" A A B B C C 1630 M Ho- A.A.Ho ur A.... Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. AM 0 9 52 0 2 8 0 9.44034 o 10.55966 9.4575 0o 10.54250 10.01716 -1 9.98284 60 SI 52 8 8 44078 I 55922 45797 1 54203 01719 0 98281 59 2 51 44 8 16 44122 1 55878 45845 2 54155 01723 0 98277 58 3 51 36 8 24 44i66 2 55834 45892 2 54108 01727 0 98273 57 4 5i 28 8 32 44210 3 55790 45940 3 54060 01730 0 98270 56 5 9 51 20 2 8 40 9.44253 4 o10.55747 9.45987 4 io.54oi3 10.01734 0 998266 55 6 51 12 8 48 44297 4 55703 46035 5 53965 01738 0 98262 54 7 5i 4 8 56 44341 5 55659 46082 5 53918 01741 o 98259 53 8 50o 56 9 4 44385 6 55615 46130 6 53870 01745 o 98255 52 9 50 48 9 12 44428 6 55572 46177 7 53823 01749 I 98251 51 10 95040o2 9 20 9.44472 7 o10.55528 9.46224 8 10.53776 10.01752 I 9.98248 50 ii 50 32 9 28 445i6 8 55484 46271 9 53729 01756 1 98244 49 12 50 24 9 36 44559 9 5544r 46319 9 53681 01760 I 98240 48 13 50o i6 9 44 44602 9 55398 46366 io 53634 01763 1 98237 47 14 50 8 9 52 44646 io 55354 46413 ii 53587 01767 I1 98233 46 15 9 5o 0 2 10 0 9.44689 II 1o.553ii 9.46460 12 io.53540o 10.01771 I 9.98229 45 16 49 52 10 8 44733 1ii 55267 46507 12 53493 01774 1 98226 44 117 49 44 io i6 44776 12 55224 46554 13 53446 01778 i 98222 43 i8 49 36 10 24 44819 13 55i8i 466oi 14 53399 01782 I 98218 42 19 49 28 io 32 44862 i4 55i38 46648 15 53352 01785 I 98215 41 20 9 49 20 2 10 4o 9.44905 14 10.55095 9.46694 i5 io.53306 10.01789 I 9.98211 40 21 49 12 io 48 44948 i5 55052 46741 i6 53259 01793 I 98207 39 22 49 4 10 56 44992 i6 55008 46788 17 53212 01796 I 98204 38 23 48 56 1i 4 45035 i6 54965 46835 i8 53165 01800oo 98200 37'24 48 48 11 12 45077 17 54923 46881 19 53119 01804 I 98196 36 25 948 40 2 11 20 9-.45120 18 o10.54880 9.46928 g19 10io.53072 1io.o8o8 2 9.98192 35 26 48 32 11 28 45i63 i8 54837 46975 20 53025 o0811 2 98189 34 27 48 24 11 36 45206 19 54794 47021 21 52979 oi8i5 2 98185 33 28 48 i6 11 44 45249 20 54751 47068 22 5293. 01819 2 98181 32 29 48 8 Ir 52 45292 21 54708 47114 22 52886 01823 2 98177 31 30 9 48 0 12 0 9.45334 21 io.54666 9.47160 23 10.52840 10.01826 2 9.98174 30 31 47 52 12 8 45377 22 54623 47207 24 52793 oi830 2 98170 29 32 47 44 12 16 45419 23 54581 47253 25 52747 01834 2 98166 28 33 47 36 12 24 45462 23 54538 47299 26 52701 01838 2 98162 27 134 47 28 12 32 45504 24 54496 47346 26 52654 o0184 2 98159 26 35 9 47 20 2 12 4o 9.45547 25 io.54453 9.47392 27 10.52608 io0.01845 2 9.98155 25 36 47 12 12 48 45589 26 544ii 47438 28 52562 01849 2 98151 24 37 47 4 12 56 45632 26 54368 47484 29 52516 01853 2 98147 23 38 Z4i6 56 13 4 45674 27 54326 47530 29 52470 01856 2 98144 22 39 46 48 13 12 45716 28 54284 47576 30 52424 oi01860 2 98140 21 4o 9 4640o a 13 20 9.45758 28 10.54242 9.47622 31 10.52378 1O. O1864 2 9.98136 20 41 46 32 i3 28 45801 29 54199 47668 32 52332 01868 3 98132 19 42 46 24 13 36 45843 30 54157 47714 32 52286 01871 3 98129 18 43 46 166 13 44 45885 3i 541i5 47760 33 52240 01875 3 98125 17 44 46 8 3 52 45927 31 54073 47806 34 52194 01879 3 98121 16 45 9 46 0 2 14 o 9.45969 32 o10.54031 9.47852 35 10.52148 io.oi883 3 9 98117 15 46 45 52 14 8 46oii 33 53989 47897 36 52103 01887 3 98113 14 47 45 44 14 I6 46053 33 53947 47943 36 52057 01890 3 98110 13 48 5 36 14 24 46095 34 53905 47989 37 52011 01894 3 98106 I12 49 45 28 i4 32 46r36: 35 53864 48035 38 51965 01898 3 98102 II 5o 9 45 20 2 14 40 9.46178 36 10.53822 9.48080 39 10.51920 10.01902 3 9.98098 io 51i 45 12 1448 46220 36 53780 48126 39 51874 6196 3 98094 9 52 45 4 14 56 46262 37 53738 48171 40 51829 o01910 3 98090 8 53 44 56 15 4 46303 38 53697 48217 41 51783 01913 3 98087 7 54 44 48 15 12 46345 38 53655 48262 42 51738 I0917 3 98083 6 55 94440 2 i5 20 9.46386 39 io.536i4 9.48307 43 10.51693 10.01921 3 9.98079 5 56 44 32 15 28 46428 40 53572 48353 43 51647 01925 3 98075 4 57 44 24- 15 36 46469 41 5353i 48398 44 51602 01929 4 98071 3 58 44 16 15 44 46511 41 53489 48443 45 51557 01933 4 98067 2 59 44/ 8 i5 52 46552 42 53448 48489 46 51511 01937 4 98063 i 60 44 o 1i6 o 465941 43 53406 48534 46 5i466 01940 4 98060 o MI Hour P.M.i HIour AI.M Cosine. iDiff. Secant. Cotangent Diff. Tangent. Cosecant. Diff.m Sine. MI 1060 A A B B C C 730 Seconds of time...... I" 28 3' 4' 5' 6' 7' {A 5 i 6 21 27 32 37 Prop. parts of cols. B 6 12 17 23 29 35 4 C 3 26 26 Page 202] TABLE XXVII. SI. Log. Sines, Tangents, and Secants. GI. 170 A A B B C C 1620 M H Aour A... our Sp. i. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 0 9 44- 2 i6 o 9.46594 o io.534o6 9.48534 o io.51466 10.01940 o 9.980601 6o i 43 52 i6 8 46635 i 53365 48579 x 51421 01944 0o 98056 59 2 43 44 i6 16 46676 i 53324 48624 i 51376 o01948 0 98052 58 3 43 36 16 24 46717 2 53283 48669 2 51331 01952 0 98048 57 4 43 281 6 32 46758 3 53242 48714 3 51286 01956 o 98044 56 5 9 43 20 2 16 4 9.46800 3 10.53200 9.48759 4 10.51241 10.01960 o 9.98040 55 6 43 12 6 48 4684i 4 53159 488o4 4 51196 01964 0 98036 54 7 43 4 i6 56 46882 5 531r8 48849 5 51151 01968 0 98032 53 8 42 56 17 4 46923' 5 53077 48894 6 51106 01971 I 9802 52 9 42 48 17 12 46964 6 53o36 48939 7 51o6i 01975 I 98025 5i o10 9 42 40 2 17 20 9.47005 7 10.52995 9.48984 7 1o.51o06 10.01979 1 9.9802i 50 ii 42 32 17 28 47045 7 52955 49029 8 50971 01983 i 98017 49 12 42 24 17 36 47086 8 52914 49073 9 50927 01987 I 98013 48 13 42 i6 17 44 47127 9 52873 49118 io 50882 01991 I 98009 47 14 42 8 17 52 47168 9 52832 49163 io 50837 01995 i 980546 I5 9 42 0 2 i8 o 9.47209 io 1o.52791 9.49207 i 10.50793 10.01o999 I 9.98001 45 16 41 52 18 8 47249 II 527 1 49252 12 50748 02003 I 97997 44 17 41 44 18 16 47290 ii 52710 49296 12 50704 02007 I 97993 43 18 4136.18 24 47330 12 52670 49341 113 50659 02011 1 9798942 19 4I 28 18 32 47371 13 52629 49385 14 50615 02014 I 97986 41 20 9 41 20 2 i8 40 9.47411 i3 10.52589 9.49430 15 10.50570 10.02018 I 9.97982 40 21 41 12 18 48 47452 14 52548 49474 15 50526 02022 I 97978 39 22 4i 4 i8 56 474921 I5 52508 49519 i6 5048I 02026 I 97974 38 23 40 56 19 4 47533 15 52467 49563 17 50437 02030 2 97970 37 24 4o 48 Ig 12 47573 16 52427 49607 18 50393 02034 2 97966 36 25 9 40 4o 2 19 20 9.47613 17 10.52387 9.49652 18 1o.50348 10.02038 2 9.97962 35 26 40 32 19 28 47654 17 52346 49696 19 50304 02042 2 97958 34 27 40 24 19 36 47694 18 52306 49740 20 50260 02046 2 97954 33 28 40 16 19 44 47734 19 52266 49784 21 50216 02050 2 97950 32 290 4o 8 19 52 47774 19 52226 49828 21 50172 02054 2 97946, 30 9 4o 0 2 20 0 9.47814 20 10.52186 9.49872 22 10.50128 10.02058 2 9.97942 30 31 39 52 20 8 47854 21 52146 49916 23 500oo84 0262 2 97938 29 32 39 44 20 i6 47894 21 52106 49960 24 50040o 02066 2 7934 28 33 39 36 20 24 47934 22 52066 5000ooo4 24 49996 02070 2 97930 27 34 39 28 20 32 47974 23 52026 5oo0048 25 49952 02074 2 97926 26 35 9 39 20 2 20 40 9.48014 23 10.51986 9.50092 26 10.49908 10.02078 2 9.97922 25 36 39 12 20 48 48054 24 51946 50o36 26 49864 02082 2 97918 24 37 39 4 20 56 48094t 25 51906 5o180 27 49820'02086 2 97914 23 38 38 56 21 4 4833 25 51867 50223 28 49777 02090 3 97910 22 39 38 48 21 12 48173 26 51827 50267 29 49733 02094 3 97906 21 40 9 38 40 2 21 20 9.48213 27 10.51787 9.50311I 29 10.49689 10.02098 3 9.97902 20 41 38 32 21 28 48252 27 51748 50355 30 49645 02102 3 97898 19 42 38 24 21 36 48292 28 51708 50398 31 49602 02106 3 97894 i8 43 38 i6 21 44 48332 29 5i668 50442 32 49558 02110 3[ 97890 17 44 38 8 21 52 48371 29 51629 50485 32 49515 02114 3 97886 16 45 938 0 2 22 0 9.48411 3o 10.51589 9.50529 33 10.49471 10.02118 3 9.97882 15 46 37 52 22 8 48450 31 51550 50572 34 49428 02122 3 97878 14 47 37 44 22.6 48490 31 51510 50616 35 49384 02126 3 97874 13 48 37 36 22 24 48529 32 51471 50659 35 49341 02130 3 97870 12 49 37 28 22 32 48568 33 51432 50703 36 49297 02134 3 97866 ii 50 9 37 20 2 22 4o 9.48607 33 ro.5393 9.50746 37 10.49254 10.0239 3.97861 0 51 37 12 22 48' 48647 34 51353 50789 37 49211 02143 3 97857 9 52 37 4 22 56 48686 35 5 03i4 5o833 38 49167 02147 3 97853 8 53 36 56 23 4 48725 35 51275 50876 39 49124 02151 4 97849 7 54 36 48 23 12 48764 36 51236 50919 4o 49081 02155 4 97845 6 55 9 3640 2 23 20 9.48803 37 10.51197 9.50962 4o 10.49038 10.02159 4 9.97841 5 56 36 32 23 28 48842 37 51158 51005 41 48995 02163 4 97837 4 57 36 24 23 36 48881 38 51119 5o1048 42 48952 02167 4 97833 3 58 36 6 23 44 48920 39 51080 51092 43 48908 02171 4 97829 2 59 36 8 23 52 48959 39 5o1041 51135 43 48865 02175 4 97825 i 60 36 o 24 0 48998 4o 51002 51178 44 48822 02179 4 97821 0 M flour P.. IHolirA.Ml. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1070r A A B B C C 721 Seconds of time...... " 28 3" 4 5 6 o 7" 3A 5 10 15 20 25 5o 35 { C o I I 2 2 3 Prop. 2 4 rt/ of3, col1 5 668j 20421 291 581 33.o 3 l9 7 9l1 TABLE XXVIL. Page 203 8'. Log. Sines, Tangents, and Secants.' 130 A A B B CC C1610 MH~our A.M. 1on-p. M. Sine. Diff. Cosecant.. Tangent. Diff. Cotangent Secant. Diff. Cosine. I 3- 9-36 0-2224 o 9.48998 o 10.51002 9.51178 0 10.48822 1o.02179 0 9.97821 6o i 35 952 24 8 49037 I 50963 51221 i 48779 02183 o 97817 59 2 3544 24 16 Z49076 i 50924 51264i 48736 02188 o 97812 58 3 35 3' 24 24 495315 2 50 5 1306 2I 48694 02o12 o 9708 157 4 3528 2432 4963 3 50847 51349 31 48651 o02196'o 97804 56 5 93520 2 24 40 9 94992 3 10.5088 9-51392 3 1.48608 10.0220 0 1o9.97800 55 6 35 12 24 48 49615 4 50769 51435 4 48565 02204 o 97796 54 7 354 24 56 49269 4 5o731 51478 5 48522 02208 o 97792 53 8 34 56 25 4 49308 5 5o692 51520 6 448480 o022 I 97788 52 9 34 48 25 12 49347 6 50653 51563 6 484371 02216 97784 51 o 934 40 2 25 20 9.49385 6I 10.50615 9.51606 7 10.48394 10.02221 1 9-97779 50 ii 34 32 25 28 49424 7 50576 52648 8 48352 02225 I 97775 49 12 34 24 25 36 49462 8 5o538 51691 8 48309 02229[ i 97771 48 i3 34 6 25 44 49500 8 5o0500oo 51734 9 48266 02233 I 97767 47 14 34 81 25 52 49539 9 5o46o 517761 io 48224 02237 i 97763 46 15 9 34 0 2 26 o 9.49577 9 10.50423 9.51819 1o 10.48785 10.02241 i 9-97750 45 26 33 52 26 8 49615 io 50385 51861 Ii 48139 02246 i 97754 44 17 33 44 27 16.49654 r1 50346 51903 12 48097 02250 i 97750 43 18 33 36 26 24 496921 i 50308 52946 13 48054 02254 I 97746 42 19 33 8 26 32 49730o 12 50270 51988 13 48o52 02258 i 97742 14 20 9 33 20 2 26 40 9.4976o8 1 10.50232 9.5203 1 410.47969 10.02262 i 9.97738 40 21 33 12 286 48 49806 13 50194 52073 I5 47927 02266 I 97734 39 22 33 4 26 56 49844 I4 50156 52115 15 47885 02271 2 97729 38 23 32 56 27 4 49882/ 142 50118 521578 16 47843 02275 2 97725 37 24 32 48 27 2 49920 15 5oo0080 52200 17 47800 02279 2 97721 36 125 9 32 40. 22720 9.49958 6j 10.50042 9.522421 17 10.47758 10.02283 2 9.97717 35 26 32 32 27 28 499960 26 50004 52284 18 47716 02287 2 977 1334 27 32 24 27 36 5oo0034 17 49966 52326 19 47674 02292 2 97708 33 28 32 i6 27 441 50072 218 49928 52368 20 47632 02296 2 97704 132 29 32 8 27 52 50110 18 49890 52410 20 47590 02300 2 97700 31 9 3 o3 2 28 o0 9.50o1 8 9 10.i49852 9.52452 21 10.47548 10.02360 2 9-97696 30 31 31 52 2 8 5081 o85 20 49815 524941 22 47506 02309 2 97691 29 32 3144i 28 16 50223 2 49777 52536 22'47464 02313 2 97687 128 33 31236 28 24 50261 21 49739 52578 23 47422 02317 2 976831 27 34 31928 28 32 50298 21 49702 52620 24 4738o 02321 2 97679 26 35 9 312 20 23 840 9.50336 22 10.49664 9.52661 24 10.47339 10.02326 2 9.97674 25 36 31 12 28 48 50374 23 49626 52703 23 47297 02330 3 97670 24 317 31 4 28 56 50o1 23 49589 52745 26 47255 02334 3 9766 2 138 30 56 29 4 50449 3 48955 52783 27 47261 02338 3 97662 22 39 30 48 29 12 50486 25 49514 52829 27 47171 02343 3 97657 21 40 930 4 2 2920.50523 25 1049477 9.52873 28 10o.4713o 10.02347 3 9.97653 20 341 8 3 29 2849 556 26 39 52912 29. 47088 02351 3 97649 14 7 30 24 29 36 5o598 26 48402 52953 29 47047 02355 3 97645 18 43 30 16 29 44 5o0635 27 49365 52995 4 47005 02360 3 97640 17 44 5308 29 52 50673 7 3 476963 02364 3 97636 16 45 9 30 0 2 3o 0 9.50710 28 180.49290 9-53078 31 10.46922 0o.o2368 3 9.976321 5 46 2952 308 50747 29 49253 53120 32 46880 02372 3 97628 14 47 29 48 30 52 50784 3 49216 53161 33 46839 02377 3 97623 1i3 48 29 36 30 24 50821 30 49179 53202 34 46798 02381 3 976 9I 12 49 29 28 30 332 50858 31 4942 532441 34 46756 02385 3 7 9764J5 _i 50 9 29 20 2 30 40 9.50896 31 cos.4914 9A 53285 35 jI.46715 10.02390 4 9-976 2o83 151 2912 30 48 50933 3 49067 53327 36 46673 02394 4 97606 o 52 29 4 30 56 5097 33 49030 53368 36 46632 02398 4 97602 8 53 28 56 31 4 51007 33 48993 53409 37 46591 02403 4 97597 7 54 28 48 3112 51043 34 48957 53450c 38 46559 6 02407 4i 97593 6 55 9 8 40 2 31 20 o.580 35 0.48920 9-53-492 38 10-I.465o8 1io0.02411 4 9-97589 5 56 2832 3128 51177 35 48883 53533 39 46467 02416 4 97584 4 57 28 24 31 36 5ii54 36 488416 53574 40 466426 02420 4 97580 3 58 2816 3 44 5II/ 317 489809 53615 41 46385 02424 4 97576 21 59 28 8 31 52 51227 37 48773 53656 41' 46344 02429 4 97571 i 60 28 o 32 o 3 512643 38 48736 53697 42 46303 02433 4 97567 o AM Honr p.1`. HourA.M. Cosine. Diff. Secant. Cotangent Diff., Tangent. Cosecant. Diff. Si "I. M 1.6)808~ """"" -~ ^ A B B ~ G C 71" Seconds of time...... Is 2o9 3' 46 59 6" 7" A 5 9 4 19 24 28 33 Prop. parts of cols. 3B 5 io I6 21 26 31 37 c 2:96 3 3 4 Page 204] TABLE XXVII. 8'. Log. Sines, Tangents, and Secants. G. 190 A A B B C C 160~ M Hour A.M. Hour.nM. Sine. Diff. Cosecant. Tangent. Dil[KGotangent Secant. Diff. Cosine. IT o 9 28 2 32 09.51264 o 10.48736 9.5369-7 0io.46303 10.02433 o 9.97567 6o i 27 52 32 8 5t3oi i 48699 53738 i 46262 02437 0 97563159 2 27 44 32 6i 5i338 I 486621 53779 46221 02442 97558 58 3 27 36 32 24 51374 2 48626 53820 2 461I8o 02446 o 97554 57 4 27 28 32 32 5141i 2 48589 5386i 3 46139 02450 0 97550 56 5 9 27 20 2 32 4o 9.51447 3 10.48553 9.53902 3 10.46098 10.02455 o 9.97545 55 6 27 12 32 48 51484 4 485i6 53943 4 46057 02459 o 97541 54 7 27 4 32 56 51520 4 4848o 53984 5 460o6 02464 I 97536 53 8 26 56 33 4 51557 5 48443 54025 5 45975 02468 I 97532 52 9 26 48 33 12 51593 5 48407 54065 6 45935 02472 8 97528 51 1O 9 26 40 2 33 20 9o.5629 6 10.48371 9.54106 7 1o.4584 10.02477 I 9.-97523 5o II 26 32 33 28 51666 7 48334 54147 7 45853 0248 1 97519 49 12 26 24 33 36 51702 7 48298 54187 8 458i3 024851 97515 48 13 26 i6 33 44 51738 8 48262 54228 9 45772 02490 T 97510 47 i4 26 8 33 52 51774 8 48226 54269 9 45731 02494 I 97506 46 15 9 26 0 2 34 0 9.51811 9 o-.48i8q 9.54309 10 10.45691 10.02499 I 9-97501 45 16 25 52 34 8 51847 0o 48i53 5435o II 45650 02503 I 97497 44 17 25 441 34 i6 51883 io 48117 54390 11 4567o 02508 I 97492 43 i8 25 36 34 24 59919 II 48o8i 54431 12 45569 02512 1 97488 42 19 25 28 34 32 51955 i1 48o45 54471 J3 45529 025r6 I 97484 4i 20 9 25 20 2 34 4o 9.5199 12 10.48009 9.54512 13 10.45488 10.02521 1 9.97479 40 21 25 12 34 48 52027 12 47973 54552 14 45448 02525 2 97475 39 22 25 4 34 56 52063 13 47937.54593 15 45407 02530 2 97470 38 23 24 56 35 4 520o99 1 47901 54633 i5 45367 02534 2 97466 37 24 24 48 35 12 52135 14 47865 54673 6 45327 02539 2 97461 36 25 9 24 40o 2 35 20 9.52171. 15 10.47829 9.54714 17 10.45286 10.02543 2 09-97457 35 26 24 32 35 28 52207 15 47793 54754 17 45246 02547 2 97453 34 27 24 24 35 36 52242 16 47758 54794 18 45206 02552 2 7448 33 28 24 16 35 44 52278 17 47722 54835 I9 45T65 02556 2 97444 32 29 24 8 35 52 52314 17 47686 54875 19 45125 02561 2 97439 31 3o 9 24 0 2 36 0 9.52350 i8 o0.47650 9.54915 20 1o.45o85 10.02565 2 9.97435 30 31 23 52 36 8 52385 18 47615 54955 21 45045 02570 2 97430 29 32 23 44.36 16 52421 19 47579 54995 21 45005 02574 2 97426 28 33 23 36 36 24 52456 20 47544 55035 22 44965 02579 2 97421 27 34 23 28 36 32 52492 20 47508 55075 23 44925 02583 3 97417 26 35 92320 236 4 9.52527 21 10.47473 9.555. 23 io.44885 10.02588 3 9.97412 25 36 23 12 36 48 52563 21 47437 55155 24 44845 02592 3 97408 24 37 23 4 36 56 52598 22 47402 55195 25 44805 02597 3 97403 23 38 22 56 37 4 52634 23 47366 55235 25 44765 02601 3 97399 22 39 22 48 37 12 52669 23 47331 55275 26 44725 02606 3 97394 21 40 9 22 402 37 20 9.52705 24 10.47295 9.55315 27 io.44685 10.02610 3 9-97390 20 41 22 32 37 28 52740 24 47260 55355 27 44645. 02615 3 97385 19 42 22 24 37 36 52775 25 47225 55395 28 44605 02619 3 97381 18 43 22 16 37 44 52811 26 47189 55434 29 44566 02624 3 97376 17 44 22 8 37 52 5284626 47154 55474 29 44526 02628 3 97372 i6 45 9 22 0 2 38 o 9.52881 27 10.47119 9.55514 30 Io0.44486 10.02633 3 9-97367 15 46 21 52 38 8 52916 27 47084 55554'31 44446 02637 3 97363 i4 47 21 44 38 16 52951 28 47049 55593 31 44407 02642 3 97358 13 48 21 36 38 24 52986 29 47014 55633 32 44367 02647 4 97353 12 49 21 28 38 32 53021 29 46979 55673 33 44327 02651 4 97349 11 50 9 21 20 23840 9.53o56 30 10.46944 9.55712 33 10.44288 10.02656 4 9.97344 io 51 2 112 38 48 53092 30 46908 55752 34 44248 02660 4 97340 9 52 21 4 38 56 53126 31 46874 55791 35 44209 02665 4 97335 8 53 20 56 39 4 53161 32 46839 5583i 35 4169 0269 4 97331 7 54 20 48 39 12 53196 32 468o4 55870o 36 44130 02674 4 97326 6 55 9 2040 2 39 20 9.53231 33 10.46769 9.55910io 37 o10.44090 10.02678 4 9.97322 5 56 20 3.2 39 28 53266 33 46734 55949 37 44o5i 02683 4 97317 4 57 20 24 39 36 5330or 34 46699 55989 38 44011 02688 4 97312 3 58 20 16 39 44 53336 34 46664 56028 39 43972 02692 4 97308 2 59 20 8 39 52 53370 35 46630 56067 39 43933 02697 4 97303 I 60 2 0 40 o 53405 36 4695 56io7 40 43893 02701 4 97299 Mji1r 1 11. H'..HourA.1. Cosine. Diff. Secant. Cotang-ent Diff. Tang-ent. I Cosecant. Diff. Sine. M, 1090 A A B B C C 70(0 Seconds of time...... Is S2^ 3 -. 4S 5s 68 7" (A 4, 9 i3 i8 22 27 31 Prop. parts of cols. jB 5 io 15 20 25 30 35 C 1 2 2 3 3 4 TABLE XXVII. [Page 205 S'. Log. Sines, Tangents, and Secants. GI 20~ A A B B C C 1590 MI HourA.eM. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff.j Cosine. M 0 9 20 0 2 4o o 9.53405 o 10.46595 9.56107 o 10.43893 10.02701 0 9.97299 6o i 19 52 4o 8 53440 I 46560 56146 i 43854 02706 0 97294 59 2 1g 44 401 6 53475 l 46525 56t85 l 438i5 02711 0 97289 58 3 1i 36 40 24 53509 2 46491 56224 2 43776 02715 0 97285 57 4 19 28 4o 32 53544 2 46456 56264 3 43736 0273201 97280 56 5 9 9 20 2 40 40 9.53578 3 10.46422 9.56303 3 10.43697 10.02724 0 9-97276 55 6 19 12 40 48 53613 3 46387 56342 4 43658 02729 0 97271 54 7 19 4 40 56 53647 4 46353 56381 4 43619 02734 I 97266 53 8 18 56 4i 4 53682 5 463i8 56420 5 4358o 02738 I 97262 52 9 i8 48 41 12 53716 5 46284 56459 6 43541 02743 i 97257 51 io9 18 40o 2 41 20 9.53751 6 10o.46240 9.56498 6 10.43502 10.02748 I 9.97252 5o ii 18 32 41 28 53785 6 46215 56537 7 43463 02752 I 97241 49 12 18 24 41 36 53819 7,461i8 56576 8 43424 02757 I 97243 48 13 i8 i6 41 44 53854 7 46i46 566i5 8 43385 02762 I 97238 47 14 188 8 41 52, 53888 8 46112 56654 9 43346 02766 I 97234 46 1i 9 18 0 2 42 o 9.53922 8 1o.46078 9.56693 10io 10.43307 10.02771 I 9.97229 45 i6 17 52 42 8 53957 9 4643 56732 10o 43268 02776 i 97224 44 17 17 44 42 i6 53991 io 46009 56771 ii 43229 02780 I 97220 43 i8 17 36 42 24. 54025 io 45975 56810 12 43190 02785 I 97215 42 19 17 28 42 32 54059 ii 4594i 56849 12 4315i 02790 1 97210 41 20 9 17 20 242 4o 9.54093 ii 10o.45907 9.56887 13 io.43ii3 10.02794 2 9.97206 40 21 17 12 42 48 54127 12 45873 56926 13 43074 02799 2 97201 39 22 174 42 56 5416i 12 45839 56965 14 43035 02804 2 97196 38 23 i6 56 43 4 54195 13 458o5 57004 15 42996 02808 2 97192 37 24 16 48 43 12 54229 I4 45771 57042 15 42958 02813 2 97187 36 25 916 40 2 43 20 9.54263 i4 10.45737 9.57081 i6 10.42919 10.02818 2 9.97182 35 26 i6 32 43 28 54297 15 45703 57120 17 42880 02822 2 97178 34 27 i6 24 43 36 54331 15 45669 57158 17 42842 02827 2 97173 33 28 i6 6 43 44' 54365 16 45635 57197 i8 42803 02832 2 97168 32 29 16 8 43 52 54399 i6 45601o 57235 19 42765 02837 a 97163 31 30 9 i6 0 2 44 0 9.54433 17 10.45567 9-57274 19 10.42726 10.02841 2 9.97159 3o 31 15 52 44 8 54466 17 45534 57312 20 42688 02846 2 97154 29 32 i5 44 44 16 54500 18 45500 57351 21 42649 0285r 3 97149 28 33 15 36 44 24 54534 19 45466 57389 21 42611 02855 3 97145 27 34 15 28 44 32 54567 19 45433 57428 22 42572 02860 3 97140 26 35 9 15 20 2 44 4o 9.546o 20 o10.45399 9.57466 22 10.42534 10.02865 3 9.97135 25 36 15 12 44 48 54635 20 45365 57504 23 42496 02870 3 97130 24 37 15 4 44 56 54668 21 45332 57543 24 42457 02874 3 97126 23 38 14 56 45 4 54702 21 45298 57581 24 42419 02879 3 97121 22 39 i4 48 45 12 54735 22 45265 57619 25 4238i 02884 3 97116 21 40 9 14 40 2 45 20 9.54769 23 10.45231 9.57658 26 10.42342 10.02889 3 9.97111 20 41 i4 32 45 28 54802 23 45198 57696 26 42304 02893 3 97107 19 42 14 24 45 36 54836 24 45164 57734 27 42266 02898 3 97102 i8 43 I4 i6 45 44 54869 24 45i3i 57772 28 42228 02903 3 97097 17 44 14 8 45 52 54903 25 45097 57810o 28 42190 02908 3 97092 i6 45 9 14 0 246 0 9.54936 25 10.45o64 9.57849 29 10.42151 10.02913 4 9-97087 I5 46 13 52 46 8 54969 26 45o3i 57887 30 42113 02917 4 97083 14 47 i3 44 46 I6 55003 26 44997 57925 30 42075 02922 4 97078 13 48 13 36 46 24 55036 27 44964 57963 31 42037 02927 4 97073 12 49 13 28 46 32 55069 28 44931 58oo001 31 41999 02932 4 97068 ii 50 9 13 20 246 409.55102 28 10.44898 9.580o39 32 10.41961 10.02937 4 9.97063,o 51 13 12 46 48 55136 29 44864 58077 33 41923 02941 4 97059 9 52 13 4 46 56 55169 29 4483i 58ii5 33 4T885 02946 4 97054 8 53 12 56 47 4 55202 30 44798 58i53 34 41847 02951 4 97049 7 54 12 48 47 12 55235 30 44765 58191 35 41809 o2956 4 97044 6 55 9 12 4o 2 47 20 9.55268 31 10.44732 9.58229 35 10.41771 10.02961 4 9.97039 5 56 12 32 47 28 553o01 32 44699 58267 36 41733 02965 4 97035 4 57 12 24 47 36 55334 32 44666 58304 37 41696 02970 4 97030 3 12 i6 47 44 55367 33 44633 58342 37 4i658 02975 5 97025 2 59 128 4752 55400oo 33 446oo00 58380 38 41620 02980 5 97020 i 6o 12 o 48 0 55433 34 44567 584i8 39 41582 o2985 5 97015 o M Hour P.M. Hour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. iDiff. Sine. M 110~ A A B B C C 69G Seconds of time...... IP" 2" 3 4' 5s 6s 7. o 4 8 13 17 21 25 30 Prop. parts of cols. 5 10 4 24 29 34 C 1 2 2 3 4 4 Page 206] TABLE XXVII. S1. Log. Sines, Tangents, and Secants. Gr. 210 A A B B C C 1580 IAI HIour Al.m.Hour p.r. Sine. Diif. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 0 9 12 0 2 48 o 9.55433 o 10.44567 9.58418 o 10.41582 10.02985 0 9.97015 60 ii 52'48 8 55466 44534 58455 i 4i545 990 0 97010 59 2 11 44 48 i6 55499 1 44501 58493 i 41507 02995 o 97005 58 3 1i 36 48 24 55532 2 44468 58531 2 41469 02999 0 970(1 57 4 1 128 48 32 55564 2 44436 58569 2 4431 03004 o 96996 56 5 9 11 202 2 48 4o 9.55597 3 1o.444o3 9.58606 3 Io.41394 o0.o3009 o 9.96991 55 6 11 12 48 48 55630o 3 44370 58644 4 4i356 o3oi4 o 96986 54 7 11 4 48 56 55663 4 44337 5868i 4 41319 03019 i 9698r1 53 8 10 56 49 4 55695 4 44305 58719 5 41281 03024 96976 52 9 10 48 49 12 55728 5 44272 58757 6 41243 03029 I 96971 51 9 9o 40 2 49 20 9.55761 5 10.44239 9.58794 6 1o.41206 o.o30o34 i 99696. 6 o0 Ii 10 32 49 28 55793 6 44207 58832 7 4ri68 03038 i 96962 49 12 Io 24 49 36 55826 6 44174 58869 7 41131 03043 i 96957 48 13 io 16 49 44 55858 7 44142 58907 8 41093 03048 96952 47 i4 1o 8 49 52 55891 7 44109 58944 9 4io56 o3053 i 96947 46 15 910 0 50 o 9.55923 8 10.44077 9.58981 9 10.41019 io.o3o58 i 9.96942 45 16 9 52 50 8 55956 9 44o44 5g909 I 40981 o3o63 i 96937 44 17 9 44 50 16 55988 9 44012 59056 io 40944 o3o68 i 96932 43 18 9 36 50 24 56021 io 43979 59094 ii 40906 03073 i 96927 42 19 9 28. 5o 32 56o53 io 43947 59131 12 40869 03078 2 96922 41 20 9 9 20 2 50 40 9.56085 ii 10.43915 9.59168 12 10.4o0832 o.o3o83 2 9.96917 40 21 9 12 50 48 561181 i 43882 59205 13 40795 03088 2 96912 39 22 9 4 50o 56 56150o 12 4385o 59243 14 40757 03093 2 96907 38 23 8 56 5i 4 56182 12 438i8 59280 14 40720 03097 2 96903 37 24 8 48 5i 12 56215 13 43785 59317 15 4o0683 o3102 2 96898 36 25 9 8 40 2 51 20 9.562471 i10.43753 9.59354 1510.40646 10.03107 2 9.96893 135 26 8 32 51 28 56279 I4 43721 59391 i6 40609 03112 2 96888 34 27 8 24'5i 36 5631i 14 43689 59429 17 4o57I o3117 2 96883 33 28 8 i6 51 44 56343 i5 43657 59466 17 4o534 03122 2 96878 32 29 8 8 51 52 56375 16 43625 59503 18 40497 o3127 2 96873 31 30 9 8 0 2 52 o 9.56408 16 10.43592 9.59540 19 1o.40460o 10.03132 2 9.96868 3o 31 7 52 52 8 5644o 17 4356o 59577 19 40423 03137 3 06863 29 33 7 44 52 i6 564,72 17 43528 59614 20 4o386 o3142 3 96858 28 33 7 36 52 24 565o4 18 43496 59651 20 40349 03147 3 96853 27 34 7 28 52 32 56536 i8 43464 59688 21 40312 03152 3 96848 26 35 9 7 20 2 52 40 9.56568 19 10o.43432 9.59725 22 10.40275 10.03157 3 9.96843 25 36 7 12 52 48 56599 19 434o0 59762 22 40238 03162 3 96838 24 37 7 4 52 56 56631 20 43369 59799 23 40201 03167 3 96833 23 38 6 56 53 4 56663 20 43337 59835 23 4oi65 03172 3 96828 22 39 6 48 53 12 56695 21 433o5 59872 24 40128 03177 3 96823 21 40 9 6 40 2 53 2o 9.56727 21 10.43273 9.59909 25 10.40091 10.03182 3 9.96818 20 41 6 32 53 28 56759 22 43241 59946 25 40054 03187 3 96813 19 42 6 24 53 36 56790 22 43210 59983 26 40017 03192 "3 968o8 i8 43 6 16 53 44 56822 23 43178 60019 27 39981 03197 4 96803 1/7 44 6 8 53 52 56854 24 43i46 6oo56 27 39944 03202 4 96798 16 45 9 6 0 2 54 o 9.56886 24 10.43114 9.60093 28 10.39907 10.03207 4 9.96793 15 46 5 52 54 8 56917 25 43083 6oi30 28 39870 03212 4 96788 14 47 5 44 54 i6 56949 25 43o5i 6o0166 29 39834 03217 4 96783 13 48 5 36 54 24 56980 26 43020 60203 3o 39797 03222 4 96778 1 2 49 5 28 54 32 57012 26 42988 60240 3o 39760 03228 4 96772 II 50 9 5 20 2 54 4o 9.57044 27 10.42956 9.60276 31 10.39724 10.03233 4 9.96767 10 5i 5 12 54 48 57075 27 42925 6o313 31 39687 03238 4 96762 9 52 5 4 54 56 57107 28 42893 60349 32 39651 03243 4 96757 8 53 4 56 55 4 57138 28 42862 60386 33 39614 03248 4 96752 7 54 4 48 55 12 57169 29 42831 6042.2 33 39578 03253 4 96747 6 55 9 4 40 2 55 o20 9.5720i 29 Il0.42799 9.60459 34 ro.39541 o.03258 5 9-96742 5 56 4 32 55 28 57232 30 42768 60495 35 39505 03263 5 96737 4 57 4 24 55 36 57264 30 42736 6o532 35 39468 03268 5 96732 3 58 4 16 55 44 57295 3i 42705 60568 36 39432 o3373 5 96727 2 59 4 8 55 52 57326 32 42674 6o6o5 36 39395 o3278 5 967221 60 4 o 56 o 57358 32 42642 6o641 37 39359 03283 5 96717 0 Hour P.M. Hour A.M.. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine.'M 1110 A A B B C C 68~ Seconds of time...... I1 2 3 4! 5" 6" 7" 4 8 12 i6 20 24 28 Prop. parts of cols. B 14 19 23 28 3 (rPPrsfos{C i i 2 1 3 4 4 TABLE XXVII. [Page 207 S'. Log. Sines, Tangents, and Secants. G'. ____A____ A B B C C 1570 M Hour A.m.Houor P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. Al 00 4 02560.53o 9.573 10.42642 9.60641 o io.39359 1o.o3283 o 9.96717 60 1 3 52 56 8 57389 i 42611 60677 1 39323 03289 0 96711 59 2 3 44 56 16 57420 i 42580 60714 1 39286 03294 o 96706 58 3 3 36 56 24 57451 2 42549 60750 2 39250 03299 o 0u 96701 57 4 3 28 56 32 57482 2 42518 60786 2 39214 o33o4 o 96696 56 5 -9 3 20 25640 9.57514 3 10.42486 9.60823 3 10.39177 1o.o339 o 9-96691 55 6 3 12 56 48 57545 3 42455 60859 4 39141 o3314 I 06686 54 7, 3 4 56 56 57576 4 42424 60895 4 39105 o3319 i 96681 53 8 2 56 57 4 57607 4 42393 6093 5 39069 03324 I 96676 52 9 2,48 57 12 57638 5 42362 60967 5 39033 o333o i 96670 51 io 9 2 40 2 57 20 9.57669 5 10.42331 9.61004 6 10.38996 io.o3335 I 9.96665 50 rit 2 32 57 28 57700 6 42300 60o4o 7 38960 o3340 96660 49 12 2 24 57 36 57731 6 42269 6I076 7 38924 o3345 i 96655 48 138 2 16 57 44 57762 7 42238 61112 8 38888 03350 i 96650 47 14. 2 8. 57 52 57793 7 42207 61i48 8 38852 o3355 i 96645 46 15 9 2 0 2 58 o 9.57824 8 10.42176 9.61184 9 io.38816 10.03360 I 9.96640 45 16C: I 52 58 8 57855 8 42145 61220 io 38780 o3366 i 96634 44 17 1 44 58 16 57885 9 42[15 61256 io 38744 03371 i 96629 43 1 8 I 36 58 24 57916 9 42084 61292 ii 38708 03376 2 96624 42 10 i 28 58 32 57947 10 42053 61328 ii 38672 03381 2 96619 4i 20 9 1 20 2 58 4o 9.57978 io 10.42022 9.61364 12 o.38636 10.03386 2 996614 4o 21' 12 58 48 58oo008 ii 41992 614oo 13 386oo00 03392 2 96608 39 22 I 4 58 56 58039 ii 41961 61436 13 38564 03397, 2 96603 38 23 0 56 59 4 58070 12 41930 61472 14 38528 03402 2 96598 37 24 048 5912 58ioi 12 41899 6150o8 14 38492 0340Z7 2 96593 36 25 9 040 259 20 9.58131 3 10.41869 9.61544 151 io.38456 10.03412 2 9.96588 35 26 0 32 5928' 58162 13 4i838 61579 15 38421 o34i8 2 96582 34 27 0 24 59 36 58192 14 4i808 6i6i5 i6 38385 03423 2 96577 33 28 0 i6 59 44 58023 14 41777 6165 i 17 38349 03428 2 96572 32.29 0 8 59 521 58253 I5 41747 61687 17 383i3 03433 3 96567 3i 30 9 003 0 o 9.58284 15 10.41716 9-61722 18 10.38278 io0.03438 3 9.96562 30 3i 8 59 52 o 8 58314 i6 4i686 61758 18 38242 o3444 3 96556 29 32 59 44 o 16 58345 16 4I655 61794 19 38206 03449 3 96551 28 33 59 36 0 24 58375 17 41625 6i830 20 38170 o3454 3 96546 27 34 59 28 o 32 58406 17 41594 6i865 20 38 r 35 o3459 3 96541 26 35 859203 0 40 9.58436 81r0.4i564 9.61901 21 10.38J099 1.0 3465 3 9.96535 25 36 5o9 12 o2 481 58467 18 41533 61936 21 3864 03470 3 96530 24 37 59 4 0 56 58497 19 4i503 61972 22 38028 03475 3 96525 23 38 58 56 I 4 58527 19 41473 62008 23 37992 o3480 3 96520 22 39 58 48 1 12 58557 20 4i443 62043 23 37957 o3486 3 96514 21 40 8 58 40o 3 1 20 9.58588 20 10.41412 9.62079 21 10.37921 10.03491 3 9.96509 20 41 58 32 o 28 58618 21 41382 621814 24 37886 03496 4 96504 19 42 588 4 2 36 58648 21 41352 62150 25 3785o 03502 4 96498 i8 43 58 i6 1 44 58678 22 41322 62185 26 37815 03507 4 96493 17 44 58 8' 1 522 58709 22 41291 62221 26 37779 o3512 4 96488 16 45 8 58 o 3 2 o 9.58739 23 10.41261 9.62256 27 10.37744 10.03517 4 9.96483 i5 46 57 521 2 8 58769 23- 41231 62292 27 37708 03523 4 96477 14 47 57 44 2 16 58799 24 41201 62327 28 37673 03528 4 96472 13 48 57 36 2 324 58829 24 41171 62362 29 37638 03533 4 96467 12 49 57 28 2 32 58859 25 41141 62398 29 37602 03539 4 9646/1 i1 50 8 57203 2 40 9.588891 25 1o.41i 9.62433 30 10.37567 io.o3544 4 9.96456'o 51 57 12 2 48 58919 26 41o08 62468 3o 37532 03549 4 96451 9 452 57 4 2 56 58949 26 410o5 629504 31 37496 o3555 5 96445 8 153 56 56 3 4 58979 27 41021 62539 32 37461 o3560 5 96440 7 154 5648 3 12 59009 27 40991 625743 32 37426 o3565 5 96435 6'55 8 56 4o 3 3 20.59039 28 10.40961 9.62609 33 0.37391 10.0o3571 5 9.96429 5 56 56 32 3 28 59069 28 40931 62645 33 37355 03576 5 96424 4 57 56 24 3 36 59098 29 40902 62680 34 37320 03581 5 96419 3 58 56 16 3 -44 59128 29 40872 62715 35 37285 03587 5 9643 2 59 56 8 3 52 5958 3o 40842 62750 35 37250 03592 5. 96408 60 56 o 4 5988 31. 40812 62785 36 37215 03597 5 96403 0 Hour p.M.I -iourA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1120 A A B B C C ~ 7' Seconds of time...... I S 3 4s 5s 6s 7" (A 4 8 11 15 19 23 27 Prop. parts of cols. B 4 13 8 22 27 31 C Ii i 2 3 3 4 5 g 8' TABLE XXVII. Log. Sines, Tangents, and Secants. 23~ ___________________ A_________A___ ________~, _________l__ C15G'. 9230 A A B B C C 1560 MA Hour A.M. Hour P.Mx. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M o8 56 3 4 09.588 o 10.40812 9.62785 o 10.372I5 10.03597 0 9.96403 60 1 55 52 48 59218 0 40782 6820 I 37180 o36o3 o 96397 59 2 55 44 46 59247 1 40753 62855 T 37145 o3608 o 96392 581 3 55 36 4 24 59277 1 40723 62890 2 37110 o363 o 96387 571 4 55 28 4 32 59307 2 40693 62926 2 37074 03619 0o 96381 561 5 855 20o 3 44o 9.59336 2 o.4664 9.6296i 3 10.37039 10.03624 0 9.96376 55 6 55 12 4 48 59366 3 4o634 62996 3 37004 03630 i 96370 54 7 55 4 4 56 59396 3 40604 63031 4 36969 03635 i 96365 53 8 54 56 54 59425 4 40575 63o66 5 36934 o3640 96360 52 9 54 48 5 12 59455 4 4o545 63ioi 5 36899 03646 i 96354 51 10 8 54 4o 3 5 20 9.59484 5 io.40516 9.63135 6 io.36865 1o.o365i i 9.96349 50 11ii 54 32 528 59514 5 40486 63170 6 36830 03657 I 96343 49 12 54 24 536 59543 6 40457 63205 7 36795 03662 I 96338 48 13 54 16 5 44 59573 6 40427 63240 7 36760 03667 1 96333 47 i4 54 8 5 52 59602 7 40398 63275 8 36725 03673 i 96327 46 i5 8 54 0 3 6 09.59632 7 io.40368 9.63310 9 io.366o 10.03678 i 9.96322 45 r6 53 52 6 8 59661 8 140339 63345 9 36655 03684 i 96316 44 17 53 44 6 6 59690 8 403o 63379 10 36621 03689 2 96311 43 i8 53 36 6 24 59720 9 40280 63414 10 36586 03695 2 96305 42 19 53 28 6 32 59749 9 40251 63449 11 36551 03700 2 96300 41 20 853 20 3 6 40 9.59778 10 1io0.40222 9.63484 12 o.3656 10.03706 2 9.96294 40 21 53 12 6 48 59808 o 40192 63519 12 3648i 03711 2 96289 39 22 53 4 6 56 59837 11 4o063 63553 13 36447 03716 2 96284 38 23 52 56 74 59866 ii 40134 63588 13 36412 03722 2 96278 37 24 52 48 7 12 59895 12 40oo5 63623 14 36377 03727 2 96273 36 25 852 4o 3 720 9.59924 12 10.40076 9.63657 14 io.36343 10.03733 2 9.96267 35 26 52 32 7 28 59954 13 4oo46 63692 15 36308 03738 2 96262 34 27 52 24 7 36 59983 13 40017 63726 6 36274 03744 2 96256 33 28 52 i6 744 6001o2 4 39988 63761 i6 36239 03749 3 96251 32 29 52 8 7 52 6oo0041 14 39959 63796 17 36204 03755 3 96245 31 30 8 52 0 3 8 o 9.60070 1 05.39930 9.63830 17 10.36170 10o.3760 3 9.96240 3o 3, S1 52 8 8 60099 15 39901 63865 18 36135 03766 3 96234 29 32 51 44 8 i6 60128 15 39872 63899 i8 36o10i 03771 3 96229 28 33 5i 36 824 60157 16 39843 63934 19 36c66 03777 3 96223 27 34 51 28 8 32 60o86 16 39814 63968 20 36032 03782 3 96218 26 35 851 20 3 840 9.60215 17 10.39785 9.64003.20 10.35997 10.03788 3 9.96212 25 36 51 12 8 48 60244 17 39756 64037 21 35963 03793 3 96207 24 37 51 4 856 60273 i8 39727 64072 21 35928 03799 3 96201 23 38 50o56 94 60302 18 39698 64io6 232 35894 0o38o4 3 96196 22 39 50o 48 912 6331 19 39669 64140 22 3586o o38io 4 96190 21 40 8 50o 4o 3 9 20 9.60359 19 10.39641 9.64175 23 10.35825 io.o3815 4 9.96185 20 41 5o 32 9 28 6o388 20 39612 64209 2' 4 35791 03821 4 96179 1 42 50 24 936 604o17 20 39583 64243 24 357 7 o3826 4 96174 1 43 50o 6 9 44 60446 21 39554 64278 25 35722 3832 4 96168 17 44 50 8 9 52 6o474 21 39526 64312 25 35688 o3838 4 96162 i6 /45 850 o 0310 o 9.6053 22 10.39497 9.64346 26 o.35654 10.03843 4 9.96157 15 46 49 52 10o 8 6o53. 22 39468 64381 26 35619 03849 4 96151 14 47 49 44 io i6 6o56i 23 39439 644i5 27 35585 o3854 4 96146 i3 48 49 36 i1 24 60589 23 39411 64449 28 35551 0386o 4 96140 12 49 49 28 io 32 606i8 24 39382 644,83 28 35517 03865 4 96135 ii 5o 849 20 3 o 40 9.60o646 24 10.39354 9.64517 29 io.35483 10.03871 5 9.96129 o10 51 49 12 io 48 60675 25 39325 64552 29 35448 03877 5 96123 9 52 494 10 56 60704 25 39296 64586 30 35414 03882 5 96118 8 53 48 56 ii 4 607o327 6 39268 64620 3i 35380 03888 5 96112 7 54 48 48 II 12 60761 26 39239 64654 31 35346 03893 5 96107 6 55 8 48 40 3 11 20 9.60789 27 10.39211 9.64688 32 10.35312 10.03899 5 9.961tI 5 56 48 32 11 28 6o80r8 27 39182 64722 32 35278 03905 5 96095 4 57 48 24 ii 36 60846 28 39154 64756 33 35244 03910 5 96090go 3 58 48 16 ii 44 60875 28, 39125 6479o 33 35210 03916 5 96084 2 59 48 8 11 52 60903 29 39097 64824 34 35176 03,21 5 96079 I 60 48 o 12 0 6o0931 29 39069 64858 35 35142 03927 6 96073 o M Hourp.Ai. HourA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. Al 1130 A A B B C C 66~ Seconds of time...... 1 2 3' 4" 5' 6' 7' oo1A 4 47 ii i8 22 o5 Prop. parts of cols. 7 22 26 31 _C i. 39 3 45 TABLE XXVII. [Page 209 Log. Sines, Tangents, and Secants. G'. 240 A A B B C C 1550 M IHour.AM. Hour P.M. Sine. Diff. Cosecant. Tang-ent. Diff. Cotangent Secant. Diff. Cosine. M o 8 48 o 3 12 o 9.609311 o 10.39069 9.64858 o 10.35142 10.03927 o 9.96073 6o 1 47 52 12 8 60960 o 390o40 64892 i 35o8 03933 o 96067 59 2 47 44 12 i6 60988i 39012 64926 I 35074 03938 o 96062 58 3 47 36 1 2 24 6ro16 38984 64960 2 35o040o 03944 0 96056 57 4 47 28 12 32 61045 2 38955 64994 2 35006 03950 0 96050 56 5 8 47 20 3 12 49.61073 2 10.38927 9.65028 3 10.34972 10.03955 o 9.96045 55 6 47 12 12 48 6io 3 38899 65062 3 34938 03961 I 96039 54 7 47 4 12 56 61129 3 38871 65096 4 34904 03966 I 96034 53 8 46 56 13 4 61158 4 38842 65i30 4 34870 03972 i 96028 52 9 46 48 13 12 61186 4 38814 65164 5 34836 03978 I 96022 51 1o 8 46 4o i3 1320 9.61214 5 10o.38786 9.65197 6 1-10o.348o3 10.039831 I 9.96017 50 ii 46 32 i3 28 61242 5 38758 65231 6 34769 03989 x o6oi 49 12 46 24 13 36 61270 6 38730 65265 7 34735 o3995 I 91-oo5 48 13 46 16 13 44 61298 6 38702 65299 7 34701 o4ooo i 96000 47 14 46 8 13 52 61326 6 38674 65333 8 34667 04006 i 95994 46 5 846 0 3 14 o 9.613547 io.38646 9.65366 810.34634 10.04012' 1 9.95988 45 16 45 52 I4 8'61382 7 38618 654oo 9 346oo o4oi8 2 95982 44 17 45 44 14 16 6i4ii 8 38589 65434 9 34566 04023 2 95977 43 18 45 36 14 24 61438 8 38562 65467 10 34533 04029 2 95971 42 19 45 28 14 32 61466 9 38534 655o0 ii 34499 04035 2.95965 4I 20 8 45 20 314 4o 9.61494 9 io.38506 9.65535 ii o10.34465 io.o4040o 2 9.95960 40 21 45 12 14 48 61522 io 38478 65568 12 34432 o4o46 2 95954 39 22 45 4 i4 56 655o0 io 3845o 65602 12 34398 04052 2 95948 38 23 44 56 i5 4 61578 11 38422 65636 13 34364 o4o58 2 95942 37 24 44 48 i5 12 6i606 ii 38394 65669 13 34331 o4o63 2 95937 36 25 8 44 40 3 520 9.6163412 10.38366 9.65703 14 10.34297 10.04069 2 9.9593 35 26 44 32 15 28 61662 12 38338 65736 15 34264 04075 2 95925 34 27 44 24 15 36 61689 12 38311 657701 5 34230 o4o8o 3 95920 33 28 44 i6 15 44 61717 13 38283 65803 16 34197 o4o86 3 95914 32 29 44 8 i5 52 61745 13 38255 65837 16 34i63 o4092 3 95908 3r 30 8 44 0l 3 i6 0o9.61773 14 10.38227 9.65870 17-10.34130 10.04098 3 9.95902 30 31 43 52 i6 8 6i8oo 14 38200 65904 17 34096 0410o3 3 95897 29 32 43 44 16 i6 61828 15 38172 65937 18 34o63 04]o9 3 95891 28 33 43 36 i6 24 6i856 15 38i44 65971 i8 34029 o4115 3 95885 27 34 43 28 16 32 6i883 i6 38117 6600oo4 19 33996 04121 3 95879 26 35 8 43 203 r6 40 9.61911 i6 10.38089 9.66038 20 10.33962 10.04127 3 9.95873 25 36 43 12 16 48 61939 17 38061 66071 20 33929 o4132 3 95868 24 37 43 4 i6 56 61966 17 38o34 66io4 21 33896 04138 4 95862 23 38 42 56 17 4 6i994 18 38oo6 66i38 21 33862 04i44 4 95856 22 39 42 48 17 12 62021 I8 37979 66171 22 33829 04150 4 95850 21 40 8 42 40 3 1720 9.62 049 i8 10.37951 9.66204 22 10.33796 10.0456 4 9.95844 20 41 42 32 17 28 62076 19 37924 66238 -23 33762 04161 4 95839 19 42 42 24 17 36 62104 19 37896 66271 23 33729 04167 4 95833 18 43 42 i6 17 44 62131 20 37869 663o4 24 33696 o4173 4 95827 17 44 42 8 17 52 62159 20' 37841 66337 25 33663 04179 4 95821 16 45 8 42 0o318 9.62186 2- 10.37814 9.66371 25 10.33629 ioo. 485 4 9.95815 15 46 41 52 18 8 62214 21 37786 6644 26 33596 04190 4 95810 14 47 4r 44 18 16 62241 22 37759 66437 26 33563 o4196 5 95804 13 48 41 36 18 24 62268 22 37732 66470 27 3353o 04202 5 95798 12 49 41 28 i8 32 62296 23 37704 66503 27 33497 04208 5 95792 II 50 841 20 3 8 4 9.62323 23 10o.37677 9.66537 28 io0.33463 10.04214 5.95786 io 51 41 12 i8 48 62350 24 37650 66570 28 33430 04220 5 95780 9 52 4' 4 8 56 62377 24 37623 66603 29 333w1 04225 5 95775 8 53 4o 56' 19 4 62405 24 37595 66636 3o 33364 o4231 5 95769 7 54 4o 48o 19 12 62432 25 37568 66669 3o 33331 04237 5 95763,6 8 40 403 19 20 9.62459 25 10.37541 9.66702 31 10.33298 10.04243 5 9.95757 5 56 40 32 19 28 62486 26 37514 66735 31 33265 04249 5 95751 4 57 40 24 19 36 62513 26 37487 66768 32 33232 04255 5 95745 3 58 40o 6 19 44 62541 27 37459 668oi 32 33199 04261 6 95739 2 59 40 8 19 52 62568 27 37432 66834 33 33i66 042671 6 95733 i 60 40 20o o 62595 28 37405 66867 33 33133 04272 6 95728 0 M Hour-P.M. Hour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1140 A A- B B C C 6~ Seconds of time....... l 29 3" 4 51 6 7' A 3 7 10 i14 17 21 24 Prop. parts of co0s. B 4 8 13 17 21 25 29 _C ___ _2 3 _ 4 5 ZT~~ T ~,'-Zq~-~ — Page 210] TBLE VII. TABLE XXVII.. Log. Sines, Tangents, and Secants. 25_ A A B B C C 1540 M Hour A.M. Hour P.M1. Sine. Diff. Cosecant. Tangent. Diff. Cotangcnt Secant. Diff. Cosine. M 8 4o0 33 o 0 9.62595 o o10.37405 9.66867 o io.33133 10.04272 0 9.95728 60 i 39 52 20 8 62622 0 37378 66oo00 i 33oo 04278 o 95722 59 2 39 44 20 i6 62649 I 37351 66933 i 33067 04284 0 95716 58 3 39 36 20 24 62676 1 37324 66966 2 33o34 0429 0 95710 57 4 39 28 20 32 62703 2 37297 66999 2 33oo00 04296 0 95704 56 5 8 39 20 3 20 4o 9.62730 2 1o.37270 9.67032 3 10.32968 10.04302 i 9.95698 55 6 39 12 20 48 62757 3 37243 67065 3 32935 o43o8 i 95692 54 7 39 4 20o 56 62784 3 37216 67098 4 32902 o4314 95686 53 8 38 56 21 4 62811 4 37189 67131 4 32869 04320 95680 52 9 38 48 21 12 62838 4 37a62 67163 5 32837 04326 i 95674 5i io 8 38 4o 3 21 20 9.62865 4 10.37o35 9.67196 5 10.32804 10.0o4332 i 9.95668 50 11ii 38 32 21 28 62892 5 37108 67229 6 32771 04337 a 95663 49 12 38 24 21 36 62918 5 37082 67262 7 32738 04343 i 95657 48 13 38 i6 21 44 62945 6 37055 67295 7 32705 04349 I 95651 47 14 38 8 21 52 62972 6 37028 67327 8 32673 o4355 i 95645 46 15 838 0 322 o 9.62999 7 io.37001 9.67360 8 10o.32640 io.o436 2 9.95639 45 i6 37 52 22 8 63026 7 36974 67393 9 32607 o4367 2 95633 44 17 37 44 22 16 63052 8 36948 67426 9 32574 04373 2 95627 43 18 37 36 22 24 63079 8 36921 67458 io 32542 04379 2 95621 42 19 37 28 22 32 63io6 8 36894 67491 10 32509 o4385 2 95615 4i 20 8 37 20 3 22 4o 9.6333 90.36867 967524 i 10.32476 10o.o.04391 2 9.95609 4 21 37 12 22 48 63159 9 3684i 67556 Ii 32444 04397 2 95603 39 22 37 4 22 55 63i86 10o 36814 67589 12 32411 04403 2 95597 38 23 36 56 23 4 63213 1o 36787 67622 12 32378 o44o9 2 95591 37 24 36 48 23 12 63239 ii 3676i 67654 13 32346 o4415 2 95585 36'25 8 36 4o 3 23 20 9.6326611 10.36734 9.67687 i4 10.32313 io.o442a 3 9.95579 35 i26 36 32 23 28 63292 ii 36708 67719 14 32281 04427 3 95573 34 27 36 24 23 36 63319 12 3668i 67752 15 32248 04433 3 95567 33 28 36 i6 23 44 63345 12 36655 67785 i5 32215 o4439 3 95561 32 29 36 8 23 52 63372 13 36628 67817 i6 32183 o4445 3 95555 31 30 8 36 o 3 24 o 9.6339813 10.36602 9.67850 i6 10.32150 o.o445i 3 9.95549 30 31 35 52 24 8 63425 14 36575 67882 17 32118 04457 3 95543 29 32 35 44 24 16 63451 14 36549 67915 17 32085 o4463 3 95537 28 [J33 35 36 24 24 63478 15 36522 67947 18 32053 04469 3 95531 27'34 35 28 24 32 635o4 15 36496 67980 i8 32020 04475 3 955I5 26 35 8 35 20 32440 9.63531 5 10.36469 9.68012 19 10.31988 10.04481 4 9.95519 25 36 35 12 24 48 63557 16 36443 68o44 20 31956 04487 4 95513 24 37 35 4 24 56 63583 i6 36417 68077 20 31923 o4493 4 95507 23 38 34 56 25 4 636o10 17 36390 68109 21 31891 o45oo 4 95500 22 39 34 48 25 12 63636 17 36364 68142 21 3i858 04506 4 95494 21 40o 8 34 40 3 25 20 9.63662 18 10.36338 9.68174 22 10.31826 10.04512 4 9.95488 20 41 34 32 25 28 63689 18 363ii 68206 22 31794 o45i8 4 95482 19 42 34 24 25 36 63715 19 36285 68239 23 31761 04524 4 95476 i18 43 34 16 25 44 63741 19 36259 68271 23 31729 04530 4 95470 17 44 34 8 25 52 63767 19 36233 683o3 24 31697 04536 4 95464 i6 45 8 34 0 3 26 0o 9.63794 20 10.36206 9.68336 24 ao.3i664 10.04542 5 9.95458 15 46 33 52 26 8 63820 20 36i8o 68368 25 31632 o4548 5 95452 14 47 33 44 26 16 63846 21 36i54 68400 25 316oo o4554 5 95446 13 48 33 36 26 24 63872J 21 36128 68432 26 3i568 o456o 5 95440 12 49 33 28 26 32 63898 22 36102 68465 27 31535 04566 5 95434 1i 50 8 33 20 3 26 4o 9.63924 22 10.36076 9.68497 27 io.3r5o3 10o.4573 5 9.95427 10io 51 33 12 26 48 63950 23 36o50 68529 28 31471 04579 5 95421 9 52 33. 4 26 56 63976 23 36024 68561 28 31439 4585 5 95415 8 53 32 56 27 4 64002 23 35998 68593 29 31407 04591 5 95409 7 54 32 48 27 12 64028 24 35972 68626 29 31374 04597 5 95403 6 55 E 32 40o 3 2720 9.64054 24 o10.35946 9.68658 3o 10.31342 1io.o46o3 6 9.95397 5 56 32 32 27 28 64080o 25 35920 68690 30 3i3io 04609 6 95391 4 57 32 24 27 36 64io6 25 35894 68722 31 31278 o46i6 6 95384 3 58 32 i6 27 44 64132 26 35868 68754 31 31246 04622 6 95378 2 59 32 8 27 52 64158 26 35842 68786 32 31214 04628 6 95372 I 60 32 0 28 0 64184 26 358i6 68818 33 31182 o4634 6 95366 o0 M Hour p.M.lHourA.M. Cosine. Diff.1 Secant. JCotangent Diff. Tangent. Cosecant.:if. Sine. M I15" A A B B C C 64" Seconds of time..... 1 9 28 3l 4 I 5' 6" 7' (A 3 7 o10 13 17 20 23 Prop. parts of cols. B 4 8 i2 16 20 24 28 C 1 2 12 3 14 5 5 TABLE XXVIa. S'. Log. Sines, Tangents, and Secants. G'' 26~ A A B B C C 1530 M Hour A.M. Hor P.M.i Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. MI o 832 0 3 28 o 9.64184 10o.35816 9.68818 o 1o,31182 10.o 4634 o 9.95366 6o I 3i 52 28 8 64210 0 35790 68850 31150 0464o o 95360 59 2 3 44 28 i6 64236 i 35764 68882 1 31118 04646 o j9,354 58 3 31 36 28 24 64262 i 35738 68914 2 3io86 04652 o 95348 57 4 3 28 28 32 64288 2 35712 68946 2 31054 04659 0 9534r 56 5 8 31 20328 4o 9.64313 2 10Io.35687 9.68978 310.31022 10.04665 i 9.95335 55 6 3i 12 28 48 64339 3 35661 69010o 3 30990 0467 1 95329 54 7 31 4 28 56 64365 3 35635 69042 4 30958 04677 I 95323 53 8 30o 56 29 4 64391 3 35609 69074 4 30926 04683 i 9531752 9 3o 48 29 12 64417 4 35583 6910o6 5 30894 o4690o, i 953o10 5 io 8 30 4o 3 29 20 9.64442 4 o10.35558 9.69138 5 10.30862 10.04696 19.95304 50 ii 30 32 29 28 64468 5 35532 69170 6 3o83o 04702 1 95298 49 12 30 24 29 36 64494 5 355o6 69202 6 30798 04708 I 95292 48 13 30 16 29 44 64519 5 35481 69234 7 30766 04714 1 95286 47 14 30o 8 29 52 64545 6 35455 69266 7 30734 04721 i 95279 46 I5 8 3o 0 330o 0 9.64571 6 10.35429 9.69298 8 10.30702 10.04727 2 9.95273 45 16 29 52 3o 8 64596 7 354o4 69329 8 3o671 04733 2 95267 44 17 29 44' 30 i6 64622 7 35378 69361 9 30639 04739 2 9526i 43 18 29 36 30 24 64647 8 35353 69393 9 30607 04746 2 95254 42 19 29 28 30 32 64673 8 35327 69425 TO 30575 04752 2 95248 41 20 8 29 20 3 3o 4o 9.64698 8 10o.35302 9.69457 ii 1io.3o543 10o.04758 2 9.95242 40 21 29 12 30 48 64724 9 35276 69488 11 3051/2 04764 2 95236 39 22 29 4 30 56 64749, 9 35251 69520 12 3o48o 04771 2 95229 38 23 28 56 3i 4 64775 - 35225 69552 12 3o448 04777 2 95223 37 24 28 48 31 12 648oo00 io 35200 695844 i3 3o046 04783 3 95217 36 25 8 28 40 3 31 20 9.64826 11 10.35174 9.69615 13 1o.30385 10.04789 3 9.95211 35 26 28 32 31 28 64851 Ii 35149 696471 i4 3o353 04796 3 95204 34 27 28 24 31 36 64877 T11 35123 69679 14 30321 04802 3 95198 33 28 28 i6 31 44 64902 12 35098 69710 15 30290 04808 3 95192 32 29 28 8 31 52 64927 12 35073 69742 15 30258 048i5 3 95185 31 30o 8 28 0 3 32 o 9.64953 3 10o.35047 9.69774 16 10.30226 10.04821 3 9.951793 31 27 52 32 8 64978 13 35022 698051 16 30195 04827 3 95173 29 32 27 44 32 i6 65003 14 34997 69837 17 3o63 04833 3 95167 28 33 27 36 32 24 65029 14 34971 69868 17 30132 o484o 3 95160 27 34 27 28 32 32 65054 14 34946 69900 8 3oioo0100 o4846 4 95154126 35 8 27 20 3 32 4o 9.65079 i 10o.34921 9.69932 8 o.3oo0068 10.04852 4 9.95148 25 36 27 12 32 48 650o4 15 34896 69963 19 30037 04859 4 95,41 24 37 27 4 32 56 65i3o i6 3487o 69995 20 3ooo5 o4865 4 95135 23 38 26 56 33 4 65155 16 34845 70026 20 29974 04871 4 95129 22 39 26 48 33 12 6518o i6 34820o 70058 21 29942 04878 4 95122 21 40 8 26 4 3 33 20 9.65205 17 10.34795 9.70089 21 10.29911 10.04884 4 9.95116 20 41 26 32 33 28 6523o 17 34770 70121 22 29879 04890 4 95110 19 42 26 24 33 36 65255 18 34745 70152 22 29848 04897 4 95o103 18 43 26 16 33 44 65281 18 34719 70184 23 29816 04903 5 95097 17 44 26 8 33 52 653o6 19 34694 70215 23 29785 04910 5 9509o 16 45 8 26 o 3 34 o 9.65331 19 10o.34669 9.70247 24 10.29753 10.04916 5 9.95084 15 46 25 52 34 8 65356 i9 34644 70278 24 29722 04922 5 95078 14 47 25 44 3.4 16 65381 20 34619 70309 25 29691 04929 5 95071 13 48 25 36 34 24 65406 20 34594 70341 25 29659 04935 5 95065 12 49 25 28 34 32 65431 21 34569 70372 26 29628 04941 5 95059 11 50o 8 25 20o 3 34 40 9.65456 21 o.34544 9.70404 26 10.29596 c.o4948 5 9.95052 10 51 25 12 34 48 65481i 22 34519 70435 27 29565 04954 5 95046 9 52! 25 4 34 56 65506 22 34494 70466 27 29534 04961 5'95039 8 53 24 56 35 4 65531 22 34469 70498 2.8 29502 04967 6 95033 7 54 24 48 35 12 65556 23 34444 70529 28 29471 04973 6 95027 6 /55 8 24 4o3 35 21 9.6558J23 10.34420 9.7056o0 29 10.29440 10.0498o 6 9.95020 5 56 24 32 35 28 656o5 24 34395 70592 3o 29408 04986 6 95014 4 57 24 24 35 36 65630 24 3437(0 70623 3o 29377 04993 6 95007 3 58J 24 i6 35 44 65655 25 34345 70654 31 29346 04999 6 95001 2 59. 24 8 35 52 65680 25 34320 70685 31 29315 05005 6 94995 60' 240o 36o0 65705 25' 34,295 7071-7 32 29283 05012 6 94988 o ou'P.M.~Hour A.m. Cosine. Diff. Secant. Cotangent Diff Tangent. Cosecant. Diff. Sine. MI 1160 A A B B C C 631 Seconds of time.. 1. 2' 3S 48 5~ 6' 7" A 3 6 10 lo i3 i6 19 22 Prop. parts of cols B 4 8 12 I 6 20 21 4 28 1 2 2 3 4 5 6 age 212] TABLE XXVII.''. Log. Sines, Tangents, and Secants. G' 9)72 A A B B C C 1520 Ai Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff.Cotangent Secant. Diff. Cosine. M 8 24 3 36 0 9.65705 o 10o.3425 9 70717 i 10.29283 10.05012 0 9.94988 6o 23 52 36 8 65729 0 34271 70748 i 29252 o5o018 o 94982 59 2 23 44 36 16 65754 I 34246 70779 / 29221 05025 0 94975 58 23 36 36 24 65779 1 34221 70810 2 291 0 o 5o3i o 94969 57 4 23 28 36 32 65804 2 34196 70841 2 29159 05038 0 94962 56 5 823 20 3 36 4 9.65828 210.34172 970873 3 10.29127 10.05044 9.94956 55 6 23 12 36 48 65853 2 34147 70904 3 29096 05051 94949 54 7 23 4 36 56 65878 3 34122 70935 4 29065 05057 I 94943 53 8 22 56 37 4 65902 3 34o98 70966 4 29034 o5o64 94936 52 9 22 48 37 12 65927 4 34073 70997 5 29003 05070 I 94930 51 io 8 22 40 3 37 20 9-65952 4 o10.3448 9.71028 5 10.28972 /1.05077 I 9.94923 50 11 22 32 37 28 65976 4 34024 71059 6 28941 05083 94917 49 12 22 24 37 36 66o00 5 33999 71090 6 28910 o5089 I 9491 48 13 22 i6 37 44 66o25 5 33975 71121 7 28879 050o96 94904 47 14 22 8 37 52 6o050 6 33950 71153 7 28847 05102 2 94898 46 15 8 22 o3l 38 9-66o75 6 10o.33925 9.71184 8 10.28816 10.05109 2 9-94891 45 16 21 52 38 8 66099 6 33901 71215 8 28785 05115 2 94885 44 17 21 44 38,6 66124 7 33876 71246 9 28754 05122 2 94878 43 18 21 36 3s 24/ 66,48 7 33852 71277 9 28723 05129 2 94871 42 19 21 28 38 32 66173 8 33827 71308 10o 28692 o5i35 2 94865 41 20 8 2120 3 38 4o 966197 8 o10.338o3 9.71339 10 i0.28661 10.05142 2 9.94858 40 21 21 12 38 48 66221 8 33779 71370 II 28630 05148 2 94852 39 22 21 4 38 56 66246 9 33754 71401 11 28599 05155 2 94845 38 23 20 56 39 4 66270 9 33730 71431 12 28569 o5i6i 3 94839 37 24 20 48 39 12 66295 io 33705 71462 12 28538 05168 3 94832 36 25 8 2040o 339 20 9.66319 i ro1.3368i 9.71493 13 10.28507 10.05174 3 9.94826 35 26 20 32 39 28 66343 ii 33657 71524 13 28476 05181 3 94819 34 27 20 24 39 36 66368 ii 33632 71555 14 28445 05187 3 94813 33 28 20 16 39 44 66392 11 33608 71586 14 28414 05194 3 94806 32 29 20 8 39 52 664i6 12 33584 71617 i5 28383 05201 3 94799 31 30 820 034o0 o9-66441 12 10.33559 9.71648 5 10.28352 10.05207 3 9.94793 3 31 19 52 40o 8 66465 13 33535 71679 i6 28321 o5214 3 94786 29.32 19 44 40 16 66489 13 33511 71709 i6 28291 05220 4 94780 28 33 19 36 40 24 665i3 13 33487 71740 17 28260 05227 4 94773 27 34 19 28 40 32j 66537 14 33463 71771 17 28229 05233 4 94767126 35 8 19 201 340 40o 9.66562 14 o10.33438 9.71802 i8 10.28198 10.05240 4 9.94760 25 36 19 12 40 48 66586 15 33414 71833 19 28167 05247 4 94753 24 37 19 4 40 56 666io 15 3339o 71863 19 28137 05253 4 94747 23 38 18 56 41 4 66634 15 33366 71894 20 28106 05260 4 94740 22 39 i8 48 41 12 66658 i6 33342 71925 20 28075 05266 4 94734 21 40 8 i8 40 3 4 20 9.66682 i6 io.333i8 9.71955 21.10.28045 10.05273 4 9.94727 20 41 18 32 41 28 66706 17 33294 71986 21 28014 05280 4 94720 19 42 18 24 4' 36 66731 17 33269 72017 22 27983 05286 5 94714 18 43 i8 16 4, 44 66755 17 33245 72048 22 27952 05293 5 94707 17 44 18 8 41 52 66779 i8 33221 72078 23 27922 05300 5 94700 i6 45 8 18 o 3 42 o 9.66803 8 1o0,33197 9.72109 23 10.27891 10.0536 5 9.94694 5 46 17 52 42 8 66827 19 33173 72140 24 27860 o5313 5 9468? 1i4 47 17 44 42 i6 6685i 19 33,49 72170 24 27830 05320 5 9468, 13 48 17 36 42 24 66875 19 33125 72201 25 27799 05326 5 94674 12 49 17 28 42 32 66899 20 331oi 72231 25 27769 05333 5 94667 11 50o 8 17 20 3 4240 9.66922 20 10,33078 9-72262 26 10.27738 o0.0534o 5 9.94660 io 51i 17 12 42 48 66946 21 33o54 72293 26 27707 05346 6 94654 9 52 17 4 42 56 66970o 21 33o3o 72323 27 27677'05353 6 94647 8 53 i6 56 43 4 66994/ 21 33oo6 72354 27 27646 o536o 6 94640 7 54 i6 48 43 12 67018 22 32982 72384 28 27616 o5366 6 94634 6 55 8 1640o3 4320o 9.67042 22 10.32958 9.72415 28 10.27585 10.05373 6 9.94627 5 56 i6 32 43 28 67066 23 32934 72445 29 27555 05380 6 94620 4 57 i6 24 43 36 67090 23 32910 -72476 299 27524 05386 6 94614 3 158 i6 i6j 43 44 67113 23 32887 72506 30 27494 o5393 6 94607 2a'591 i6 8 43 52 67137 24 32863 72537 3o 27463 o54oo 6 94600 I 6o i6 o01 44 o 67161 24 32839 72567 3 - 27433 o540o7 7 94593 o fM Hour P.. HIour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1170 A A B B C C 62 Seconds of time...... Is 2 3s" 4 5- 6- 77 (A 3 6 9 12 i5 i8 21 Prop. parts of cols. B 4 8 12 i5 19 23 27 C. 1 2 2 3 4 5 6 TABLE XXVII. [Page 213 Log. Sines, Tangents, and Secants. a'. 280 A A B B C C 151~ AT IHour A.M. Hour p.i. Sine. Diff.Cosecant. Tangent. DiffiCotangenti Secaf. Dififi Cosine. II o 8 16 o 3 44 o 9.6716I1 o 1o.32839 9.72567 0o10.27433 10.05407 o 9.94593 60 15 52 44 8 67185 o 32815 72598 I 27402 o5413 o 94587 59J 2 15 44 44 16 67208 i 32792 72628 i 27372 05420 0 94580 58 3 5 36 44 24 67232 1 32768 72659 2 27341 05427 0 94573 57 4 15 28 44 32 67256 2 32744 72689 2 27311 05433 o 94567 56 5 8 15 20 3 44 4o 9.67280 2 10.32720 9.72720 3 10.27280 i0o. 0544o i 9.94560 55 6 5 i12 44 48 67303 2 32697 72750 3 27250 05447 I 94553 54 7 15 4 44 56 67327 3 32673 72780 4 27220 05454 I 94546 53 8 4 56 45 4 67350 3 32650 72811 4 27189 05460 i 94540 521 9 i4 48 45 12 67374 3 32626 72841 5 27159 05467 1 _ 94533 5S 10 8 14 40 3 45 20 9.67398 4 o.32602 9.72872 5 0.2712810.05474 9.9452650 11 14 32 45 28 67421 4 32579 72902 6 27098 o5481 i 94519 49 12 14 24 45 36 67445 5 32555 72932 6 27068 05487 1 94513 48 13 14 16 45 44 67468 5 32532 72963 7 27037 05494 1 94506 47 14 14 8 45 52 67492 5 32508 72993 7 27007 o5501 2 94499 46 15 8 i4 o 3 46 0 9.67515 6 10.32485 9.73023 8 10.26977 1o.o055o8 2 9.9449 45 16 13 52 46 8 67539 6 32461 73054 8 26946 o5515 2 94485 44 17 13 44 46,6 67562 7 32438 73084 9 2696 05521 2 94479 43 18 i3 36 46 24 67586 7 32414 73114 9 26886 o5528 2 94472 42 19 13 28 46 32 67609 7 32391 73,44 10 26856 05535 2 94465 4i 20 8 i3 203 46 4o 9.67633 8i10.32367 9-73175 10 10.26825 10o.5542 2 9.94458 40 21 i3 12 46 48 67656 8 32344 73205 II 26795 05549 2 94451 39 22 13 4 46 56 67680 9 32320 73235 II 26765 05555 3 94445 38 23 12 56 47 4 67703 9 32297 73265 12 26735 05562 3 94438 37 24 12 48 47 12 67726 9 32274 73295 12 26705 05569 3 94431 36 25 8 12 40 3 47 20 9.67750 io 10.32250 9.73326 13 10.26674 10.5576 3 9.94424 35 26 12 32 47 28 67773 io 32227 73356 13 26644 05583 3 94417 34 27 12 24 47 36 677961 O 32204 73386 14 26614 05590 3 94410 33 28 12 16 47 44 67820 ii 32180 73416 i4 26584 05596 3 94404 32 29 12 8 47 52 67843 ii 32157 73446 15 26554 o56o3 3 94397 31 30 8 12 0 3 48 0 9.6786612 10.32134 973476 5 1 i.26524 o0.05610o 3 9-94390 30 31 11 52 48 8 67890 12 32110 73507 16 26493 05617 4 94383 29 32,i 44 48 16 67913 12 32087 73537 16 26463 05624 4 94376 28 33 ii 36 48 24 67936 I3 32064 73567 17 26433 o5631 4 94369 27 34 11 28 48 32 67959 53 32041 73597 1'7 26403 o5638 4 94362 26 35 8 11 203 48409.67982 14 10.32018 9.73627. 18 0.26373 io.5645 4 9.94355 25 36 11 12 48 48 68006 14 31994 73657 s8 26343 o5651 4 94349 24 37 ii 4 48 56 68029 14 31971 73687 19 26313 o5658 4 94342 23 38 10o 56 49 4 680o52 15 31948 73717 19 26283 05665 4 94335 22 39 0o 48 49 12 68075 15 31925 73747 20 26253 05672 4 94328 21 40 8 10 40o'3 49 20 9.68098 16 10.31902 9-73777 20 10.26223 10.05679 5 9.94321 20 41 io 32 49 28 68121 16 31879 73807 21 26193 05686 5 94314 19 42 io 24 49 36 68i44 16 31856 73837 21 26163 05693 5 94307 18 43 ioi6 49 44 68167 17 3i833 73867 22 26133 05700 5 94300 17 44 1o 8 49 52 68690 17 31870 73897 22 26103 05707 5 94293 16 45 8 io o 3 50 o 9.68213 1710.31787 9-73927 23 10.26073 1 o.o5714 5 9.94286 15 46 9 52 50 8 68237 18 31763 73957 23 26043 05721 5 94279 i4 47 9 44 50 16 68260o 18 374o0 73987 24 26o013 05727 5 94273 13 48 9 36 50 24 68283 19 31717 74017 24 25983 05734 5 94266 12 419 9 28 5o 32 68305 19 31695 74047 25 25953 0574i 6 94259 ii 50 8 9 20 3 5o 40.68328 19 10.31672 9-74077 25 10.25923 10.05748 6 9.94252 1io 51 9 12 50 48 68351 20 31649 74107 26 25893 05755 6 94245 9 52 9 4 50 56 68374 20.31626 74137 26 25863 05762 6 94238 8 53 8 56' 5 4 68397 21 3i6o3 74166 27 25834 05769 6 94231 7 54 8 48 5i 12 68420o 21 31580 74196 27 25804 05776 6 94:24 6 55 8 8 40o 3 51 20 9.68443 21 Io.31557 9.74226 28 10.25774 10.05783 6 9.94217 5 56 8 32 5i 28 68466 22 31534 74256 28 25744 05790 6 94210 4 57 8 24 51 36 68489 22 3i51s 74286 29 25714 05797 7 94203 3 58 8 16 51 44 68512 22 3i488 74316 29 25684 05804. 7 94196 2 59 8 8 5, 521 685341 23 31466 74345 3o 25655 05811 7 94189 i 60 8 o 52 o 68557 23 31443 74375 3o 25625 o58i8 7 94182 o i Hour P.M1. Hour A.M.I Cosine. Diff. Secant. Cotang-ent Diff. Tangent. Cosecant. Diff. Sine. MI 1186: A A B B C C 61 Seconds of time.... 1.'1 3' 4' 5 6 7 A 3 6 9 12 15 17 20 Prop. parts of cols. B 4 8 I 19 23 26 C 1 2 3 3 4 5_ 6 Page 2141 TABLE XXVII. S'. Log. Sines, Tangents, and Secants. GW. 290 A A B B C C 1500 M Hour A.M. HIoornPM. Sine. Diff. Cosecant. Tangent. Diff.Cotangent Secant. Diff. Cosine. III 8 8 o 3 52 0 9.68557 o ro.31443 9.74375 0o 10.25625 io.058I8 o 9.94182 6o 7 52 52 8 6858o o 31420o 74405 o 25595 o5825 94175 59 2 7 44 52 i6 68603 i 3397 74435 i 25565 05832 o 94168 58 3 7 36 52 24 68625 i 31375 74465 i -25535 05839 o 94161 57 4 7 28 52 32 68648 i 31352 74494 2 25506 05846 o 94154 56 5 8 7 203524 9.68671 2 io0.3i329 9-74524 210.25476 io.058'3 9.94147 55 6 7 12 52 48 68694 2 3i306 74554 3 25446 0o5860 94140 54 7 7 4 52 56 68716 3 31284 74583 3 25417 05867 I 94133 53 8 6 56 53 4 68739 3 31261 74613 4 25387 05874 I 94126 52 9 6 48 53 12 68762 3 3138 74643 4 25357 o5881 I 94119 5 Io 8 6 40o3 53 20 9.68784410.31216 9.74673 510.25327 1I0. 05888 i 9.94112 50 i 6 32 53 28 68807 4 31193 74702 5 25298 05895 i 94105 49 12 6 24 53 36 68829 4 31171 74732 6 25268 05902 I 94098 48,13.6 6 53 44 68852 5 31148 74762 6 25238 05910 2 94090 47 14 6 8 53 52 68875 5 31125 74791 7 25209 05917 2 940)83 46 15 8 6 0 3 54 0 9.68897 61o.311o 3 9.74821 7 10.25179 10.05924 2 9.94076 45 16 5 52 54 8 68920 6 3io8o 74851 8 25149 o5931 2 94069 44 17 5 44 54 16 68942 6 3io58 74880 8 25120 05938 2 94062 43 18 5 36 54 24 68965 7 3I035 74910 9 25090 o5945 94055 42 19 5 28 54 32 68987 7 3io03 74939 9 25061 05952 2 94048 41 20 8 5 20 3 54 4o 9.69o010 7 10o.30990 9.74969 010.25031 o0.0o595 2 9.94041 40 21 5 12 54 48 69032 8 30968 74998 1o 25002 05966 3 94034 39 22 5 4 54 56 69055 8 30945 75028 i 24972 05973 3 94027 38 23 4 56 55 4 69077 9 30923 75058 i i 24942 05980 3 94020 37 24 4 48 55 12 69100 9 30900 75087 12 24913 05988 3 94012 36 25 8 4 4 3 55 20 9.69122 9 o10.3o878 9.75117 12 10.24883 10.05995 3 9.94005 35 26 4 32 55 28 69144 io 30856 75146 13 24854 0600. 3 93998 34 27 4 24 55 36 69167 3833 75176 13 2484 06009 3 93991 33 28 4 16 55 44 69189 io 3o8tt 75205 14 24795 o06o6 3 93984 32 29 4 8 55 52 69212 ii 30788 75235 14 24765 o6o23 3 93977 31 30 8 4 0 3 56 0 9.69234 ii 10.30766 9.75264 15 10.24736 o10.06030 4 9.93970 3 31 3 52 56 8 69256 12 30744 75294 15 24706 06037 4 93963 29 32 3 44 56 16 69279 12 30721 75323 16 24677 o6045 4 93955 28 33 3 36 56 24 69301 12 30699 75353 16 24647 06052 4 93948 27 34 3 28 56 32 69323 13 30677 75382'7, 24618 06059 4 93941 26 35 8 3 20 3 56 40 9.69345 13 1o.3o655 9.75411 17 10o.4589 1o.o6o66 4 9.93934 25 36 3 12 56 48 69368 13 3o632 75441 18 24559 06073 4 93927 24 37 3 4 56 56 69390 14 3o6o10 75470 8 24530 o6o8o 4 93920 23 38 2 56 57 4 69412 14 30588 75500 19 24500 o6o88 5 93912 22 39 2 48 57 12 69434 15 30566 75529 19 24471 06095 5 93905 21 40 8 2 40 3 57 20 9.69456 i5 o0.3o544 9.75558 20 10.24442 10.06102 5 9.93898 20 41 2 32 57 28 69479 i5 30521 75588 20 24412 06109 5 93891 19 42 2 24 57 36 69501 i6 30499 75617 21 24383 o61i6 5 93884 18 43 2 16 57 44 69523 16 30477 75647 21 24353 06124 5 93876 17 44 2 8 57 52 69545 16 30455 75676 22 24324 o6031 5 93869 16 45 8 2 0 3 58 o 9.69567 17 ro.3o433 9.75705 22 10.24295 o.o6138 5 9.93862 15 46 1 52 58 8 69589 17 3o41i 75735 23 24265 o6t45 5 93855 14 47.1r 144 58 16 69611 17 30389 75764 23 24236 06153 6 93847 13 48 136 58 24 69633 18 30367 75793 24 24207 o6160 6 93840 12 49 1 28 58 32 69655 18 30345 75822 24 24178 06167 6 93833 it 50 8 i 20 3 58 40 9.69677 191-o.30323 9.75852 25 10o.2448 10.06174 6 9.93826 io 5t 1 12 58 48 69699 19 3o3o0 75881 25 24119 06181 6 93819 9 52 1 4 58 56 69721 19 30279 759to10 26 24090 o06189 93811 8 53 0 56 59 4 69743 20 30257 75939 26 24061 06196 6 93804 7o 54 048 59 12 69765 20 30235 75969 27 24031 06203 6 93797 6 55 8 0 4o 3 59 20 9.69787 20 10.30213 9.75998 27 10.24002 10.06211 7 9-93789 5 56 0 32 59 28 69809 21 30191 76027 28 23973 06218 7 93782 4 57 0 24 59 36 69831 21 30169 76056 28 23944 o06225 7 93775 3 58 o 16 59.44 69853 22 30147 76086 29 23914 06235 7 93768 2 59 o 8 59 52 69875 22 30125 76115 29 23885 06240 7 93760 1 6o 0 0 4 0 69897 22 3o010o3 76144 29 23856 06247 7 93753 l Hour P.M. Hour A.M. I Cosine. Diff-. Secant. Cotangent Diff, Tangent. Cosecant. Diff S.in 1190 A A B B C C 60' Seconds of time...... I". 2 3S 48 5's 6' 7' A 3 6 8 14 17 20 Prop. parts of cols. B 4 7 i i5 18 22 26 C i 2 3 4 4 5 6 TABLE XXVII. [Page SI. Log. Sines, Tangents, and Secants. G'. 300 A A B B C C 1490 M Iour A.M. Hour P.n. Sine. Difl. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine..M 0 8 o o 4 o o 9.69897 o o.30oo3 9.76144 o 10.23856 10.06247 0 9.93753 60 1 7 59 52 0 8 69919 0 3oo8I 76173 o 23827 o6254 o 93746 59 2 59 44 0 16 69941 30059 76202 1 23798 06262 o 93738 58 3 59 36 0 24 69963 30037 76231 I 23769 06269 o 93731 57 4 59 28 0 32 69984 3oo6 76261 2 23739 06276 o 93724 56 5 7 59 2o 4 0 40 9.70006 2 10.29994 9.76290 2 10.23710 10.06283 i 9.93717 55 6 59 12 0 48 70028 2 29972 76319 3 2368 06291 I 93709 54 7 59 4 0 56 70050 3 29950 76348 3 23652 06298 i 93702 53 8 58 56 1 4 70072 3 29928 76377 4 23623 o63o5 i 93695 52 9 58 48 I 12 70093 3 29907 76406 4 23594 06313 I 93687 51 o10 7 58 40 4 I 20 9.70115 4 0.29885 9.76435 5 10.23565 10.06320 I 9.93680 5o ii 58 32 I 28 70137 4 29863 76464 5 23536 06327 1 93673 49 12 58 24 36 70159 4 29841 76493 6 23507 o6335 i 93665 48 13 58 6 1 44 7018o 5 29820 76522 6 23478 06342 2 93658 47 14 58 8 52 70202 5 29798 76551 7 23449 o635o 2 93650 46 15 7 58 0 4 2 0o9.70224 5 10.29776 9.76580 7 10.23420 10.06357 2 9.93643 45 16 57 52 2 8 70245 6 29755 76609 8 23391 06364. 2 93636 44 17 57 44 2 i6 70267 6 29733 76639 8 23361 06372 2 93628 43 18 57 36 2 24 70288 6 29712 76668 9 23332 06379 2 93621 42 19 57 298 2 32 70310 7 29690 76697 9 23303 06386 2 93614 4" 20 7 57 20 4 2 40 9.70332 7 10.29668 9-76725 o 10.23275 10.06394 2 9.93606 4o 21 57 12 2 48 70353 8 29647 76754 10 23246 o64oi 3 93599 3o9 22 57 4 2 56 70375 8 29625 76783 II 23217 06409 3 93591 38 23 56 56 3 4 70396 8 29604 76812 23188 06416 3 93584 37 24. 56 48 3 12 70418 9 29582 76841 12 23159 06423 3 93577 36 25 7 56 4o4 3 20 9.70439 9 10io.29561 9.76870 12 I10.2330o io.o643i 3 9.93569 35 26 56 32 3 28 70461 9 29539 76899 13 23101 06438 3 93562 34 27 56 24 3 36 70482 10 29518 76928 13 23072 o6446 3 93554 33 28 56 16 3 44 70504 0o 29496 76957 13 23043 o6453 3 93547 32 29 56 8 3 52 70525 10 29475 76986 14 23o014 o646i 4 93539 31 30 756 0 4 4 0 9-70547 II 10.29453 9.77015 14 10.22985 10.06468 4 9.93532 30 31 55 52 4 8 70568 II 29432 77044 15 22956 06475 4 93525 29 3 55 44 4 i6 70590 II 29410 77073 15 22927 o6483 4 93517 28 33 55 36 4 24 70611 12 29389 77101 16 22899 06490 4 93510 27 34 55 28 4 32 70633 12 29367 77130 i6 22870 06498 4 93502 26 35 7 55 20 4 4 4o 9.70654 13 10.29346 9.77159 i7 10.22841 10.065o5 4 9.93495 25 36 55 12 4 48 70675 13 29325 77188 17 22812 o65i3 4 93487 24 37 55 Z4 4 56 70697 I3 29303 77217 18 22783 06520 5 93480 23 38 54 56 5 4 70718 14 29282 77246 18 22754 06528 5 93472 22 39 54 48 5 12 70739 i4 2926I 77274 19 22726 06535 5 93465 21 4o 7544o4 52 9.70761 14 10.29239 9.77303 19 10.22697 io.06543 5 9.93457 20 41 54 32 5 28 70782 15 29218 77332 20 22668 06550 5 93450 19 42 54 24 5 36 7o803 15 29197 77361 20 22639 06558 5 93442 18 43 54 i6 5 44 70824 15 29176 77390 21 22610 06565 5 93435 17 44 54 8 5 J2 70846 16 29154 77418 21 22582 06573 5 93427 16 45 7 54 0 4 6 9.70867 16 10.29133 9.77447 22 10.22553 io.o658o 6 9.93420 15 46 53 52 6 8 70888 16 29112 77476 22 22524 o6588 6 93412 14 47 53 44 6 16l 70909 17 29091 77505 23 22495 06595 6 93405 13 48 53 36 6 24 7093i 17 29069 77533 23 22467 o66o3 6 93397 12 49 53 28 6 32 70952 i8 29048 77562 24 22438 o66io 6 93390 II 50 7 53 20 4 6 40 9.70973 i8 10.29027 9-77591 24 10.22409 o.o66i8 6 9.93382 10 51 53 12 6 481 709941 8 29006 77619 25 22381 06625 6 93375 9 52 53 4 6 56 71015 19 28985 77648 25 22352 o6633 6 93367 8 53 52 56 7 4 710361 19.28964 77677 26 22323 06640 7 93360 7 54 52 48 7 712 58 19 28942 77706 26 22294 o6648 7 93352 6 55 7 52 40 4 7 20 9.71079 20 10.28921 9.77734 26 10.22266 10.06656 7 993344 5 56 52 32 7 28 71100 20 28900 77763 27 22237 o6663 7 93337 4 57 52 24 7 36 71121 20 28879 77791 27 22209 06671 7 93329 3 58 52 16 7 44 71142 21 28858 77820 28 22180 06678 7 93322 2 59 52 8 7 52 71163 21 28837 77849i 28 22151 06686 7 93314 1 60o 52 0 8 0 71184 21 2886 778771 29 22123 06693 7 93307 0 M Hour P.1. Hour A.M. Cosine. Diff. Secant. CotangentDiff. Tangent. Cosecant. Diff. Sine. M 1200 A A B B C C 9 Seconds of time...... 1 2 3 4 5 6 7 A 3 5 8 11 i3 16 19ig Prop. parts of cols. B 4 7 11 14 18 22 25 C I I J 3 4 5/6 7 Page 216] TABLE XXVII.'t. Log. Sines, Tangents, and Secants, 310 A A B B C C 1480 I IHour A.-M. Hourp.m. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. 9 i 0 752 0 4 8 0 9-71184 0o 10.28816 9-77877 0 10.22123 10.0663 0 9.93307 6 1 51 52 8 8 712o5 o 28795' 77906 0 22094 06701 0 93299 59 2 51 44 8 16 71226 i 28774 77935 1 22065 06709 0 93291 58 3 5 36 8 24 71247 1 28753 77963 1 22037 06716 o 93284 57 4 51 28 8 32.71268 I' 28732 77992 2 22008 06724 I 93276 56 5- 75 204 8 40 9.71289 2 10.28711 9.78020 2 10.21980 10.06731 i 9.93269 55 6 51 12 8 48 71310 2 28690 78049 3 21951 06739 i 93261 54 - 51 4 856 71331 2 28669 78077 3 21923 06747 I 93253 53 8 5o 56 9 4 71352 3 28648 78106 4 21894 06754 / 93246 52 9 1 48 9 2 71373 3 28627 78135 4 21865 o6762i, 93238 51 10 7o50 40 4 9 20 9-71393 3 10.28607 9.78163 5 10.2-837 1 o.o677o I 9.93230 5o 11 5o 32 9 28 71414 4 28586 78192 5 21808 06777 I 93223 49 12 So 24 9 36 71435 4 28565 78220 6 21780 06785 2 93215148 13 5o 16 9 44 7i456 4 28544 78249 6 21751 06793 2 93207 471 14 50 8 9 52 71477 5 28523 78277 7 21723 o6800 2 93200 46 15 750 4 10 0 9-71498 5 10.28502 9.78306 7 10.21694 1o.o68o8 2 9.93192 45 16 4952 10 8 71519 5 28481 78334 8 21666 o68i6 2 93184 44 17 4944 io 16 71539 6 2846i 78363 8 21637 06823 2 93177 43 18 49 36 10o 24 756o 6 28440 78391 9 21609 o683i 2 93169 42 19 49 28 io 32 71581 7 28419 78419 9 21581 06839 a 3i61 41 20 7 4 20 4 10 40 9.71602 710.28398 9.78448 910.21552 o.o6846 3 9.93154 4 21 49 12 10 48 71622 7 28378 78476 10 21524 06854 3 93146. 39 22 49 4 10o 56 71643 8 28357 78505 io 21495 06862 3 93138 38 23 48 56 11 4 71664 8 28336 78533 ii 21467. 06869 3 93131 37 24 48 48 11 12 71685 8 28315 78562 iI 21438 06877 3 93123 36 257 48 40 4 1120 9.71705 9 10o.8295 9.78590 12 10.214ol0 o.o6885 3 9.93115 351 26 48 32 Ii 28 71726 9 28274 78618 12 21382 06892 3 93108 34 27 48 24 iI 36 71747 9 28253 78647 13 21353 06900oo 3 93100 33 28 48 i6 ii 44 77671 o0 28233 78675 13 21325 06908 4 93092 32 29' 48 8 1.52. 717881 io 28212 78704 1I4 21296 06916 4 93084 3i 30 7 48 o 4 12'0o 9.71809 1io 10i.28i9i 9-78732 14 10.21268 10.06923 4 9.93077 30o 31 47 52 12 8 71829 ii 28171 78760 15 21240 06931 4 93069 29 32 47 44 12 16 7185o0 II 28150 78789 i5 21211 06939 4 93061 28 33 47 36 12 24 71871 11 28130 78817 i6 21183 06947 4, 93053 27 34 47 28 12 32 71891 12 2809 78845 i6 21155 06954 4 93046 26 35 77 20 4 12 4o 9.7191 2 10o.28o089 9-78874 17 10.21126 10.06962 5 9.93o38 25 36 47 12 12 48 71932 12 28(o68 78902 17 21098 06970 5 93030 24 37 47 4 12 56 719521 3 28048 78930 17 21070 06978 5 93022 23 38 46 56 13 4 71973 13 28027 78959.8 21041 06986 5 93014 22 39 46 48 13 12 71994 13 28oo006 78987 8 21013 06993 5 93007 21 40 746 40 4 I3 20 9,72014 14 10.27986 9.79015 19 10.20985 10.07001 5 9.92999 20 41 46 32 i3 28 72034 14 27966 79043 19 20o57 07009 5 92991 19 42 46 24 13 36 72055 14 27945 79072 20 20928 07017 5 92983 18 43 46 16 13 44 72075 15 27925 79100 2(0 20900 07024 6 92976 17 44 46 8 13 52 72096 15 27904 79128 21 20872 07032 6 92968 i6 45 746 0 4 14 0 9.7216 15 10.27884 9-79156 21 10.20844 10.07040 6 9.92960 I5 46 45 52 14 8 72137 16 27863 79185 22 20815 07048 6 92952 14 47 45 44 14 16 72157 16 27843 79213 22 20787 07056 6. 92944 13 48 45 36 14 24 72177 i6 27823 79241 23 20759 07064 6 92936 12 49 45 28 14 32 72198 17 27802 79269 23 20731 07071 6 92929 1I 50o 745 20 4 i4 4o0 9.72218 17 10.27782 9.79297 24 10.20703 10.07079 6 9.9292I 10 51 45 12 14 48 72238 18 27762 79326 24 20674 07087 7 92913 9 52 45 4 I'4 56 72259 i8 27741 79354 25 20o646 07095 7 92905 8 53 44 56 15 4 72279,8 27721 79382 25 20618 07103 7 92897 7 54 4 48 15 12 72299 19 27701 79410 26 20590 07111 7 92889 6 55 7 44 40o 4 15 20 19.7320 19 10.27680 9.79438 26 o10.20562 10o 07oI 7 9.92881 5 56 44 32 15 28 72340 19 27660 79466 26 20534 07126 7 92874 4 57 44 24 15 36 72360 20 27640 79495 27 20505 07134 7 92866 3 58 44 16 15 44 72381 20 27619 79523 27 20477 07142 7 92858 2 59 44 8 i5 52 72401 20 27599 79551 28 20449 07150 8 92850 i 6o 44 o 16 0 72421 21 27579 795791 28 20421 07158 8 92842 0 M Hour P.M. HOUnrA.M. Cosine. (Diff. Secant. ICotangent[ Diff. Tangent. T Cosecant. Diff. Sine. I 1210 A A B B C C 58 Seconds of time.,,. 1J 2~ 3s 4~ 5u 6" 7I -A 3 5 8 10 13 1 1 -8 Prop. parts of cols B 4 7 14 1 8 21 25 C i 2 3 4 5 6 7 TABLE XXVII. [Page 217 $' " Log. Sines, Tangents, and Secants. G'. 32c______ _ A A B B C C 147~ M Hour A.M Hour P.M. Sine. IDiff. Cosecant. Tangent. Diffl Cotangent Secant. Diff. Cosine. M o 7 44 o 4 i6 o 9.72421 o 10.27579 9.79579 0o10.20421 10.07158 o 9.92842 6o I 43 52 i6 8 72441 0 27559 79607 0 20393 07166 o 92834 59 2 43 44 16 i6 72461 I 27539 79635 I 20365 07174 o 92826 58 3 43 36 i6 24 72482 I 27518 79663 I 20337 07182 o 92818 57 4 43 28 16 32 72502 1 27498 79691 2 20309 07190 I 92810 56 5 7 43 20 4 i6 40 9.72522 2 10.27478 9.797i9 2 10.20281 10.07197 I 9.92803 55 6 43 12 i6 48 72542 2 27458 79747 3 20253 07205 I 92795 54 7 43 4 i6 56 72562 2 27438 79776 3 20224 07213 I 92787 53 8 42 56 17 4 72582 3 27418 79804 4 20196 07221 i 92779 52 9 42 48 17 12 72602 3 27398 79832 4 2o0168 07229 1 92771 5i TO 7 42 40 4 17 20 9.72622 3 10.27378 9.79860 5 10.20140 10.07237 I 9.92763 50 ii 42 32 17 28 72643 4 27357 79888 5 20112 07245 I 92755 49 12 42 24 17 36 72663 4 27337 79916 6 2o0084 07253 2 92747 48 13 42 i6 17 44 72683 4 27317 79944 6 20056 07261 2 92739 47 14 42 8 17 52 7270)3 5 27297 79972 7 20028 07269, 2 92731 46 15 7 42 0 4 i8 0 9.72723 5 10.27277 9.80000 7 10.20000 10.07277 2 9.92723 45 i6 41 52 I8 8 72743 5 27257 80028 7 19972 07285 2 92715 44 17 41 44 18 i6 72763 6 27237 80056 8 19944 07293 2 92707 43 18 41 36 i8 24 72783 6 27217 80084 8 i9916 07301 2 92699 42 19 41 28 18 32 72803 6 27197 80112 9 19888 07309 3 92691 41 20 7 41 20 4 18 40 9.72823 7 10.27177 9-80140 9 10.19860 10.07317 3 9.92683 40 21 41 12 x8 48 72843 7 27157 80168 io 19832 07325 3 92675 39 22 41 4 i8 56 72863 7 27137 80195 io 19805 07333 3 92667 38 23 40 56 19 4 72883 8 27117 80223 II 19777 07341 3 92659 37 24 40 48 19 12 72902 8 27098 80251 II 19749 07349 3 92651 36 25 7 40 40'4 19 20 9.72922 8 10.27078 9.80279 12 10.19721 10.07357 3 9.92643 35 26 40 32 19 28 72942 9 27058 80307 12 19693 07365 3 92635 34 27 40 24 19 36 72962 9 27038 80335 13 19665 07373 4 92627 33 28 4o i6 19 44 72982 9 27018 80363 13 19637 07381 4 92619 32 29 40 8 19 52 73002 10 26998 80391 13 19609 07389 4 92611 31 30 74 0 4 20 0o 9.73022 10 10.26978 9.80419 4 10.19581 10.07397 4 9.92603 3o.31 39 52 20 8 73041 io 26959 804471 4 19553 07405 4 92595 29 32 39 44 20 i6 73061 II 26939 804741 5 19526 07413 4 92587 28 33 39 36 20 24 73081 11 26919 80502 15 19498 07421 4 92579 27 34 39 28 20 32 73101o I 26899 80530 16 19470 07429 5 92571 26 35 7 39 20 4 20 40 9.73121 2 10.26879 9.80558 i6 10.19442 10.07437 5 9.92563 25 36 39 12 20 48 73140 12 26860 80586 17 19414 07445 5 92555 24 37 39 4 20o 56 73160o 12 26840 8o6i4 17 19386 07454 5 92546 23 38 38 56 21 4 73180 13 26820 80642 18 19358 07462 5 92538 22 39 38 48 21 12 73200 13 26800 80669 18 19331 07470 5 92530 21 40 7 38 4o 4 21 20 9.73219 13 10.26781 9.80697 19 10.19303 10.07478 5 9.92522 20 41 38 32 21 28 73239 i4 26761 80725 19 19275 07486 6 92514 19 42 38 24 21 36 73259 14 26741 80753 20 19247 07494 6 92506 18 43 38 16 21 44 73278 14 26722 80781 20 19219 07502 6 92498 17 44 38 8 21 52 73298 15 26702 80808 20 19192 07510 6 92490 16 45 7 38 o 22 9.73318 10.26682 9.80836 21 o10.1i9164 10.0o.o 7518 6 9.92482 15 46 37 52 22 8 73337 15 26663 80864 21 19136 07527 6 92473 14 47 37 44 22 16 73357 16 26643 80892 22 19108 07535 6 92465 13 48 37 36 22 24 73377 i6 26623 80919 22 i9081 07543 6 92457 12 49 37 28 22 32 73396 I6 26604 80947 23 19053 07551 7 92449 11 50 7 37 20 4 22 40o9.7341 17 10.26584 9.80975 23 10.19025 10.07559 7 9.92441 10 51 37,12 22 48 73435 17 26565 81003 24 18997 07567 7 92433 9 52 37 4 22 56 73455 17 26545 81030 24 18970 07575 7 92425 8 53 36 56 23 4 73474 18 26526 8o1058 25 18942 07584 7 92416 7 54 36 48 23 12 73494 18 26506 8o1086 25 i8914 075929 7 92408 6 55 7 36 4 4 23 20 9.73513 I8 10.26487 9.81113 26 I0.18887 10.07600 7 9.92400 5 56 36 32 23 28 73533 19 26467 81141 26 i8859 07608 8 92392 4 57 36 24 23 36 73552 19 26448 81169 26 18831 07616 8 92384 3 58 3616 23 44 73572 19 26428 81196 27 18804 07624 8 92376 2 59 36 8- 23 52 73591 20 26409 81224 27 18776 07633 8 92367 1 6o 36 o 24 o0 73611 20 26389 81252 28 18748 07641 8 92359 0 M Hour P.M.Hour A.M. Cosine. Diff. Secant. Cotangenta Diff. Tangent. Cosecant. Diff. Sine. M 1220 A A B B C C 570 Seconds of time...... 2l 3 4' 5' 6' 7' (A 2 5 7 10 12 i5 17 Prop. parts of cols. B 3 7 710 o 14 17 21 I 7 1 i4 17 4 1 246 7 ________ (0 23 4 5 6 7 arce ~1s] TABLE XXVII. Log. Sines, Tangents and Secants. G'. 333 A A B B C C 14G' HIour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 736 04i 24 o 9.736i 1 i 10.26389 9.81252 o 10-18748 10. 0764 o 09.9235 60 1 35 52 24 8 73630 o 26370 81279 0 18721 076491 92351 59 2 35 44 24 16 73650 i 26350 81307 I 18693 07657 o 92343 58 3 35 36 24 24 73669 I 2633I, 8i335 I 18665 07665 o 92335 57 4 35 28 24 32 73689 I 26311 81362 2 18638 07674 I 92326 56 5 3520 42440 9.73708 2 10.26292 9.81390 2 0.86o 10.0.o 7682 I 9.92318 55 6 35 12 24 48 73727 2 26273 81418 3 i8582 07690 I 92310 54 7 35 4 2456 73747 2 6253 81445 3 i8555 07698 I 92302 53 8 3456 25 4 73766 3 26234 81473 4 18527 07707 1 92293 5c 9 34 48 25 12 73785 3 26215 815oo00 4 i8500 07715 I 92285 5i 344 7 4 25 20 9.73805 3.26195 9.81528 5 10.18472 10.07723 i 9.92277 50 II 3432 25 28 73824 3 26176 81556 5 i8444 07731 2 92269 49 12 34 24 25 36 73843 4 26157 81583 5 18417 07740 2 92260 4813 34 i6 25 44 73863 4 26137 81611 6 18389 07748 2 92252 47 14 34 8 25 52 73882 4 26118 81638 6 18362 07756 2 92244 46 15 34 4 26 o 9.739o 5 10.26o099 9.81666 7Iio.i8334 10.07765 2 9.92235 45 16 33 52 26 8 73921 5 26079 81693 7 18307 07773 2 92227.44: 17 33 44 26 16 7394o 5 26060 81721 8 18279 07781 2 92219 43 18 33 36 2624 73959 26041 81748 8 18252 07789 3 92211 42 19 33 28 26 32 73978 6 26022 81776 9 18224 07798 3 92202 4i1 20 7 3320426409.73997 6 10.26003 9.81803 9 1o.18197 10.07806 3 9.92194 40 21 33 12 26 48 74017 7 25983 8i83 io. 18169 07814 3 9218639 22 33 4 26 56 74036 7 25964 81858 io 18142 07823 3 92177 386 23 32 56 27 4 74055 7 25945 81886 ii 18iI4 07831 3 92169 37 24 32 48 27 12 74074 8 25926 81913 ii 18087 07839 3 92161 36 25 73240 4 27 20 9.74093 8 10.25907 9.81941 ii 10.18059 10.07848 3 9.92152 35 26 32 32 27 28 74113 8 25887 81968 12 18032 07856 4 92144 34 27 32 24 27 36 74132 9 25868 81996 12 i8o04 07864 4 92I36 33 28 32 16 27 44 74151 9 25849 82023 13 17977 07873 4 92127 32 29 32 8 27 52 74170 9 25830 82051 13 17949 07881 4 9211 1i 30 7 32 0 4 28 o 9.74189 io 10.25811 9.82078 14 10.17922 10.07889 4 9.92111 3o 31 31 52 28 8 74208 10 25792 82106 14 17894 07898 4 92102 290 32 31 44 28 i6 742271 I0 25773 82133 15 17867 07906 4 92094 28 33 31 36 28 24 74246 10 25754 8216 15 17839 07914 5 92086 27 34 31 28 28 32 74265 II 25735 82188 16 17812 07923 5 92077 26 357 3 2(1 4 28 40 9.74284 II 10.25716 9.82215 16 10.17785 10.07931 5 9.92069 25 36 31 12 28 48 74303 Ii 25697 82243 i6 17757 07940 5 92060 24 37 31 4 28 56 74322 12 25678 82270 17 17730 07948 5 92052 23' 38 30 56 29 4 74341 12 25659 82298 17 17702 07956 5 92044 22 39 30 48 29 12 74360 12 25640 82325 18 17675 07965 5 92035!21 40 7 30 4o 4 29 20 9.74379 I3 10.25621 9.82352 i8 10.17648 10.07973 6 9.92027 2:0 41 30 32 29 28 74398 13 25602 82380 19 17620 07982 6 92018 19 42 30 24 29 36 74417 13 25583 82407 19 17593 07990 6 92010 18 43 30 16 29 44 74436 14 25564 82435 20 17565 07998 6 92002 17 44 30 8 29 52 74455 14 25545 82462 20 17538 08007.6 91993 i6 45 7 3o0 4 3o 0 9.74474 14 10.25526 9.82489 21 10.17511 o.o8o5 6 9.9985 15 46 29 52 3o 8 74493 15 25507 82517 21 17483 08024 6 91976 14 47 29 44 30 16 74512 I5 25488 82544 22 17456 08032 7 91968 13 48 29 36 30 24 7453i 15 25469 82571 22 17429 08041 7 91959 I 49 29 28 30 32 74549 i6 25451 82599 22 17401 08049 7 9195i II 50 7 2920 43040 9.74568 i6 10.25432 9.82626 23 10.17374 1o.o8o58 7 9.91942 10 51 29 12 30 48 74587 16 25413 82653 23 17347 08066 7 91934 9 52 29 4 30o 56 74606 I7 25394 82681 24 17319 08075 7 91925 8 53 28 56 31 4 74625 17 25375 82708 24 17292 08083 7 91917 7 54 28 48 31 12 74644 17 25356 82735 25 17265 08092 8 91908 6 55 7 28 40 4 31 20 9.74662 17 10.25338 9.82762 25 10.17238 io.o8ioo 8 9.91900 5 56 28 32 31 28 74681 i8 25319 82790 26 17210 08109 8 91891 4 57 28 24 31 36 74700 I8 25300 82817 26 17183 08117 8 91883 3 58 28 i6 3144 74719 18 25281 82844 27 17156 08126 8 91874 2 59 28 8 31 52 74737 19 25263 82871 27 17129 o8134 8 91866 i 6o 28 0 32 0 74756 19 25244 82899 27 17101 08143 8 91857 0 I Hour P.i. IlHour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1230 A A B B C C 5 6 Seconds of time...... I 28 3" 4s 5s 6" 7 A 2 5 7 10 12 1 i4 17 Prop. parts of cols. B 3 7 i4 17 21 24 C 1 2 3 4 5 6 7 TABLE XXVII. [Page 219 S'1. Log. Sines, Tangents, and Secants. G. 340 A A B B C C 1450 M Hfour A. m. Hour p.m. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. A o0 7 28 04 32 o0 974756 0 10.25244 9.82899 o10.17101 10.08143 o 9.91857 60 27 52 32 8 74775 o 25225 82926 o 17074 08151 0 91849 59 2 27 44 32 16 74794 I 25206 82953 17047 0860o 0 91840 58 3 27 36 32 24 74812 1 25188 82980 17020 0o868 o 91832 57 4 27 28 32 32 74831 25169 83oo008 2 16992 08177 1 91823 56 5 72720 4 3240 9.74850' 2 10.25150 9.83035 2 10.16965 o10 8185 i 9.91815 55 6 27 12 32 48 74868 2 25132 83062 3 6938 08194 I 91806 54 7 27 4 32 56 74887 2 25113 83089 3 16911 08202 i 91798 53 8 26 56 33 4 7490o6 2 25094 83117 4 16883 08211 I 91789 52 9 2648 33 12 74924 3 25076 83i44 4 16856 08219 I 91781 5i 10 7 26 40 4 33 20 9.74943 3 10.25057 9.83171 5 io.16829 10.082281 I 9.91772 50 1i 26 32 33 28 74961 3 25039 83198 5 16802 082371 2 91763 49 12 26 24 33 36 74980 4 25020 83225 5 16775 08245 2 91755148 13 26 i6 33 44 74999 4 25o00 83252 6 16748 08254 2 91746 47 14 26 8 33 52 75017 4 24983 83280 6 16720 08262 2 91738 46 i5 726 o 4-34 o9-75o36 510.24964 9.83307 710o.16693 10.08271 2 9.91729 45 i6 25 52 34 8 75054 5 24946 83334 7 16666 08280 2 91720 44 17 25 44 34 16 75073 5 24927 8336i 8 16639 08288 2 91712 43 18 25 36 34 24 75091 6 24909 83388 8 16612 08297 3 91703 42 19 25 28 34 32 75110 6 2489( 834r5 9 16585 o83o5 3 91695 41 20 7 25 20 4 34 40 9.75128 6 10.24872 9.83442 9 1o.i6558 1o.o83i4 3 9.91686 40 21 25 12 34 48 75147 6 24853 83470 9 i653o 08323 3 91677 39 22 25 4 34 56 75165 7- 24835 83497 Io i65o3 o833i 3 91669 38 23 24 56 35 4 75184 7 24816 83524 io 16476 o834o 3 91660 37 24 24 48 35 12 75202 7 24798 83551 ii 16449 08349 3 91651 36 25 7 4 35 20 9.75221 8 10.24779 9.83578 ii 10.16422 10.08357 4 9.9i643 35 26 24 32 35 28 75239 8 24761 83605 12 16395 08366 4 91634 34 27 24 24 35 36 75258 8 24742 83632 12 i6368 08375 4 91625 33 28 24 16 35 44 75276 9 24724 83659 13 i634i 08383 4 91617 32 29 24 8 35 52 75294 9 24706 83686 13 i6 i4 08392 4 91608 31 30 7 24 4 36 0o9.75313 9 10.24687 9.83713 14 10o.16287 0o.8401 4 9.9i599 3o 31 23 52 36 8 75331 9 24669 83740 14 16260 08409 4 91591 29 32 23 44 36 i6 753501 o 24650 83768 14 16232 o84i8 5 91582 28 33 23 36 36 24 75368 io 24632 83795 15 16205 08427 5 91573 27 34 23 28 36 32 7538.6 io 24614 83822 i5 16178 08435 5 91565 26 35 7 23 20 4 36 4o 9.75405 1 10.24595 9.83849 6 10.16151 io.08444 5 9.9556 25 36 23 12 36 48 75423 II 24577 83876 16 16124 08453 5 91547 24 37 23 4 36 56 75441 II 24559 83903 17 16097 08462 5 9153823 38 22 56 37 4 754591 12 24541 83930 17 16070 08470 5 91530 22 39 22 48 37 12 75478 12 24522 83957 18 16043 08479 6 91521 21 40 7 22 40 4 3720 9.75496 12 10.24504 9.83984 18 10.16016 o.o8488 6 9.91512 20 41 22 32 37 28 75514 i3 24486 84011 18 15989 08496 6 9o504 19 42 22 24 37 36 75533 i3 24467 84038 1 9 5962 o85o05 6 91495 8 43 22 i6 37 44 75551 13 24449 84o65 19 15935 o85i4 6 91486 17 44 22 8 37 52 75569 13 24431 84092 20 15908 08523 6 91477 16 45 7 22 0 4 38 09.75587 1i4 10.24413 9.84119 20 o10.15881 10io.o8531 7 9.91469 iS 46 21 52 38 8 75605 i4 24395 84i46 21 15854 08540 7 91460 14 47 2144 3816 75624 14 24376 84173 21 15827 08549 7 91451 13 48 21 36 38 24 75642 15 24358 84200 22 158oo00 08558 7 91442 12 49 21 28 38 32 75660 15 24340 84227 22 15773 08567 7 91433 ii 50o 7 21 20 4 38 40 9.75678 i5 10.24322 9.84254 23 10.15746 10.08575 7 9.91425 io 5i 21 12 38 48 75696 i6 24304 84280 23 15720 08584 7 91416 9 52 21 4 38 56 75714 i6 24286 84307 23 15693 08593 8 91407 8 53 20 56 39 4 75733 16 24267 84334 24 i5666 08602 8 91398 7 54 2048 39 12 75751 17 24249 8436i 24 15639 o8611 8 91389 6 55 7 20 40 4 39 20 9.75769 17 10.24231 9.84388 25 o10.15612 10.08619 8 9.91381 5 56 20 32 39 28 75787 17 24213 84415 25 i5585 08628 8 91372 4 57 20 24 39 36 75805 17 24195 84442 26 i5558 08637 8 91363 3 58 20 i6 39 44 75823 18 24177 84469 26 15531 08646 8 91354 2 59 20 8 39 52 7584i i8 24159 84496 27 i5504 08655 9 9345 i 60 20 o 40 0 75859 i8 24141 84523 27 15477 o8664 9 91336 o M IHour P.m.JIHOUrA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. M 1240 A A B B C C 550 Seconds of time...... 1 3S 4S 54 63o 7 A 2 5 7 9 11 I i6 Prop. parts of cols. B 3 7 10 14 17 20 24 C 1 3 4 5 7 8 Page 220] TABLE XXVII. s'. Log. Sines, Tangents, and Secants. GI. 350 A A B B,C C 144~ M Hour A.M.' Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine..M 0 7 20 o44o 019.75859 o10.24141 9.84523 o io0.5477 o.o8664 o 9.91336 6o 19 52 4o 8 75877 0 24123 84550 o i545o 08672 o 91328 59 2 1944 40o 6 75895 I 24105 84576 1 15424 o868i o 91319 58 3 9 36 40 24 75913 i 24087 84603 i 15397 0869o 0 o 91310 57 4 19 28 4o 32 75931 i 24069 8463o 2 15370 08699 1 91301 56 5 7 19 20o4 40 40 9.75949 1 10.24051 9.84657 2 10o.5343 10o.08708 9.91292 55 6 19 12 40 48 75967 2 24033 84684 3 15316 08717 1 91283 54 7 19 4 4o 56 75985 2 24015 84711 3 15289 08726 I 91274 53 8 i8 56 4 41 76003 2 23997 84738 4 15262 08734 i 91266 52 9 i8 48 41 12 76021 3 23979 84764 4 15236 o8743 1 i 91257 51 10 7 18 40 4 41 20 9.76039 3 10.23961 9.84791 41 o.i5209 10.08752 2 9.91248 50 II 18 32 41 28 76057 3 23943 848i8 5 15182 08761 2 91239 49 12 i8 24 41 36 76075 4 23925 84845 5 15155 08770 2 91230 48 13 i8 i.6 41 44 76093 4 23907 84872 6 15128 08779 2 91221 47 14 18 8 4/ 52 76111 4 23889 84899 6 i5ioi 08788 2 91212 46 15 7 i8 0 4 42 0 9.76129 4 10.23871 9.84925 7 10.1-5075 10.08797 2 9.91203 45 i6 17 52 42 8 76,46 5 23854 84952 7 i5048 08806 2 ii94 44 17 I7 44 42 i16 76164 5 23836 84979 8 15021 08815 3 91185 43 i8 17 36 42 24 76182 5 23818 85oo6 8 14994 08824 3 91176 42 19 17 28 42 32 76200 6 23800 85o33 8 o 4967 o8833 3 91167 41 20 7 17 20 4 42 40 9.76218 6 10.23782 9.85059 9 10.14941 10.08842 3 9.9158 140 21 17 12 42 48 76236 6- 23764 85o86 9J 14914 o885i 3 91149139 22 17 4 42 56 76253 6 23747 85113 io 14887 08859 3 91141 38 23 i6 56 43 4 76271 7 23729 85I40 I0 i 486o 08868 3 91132 37 24 16 48 43 12 76289 7 23711 85166 ii i14834 08877 4 91123 36 25 7 i6 40o 4 43 20 9.76307 7 10.23693 9.85193 11 10.14807 io.o8886 4 9.911o4 35 26 i6 32 43 28 76324 8 23676 85220 12 14780o 08895 4 9110o5 34 27 i6 24 43 36 76342 8 23658 85247 12 14753 08904 4 91096 33 28 i6 16 43 44 76360 8 23640 85273 12 14727 08913 4 91087 32 29 16 8 4352 76378 9 23622 853oo 13 14700 08922 4 91078 3A 30 7 6 4 44 0 9.76395 9 10.23605 9.85327 13 10.14673 o10.0893 5 9.91069 3o 3i 1552 44 8 76413 9 23587 85354 i4 14646 08940 5 91060 29 32 15 44 44 i6 76431 9 23569 8538o 14 14620 08949 5 91051 28: 33 5 36 44 24 76448 o10 23552 854o7 15 14593'08958 5 91042 27 34 15 28 44 32 76466 10 23534 85434 15 14566 08967 5 91033 26 35 7 5 20 4- 44 4o 9.76484J 1 10.23516 9.85460 16 0o.i454o 10.08977 5 9.91023 25 36 I5 12 44 48 76501 11ii 23499 85487 i6 i45i3 08986 5 91014 24. 37 i5 4 44 56 76519 ii 23481 85514 16 i4486 08995 6 9100oo5 23 38 14 56 45 4 76537 Ii 23463 8554o 17 1446o 09004 6 90996 22 39 4 48 45 12 76554 1 2 23446 85567 17 14433 09013 6 90987 21 40 7 14 40445 20 9.76572 12 10.23428 9.85594 18 10o.44o6 10.09022 6 9.90978 20 41 i4 32 45 28 76590 12 23410 85620 i8 i438o 09031 6 90969 19 42 14 24 45 36 76607 12 23393 85647 19 4353 09040 6 go9096 18 43 14 i6 45 44 76625 13 23375 85674 19 14326 09049 6 9095, 17 44 14 8 45 52 76642 13 23358 85700 20 143o00 09058 7 90942 i6 45 7 i4 0 446 0 97666o 13 10.23340 9.85727 20 10.14273 o10.090o67 7 9.9o933 15 46 13 52 46 8 76677 14 23323 85754 20 14246 09076 7 90924 14 47 13 44 46 16 76695 14 23305 85780 21 14220 09085 7 90915 13 48 13 36 46 24 76712 I4 23288 85807 21 14193 09094 7 9o9o6 1.2 49 13 28 46 32 7573o 14 23270 85834 22 14166 09104 7 90896 I1 50 7 13 20 4 46 4o 9.76747 15 10.23253 9.85.860 22 10.14140 10.09113 8 9.90887 10 51 13 12 46 48 76765 15 23235 85887 23 14113 09122 8 90878 9 52 13 4 46 561 76782 15 23218 85913 23 14087 09131 8 90869 8 53 12 56 47 4 768oo0 6 23200 85940 24 i4o6o 09140 8 9o860 7 54 12 48 47 12 76817 16 23183 85967 24 14o33 09149 8 9085,1 6 55 7 12 40 4 47 20 9.76835 16 10.23165 9.85993 24 10.14007 10.09158 8 9.9842 5 56 12 32 47 28 76852 17 23148 86020 25 I3980 o0968 8 90832 A 57 12 24 47 36 76870 17 23130 86046 25 13954 09177 90823 3 58 12 16 47 44 76887 17 23113 86073 26 13927 09186 9 90814-2 59 12 8 47 52 76904 17 23096 861oo 26 13900 09195 9 90805 1 60 12 0 480 76922 18 23078 86126 27 13874 09204 9 90796 o HM Hourp.M. HourA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. 1250 A A B B C C 5 Secondsiof time...... 1i 2[ 31 4i 5s ]" 6 7' jA 2.4 7 9 1 3 i6 Prop. parts of ols. B 3 7 10 3 I7 20 23 C II__2 ( 1 3 5 6 7 8 TABLE XXVII. [Page 221 SI. Log. Sines, Tangents, and Secants. G'. 36" A A B B C C 1430 MA Hour A.M. Hour p.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 0 7 12 0 4 48 0 9.76922 o 10.23078 9.86126 o 1o.13874 10.09204 0 9.90796 60 I 1152 48 8 76939 o 2306I 86153 o 13847 09213 o 90787 59 2 ii 44 48 i6 76957'I 23043 86179 I 13821 09223 o 90777 58 3 11 36 48 24 76974 1 23026 86206 i 13794 09232 0 90768 57 4 11 28 48 32 7699I I 23009 86232 9 13768 09241 r 90759 56 5 7 11 20 4 48 40 9.77009 i 10.2299I 9.86259 2 10.13741 10.09250 I 9.90750 55 6 11 12 48 48 77026 2 22974 86285 3 13715 09259 I 90741 54 7 It 4 48 56 77043 2 22957 86312 3 13688 09269 I 90731 53 8 10 56 49 4 77061 2 22939 86338 4 13662 09278 I 90722 52 9 10 48 49 12 77078 3 22922 86365 4 i3635 09287 I 90713 5i 10 7 io 40 4 49 20 9.77095 3 10.22905 9.86392 4 io.136o8 10.09296 2 9.90704 50 11 10 32 49 28 77112 3 22888 86418 5 13582 09306 2 90694 49 12 10 24 49 36 77130 3 22870 86445 5 13555 o9315 2 90685 48 13 10 i6 49 44 77147 4 22853 86471 6 13529 09324 2 90676 47 14 10o 8 49 52 77164 4 22836 86498 6 13502 09333 2 90667 46 15 7 10 0 4 o 0977181 4 10.22819 9.86524 7 o10.13476 10.09343 2 9.90657 45 16 9 52 50 8 77199 5 22801 86551 7 13449 09352 2 90648 44 17 9 44 50o 16 77216 5 22784 86577 7 13423 09361 3 90639 43 1i8 936 50 24 77233 5 22767 86603 8 13397 09370 3 90o630 42 19 928 So5032 77250 5 22750 86630 8 13370 09380 3 90620 4i 20 7 9 204 50 40 9.77268 6 10.22732 9.86656 9 1ro.3344 10.09389 3 9.90611 40 12.1 9 12 50 48 77285 6 22715 86683 9 13317 09398 3 90602 39 22 9 4 5o 56 77302 6 22698 86709 1o 13291 09408 3 90592 3-8 123 8 56 51 4 77319 7 22681 86736 io 13264 09417 4 90583 37 124 8 48 51i 12 77336 7 22664 86762 ii 13238 09426 4 90574 36 25 7 8 40 4 51 20 9.77353 710.22647 9.86789 ii 10.13211 10.09435 4 9.90565 35 26 8 32 5128 77370~ 7 22630 86815 11 I3i85 09445 4 90555 34 27 8 24 51i 36 77387: 8 22613 86842 12 13r58 09454 4 90546 33 28 8 16 51 44 77405 8 22595 86868 12 13132 09463 4 90537 32 29 8 8 51 52 77422 8 22578 86894 13 1310O6 09473 5 90527 31 30 7 8 0 4 52 0 9.77439 9 10.22561 9.86921 13 10.13079 10.09482 5 9.90518 3o 3r 7 52 52 8 77456 9 22544 86947 14 i3o53 09491 5 90509 29 33 7 36 52 24 77490 9 225o10 87000 15 13000 09510 5 90490 27 34 7 28 52 32 77507 101 22493 87027 15 12973 09520 5 90480 26 35 7 7 20 4 52 40 9.77524 10o10.22476 9.87053 15 10.12947 10.09529 5 9.90471 25 36 7 12 52 48 77541 io 22459 87079 i6 12921 09538 6 90462 24 37 7 4 52 56 77558 1i 22442 87106 i6 12894 09548 6 90452 23 138 6 56 53 4 77575 II 22425 87132 17 12868 09557 6 90443 22 39 6 48 53 12 77592 II 22408 87158 17 12842 09566 6 90434 21 40 7 6 40 4 53 20 9.77609 11 10.22391 9.87185 i8 10o.12815 10.09576 6 9.90424 20 41 6 32 53 28 77626 12 22374 87211 i8 12789 09585 6 90415 19 42 6 24 53 36 77643 12 22357 87238 18 12762 09595 7 90405 i8 43 6 16 53 44 7766(0 12 22340 87264 19 12736 09604 7 90396 17:44 6 8 53 52 77677 13 22323 87290 19 12710 09614 7 90386 s6 45 7 6 04 54 9.77694 13 10.22306 9.87317 20o 10.12683 o0.09623 7 9.90377 15 46 5 52 54 8 777I11 13 22289 87343 20 12657 09632 7 go90368 14 47 5 44 54 16 77728 13 22272 87369 921 12631 09642 7 90358 13 48 5 36 54 94 77744 14 22256 873961 i I260O4 o9651 7 90349 12 49 5 28 54 32 77761 14 22239 87422 22 12578 09661 8 oo339 II 50 7 5 20 4 54 40 9.77778 14 10.22222 9.87448 22 10.12552 10.09670 8 9.90330 10 51 5 12 54 48 77795 25 22205 87475 22 12525 09680 8 90320 9 52 5 4 54 56 778i4 15 22188 87501 23 12499 09689 8 9c3ii 8 53 4 56 55 4 77829 I5 22171 87527 23 12473 09699 8 90o30 7 54 4 48 55 12 77846 I5 22154 87554 24 12446 09708 8 90292 6 55 7 4 4o 4 55 20 9-77862 16 10.2238 9-.8758 24 10.12420 10.09718 9 9.90282 5 56 4 32 55 28 77879 i6 22121 87606 25 12394 09727[ 9 90273 4 5 4 24 55 36 77896 16 22104 876331 25 12367 097371 9 90263 3 58 4 16 55 44 77913 16 22087 87659; 26 12341 09746 9 90254 2 5. 4t 8 55 52 77930 17 22070 87685 26 12315 09756 9 90244 i 60o 40o 560 77946 17 220o54 87711 26 12289 09765 9 90235 o M1 Hour P.M.ilour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. MA W126 A A B B C C 53" Seconds of time...... I 2' 38 4s 5. 6' 7' (A 2 4 6 9 ii 1 3 i5 Prop. parts of cols. B 3 7 10 3 17 20 23 C 1 2 4 5 6 8 Page 233] TABLE XXVII. S'. Log. Sines, Tangents, and Secants. G'. 37~. A A B B C C 142C Al Hour A.M. Hour PA. Sine. Diff. Cosecant. Tangent. Diffl Cotangent Secant. Diff. Cosine. 31 0 7 4 0 4 560 9.77946 0 10.22054 9.87711 0 10.12289 10.09765 0 9.90235 6o0 1 352 56 8 77963 o 22037 87738 o 12262 09775 0 90225 59 2 3 44 56 i6 77980 I 22020 87764 1 12236 09784 o 90216 58 3 3 36 56 24 77997 I 22003 87790 1 12210 09794 0 90206 57 4 3 28 56 32 78013 I 21987 87817;.2 12183 09803 I 90197 56 5 7 3 20 4 56 40o 9.78030 10.21970 9.87843 2 10.12157 1.o09813 1 9.90187 55 6 3 12 56 48 78047 2 21953 878691 3 12131 09822 1 90178 54 7 3 4 56 56 78063 2 21937 87895 3 12105 09832 I 90168 53 8 2 56 57 4 78080 2 21920 87922 3 12078 0984 1 I go r 52.9 2 348 57 12 78097 2 21903 87948 4 12052 o985I 1 o90149 5 10 7 2 40o4 5720 9.78113 3 10.21887 9.87974 4 10.12026 10.09861 2 9.90139 50 ii 2 32 57 28 78130 3 21870 88oo000 5 12000 09870 2 90130 49 12 2 24 57 36 78147 3 21853 88027 5 11973 09880 2 90120 48 13 2 i6 57 44 78163 4 21837 88o53 6 11947 09889 2 90111 47 14 2 8 57 52 78180 4 21820 88079 6 11921 098992 90101 46 15 7 2 o 4 58 o09.78197 4 10.2183 9.88105 7 10.11895 10.0o 909 2 9.90091 45 16 52 58 8 78213 4 21787 8813r 7 11869 09918 3 90082 44 17 I 44 58 i6 78230 5 21770 88i58 7 11842 09928 3 90072 43 I8 I 36 58 24 78246 5 21754 88184 8 11816 09937 3 90063 42 ig 1 28 58 32 78263 5 21737 88210 8 11790 09947 3 90053 4i 20 7 I 20 4 58 40o 9.78280 5 10.21720 9.88236 9 10.11764 10.0o.9957 3 9.90043 4o 21 1 12 58 48 78296 6 21704 88262 9 11738 09966 3 90034 39 22 1 4 58 56 78313 6 21687 88289 10 11711 09976 4 90024 38 23 0 56 59 4 78329 6 21671 883i5 iO 1i685 09986 4 90014 37 24 0 48 59 12 78346 7 21654 8834, io 11659 09995 4 90005 36 25 7 o 4045920 9.78362 7 10.21638 9.88367 ii to.1i633 1o.i ooo5 4 9-89995 35 26 0 32 59 28 78379 7 21621 88393 II 11607 1ooI5 4 89985 34 27 0 24 59 36 78395 7 21605 88420 12 11580 10024 4 89976 33 28 o 16 59 44 78412 8 21588 88446 12 11554 1oo34 5 89966 32 29 o 8 59 52 78428 8 21572 88472, 13 11528 I0044 5 89956 31 30 7 0 0 5 o o9.78445 8 10.21555 9.88498 13 10.115o2 1o.ioo53 5 9.89947 3o 31 6 59 52 0 8 78461 9 21539 88524 I4 11476 ioo63 5 89937 29 32 59 44 016 78478 9 21522 88550 14 i145o 10073 5 89927 28 33 59 36 0 24 78494 9 21506 88577 14 11423 10082 5 89918 27 34 59 28 0 32 78510 9 21490 886o3 i5 11397 10092 5 89908 26 35 659 20o 5 0 40 9.78527 Io 10.21473 9.88629 i5 10.11371 10.10102 6 9.89898 25 36 59 12 0 48 78543 io 21457 88655 16 11345 10112 6 89888 24 37 594 056 78560 I0 21440 8868r 16 I1319 ii2 89879 23 38 58 56 4 78576 Io 21424 88707 17 11293 o10131 6 89869 22 39 58 48 1 12 78592 I1 21408 88733 17 11267 I014i 6 89859 21 40 6 58 405 20 9.78609 ii 10o.21391 9.88759 17 10.11241 io.ioi5 6 9.89849 20 41 58 32 I 28 78625 II 21375 8878 18 11214 1o160 7 89840 19 42 58 24 I 36 78642 12 21358 88812 18 uIi88 10170 7 898301 8 43 58 6 i 44 78658 12 21342 88838 19 11162 ioi80 7 89820 17 44 58 8 52 78674 12 21326 88864 g19 1136 10190 7 89810 I6 45 6 58 o 5 2 o0 9.78691 12 10.21309 9.88890 20 10.11110 10.-10199 7 9.89801 15 46 57 52 2 8 78707 I3 21293 88916 20 Iio84 10209 7 89791 14 47 57 44 2 i6 78723 13 21277 88942 20 iio58 10219 8 89781 13 48 57 36 2 24 78739 13 21261 88968 21 11032 10229 8 89771 12 49 57 28 232 787561 3 21244 88994 21 11006 10239 8 89761 11 50 6 57 20 5 240 9.78772 I4 10.21228 9.89020 22 10.10980 10.10248 8 9.89752 10 5 57 12 248 78788 14 21212 89046 22 10954 10258 8 89742 9 52 57 4 2 56 78805 14 21195 89073 23 10927 10268 8 89732 8 53 56 56 34 7882I I5 21179 89099 23 I0901 10278 9 89722 7 54 56 48 3 12 78837 15 21163 89125 24 10875 10288 9 89712 6 55 6 56 40 5 3 20 9.78853 15 10.21147 9.89151 24 10.10849 10.10298 9 9.-89702 5 56 56 32 3 28 78869 15 21131.89177 24 10823 10307 9 89693 4 57 56 24 3 36 78886 16 21114 89203 25 10797 10317 9 89683 3 58 56 16 344 78902 i6 21098 89229 25 10771 10327 9 89673 2 59 56 8 3 52 78918 I6 21o82 89255 26 10745 10337 io 89663 i 60 56 o 4 o0 78934 16 21066 89281 26 10719 10347 io 89653 o l Hourp.rM.I.Hournr.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant.IDif. Sine. Mi 1270 A A B B C C %. Seconds of time...... I" 2' 3s 4' 5' 6' 7 A 2 - 4 6 8 10 12 14 Prop. parts of cols B 3 7 1O I3 i6 20 23 C t 2 4 5 6 7 8 TABLE XXVII. [Pae 223 S'. Log. Sines, Tangents, and Secants. G'I 380 A A B B C C 1410 M11 Houl-A.M. Hourp.M. Sine. IDiff. Cosecant. Tangent. Diff. Cotangent Secant. Diff.1 Cosine. M o6 560 5 4 09.78934 1 0.2066 9.89281 o 1o.10719 10.10 347 09.89653 60 i 55 52 48 78950 0 21050 89307 o 10693 10357 o 89643 59 2 55 44 4 16 78967 i 21033 8933 I 1o667 10367 o 89633 58 3 55 36 424 78983 I 21017 89359 i o641 10376 I 89624 57 4 55 28 4 32 78999 I 21001 89385 2 o10615 10386 89614 56 5 6 55 20 5 4 40 9.79015 i 10.20985 9-89411 2 10.10589 10.10396 I 9.89604 55 6 55 12 4 48 7903 2 20969 89437 3 10563 10406 I 89594 54 7 55 4 4 56 79047 2 20953 89463 3 10537 Io476 I 89584 53 8 54 56 54 79063 2 20937 89489 3 o10511ii 10426 89574 52 9 54 48 5 12 79079 2 20921 89515 4 10485 10o436 2 89564 5I o 6 54 4o 5 5 20 9.79095 3 10.20905 9.89541 4 o10.10o459 o.o446 2 9.89554 50 ii 54 32 5 28 79111 3 2o8&9 89567 5 10433 10456 2 89544 49 12 54 24 536 79128 3 20872 89593 5 10407 o10466 2 89534 48 13 54 i6 544 79144 3 20856 89619 6 10381 10476 2 89524 47 14 54 8 552 79160 4 2o84o 89645 6 o10355 10486 2 89514 46 15 540 5 6 059.797o 6 4 10.20824 9.89671 6 10.10329 10.10496 3 9.89504 45 6 53 52 6 8 79192 4 20808 89697 7 o10303 10505 3 89495 44 17 53 44 6 16 79208 5 20792 89723 7 10277 io5i5 3 89485 43 i8 53 36. 6 24 79234 5 20776 89749 8 10251 10525 3 89475 42 19 53 28 6 32 79240 5 20760 8977 8 10225 o10535 3 89465 41 20 653 205 640o 979256 5 10.20744 9.89801 9 10.10199 io.io545 3 9.89455 40 21 53 12 648 79272 6 20728 89827 9 10173 o10555 4 89445 39 22 53 4 656 79288 6 20712 89853 1io 10147 10565 4 89435 38 23 52 56 74 79304 6 20696 89879 10 10121 10575 4 89425 37 24 52 48 712 79319 6 20681 89905 io 10095 o10585 4 89415 36 25 652 40o5 720 9.79335 7 10.20665 9.89931 ii 10.10069 10.10io 595 4.89405 35 26 52 32 7 28 79351 7 20649 89957 11 ioo43 10605 4 89395 34 27 52 24 7 36 79367 7 20633 89983 12 10017 o06i5 5 89385 33 28 52 16 7 44 79383 7 20617 90009 12 09991 10625 5 89375 32 29 52 8 7 52 79399 8 20601 90035 13 09965 10636 5 89364 31 30 6 52 0 5 8 o 9.79415 8 10.20585 9.90061 13 o10.099ogg39 io.io646 5 9.89354 30 31 51 52 88 79431 8 20569 90086 13 09914 10656 5 89344 29 332 51 44 8 16 79447 8 20553 90112 14 09888 10666 5 89334 28 33 51i 36, 824 79463 9 20537 o90138 4 09862 10676 6 89324 27 34 51 28 832 79478 9 20522 90164 15 09836 o10686 6 893i4 26 135 651 o205 8409.79494 9 10.20506 9.90190 i5 10.09810 10.10696 69.89304 25 36 51 12 848 79510 o10 20490 90216 16 09784 10706 6 89294 24 37 51 4 8 56 79526 10io 20474 90242 16 09758 1076 6 89284 23 38 50o 56 94 79542 10 20458 90268 16 09732 10726 6 89274 22 39 50 48 9 12 79558 io 20442 90294 17 09706 10736 7 89264 21 40 6 50 4 5 9 20 9.79573 i 110.20427 99032017 170.096801 0.10746 7 9.89254 20 41 50o 32 928 79589 ii 204i 90346 i8 09654 10756 7 89244 19 42 50 24 9 36 79605 II 20395 90371 18 09629 10767 7 89233 i8 43 50o 6 944 79621 ii 6o379 90397 19 09603 10777 7 89223 17 44 50 8 9 52 79636 12 20364 904231g 09577 1 0787 7 89213 16 45 6 50 0 5 10 o 9.79652 12 10.20348 9.90449 i9 io.o955i 10.10797 8 9.89203 15 46 49 52 10 8 79668 12 20332 90475 20 09525 10807 8 89193 14 47 49 44 o10 16 79684 12 20316 90501 20 09499 10817 8 89183 13 48 49 36 io 24 79699 3 2o31 90527 21 09473 10827 8 89173 12 49 49 28 io 32 79715 13 20285 9o553 21 09447 io838 8 89162 II 50 6 49 20 5 o 40 9.79731 13 10.20269 9.-o578 22 10.09422- 0.10848 8 9.89152 io 51 49 12 10 48 79746 4 20254 o90604 22 09396 o10858 9 89142 9 52 49 4 10 56 79762 4 20238 o90630 22 09370 o10868 9 89132 8 53 48 56 11 4 79778 14 20222 90656 23 09344 10878 9 89122 7 54 48 48 1112 79793 4 20207 90682 23 09318 o888 9 89112 6 55 6 48 40 5 i 20 9.79809 15 10.20191 9.9070824 10.09292 10.10899gg 9 9.89101 5 56 4 I832 if128 79825 15 20175 90734 24 09266 g0909 9 89091 4 5i7 4 89 24 ii 36 79840 61 20160 90759 25 09241 10919 io 89081 3 5S 48 i6 11 44 79856 15 20144 90785 25 09215 10929 io 89071 2 59 48 8 11 52 79872 16 20128 o90811 26 o09189 10940 10io 89060 6o 48 o 12 0 79887 i6 20113 90837 26 09163 og95o io 89o5o o LM Hour P.Mi. I-our A.M. Cosine. jDiff Secant. Cotangent Diff. Tangent. Cosecant. Diff Sine. M 1280 A A B B C C 5l Seconds of time...... P 2 3" 4" 58 6" 7" (A 2 4 6 8 10o 31 i4 Prop. parts of cols. B 3 6 o 3 6 19 23 C I 3 4 5 6 8 9 Page 224] TABLE XXVII. SI. Log. Sines, Tangents, and Secants. G' 390 A A B B C C 1400 M Hour A.SM.Hour'.nL. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M o 648 o5 12 0 9.79887 o 010.20113 9.90837 o 10.0og163 10.10950 o09.8905o 60 47 52 12 8 79903 0 20097 90863 o 0og137 106og 0 89040 5 2 47 44 12 I16 79918 I 20082 90889 I 09111 10970 0 89030 58 3 47 36 12 24 79934 1 20066 90914 1 09086 10980 i 89020 57 4 47 28 12 32 79950 I 20050 90940 2 09060 o10991 I 89009 56 5 6 47 20 5 12 40 9-79965 10.2o0035 9.90966 2 10.09034 io.iooi 1 9.88999 55 6 47 12 12 48 79981 a 20019 90992 3 09008 iioi I 88989 54 7 47 4 12 56 79996 a 20004 91018 3 08982 11022 i 88978 53 8 46 56 13 4 80012 2 1/9988 91043 3 08957 11032 1 88968 52 9 46 48 13 12 80027 2 19973 91069 4 08931 11042 2- 88958 51 io 6 46 4 5 3 20 9.8oo43 3 Io.19957 9.91095 4 10o.o8905 10.11052 2.88948 50o ii 46 32 I3 28 80058 3 19942 91121 5 08879 1Io63 2 88937 49 12 46 24 13 36 80074 3 19926 91147 5 08853 11073 2 88927 48 13 46 i6 13 44 80089 3 19911 91172 6 08828 11083 2 88917 47 4 46 8 13 52 8oo0105 4 19895 91198 6 08802 11094 2 88906 46 15 6 46 0 5 14 o 9.80120 4 o10.19880 9.91224 6 10io.08776 1o.iiio4 3 9.88896 45 16 45 52 14 8 8oi036 4 19864 91250 7 08750 11114 3 88886 44 17 45 44 i4 16 80151 4 19849 91276 7 08724 11125 3 88875 43 18 45 361 4 24 8o0166 5 19834 9130o 8 o8699 11135 3 88865 42 g19 45 28 14 32 80182 5 19818 9,1327 8 08673 11145 31 88855 14 20 6 45 20 5 1440 9.80197 5 o10.19803 9.91353 9 io.o8647 io.iii56 3 9.88844 40o 21 45 12 4 48 8o0213 5 19787 91379 9 08621 166 4 88834 39 22 45 4 14 56 80228 6 19772 91404 9 08596 11176. 4 88824 38 23 44'56 15 4 80244 6 19756 91430 io 08570 11187 4 888i3 37 24 44' 48 i5 12 80259 6 19741 91456 io o8544 11197 4 888o3 36 25 64440515 209.80274 6 10.19726 9.91482 ii io.o0858 10.11207 4 9.88793 35 26 44 32 15 28 80290 7 19710 91507 ii 08493 11218 5 88782 34 27 44 24 i5 36 80305 7 19695 91533 12 08467 11228 5 88772 33 28 44 6 i5 44 80320 7 19680 91559 12 o844, 11239 5 88761 32 29 44 8 15 52 80336 7 19664 91585 12 o84i5 11249 5 88751 3i 30 6 44 0 5 16 o 9.80351 8 o10.19649 9.91610io 13 10.08390 10,I11259 5 9.88741 30 31 43 52 i6 8 8o366 8 19634 91636 13 o8364 11270 5 88730 29 32 43 44 i6 16 80382 8 19618 91662 14 o8338 11280 6 88720 28 33 43 36 16 24 80397 8 19603 91688 14 08312 11291 6 88709 27 34 43 28 16 32 80412 9 19588 91713 15 08287 113o01 6 88699 26 35 6 43 20 5 16 4o 9.80428 9 10.19572 9.91739 15 10.08261 10.11312 6 9.88688 25 36 43 I1 i6 48/ 8o443 9 19557 91765 15 08235 11322 6 88678 24 37 43 4 16 56 80458 9 19542 91791 i6 08209 11332 6 88668 23 38 42 56 17 4 80473 I1o 19527 91816 16 o8i84 11343 7 88657 22 39 42 48 17 12 80489 io 19511 9184a2 17 08158 11353 7 88647 21 40 6 42 40 5 17 20 9.80504 1io 10.19496 9.91868 17 10.08132 1o.11364 7 9.88636 20 41 42 32 17 28 80519 Io 19481 91893 I8 o8107 11374 7 88626 19 42 42 24 17 36 8o534 ii 19466 91919 I8 o8o8i 1I385 7 886I5 i8 43 42 i6 17 44 8o55o II 19450 91945 8 o08055 11395 7 886o5 17 44 42 8 17 52 8o565 II 19435 91971 19 08029 1I4o6 8 88594 i6 45 6 42 o 9.80580 12 10.19420 9.91996 19 10.08oo004 o.1416 8 9.88584 15 46 4i 52 18 8 80595 12 19405 92022 20 07978 11427 8 88573 14 47 4, 44 18 16 8o6io I2 19390 92048 20 07952 11437 8 88563 13 48 41 36 18 24 80625 12 19375 92073 2I 07927 ii448 8 88552 12 49 41 28 i8 32 80641 13 19359 92099 21 07901 11458 9 88542 [II 50 641 20 5 i8 4o 9.80656 13 10.19344 9.92125 21 10.07875 10.11469 9 9.88531 Io 51 41 12 18 48 48 80671 13 19329 92150 22 07850 11479 9 88521 9 52 41 4 18 56 8o686 13 19314 92176 22 07824 11490 9 88510o 8 53 4o 56 19 4 80701 14 19299 92202 23 07798. i1501 9 88499 7 54 40 48 19 12 80716 i4 19284 92227 23 07773 ii5iI 9 88489 6 55 6 4o 40 5 19 20 9.80731 I4 10o.9269 9.92253 24 10.077471Z. 11522 1o.88478 5 56 4o 32 19 28 80746 14 19254 92279 24 07721 11532 io 88468 4 57 40 24 19 36 80762 15 19238 92304 24 07696 1.1543 Io 88457 3 58 4o i6 19 44 80777 15 19223 92330 25 07670 11553 io 88447 2 59 4o 8 19 52 80792 i5 19208 92356 25 07644 ii564 Io 88436 I 60 4o 0 20 0 80807 15 19193 92381 26 07619 11575 Io 88425 0 I Hour pM. Hour A.M.I Cosine. Diff. Secant. Cotangent!Diff. Tangent. Cosecant. DiffL. Sine. 129" A A B B C C 50T Seconds of time... 1 2 3S 4s 5s 6s 7s (A 2 4 6 8 10 12 13 Prop. parts of cols. B 33 i6 19 23 C 3 4 5 8 9 TABLE XXVII. [Page 25 S'. Log. Sines, Tangents, and Secants. G. 400 A A B B C C 1390 M Hour A.M. Ilour P.M. Sine. Diff. Cosecant. Tangent. Diff.iCotangent Secant.: Diff. Cosine. MI o 6 40 o 5 20 o 9.80807 o 10o.9193 9.92381 0 10.07619 10.11575 0 9.88425 60 I 39 52 20 8 80822 0 1I9178 92407 0 07593 11585 0 88415 59 2 39 44 20 i6 80837 0 19r63 92433 I 07567 11596 0 88404 58 3 39 36 20 24 80852 I 19148 92458 I 07542 i16o6 I 88394 57 4 39 28 20 32 80867 I 19133 92484 2 07516 11617 I 88383 56 56 39 20, 520 4o 9.80882 10o.19118 9.92510 2 10.07490 10.11628 I 9.88372 55 6 39 12j 2o 48 80897 I 19ro3 92535 3 07465 11638 i 88362 54 7 39 4 20 56 80912 2 19088 92561 3 07439 11649 i 88351 53 8 38 561 2I 4 80927 2 19073 92587 3 07413 I166o I 88340o 52 9 38 48 2 1 12 80942 2 I 19058 92612 4 07388 11670 2 883305i 10 6 38 40 5 21 20 9.80957 2 10.19043 9.92638 4 10.07362 o0.11681 2 9.88319 50 ii 38 32 21 28 80972 3 19028 92663 5 07337 11692 2 88308 49 12 38 24 21 36 80987 3 1901o3 92689 5 07311 11702 2 88298 48 13 38 16 21 44 81002 3 18998 92715 6 07285 11713 2 88287 4 14 38 8 21 52 81017 3 18983 92740 6 07260 11724 3 88276 46 15 6 38 0 5 22 0 9.81032 4 10.18968 9.92766 10o.07234 10.11734 3 9.88266 45 16 37 52 22a 8 81047 4 18953 92792 7 07208 11745 3 88255 44 17 37 44 22 16 81o6i 4 18939 92817 7 07183 11756 3 88244 43 i8 37 36 22 24 81076 4 18924 92843 8 07157 11766 3 88234 42 19 37 28 22 32 81091 5 18909 92868 8 07132 11777 3 88223 41 20 6 37 20 5 22 40 9.81106 5o10.18894 9.92894 9 10.07106 10.11788 4 9.88212 4o 21 37 12 22 48 81121 5 18879 92920 9 07080 11799 4 88201 39 22 37 4 22 56 81136 5 18864 92945 9 07055 11809 4 88191 38 23 36 56 23 4 81151 6 18849 92971 10 07029 11820 4 88180 37 24 36 48 23 12 81166 6 i8834 92996 io 07004 11831 4 88169 36 25 6 36 4o 5 23 20 9.8118o 6z10.18820 9.93022 io0.0o6978 10.11842 4 9.88158 35 26 36 32 23 28 81195 6 188o5 93048 i 06952 11852 5 88148 34 27 36 24 23 36 81210 7 18790 93073 12 06927 11863 5 88137 33 28 36 16 23 44 81225 7 18775 93099 12 069o01 11874 5 88126 32 29 36 8 23 52 81240 7 18760 93124 12 06876 11885 5 88115 31 30o 6 36 o 5 24 o 9.81254 7 10.18746 9.93150 13 io.o685o 10.11895 5 9.88105 30 31 35 52 24 8 81269 8 18731 93175 13 06825 11906 6 88094 29 32 35 44 24 i6 81284 8 18716 93201 14 06799 11917 6 88083 28 33 35 36 24 24 81299 8 18701 93227 14 66773 11928 6 88072 27 34 35 28 24 32 8i314 8 18686 93252 14 06748 11939 6 88o61 26 35 6 35 20 5 24 4o 9.81328 9 10.18672 9.93278 15 0.06722 10.11949 6 9.88051 25 36 35 12 24 48 8i343 9 18657 93303 i5 06697 11960 6 88040 24 37 35 4 24 56 81358 9 18642 93329 16 06671 11971 7 88029 23 38 34 56 25 4 81372 9 18628 93354 16 06646 11982 7 88o018 22 39 34 48 25 12 81387 10o 8613 93380 17 06620 11993 7 88007 21 40 6 34 4o 5 25 20 9.81402 io 10.18598 9.93406 17 10o.o6594 10.1200oo 4 7 9.87996 20 41 34 32 25 28 81417 io i8583 93431 17 06569 12015 7 87985 19 42 34 24 25 36 81431 io 18569 93457 18 06543 12025 8 87975 18 43 34 16 25 44 81446 iI i8554 93482 18 o6518 12036 8 87964 17 44 34 8 25 52 8146i Ii 18539 93508 19 06492 12047 8 87953 16 45 6 34 o 5 26 o 9.81475 II 10.18525 9.93533 19 10.06467 10.12058 8 9.87942 15 46 33 52 26 8 849o0 ii i85ro 93559 20 06441 12069 8 87931 14 47 33 44 26 i6 815o5 12 18495 93584 20 06416 12080 8 87920 13 48 33 36 26 24 81519 12 1848i 93610 20 06390 12091 9 87909 12 49 33 28 26 32 81534 12 18466 93636 21 06364 12102 9 87898 Ii 50 6 33 20 5 26 40 9.81549 12 10.18451 9.93661 21 0o.o6339 10.12113 9 9.87887 10 51 33 12 26 48 81563 13 18437 93687 22 06313 12123 9 87877 9 52 33 4 26 56 81578 13 18422 93712 22 06288 12134 9 87866 8 53 32 56 27 4 8r592 13 18408 93738 23 06262 12145i io 87855 7 54 32 48 27 12 81607 13 18393 93763 23 06237 1256 10o 87844 6 55 6 32 40 5 27 20 9.81622 14 10.18378 9.93789 23 o0.06211 10.12167 io 9.87833 5 56 32 32 27 28 81636 14 i8364 93814 24 o6i86 12178 Io 87822 4 57 32 24 27 36 81651 14 18349 93840 24 06160 12189 io 87811 3 58 32 16 27 44 81665 14 18335 93865 25 o6r35 12200 10 87800 2 59 32 8 27 52 8i68o 15 18320 93891 25 o6109o 1221i1 z 87789 I 60 32 0 28 0 81694 5 i83o06 93916 26 06084 12222 11 87778 0 II Hour P.Mr. HourA.ll. Cosine. Diff. Secant. Cotangent DiffJ Tangent. Cosecant. Diff. Sine. M 1300 A A B B C 0C 49' Seconds of time...... I' 2 3s 4! 5" 6~ 7" 5 C 3 3 4 5 7 8 ^~~~~~~~'{4I5{6{7 Page 226] TABLE XXVII. Log. Sines, Tangents, and Secants. G 410 A A B B C C 1380 M Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff.lCotangent Secant. Diff. Cosine. M o 6 32 o 5 28 o 9.81694 om10.18306 9.93916 o 1o.o6o84 10.12222 0 9.87778 6Q i 31 52 28 8 81709 o 18291 93942 0 o6o58 12233 o 87767 59 2 31 44 28 16 81723 o 18277 93967 I o6o33 12244 o 87756 58 3 31 36 28 24 81738 I 18262 93993 i 06007 12255 i 87745 57 4 31 28 28 32 81752 1 18248 94018 2 o5982 12266 I 87734 56 5 6 31 20 528 4o 9.81767 i10.18233 9.94044 2 10.05956 10.12277 I 9.-87723 55 6 31 12 28 48 81781 i 18219 94069 3 05931 12288 I 87712 54 7 3i 41 28 56 81796 2 18204 94095 3 05905 12299 I 87701 53 8 30 56 294 81810 2 18190 94120 3 05880 12310 i 87690 52 9 30 48 29 12 81825 2 18175 94146 4 o5854 12321 2 87679 51 o10 6 30 4 5 29 20 9.81839 2o10.18161 9-94171 4 10.05829 10.12332 2 9.87668 5o II 30 32 29 28 81854 3 i8i46 94197 5 o58o3 12343 2 87657 49 12 30 24 29 36 81868 3 18132 94222 5 05778 12354 2 87646 48 13 30 i6 29 44 81882 3 18118 94248 6 05752 12365 2 87635 47 14 30 8 29 52 81897 3 181o3 94273 6 05727 12376 3 87624 46 15 6 30 53o o 9.81911 4 10.18089 9.94299 6 10.05701 10.12387 3 9.87613 45 16 29 52 30 8 81926 4 18074 94324 7 05676 12399 3 87601 44 17 29 44 3o i6 81940 4 18o6o 94350 7 o565o 12410 3 87590 43 i8 29 36 30 24'81955 4 18045 94375 8 05625 12421 3 87579 42 19 29 28 3o 32 81969 5 18031 94401 8 05599 I2432 4 87568 41 20 629 20 5 30 40 9.81983 510.18017 9.94426 8 10o.05574 IO.12443 4 9.87557 40 21 29 12 3o 48 81998 5 18002 94452 9 o5548 12454 4 87546 39 22 29 4 30 56 82012 5 17988 94477 9 05523 12465 4 87535 38 23 28 56 31 4 82026 5 I7974 94503 Io0 05497 12476 4 87524 37 24 28 48 31 12 82o41 6 17959 94528 Io 05472 12487 4 87513 36 25 62840 5 31 20 9.82055 6 10.17945 9.94554 ii o0.05446 10.12499 5 9.87501 35 26 28 32 3, 28 82069 6 17931 94579 II o5421 12510 5 87490 34 27 28 24 31 36 82084 6 17916 94604 ii 05396 12521 5 87479 33 28 28 16 31 44 82098 7 17902 94630 12 05370 12532 5 87468 32 29 28 8 31 52 82112 7 17888 94655 12 o5345 12543 5 87457 31 30o6 28 o 532 o 9.82126 710.17874 9.94681 3 10.05319 10.12554 6 9.87446 3o 31 27 52 32 8 82141 7 17859 94706 13 05294 12566 6 87434 29 32 27 44 32 16 82155 8 17845 94732 14 05268 12577 6 87423 28 33 27 36 32 24 82169 8 17831 94757 14 05243 12588 6 87412 27 34 27 28 32 32 82184 8 17816 94783 14 05217 12599 6 87401 26 35 627 20 532 40 9.82198 8 10.17802 9.94808 15 10.05192 10.12610 7 9.87390 25 36 27 12 32 48 82212 9 17788 94834 15 o05166 12622 7 87378 24 37 27 4 32 56 82226 9 17774 94859 16 o5141 12633 7 87367 23 38 26 56 33 4 82240 9 17760 94884 16 o5116 12644 7 87356 22 39 26 48 33 12 82255 9 17745 94910 17 05090 12655 7 87345 21 40 626 4o 5 33 20 9.82269 iO 10.17731 9.94935 17 1o.o5065 10.12666 7 9.87334 20 41 26 32 33 28 82283 Io 17717 94961 17 05039 12678 8 87322 19 42 26 24 33 36 822971 o 17703 94986 18 0o5o4 12689 8 87311 18 43 26 16 33 44 82311 IO 17689 95012 18 04988 12700 8 87300 17 44 26 8 33 52 82326 Io 17674 95037 19 04963 12712 8 87288 16 45 6 26 0 5 34 0 9.82340 I 10.17660 9.95062 19 10.04938 10.12723 8 9.87277 15 46 25 52 348 82354 II 17646 95088 20 04912 12734 9 87266 14 47 25 44 34 16 82368 II 17632 95113 20 04887 12745 9 87255 13 48 25 36 34 24 82382 II 17618 95139 20 0486i 12757 9 87243 12 49 2-5 28 34 32 82396 12 17604 95164 21 o4836 12768' 9 87232 11 50 625 20 5 34 409.82410 12 10.17590 9.95190 21 10.04810 10.12779 9 9.87221 10 51 25 12 34 48 82424 12 17576 95215 22 04785 12791 10 87209 9 52 25 4 34 56 82439 12 17561 95240 22 04760 12802 Io 87198 8 53 24 56 35 4 82453 13 17547 95266 22 04734 12813 Io 87187 7 54 24 48 35 12 82467 i3 17533 95291 23 04709 12825 io 87175 6 55 624 40 535 2 9.82481 3 10. I7519 9.95317 23 10o.o4683 10.12836 O 19.87164 5 56 24 32 35 28 82495 i3 17505 95342 24 04658 12847 10o 87153 4 57 24 24 35 36 825095 14 17491 95368 24 04632 128F9 ii 87141 3 58 24 16 35 44 82523 14 17477 95393 25 04607 12870 ii 87130 2 59 24 8 35 52 82537 14 17463 954i8 25 04582 12881 ii 87119 I 60 24 0 36 o 82551 14 17449 95444 25 04556 12893 ii 87107 0 M Hour P.e. Hour A.M. Cosine. Diff. Secant. Cotangent Diff.Tang-ent. Cosecant. Diff. Sine. M 1310 A A B B C C 48~ Seconds of time...... I" 2 35 4s 5' 68 7' A 2 14 5 7 9 II 12 Prop. parts f cols. B 3 6 o10 3 16 19 22 (C 0 3 4 6 7 8 Io TABLE XXVII. LPage 27 S'. Log. Sines, Tangents, and Secants, G'. 420 A A B B C C 137~ mHou PA. IHour pi. Sine. Diff.Cosecant.. Tangent. Diff. Cotangent Secant. DIff. Cosine. M -6 24 0 5 36 o 9.82551 o IO-.7449 9.95444 0 1o.o4556 10.12893 9.87107 60 1 23 52 36 8 82565 o 17435 95469 0 o4531 12904 o 87096 59 2 23 441 36 i6 82579 o I7421 95495 i 04505 12915 0 87085 58 3 23 36 36 24 82593' 17407 95520 i 4480 12927 I 87073 57 4 23 28! 36 32 82607 I 17393 95545 2 4455 12938 I 87062 56 5 6 23 20o5 36 4o 9.82621 i 10.17379 9.95571 2 o.o4429 o.I2950 i 9.87050o 55 6 23 12 36 48 82635 i 17365 95596 3 o44o4 I2961 i 87039 54 7 23 4 36 56 82649 2 17351 95622 3 04378 12972 I 87028 53 8 22 56 37 4 82663 2 17337 95647 3 04353 12984 2 87016152 9 22 48 37 12 82677 2 17323 95672 4 04328 12995 2 87005 51 10 6 22 40 5 37 20 9.82691 2 10.17309 9.95698 4 10o.4302 10.13007 2 9.86993 o 11 22 32 37 28 82705 3 17295 95723 5 04277 13018 2 86982 49 12 2.2 24 37 36 82719 3 17281 95748 5 04252 I3o3o 2: 86970 48 13 22 16 37 44 82733 3 17267 95774 5 04226 i304i 3'86959 47 144 22 8 37 52 82747 3 17253 95799 6 04201 13053 3 86947 46 15 6 22 0o 5 38 0o 9.82761 3 10.17239 9.95825 6 10.04175 io.i3o.13 064 3 9.86936 45 6 21 52 38 8 82775 4 17225 95850 7 o4i50 13076 3 86924 44 17 21 44 38 i6 82788 4 17212 95875 7 04125 13087 3 86913 43 18 21 36 38 24 82802 4 17198 95901 8 04099 13098 3 86902 42 19 21 28 38 32 82816 4 17184 95926 8 04074 1311o 4 86890 41 20 6 21 20 5 38 o 9.82830 5 10o.17170 9.95952 8 io.o4048 10.13121 4 9.86879 40' 21 21 12 38 48 82844 5 17156 95977 9 34023 13133 4 86867 39 22 21 4 38 56 82858 5 17142 96002 9 03998 13145 4 86855 38 23 20 56 39 4 82872 5 17128 96028 io 03972 13156 4 86844 37 24 20 48 39 12 82885 6 17115 96053 io 03947 13168 5 86832 36 25 620 40 5 39 20 9.82899 6 10.17101 9.96078 ii 10.03922 10o.3179 5 9.86821 35 26 20 32 39 28 82913 6 17087 96104 ii 03896 i3191 5 86809 34 27 20 24 39 36 82927 6 17073 96129 ii 03871 13202 5 86798 3\3 28 20 i6 39 44 82941 6 17059 96155 12 o3845 13214 5 86786 32 29 20 8 39 52 82955 7 17045 96180 12 03820 13225 6 86775 31 30 6 20 0 5 4o 0 9.82968 7 10.17032 9.96205 i3 10.03795 10.13237 6 9.86763 3o 31 19 52 40 8 82982 7 I17018 96231 i3 03769 13248 6 86752 29 32 19 44 4o i6 82996 7 17004 962561 4 03744 13260 6 86740 28 33 19 36 40 24 8360o 8 16990 96281 14 03719 13272 6 86728 27 34 19 28 4o 32 83023 8 16977. 96307 T4 03693 13283 7 86717 26 35 619 20 5 40 40 9.83037 8 10.16963 9.96332 15 io.o3668 10.13295 7 9.86705 25 36 19 12 40 48 83051 8 16949 96357 15 03643 1i3306 7 86694 24 37 19 4 40 56 83o65 8 16935 96383 16 03617 13318 7 86682 23 38 i8 56 41 4 83078 9 16922 96408 16 03592 i3330 7 86670 22 39 i8 48 41 12 83092 9 16908 96433 16 03567 i334i 8 86659 21 40 6 18 40o541 20 9.83106 9I10.16894 9.9645917 1ioo354r 1i0.13353 8 9.86647 20 41 18 32 41 28 83120 9 1688o 96484 17 o3516 i3365 8 86635 19 42 18 24 41 36 83i33] Io 16867 96510 18 03490 13376 8 86624 i8 43 18 16 41 44 83147 10o i6853 96535 i8 o3465 r3388 8 86612 17 44 18 8 41 52 83i6i Io 16839 96560 19 o344o0 34oo 8 866oo00 16 45 6 i8 0 5 42 o 9.83174 I 10.16826 9.9658619 io.o34r4 1o.i341' 9 9.86589 15 46 17 52 42 8 83i88 II 16812 96611 19 03389 13423 9 86577 14 47'7 44 42 16 83202 ii 16798 96636 20 o3364 13435 9 86565 13 48 17 36 42 24 83215 ii 16785 96662 20 03338 13446 9 86554 12 49 17 28 42 32 83229 II 16771 96687 21 o33i3 i3458 9 86542 II 50 6 17 20 5 42 40 9.83242 II 10o.6758 9.96712 21 1o.03288 io.13470 io 9.8653o io s5 17 12 42 48 83256 12 16744 96738 22 03262 13482 10o 86518 9 52 17 4 42 56 83270 12 16730 96763 22' 03237 13493 10 86507 8 53 i6 56 43 4 83283 12 16717 96788 22 o3212 135o051 o 86495 7 54 i6 48 43 12 83297 12 16703 96814 23 0o386 13517 10 86483 6 55 6 16 4o'5 43 20 9.83310o 131o.1669o 9.9683923 o10.03i6 Io.1352811 i9.86472 51 56 i6 32 43 28 83324 13 16676 96864 24 03i36 i3540 ii 86460 41 57 16 24 43 36 83338 13 16662 96890 24 03110' 235512 i 86448 3 58 16 16 43 44 83351 13 16649 96915 25 03085 13564 IT 86436 2 59 i6 8 43 52 83365 14 16635 96940 25 o30;60 13575 ii 86425 1 6o 160 440 83378 14 16622 96966 25 03034 53587 12 864i3 r MI Hour Pni. Hour A.. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Difli. Sine. - IM[ 132~ A A B B C C 47 Seconds of time..... 1. I 2 3s 4s 5S 6" 78 A 2 3 5 7 9 10 12 Prop. parts of cobl. B 3 6 1 o 3 i6 19 22 C 3 4 6 7 9 10 Page T28] TABLE XXVII.'Si. Log. Sines, Tangents, and Secants. G'. 430 A A B B C 2(' 1360 M Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cone. M o 6 16 0-5 44 o 9.83378 o 1o.16622 9.969661 o io.o3o34 10.13587 o 9.86413 6o i5 52 44 8 83392J o 166o8 96991 oj 03009 13599 o 86o40 59 2 15.44 44 i6 83405 o 16595 97016 1 02984 i36i'i o 86389 58 3 15 36 44 24 83419 i i658i 97042 1 02958 13623 i 86377 57 4 15 28i 44 32 83432 1 i6568 97067 2 02933 i3634 i 86366 56 5 6 15 20 5 44 40 9.83446 i 10.16554 9.97092 2 10.02908 i.I3646 i 9.86354 55 6 5 152 44 48 834591 i654 97118 3 02882 13658 i 86342 54 7 i5 4 44 56 83473. 2 16527 97143 3 02857 13670 i 8633o 53 8 14 56 45 4 83486 2 16514 97168 3 02832 13682 2 86318 52 9 14 48 45 12 83500 2 165oo00 97193 4 02807 3694 2 863o6 5r io 6 i4 4o 5 45 20 9.83513 2 10.16487 9.97219 4 10.02781 10.13705 2 9.86295 5 11 14 32 45 28 83527 2 16473 97244 5 02756:3717 2 86283 49 12 14 24 45 36 8354o 3 i646o 97269 5 02731 13729 2 86271 48 13 i4 16 45 44 83554 3 i6446 97295 5 02705 13741 3 86259 47 14 i4 8 45 52 83567 3 i6433 97320 6 02680 13753 3 86247 46 15 6 14 0 5 46 o 9.83581 3 10.16419 9-97345 6 10.02655 10.13765 3 9.86235 45 i6 13 52 46 8 83594 4 i64o6 97371 7 02629 13777 3 86223 44 17 13 44 46 16 836o8 4 16392 97396 7 02604 13789 3 86211 43 I8 13 36 46 24 83621 4 16379 97421 8 02579 i38oo00 4 86200 42 19 13 28 46 32 83634 4 i6366 97447 8 02553 13812 4 86i88 4r 20 6 13 20 5 46 4o 9.83648 4 10.16352 9.97472 8 10.02528 10.13824 4 9.86176 40 21 13 12 46 48 8366i 5 16339 97497 9 02503 13836 4 86r64 39 22 i3 4 46 56 83674 5 16326 97523 9 02477 i3848 4 86152 38 23'1 2 56 4 74 83688 5 16312 97548 io 02452 i386o 5 (86i40 37 24 12 48 47 12 83701 5 16299 97573 io 02427 13872 5 86128 36 25 6 12 4o 5 47 20 9.83715 6 10.16285 9.97598 ii 10.02402 io.13884 5 9.86116 35 26 12 32 47 28 83728 -6 16272 97624 Ii 02376 13896 5 86o04 34 27 12 24 47 36 83741 6 16259 97649 Ii 02351 13908 5 86092 33 28 12 16 47 444 83755 6 16245 97674 12 02326 13920 6 86o8o 32 29 12 8 47 52 83768 6 16232 97700 12 02300 13932 6 86o68 31 30 6 12 0 5 48 o 9.83781 7 10.16219 9.97725 13 10.02275 o10.13944 6 9.86056 3 31 I 52 48 8 83795 71 6205 97750 13 02250 13956 6 86044 29 32 11 44 48 16 838o8 7 16192 97776 13 02224 13968 6 86032 28 33 ir 36 48 24 83821 7 16179 97801 14 02199 13980 7 86020 27 34 r 1128 48 32 83834 8 16166 97826i 14 02174 13992 7 86008 26 35 6 ii 20 5 48 40o 9.83848 8 10.16152 9.97851 i5 10.02149 10.14004 7 9.85996 25 36 11 12 48 48 8386i 8 16139 97877 15 02123 i4oi6 7 85984 24 37 11 4 48 56 83874 8 16126 97902 i6 02o98 14028 7 85972 23 38 10o 56 49 4 83887 8 16113 97927 i6 02073 14o4o 8 85960 22 39 io 48 49 12 83901 9 16099 97953 16 02047 14052 8 85948 21 40o 6,0 40o 5 49 20 9.83914 9 10.16086 997978 17 10.02022 io.i4o64 8 9.85936 20 41 10 32 49 28 83927 9 16073 98003 17 01997 14076 8 85924 19 42 io 24 49 36 83940 9 i6060. 98029 18 01971 i4088 8 85912 18 43 io i6 49 44 83954 Io I6046 98054 8 o01946 141oo 9 859oo00 17 44 io 8 49 52 83967 JO 16033 98079 19 01921 14112 9 85888 i6 45 6 0 5 55o0 o9.83980 io 10.16020 9.98104 19 10.01896 10.14124 9 9.85876 15 46 9 52 50 8 83993 io 16007 98130 19 01870 14136 9 85864 14 47 9 44 5o 16 84oo6 io 15994 98155 20 oi845 14,49 9 85851 13t 48 9 36 50 24 840oo ii 15980 98180 20 01820 i4i6i io 85839 12 49 9 28 50o 32 84o33 ii 15967 98206 21 01794 14173 io 85827 11 50 6 9 20 5 50 40 9.84046 ii i0.15954 9.98231 21 10.01769 io.14185 io 9.85815 J1 51 9 12 50 48 8459 IIJ 15941 98256 22 01744 14197 10 858o3 9 52 9 4 50 56 84072 12 15928 98281 22 01719 14209 i 85791 8 53 8 56 51 4 84o85 12 15915 98307 22 o0693 14221 11 85779 7 54 8 48 5i 12 84098 12 159o2 98332 23 01668 i4234 ii 85766 6 55 6 8 4o 5 51 20 9.84112 12 io.i5888 9.98357 23 1o.oi643 10.14246 ii 9.85754 5i 56 8 32 5i 28 84125 12 15875 98383 24 01617 14258 ii 85742 4 57 8 24 5i 36 84i38 13 15862 984o08 24 01592 14270 ii 8573o 3 58 8 16 51 44 84i5r 13 15849 984331 24 01567 14282 12 85718 2 59 8 8 50 52 84164 13 15836 98485 25 0or5r 14294 12 85706 1 S6 80 520 84177 13 i5823 98484/ 25 01546 14307/ 12 85693 o M I-our P.M. lHourA.M. Cosine. Diff. Secant. Cotangent ff. Tangent. Cosecant Diff. Sine. 0A A A B B C C 460 Seconds of time...... 1' 2" 3' 4' 5s 6 7" A 2 3 5 7 8 10 12 Prop. parts of cols. _B 3 6 9_3_ _ 6_ 9 22" (C 2 3 5 6 8 9 ii i~~~~~~~~~~~~~~~~~~~~~~ TABLE XXVII. [Vage 229 S'. Log. Sines, Tangents, and Secants. G'. 440 A A B B C C 1350 [M Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff.Cotangent Secant. Diff. Cosine. M o 6 8 0 5 52 9.84177 10.15823 9.98484 o0 o.oi56 10.1o4307 19.85693 60 I 7 52 52 8 8419o 0 i58o10 98509 o 01491 14319 o 85681 59 2 7 44 52 16 84203 0 15797 98534 i 0o466 14331 o 85669 58 3 7 36 52 24 84216 i 15784 98560 i 01440 4343 i1 85657 57 4 7 28 52 32 84229 I 15771 98585 2 014I5 i4355 i ~5645 56 5 6 7- 20552 40 9.84242 1i 10.15758 9.98610io 2 10.01390 o10.14368 i 9.85632 55 6 7 12 52 48 84255 i 15745 98635 3 oi365 1438o0 i 85620 54 7 7 4 52 56 84269 2 z5731 98661 3 01339 4392 I 856o8 53 8 6 56 53 4 84282 2 15718 98686 3 o01314 14404 2 85596 52 9 6 48 53 I2 84295 2 15705 98711 4 o01289 4417 2 85583 51 io 6 6 4o 5 53 20 9.84308 2 10.15692 9.98737 4 10.01263 10.14429 2 9.85571 50 ii 6 32 53 28 84321 a2 175679 98762 5 01238 1'4441 2 85559 49 12 6 24 53 36 84334 3 15666 98787 5 01213 i4453 2 85547 48 13 6 16 53 44 84347 3 15653 98812 5 0o1188 14466 3 85534 47 14 6 8 53 52 84360 3 15640 98838 6 01162 14478 3 85522 46 156 6 0 5 54 0 9.84373 3 3 10.15627 9.98863 6 o10.011oii37 1io.1449o 3 9.8551o 45 16 5 52 54 8 84385 3 i56i5 98888 7 01112 i45o3 3 85497 44 17 5 44 54 16 84398 4 15602 98913 7 01087 14515 4 85485 43 18 5 36 54 24 8441, 4 15589 98939 8 o0o61 14527 4 85473 42 19 5 28 54 32 84424 4 15576 98964 8 o01036 14540 4 85460 41 20 6 5 20 5 54 40 9.84437 410o.15563 9.98989 8 io.oloii 10.14552 4 9-85448 40 21 5 12 54 48 84450 5 15550 99015 91 00985 14564 4 85436 39 22 5 4 54 56 84463 5 15537 99040 9 00960 14577 5 85423 38 23 4 56 55 4 84476 5 I5524 99065 io 00935 14589 5 854 1137 24 4 48 55 12 84489 5 15511 99090 io 00910 i46oi 5 85399 36 25 6 4 45 55209.84502 5 10.15498 9.996 10o.oo00884 o.i46i4 5 9.85386 35 26 4 32 55 28 84515 6 15485 99141 11 00859 14626 5 85374 34 27 4 24 55 36 84528 6 15472 99166 ii oo834 14639 6 8536i 33 28 4 16 55 44 84540 6 1546o 99191 12 00809 i465i 6 85349 32 29 4 8 55 52 84553 6 15447 99217 12 00783 i4663 6 85337 31 30 6 4 0 5 56 0 9.84566 6 10.15434 9.99242 13 10.00758 10.14676 6 9.853324 30 31 3 52 56 8 84579 7 15421 99267 I3 00733 14688 6 85312 29 32 3 44 56 i6 84592 7 i54o8 99293 13 00707 14701 7 85299 28 33 3 36 56 24 846o5 7 75395 99318 14 oo682 14713 7 85287 27 34 3 28 56 32 846i8 7 15382 99343 i4 00657 14726 7 85274 26 35 6 3 20 5 56 4o 9.84630 8 10o.5370 9.99368 5 o10.00oo632 10.14738 7 9.85262 25 36 3 12 56 48 84643 8 15357 99394 15 00606 14750 7 85250 24 37 3 4 56 56 84656 8 15344 99419 i6 oo581 14763 8 85237 23 38 2 56 57 4 84669 8 i5331 99444 16 00556 14775 8 85225 22 39 2 48 57 12 84682 8 i53i8 99469 i6 oo0053 14788 8 85212 21 40 6 2 40 5 57 20 9.84694 9 i0o.5306 9.99495 17 10o.oo00505 10o.i4800 8 9.85200 20 41 2 32 57 28 84707 9 15293 99520 17 oo0048o 480 1 8 85187 19 42 2 24 57 36 84720 9 15280 99545 i8 oo455 14825 9 85175 18 43 2 16 57 44 84733 9 15267 99570 i8 oo43o 14838 9 85162 17 44 2 8 57 52 84745 9 15255 99596 19 00404 i485o0 9 85150o 6 45 6 2 o 5 58 0 9.84758 -o 10.15242 9.99621 9 10o.00379 io.i4863 9 9.85137 15 46 1 952 58 8 84771 io I5229 99646 19 oo354 14875i o0 851 25 14 47 1 44 58 16 847841 o10 15216 99672 20 00328 14888 io 85112 13 48 1 36. 58 24 84796 o 5204 9969720 o00303 14900 io 85oo0012 49 I 28 58 32 84809 ii 15191 99722 21 00278 14913 io 85087 II 0 20 5 58 40 9.84822 I 10.15178 9.99747 21 io.00253 o0.14926 io 9.85074 io 51 112 58 48 84835 ii 15165 99773 21 00227 14938 ii 85o62 9 59, 4 58 56 84847 ii 15153 99798 22 00202 14951 ii 85049 8 53 o 56 59 4 84860o ii 15i4o 99823 22 00177 14963 ii 85037 7 S4 0 48 59 12 84873 12 15127 99848 23 oo00152 14976 ii 85024 6 556 o 40o 5-59 20 9.84885 12 io0.15115 9.99874 23 10o.oo00126 10.49881 ii 9.85012 5 56 o 32 59 28 84898 12 i5102 99899 24 ooIoi 15001 12 84999 4 57 o 24 59 36 8491l 12 15089 99924 24 00076 i5oi4 12 84986 3 6o 6 59 44 84923 12 15077 99949 24 00051 15026 12 84974 2 S9 o 8 59 52 84936 13 i5064 99975 25 00025 15039 12 84961 1 o 0 6 o 0 84949 13 i505i 10.00000 25 00000 15051 12 84949 0 M Hour P.m.I1Hour A.M. Cosine. Diff. Secant. lCotangent Diff. Tangent. Cosecant. Diff. Sine. IMi 1340 A A B B C C 450 Seconds of time...... I' 2 3 4s_ 5' 6" 7' A 2 3 5 6 8 io ii Prop. parts of cols. B 3 6 9 3 6 19 22 9C 2a 3 5 6 8 9 230 TABLE XXVIII. A TABLE OF RHUMBS, SHOWING THE POINTS AND QUARTER-POINTS, AND THE DEGREES, MINUTES, AND SECONDS, CORRESPONDING TO ANY COURSE. NORTH. Pts. qr. o / Pts. qr. SOUTH. 0 1 2 48 45 0 1 0 2 5 37 30 0 2 0 3 8 26 15 0 3 N. byE. N. byW. 1 0 11 15 0 1 0 S. byE. S. by W. 1 14 345 11 12 16 52 30 1 2 1 3 19 41 15 1 3 N.N.E. N.N.W.. 2 0 22 30 0 2 0 S.S.E. 8.S.W. 2 1 25 18 45 2 1 2 2 28 7 30 2 2 2 3 30 56 15 2 3 N.E.byN. N.W.by N. 3 0 33 45 0 3 0 S.E.byS, S.W, by S. 3 1 36 33 45 3 1 3 2 39 22 30 32 3 3 42 11 15 3 3 N.E. N.W. 4 0 45 0 0 4 0 S.E. S.W. 4 1 47 48 45 4 1 4 2 50 37 30 4 2 4 3 53 26 15 4 3 N.E.byE. N.W.byW. 5 0 56 15 0 5 0 S.E. by E. S.W. by W. 5 1 59 345 5 1 5 2 61 5230 52 5 3 64 41 15 5 3 E.N.E. W.N.W. 6 0 67 30 0 6 0 E.S.E. W.S.W. 6 1 70 18 45 6 1 6 273 7 30 6 2 6 3 75 56.15 6 3 E. by N. W.byN. 7 0 7845 0 7 0 E.byS. W.byS. 7 1 81 33 45 7 1 7 2 84 22 30 72 7 3 87 11 15 7 3 East. West. 8 090 0 0 8 0 East. West.. o, - TABLE XXI., WORKMAN'S TABLE, FOR CORRECTING THE MIDDLE LATITUDE 232 DIFFELCENCE OF LATITUDE. [Table xxIx. I.t. 30 40 5o 60 70 80 90 10o 110 0 0 / 0 1 01 o 0 0 0 1 o0 1 01 0 15 0 02 003 004 006 0 09 012 015 019 023 16 002 0 03 004 006 0 09 012 15 0 18 0 22 17 002 003 004 06 0 08 011 014 0 17 021 18 002 0 03 004 006 0 08 0 11 0 14 0 17 0 20 19 002 0 03 004 006 0 07 010 0 13 0 16,0 19 20 002 0 03 004 006 007 009 0 12 0 15 0 18 21 002 003 004 0 06 007 009 0 12 015 018 22 002 003 004 0 06 007 009 0 12 015 017 23 002 0 03 004 0 06 007 009 0 12 015 017 24 002 0 03 004 006 007 009 0 1 014 016 25 002 0 03 004 0 05 007 009 1 0 14 016 26 002 0 03 0 04 005 0 07 009 0 11 0 14 0 16 27 002 0 03 004 005 0 07 008 0 11 0 14 0 16 28 002 0 03 004 005 006 008 010 013 015 29 002 003 004 005 006 008 010 013 015 30 002 003 004 005 006 008 010 013 015 31 002 0 03 004 0 05 0 06 008 0 10 0 13 0 15 32 002 0 03 004 0 05 0 06.0 08 0 10 0 13 0 15 33 002 0 03 004 0 05 0 06 008 010 0 13 015 34 002 003 004 0 05 0 06 008 010'013 015 35 002 003 004 0 05 006 008 010 0 13 0 15 36 002 003 004 0 05 006 008 010 013 015 37 002 003 0 04 005 0 06 008 0 10 013 0 15 38 002 003 004 0 05 006 008 0 10 013 015 39 002 0 03 004 005 006 008 0 10 013 0 15 40 002 003 004 05 0 06 008 010 013 0 15 41 002 003 004 005 006 008 1 0 0 13 0 15 42 002 003 004 0 05 006 008 010 0 13 0 15 43 002 003 004 0 05 007 009 0 11 0 14 0 16 44 002 003 004 005 007 009 0 11 014 016 45 002 0 03 004 0 05 007 009 0 11 0 14 0 16 46 0 02 0 03 004 0 05 0 07 009 0 11 0 14 0 16 47 002 003 004 005 0 07 009 0 11 0 14 0 16 48 0 02 0 03 004 0 05 0 07 009 0 11 0 14 0 16 49 0 02 0 03 004 0 05 0 07 009 0 11 0 14 0 17 50 002 0 03 004 0 05 007 009 0 11 014 017 51 002 0 03 004 0 05 0 07 0 09 0 11 0 14 017 52 002 0 03 004 0 05 0 07 009 0 12 0 15 0 18 53 002 0 03 004 0 06 0 07 i 0 09 0 12 0 15 0 18 5 002 0 03 004 006 0 08 010 0 13 0 16 0 19 55 002 0 03 004 0 06 0 08 010 0 13 0 16 0 19 56 002 0 03 004 0 06 0 08 010 0 13 0 16 0 20 57 002 0 03 004 0 06 0 08 011 0 14 0 17 0 20 53 002 003 004 0 06 009 00 11 0 14 0 17 0 21 59 002 0 03 004 0 06 0 09 012 0 15 0 18 0 22 60 002 0 03 004 0 06 0 09 0'12 0 15 0 19 0 23 61 002 003 00 05 0 07 0 09 0 12 015 0 19 023 62 002 003 005 007 0 09 012 016 0 20 0 24 63 002 0 04 005 0 07 0 09 013 0 16 20 0 24 64 002 0 04 006 0 08 0 09 013 0 17 0 21 0 25 65 002 0 04 006 0 0 010 013 01 17 0 21 0 25 66 002 0 04 006 0 0 10 100 14 018 0 22 0 26 67 002 0 04 0 06 008 011 015 018 023 027 68 0 02 004 006 0 0 00 0 11 015 19 0 24 0 28 69 002 005 006 0 09 012 016 0 20 0 25 0 30 70 003 0 05 006 0 09 0 13 0 17 0 21 026 031 71 004 0 06 007 0 09 0 13 018 1 22 0 27 0 33 72 004 0 06 008 010 ) 14 019 023 29 0 035 Table xxix.] DIFFERENCE OF LATITUDE. 233 id 12 130 140 150 160 170 180 190 200 o0 o 0 O/ Oi 0 0/ 0/ 0 15 0 7 0 1 35 00 40 045 051 058 106 114 16 0 26 0 30 0 34 0 38 0 43 0 49 0 56 1 03 1 11 17 0 25 0 28 0 32 0 37 0 42 0 48 0 54 1 01 1 08 18 0 24 0 27 0 31 0 36 0 41 0 46 0 52 0 58 1 06 19 0 23 0 26 0 30 0 34 0 40 0 45 0 50 0 56 1 03 20 022 025 0 29 0 33 0 38 0 43 0 48 0 54 1 00 21 021 025 029 033 0 37 0 42 0 47 0 53 0 58 22 020 024 028 032 036 041 046 051 056 23 020 0 24 0 28 032 0 36 040 0 45 050 055 24 19 0 23 0 27 031 035 039 044 048 053 25 09 9 023 027 031 035 039 043 047 052 26 0 19 022 0 26 030 034 038 042 047 052 27 0 19 0 22 0 26 0 30 0 33 0 38 0 42 0 46 0 51 28 018 021 025 029 033 037 041 046 051 29 0 18 0 21 0 25 0 29 0 32 0 36 0 41 0 45 0 50 30 018 021 025, 028 032 036 041 045 050 31 018 021 025 028 0 32 0 36 0 41 045 050 32 0 18 0 21 025 0 28 0 31 036 041 045 050 33 0 18 0 21 0 24 0 27 0 31 0 35 0 40 0 44 0 49 34 0 18 021 024 0 27 31 0 35 040 044 049 35 018 021 024 0 27 0 31 035 040 0 44 049 36 0 18 0 21 0 24 0 27 0 31 0 35 0 40 0 44 0 49 37 0 18 0 21 0 24 0 27 0 31 0 35 0 40 0 44 0 49 38 0 18 0 21 0 24 0 27 0 31 0 36 0 40 0 45 0 50 39 0 18 0 21 0 25 0 28 0 32 0 36 0 41 0 45 0 50 40 018 022 025 0 28 032 0 36 041 0 45 050 41 018 0 22 025 0 28 032 0 37 041 045 050 42 018 022 0 26 029 033 037 042 046 051 43 019 023 026 030 034 038 042 046 051 44 0 19 023 027 030 034 0 38 043 047 052 45 019 023 27 031 0 35 0 39 043 047 052 46 0 19 023 0 27 0 31 03,5 0 39 044 048 053 47 0 20 0 23 027 031 035 0 40 044 049 054 48 0 20 023 0 27 0 31 0 351 0 40 0 45 0 50 0 55 49 0 21 0 24 0 28 032 036 041 045 0 51 057 50 021 024 028 032 036 0 41 046 0 52 058 51 0 21 0 24 0 28 0 32 0 37 0 42 0 47 0 53 0 59 52 0 22 0 25 0 29 0 33 0 37 0 42 0 48 0 54 1 00 53 0 22 025 0 29 033 038 043 049 055 101 54 0 23 0 26 030 0 34 0 39 044 0 50 056 102 55 0 23 0 26 0 0 0 35 0 40 0 45 0 51 0 57 1 03 56 024 0 27 0 31 0 36 041 046 052 058 104 57 0 24 0 28 0 32 0 37 042 048 054 1 00 106 58 0 25 0 29 0 33 0 38 0 44 0 50 0 55 1 02 1 08 59 0 26 0 0 0 34 0 39 0 45 0 51 0 57 104 1 10 60 0 27 0 31 0 35 0 40 0 46 0 52 0 59 1 06 1 13 61 0 27 0 31 0 36 0 41 0 47 0 54 1 01 1 08 1 15 62 0 28 0 32 0 37 0 42 0 49 0 56 1 03 1< 10 1 18 63 0 29 0 33 0 39 0 44 0 51 0 58 1 05 1 12 1 21 64 0 29 034 040 0 46 0 53 00 1 07 114 124 65 0 30 035 041 0 48 1055 1 02 1 09 1 17 1 27 66 031 0 37 0 43 050 0 58 1 05 112 1 21 131 67 0 33 0 38 0 45 0 53 1 00 1 07 1 16 1 25 1 35 68 0 34 0 40 0 48 0 55 1 02 1 10 1 19 1 30 1 39 69 1036 0 42 0 50 0 58 1 05 1 13 1 23 1 34 1 44 70 0 38 0 44 0 52 1 00 1 08 117 1 28 1 39 1 50 71 040 0 46 0 55 103 112 122 1 32 144 1 56 72 042 049 058 106 116 1 27 1 3 150 204 234 TABLE XXX. TABLE OF REFRACTIONS. Refr. Diff. Bff Diiff. Refr. Diff: Diff. Dif. App. Br. 30. for for for App. Br. 30. for for for Altitude. Th.T 500. h lt 1 B.5 I -1 Fal. Altitude. Th. 500. I1 Alt. + 1 B. o Fah. _.0 / 1. a i _ i ii._ 4 0 11 52 2.2 24.1 1.70 10 0 520.5 10.8.69 10 11 30 2.1 23.4 1.64 10 5.5 10.6.67 20 11 10.0 22.7 1.58 20 5 10.5 10.4.65 30 10 50 1.9 22.0 1.53 30 5.5 10.2.64 40 10 32 21.8 21..48 40 0.5 10.1.63 50 10 15 1.7 20.7 1.43 50 4 56 4 9.9.62 5 0 958 1.6 20.6 1.38 11 0 4 51.4 9.8.60 10 9 42 1.5 19.1 1.34 10 4 47.4 9.6.59 20 9 27 1.5 19.1 1.30 20 4 43.4 9.5.58 30 9 11 1.4 18.6 1.26 30 4 9.4 9.4.57 40 8 58 1.3 18.1 1.22 40 4 35.4 9.2.56 50 8 45 1.3 17.6 1.19 50 4 31.4 9.1.55 6 0 8 32 1.2 17.2 1.15 12 0 428.1.38 9.00.556 10 8 20 1.2' 16.8 1.11 10 4 24.4.37 8.86.548 20 8 9 1.1 164 1.09 20 4 20.8.36 8.74.541 30 7 58 1.1 16.0 1.06 30 4 17.3.35 8.63.533 40 7 47 1.0 15.7 1.03 40 1 13.9.33 8.51.524 50 7 37 1.0 15.3 1.00 50 4 10.7.32 8.41.517 7 0 7 27 1.0 15.0.98 13 0 4 7.5.31 8.30.509 10 7 17.9 14.6.95 10 4 4.4.31 8.20.503 20 7 8.9 14.3.93 20 41.4.30 8.10.496 30 6 59.8 14.1.91 30 3 5.4.30 8.00.490 40 651.8 13.8.89 40 3 55.5.29 7.89.482 50 643.8 13.5.87 50 3 2.6.29 7.79.476 8 0 6 35.7 1.3.3.85 14 0 13.9 3 7.76.469 10 628.7 13.1 *.83 10 3 7.1.28.61.464 20 621.7 12.8.82 20 4.4.27 7.52.458 30 6 14.7 12.6.80 30 341..26 7.43.453 40 6 7.7 12.3.79 40 3 39.2.2 7.4.448 50 6 0.6 12.1.77 50 3 3.7. 7..444 9 0 55 54.6 11.9.76 15 0 34.3.24 7.18.433 10 5 47.6 11..74 30 3 27.3.22 6.5.4220 5 41.6 11.5.73 16 0 3 20.6.21 6.73.411 30 5 36.6 11.3.71 30 3 14.4.20 6.51.399 40 5 30.5 11.1.71 17 0 3 8.5.19 6.31.386 50 5 25.5 11.0.70 30 3.9.8.18 6..374 18 0 2 57.6.17 5.98.362 __ ___ _l l 19 0 2 47.7.16 5.61.340 L_____ 235 TABLE XXX. TABLE OF REFRACTIONS. Refr. Diff. 3iff. - Di: lefr. Diff. Diff. Diff. App. Br. 30. for fr for App. Br. 30. for for for Altitude. Th. 500. 11 Al. + 1 B. -10 ah. Altitude. TO. 500. I Alt. + 1 B. -10 Fah. o0 / /1 ii i t / 0 1/ i l /11 1 0 2 38.7.15 5.31.322 55 40.8.025 1.36.082 21 2 30.5.13 5.04.305 56 39.3.025 1.31.079 22 2 23.2.12 4.7.9.290 57 37.8.025 1.26.076 23 2 16.5.11 4.57.276 58 36.4.024 1.22.073 24 2 10.1.10 4.35.264 59 35.0.024 1.17.070 25 2 4.2.0 4.16.252 60 33.6.023 1.12.067 26 1 58.8.09 3.97.241 6 1.3.022 1.08.065 27 1 53.8.08 3.81.230 62 31.0.022 1.04.062 28 1 49.1.08 3.65.219 63 29.7.021.99.060 29 1 44.7.07 3.50.209 64..021.9.057 30 1 40.5.07 3.36.201 65 27.2.020.91.055 31 1 36.6.06 3.23.193 66 25.9.020.87.052 32 1 33.0.06 3.11.186 67 24.7.020.83.050 33 1 29.5.06 2.99.179 68 23.5.020.79.047 34 1 26.1.05 2.88.173. 69 22.4.020.75.045 35 1 230.05 2.78.167 70 21.2.020.71.043 36 1 20.0.05 2.68.161 71 19.9.020.67.040 37 1 17.1.05 2.58.155 72 18.8.019.63.038 38 1 14.4.05 2.49.149 73 17.7.018.59.036 39 1 11.8.04 2.40.144 74 16.6.018.56.033 40 1 9.3.04 2.32.139 75 15.5.018.52.031 41 1 6.9.04 2.24.134 76 14.4.018.48.029 42 1 4.6.038 2.16.130 77 13.4.017.45.027 43 1 2.4.036 2.09.125 78 12.3.017.41.025 44 1 0.3.034 2.02.120 79 11.2.017.38.023 45 58.1.034 1.94.117 80 10.2.017.34.021 46 56.1.033 1.88.112 81 9.2.017.31.018 47 54.2.032 1.81.108 82 8.2.017.27.016 48 52.3.031 1.75.104 83 7.1.017.24.014 49 50.5.030 1.69.101 84 6.1.017.20.012 50 48.8.029 1.63.097 85 5.1.017.17.010 51 47.1.028 1.58.094 86 4.1.017.14.008 52 45.4.027 1.52.090 87 3.1.017.10.006 53 43.8.026 1.47.088 88 2.0.017.07.004 54 142.2.026 1.41.085 89 1.0.017.03.002 236 TABLE XXXL TABLE XXXII. TABLE XXXIII. w1 ), lft-he | lilMi.t (in tllhe Aulllrentation of I lcrzoi:., veirtic l semllliildin. O's semidiam. of' ~ or ~, oIn ucIHeighlit lip. j colut ol hKefiactiou. Alt. Aug. 1.. 08 At lh. of 00 0// l1t. O' 08' Alt. seitclidi. 5 1 3 12 4 2 D5 25" 10 3 419 566 19 15 4 1 97 14 20 6 8 1 1 25 7 6 Q2 9 99 30 8 I 10 8 35 9 9 253 11 7 40 10 1 32 12 6 45 11 1 13 5 50 12 1 ) 3414 4 55 1 3 19 319 "7 9 15 4 60 14 1 4, 168 3 70 15 1 9 41 I 19 43 7n 20 5 4 1790 22 4 3 23 4 36 24 4 42 26 452 28 5 5 30 5 15'353 5 9 40 6 4 TABLE XXXV. 45 6 27 70 81 4 &1 oriizont.l Pal.lax.. 80 834 5 6 58 60 90 9 6 ~ _____ ~ -. ____ _~54' 5 100 935 0 a (l a -~ ^ 03.0 0.0 0.0 0.0 0.0 fi 0.2 0.2 0.2 0.2 0.2 TABLE XXXIV, 16 0.8 0.8 0.9 0.9 0.9 I' S1 par in Alt. 20 1.3 1.3 1.4 1.4 1.5 90 ~70 1 2 9 ol 4 1.8 1.9 1.9 2.0 2.0 lt. p~r. 28 2.4 2.5 2.6 2.6 2.7 00 911 33 3.0 3.1 3.3 3.4 3.5 10 936 3.7 3.9 4.0 4.1 4.3 20 8 40 4.5 4.6 4.8 5.0 5.1 30 8 44 5,2 5.4 5.6 5.8 6.0 ~40 7 I YO48 ICIS6.0 6.2 6.3 6.6 6.8 ~50 6 ~ 52 6.7 7.0 7.2 7.4 7.6 ~~55 * 5 56 7.4 7.7 8.0 8.2 8.5 60 4 6 0 8.1 8.4 8.7 9.0 9.3 ~~65 4 164 8.7 9.1 9.4 9.7 10.0 ~70 3 i~68 9.3 9.6 10.0 10.3 10.6 7 7,9.8 10.1 10.4 10.8 11.2 80 2 76 10.2 10.6 10.9 11.3 11.7 ~85 1,184 10.7 11.1 11.5 11.9 12.0 190 0 90 1 G0.8 11.2 11.6 12.0jl2.4 237 TABLE XXXVI. Reduction to the Meridian. Argument n. the Hour Angle from the Meridian. S (m Im 2 m 13m1 41 5m m 7im 8i 9m O1m 11Im 121' 13m 14m 0 0 9 3* 86 159 6238 343 466 609 771 952 11552 137U0 1(08 1865 1 0 10 391 87 153 240) 345 468 612 774 955 1155 1374 1612 1870 2 0 10 3!9 88 1 "55 241 346 471 614 777 958 1159 1378 1617 1874 3 0 100 40 89 156 ( 243 318 4731 617 780 9(;i 1162 1382 1621 1879 4 01 1 41 90 { 157 244 350 475 619 782 964 1166 1386 1625 1883 5 ( l1 41 91 159 2'16 352 478 62-2 785 968 1169 1390 1629 1887 6 0 12 40 9! 1 60 241 354 4801 624 788 971 1173 1393 1633 1892 7 0 12 4'1 93 161 54! 351; 482 6(27 791 974 1176 1397 1637 1896 8 0 1 1 43 93 163 21 358 484 630 794 977 1180 1401 (1 1 1931 9 0 13 44 94 164 253 3610 487 632 797 981 1183 14(15 1646 1905 10 0 13 45 95 165 254 3621 489 635 800 984 1187 1409 16503 1910 11 0 13 45 96 167 256 364 4.91 637 803 987 11903 1413 1654 1914 12 o 14 4| 97 1368 2657 366( 493 6(40 8(06 9903 1194 1416 165S 191 13 0 14 47 98 1369 2535 3338 4!t(i 64:3 809 993 1197 143201 1(6 19231 14 1 14 47 99 171 2681 370 4398 645 811 997 1201 1424 166i7 1928 15 1 15 4.1 100 17]2 6 l_ 372 5 0' 648 814 1(000 1205 1428 1671 1932 16 1 15 4 10/2 1731 2i64 374 5033 650 817 10(03 1208 1432 1675 1937 17 1 16 50 103 175 266 376 5305 653 8201 1006 1212 1436 1679 1941. 18 1 16 50 104 176t 267 378 507 656 823 I1016 1215 1440 1683 194( 19 1 17 51 105 177 269 480 510 658 826 1013 1219 1444 1688 19503 20i 1 17 52 106 179 271 382 5123 661 809 G101 1222 1448 1696 1954 21 1 17 53 107 1803 272 384 514 664 832 10201 1226( 1451 1696i 1960 22 1 18 53 108 182 274 386( 517 6i66 835 1023 1231) 1455 1700 1964 23 1 18 54 19 109 183 276 388 519 669 838 1'26 1233 1459 1705 1968 4i 2 19 55 110 184 278 390 521 6726 841 1029 1237 1463 1709 1973] 51 2 19t 56) 11 1851 279{ 392 524 67-4 844 1033 1241 1467 1713 19781 26 216 260 561 112 1871 281 394 526G 677 847 1036 1244 1471 1717 1982] 27' 20 57 113 188 283 3396 528 6801 850 10391 1248 1475 1722 19871 28l i 20 58 114 1901 284 398 531 (682 853 1(043 1251 1479 1726 1992 29 2 21 59 116 191 286 400 533 685 856 1046f 1255 1483 1730 1997 30 3 21 59 117 193 288 402 535 6881 859 1049 1259 1487 1734 2001 31 3 2-2 601 118 1946 289 404 538 690 862Q 1053 12632 1491 1739 2005 32 3 22 61 119 1963 291 406 5403 693 865 11563 1266( 1495 1743 2010 33 3 23 62 120 197 293 408 5431 6396 868 1059 1270 1499 1747 2014 34 31 23 63 121 198 2095 410 5451 699 871 1062) 1273 15"31 1751 2019 35 31 24 641 126 26)00 297 412 547 701 874 1066( 1277 1507 1756( 2024 36 3 24 64 123 2016 299 415 550 704 877 1069 1281 1511 1760 2028i 37 4 25'65 124 2031 300 41715 5-21 707 880 1073 1284. 1515 1764 203:3 38 4 25 66 126 204 302 419 555 709 88:3 1076 1288 1519 1669 2038 39 4 26 67 127 206 304 421 551 712 886 1079 12926 1523 1773 20142 40 4 26 68 128 2071 3061 423 5593 715- 889 1083 12)95 1527 177-7 2047 41 4 27 68 129 209 307 425 5 562 718 892 1086 1298 15:31 1721 220521 42 5, 28 69 130 2101 3091 427 564 7201 896( 1(903 13(): 1535 1786 2056 43 51 28 701 131 2126 311 429/ 567 723 899 1093 1307 1539 17901 2061 44 51 29 71 133 213 313 43'. 569 726i 9302 1096 1310 15431 1795 2066 45 5 29 72 134 615 315 434 5721 729 9:05 1131u 1314 1547 1799 2070 46 63 30 73 135 216 316 436 541 732 908 1103 1318 1551 18041 2075 47 6 30 74 136 218. 31i 438 5771 734 911 1107 1321 1555 1808 50180 48 6 31 75 137 219 320 440 579 7371 914 1110[ 1:325 1559 1812 2084 49 6 31 75 139 221 322 442 582", 710 917.1114 1:329 156i3 1817 20189 50 7 32 76 140 222 324 44, 584 743 920 1117 1333 1567 1821 21094 51 7 33 77 -141 225 326 447 587 745 923 1120 13363 1571 18251 2099 52 7 33 78 1421 226 328 449 589 748 927 1124 1340( 1.575 18311 2103 53 7 34 79 144 227 32,9 4511 5;)2 751 930 1127 1344 1580 1834 2(1)8 54 8 34 801 145 229 331 453 54 754 933 1131 1348 1584 1839) 2113! 55 8 35 81 146 230 333 455 597 757 936 1134 1352 1.588 1843 2117; 56 8 36 82 147 2321 335 458 599 760i 9:39 1138 1355 1;592 1847 2122 57 8 36 83 149 233 337 4601 601 763 942 1141 1359 1596,1852 2127 58 9 37 84 1501 235 339 462 60414 7i65 945 11451 1363 160 18356 2132 59 9 37 85 1511 236 341 464 607 768 949 1148 1367 1604] 1861 2136 60 9 38 86 152 238 343 466 6U09 771 952 1152 1370 16081 1865 2141 L r ~ ~ ~ I ~_ 238 TABLE XXXVI. Reduction to the Meridian. Argument =the Hour Angle from the Meridian. 15min 16m 17l 18im m19n 20m 11m 2 nm 23m 24m 0 2141 2436 2750 3083 3434 3805 4195 4604 5031 547 1 2146 2441 2755 3088 3441 3812 4202 4611 5039 5486 2 2151 244 2761 33 3447 3981 4208 4618 5046 5494 3 2155 2451 2 766 3099 3453 3824 4215 4625 5053 5501 4 1601 452 2 771 2 316 3906 349 3831 422 4632 5061 5509 5 2165 2461 2777 3111 3465 3837 4228 4639 5068 5516 6 2170 2466 2782 3117 3471 38431 4235 4646 5075 5524 7 2175 0 2472 2788 3123 3477 3850 4242 453 5083 5531 8 2179 2477 2793 3128 3483 3856 4248 4660 5090 5539 9 2184 2485 2799 3134 3489 3863 4255 4667 5 597 547 10 2189 2487 2804 314 31 49 3 869 4262 4674 5105 5554 11 2194 2492 2809 3146 3501 3875 4269) 4681 5112 5562 12 1 2198 2497 2815 3151 3507 3882 475 4688 5119 5570 13 2203 2502 2820 3157 3513 3888 4282 4695 5127 5577 14 208 2507 29S26 3163 3519 3895 4289 4702 5134 5585 15 2213 2513 2831 3169 3525 3901 4295 4709 5141 5593 16 2218 2518 2837 31 7 5 353 39017 4302 4716 5149 5600 17 22323 2523 2842 3180 3538 3914 4309 4723 5156 5(08 18 2.2327 2528 2848 3186 3544 3920 4316 4730.5163 5616 19 2232 2533 2853 3192 3550 3927 4322 4737 5171 5623 20 2237 2538 2859 3 3198 356 3933 4329 4744 5178 5631 321 2242 2544 2864 3204 3562 3940 4336 4751 5186 5639 22 2247 2549 2870 3209 3568 3946 4343 4758 5193 5647 23 2252 2554 2875 3215 3574 3952 4350 4766 5200 5654 24 2357 2559 2881 32211 3581 3959 4 4356 4773 5208 5662 25 2262 2564 2886 32-27 3587 3965 4 4363 780 5215 5670 26 2266 26570 2892 3233 59 39 3972 4370 4787 5223 5678 27 22732 275 2897 323939 3599 3978 4377 4794 5230 5685 28 23276 2580 2903 3244 360)5 3985 4383 4801 5237 5693 29 2281 2585 2908 33'50 3611 3 991 4390 4808 5245 5701 430 2386 290 2914 325) 3618 3998 4397 4815 5 52 5709 31 2291 2596 2913 32i2 3624 1404 4404 4822 5260 5716 3 22 2929 2691 2325 3268 36:30 4011 44112 482'9 5267 5724 33 2301 2606 2930 3274 3636 4017 4416 4837 5375 5732 34 5 23026 2611 2936 3230 36fi4 424 44284 4844 5282 5740 35 2311 261 37 1 291 36 36 48 4030 4431 4851 5290 5747 364 23106 71 6321 2947 3391 30655 4037 4438 4858 5297 5755 37 2321 2('37 2952 3397 3661 40423 4445 4865 5305 5763 38 2326 2632 31958 33)03 3667 405 4 4452 4872 5312 5771 39 2331 2638. 2964 3309 3673 4056 4459 4880 5320 5779 40 2 335 2f643 32970 3315 3680 4063 4465 4887 5327 5786 41 2340 1 2648 2975 3321 3686 407 0 4472 4894 5335 5794 42 2345 1 2654 2981. 3327 3i92 4076 4479 4901 5342 5802 43 2350 2659 2986 3333 3698 4083 4486 4908 5350 5810 44 2355 2664 2 992 3339 3705 4089 4493 4916 5357 5818 45 2360 2669 2998 3345 3711 4096 4500 4923 5365 5826 46 2365 2675 3003 3351 3717 4lu2 4507 4930 53723 5833 4-7 2370 2680 3009 3357 3723 4109 4514 4937 5380 i 5841 48 2375 2685 3015 3363 3730 4116 4520 4944 5387 5849 49 2380 2690l 3020 3369 3736 4122 4527 4952. 5395 5857 50 2385 -2696 3026 33 33742 4191 4534 49590 5402 5865 51 23390 2701 3031 3380 3749 4135 4541 4966 5100 5873 52 2395 2707 3037 3386 3755 4142 4548 4973 5417 5881.53 2401 2712 3043 3392 3761 4149 4555 4981 5425 5888. 54 2406 2718 3049 3398 3767 4155 4562 4988 5433 5896 55 2411 2723 3054 3404 3774 4162 4569 4995 5440 5904 56 2416 2728 1 3060 3410 3780 4168 4576 5002 5448 5912 57 2421 2734 3066 3116 3786 4175 4583 5010 5455 5920 58 | 2426 2739 3071 3422 3793 4182 4590 5017 5463 5928 59 2431 2744 3077 3428 3799 4188 4597 50324 5470 5936 60 2436 2750 3083 3434 1 3-05 | 4195 4604 5031 5478 5.944 bk-.. _ __ _.