TR T I S E SURVEYING AND NAVIGATION: UNITING THE THEORETICAL, PRACTICAL, AND EDUCATIONAL FEATURES OF THIESE SUBJECTS BY IIORATIO N. ROBINSON, A. M. FOiRMERLY PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY; AUTHOR OF MATHEMATICAL, PHILOSOPHICAL, AND ASTRONOMICAL WORKS. THIRD EDITION, CINCINNATI: JACOB ERNST, 112 MAIN STREET. BOSTON: B. B. I1USSEY & CO.; ROBERT S. DAVIS & CO.; W. J. REYNOLDS & CO. NEW YORK: MASON BROTHERS; D. BURGESS & CO.; A. S. BARNES & CO.; NEWMAN & IVISON; PRATT, WOODFORD & CO. P H I L A D E L P I I A: LIPPINCOTT, GRAMBO & CO.; THOMAS, COWPERTHWAITE & CO.; E. H. BUTLER & CO.; URIAH HUNT & SON. 1853. ENTERED, according to act of Congress, in the year 1852, By II. N. ROBINSON, in the Clerk's Office of the District Court for the Northern District of New York. A. C. JAMES, STEREOTYPER, 167 WALNUT ST, CINCINNATI. PREFACE. THIS book is more than its title page proclaims it to be: it is the practical application of the Mathematical Sciences to Mensuration, to Land Surveying, to Leveling, and to Navigation. Nor is the work merely practical. Elementary principles are here and there brought before the mind in a new light; and original investigations will be found in many parts of the work. To show the reader how a thing is to be done, is but a small part of the object sought to be obtained: the great stress is put upon the reasons for so doing, which gives true discipline to the mind, and adds greatly to the educational value of any book. We have illustrated the subject of -logarithms, and their practical uses, the same in this book as is common to: be found in other books, and this is sufficient for the common pupil, or the ordinary practical man, whether surveyor or navigator; but in addition to this,. we have carried logarithms much further in this work than in any I have seen. I do not mean by this that we have more voluminous tables than others. Such is not the fact. Voluminous tables are not necessary for those who really understand the nature of logarithms, and such are mainly intended for those who are not expected to understand principles. To give a more practical illustration of logarithms, and to suggest artifices in using logarithms generally, we have given Table III and its auxilliaries, on page 70 ot tables, showing logarithms to twelve places of decimals, a degree ol accuracy which practice never demands. By the help of this table combined with a true knowledge of the subject, the logarithm of any number may be readily found true to ten places of decimals, or, conversely, the number corresponding to any given logarithm may be found to almost any degree of accuracy. Our Traverse Table is not so full as in some other books, but it iv PREFACE. is full enough to answer every purpose; and latitude and departure, corresponding to any course and distance, can be found by it, provided the operator's good judgment is awake. Indeed a contracted table, in an educational point of view, is better than a full one; for the former calls forth and cultivates tact in the student, but the latter is best for the unanimated plodder. In running lines, and computing the areas of surveys, we have endeavored to present the subject in such a manner that the reader must constantly keep Elementary Geometry in view, and the whole is so clear and simple, that many will think it unworthy of the rank that it seems to hold in the public estimation, but there are other reasons for this. The chapter on surveys and surveyors will be found to be a little peculiar, but the information there given, will be highly useful to all those who are inclined to look upon a survey as a mathematical problem only. On the compass, and the declination of the needle, we have been very full: the subject embraces meridians and astronomical lines drawn on the earth. The manner in which we should proceed to make a survey, provided no such instrument as the compass existed, and there were no such thing as a magnetic needle, is taken up and illustrated in this work. The subject of dividing lands is fully discussed and illustrated, and if any one has occasion to complain of mathematical abstrusity in this work, it will be found in this connection; yet there is nothing here above elementary algebra and geometry. The method of taking levels and making a profile of the vertical section of a line for rail roads, is set forth in this work. The profile shows -the necessary excavation or embankment, which it is necessary to cut down or build up at any particular point, to conform to any proposed grade that may be contemplated. To determine the elevation of any place above the level of the sea, by means of the barometer, has been, and now is, a very interesting problem to all philosophical students, yet very few of them have been able to comprehend it beyond its first great principle, the variation of atmospheric pressure. To trace, or rather to discover the mathematical law which connects the elevation of any locality with the mean hight of the barometer at the same place, has been an obscure problem, and we have taken hold of it with a determination to break open some avenue of light (if such were possible) by which the simplicity of the problem might be brought to the comprehension of the every-day mathematical student, and we believe that we have succeeded in the undertaking. PREFACE. v The part on Navigation, might be regarded, at first view, an abridgment of that subject, and in one sense it is, for we have studied to be as brief as possible, but we would never let brevity stand in the place of perspicuity; and however it may appear, we have given all the mathematical essentials of the subject, and whoever acquires what is here given, will find very little necessity of looking elsewhere for the continuation of the study, unless it is for sea terms and seamanship; but these have nothing to do with Navigation as a science. Our method of working lunars is more brief than any other, where auxilliary tables and methods of approximation are not resorted to, but to attain this brevity, we have been compelled to use Natural Sines in part of the operation; but on the other hand, this should be no objection, for it gives us a clearer view of the unity and harmony of the mathematical sciences. CONTENTS. INTRO DUCTION. CHAPTER I. PAGE Introduction..................... 9 Construction of Geometrical Problems, with the use of instruments.......................... 12 CHAPTER II. Logarithms...................... 22 Application of Logarithms to Multiplication, Division, and the extraction of Roots................. 29-32 Artifices in the use of Logarithms (Art. 12)....... 32-34 Logarithms extended to a greater number of Decimals (Art. 13) 34 —43 CHAPTER III. Plane Trigonometry................ 44 Explanation of Tables................. 58 Oblique-angled Plane Trigonometry........... 64 SURVEYING. Introductory Remarks................. 70 Finding Areas in general.............. 70-79 Mensuration of Solids................. 79 CHAPTER I. Mensuration of Lands................ 80 To measure a Line................... 81 Surveyor's Compass.............. 82-85 Vernier Scales in general............... 85-86 CHAPTER II. Latitude and Departure................ 87 Taking angles by the Compass.............. 89 CONTENTS, vii PAGE. To close a Survey........ 89 To find the true from a random Line......... 90 Computation of Areas by trapezoids......... 91 —103 CHAPTER III, To find Meridian Lines......... 103 Variation of the Compass... 104.-112 Practical Difficulties.......... 112 CHAPTER IV. To Survey without a Compass..... -. 113 The Circumferentor............. 114 CHAPTER V. Original and subsequent Surveys'.... 116 Difficulties and Duties of a Surveyor........... 118 United States' Land..,,........, 121 CHAPTER VI. Very irregular Figures............. 123 Division of Lands —a variety of Problems........ 124 —145 CHAPTER VII, Triangular Surveying................ 146 The Plane Table-its uses &c............ 147-152 Marine Surveying.................. 152 Piloting Ships............. 154 CHAPTER VIII. Leveling......,........ 155 Description of the Level................ 158 Adjusting the Level........... 159-161 Keeping Book........ 163 Contour of Ground...... 165 Elevation determined by atmospheric Pressure, as indicated by the Barometer........... 166 —-174 NAVIGATION. CHAPTER I. Introduction.................. 175 The Log-line.................. 176 The Mariner's Compass........ 177 viii CONTENTS. CHAPTER II. PAGE. Plane Sailing.............. 180-186 Middle Latitude Sailing............. 186 Traverse Sailing.................. 186 Sailing in Currents.............. 191 CHAPTER III. Mercator's Chart —its construction............ 194-196 Mercator's Sailing.................. 196 CELESTIAL OBSERVATIONS. CHAPTER I. Definition of Terms......... 198 —199 Quadrant and Sextant................. 202 Construction of the Sextant.............. 204 The adjustments of the Sextant............. 205 To take an Altitude of the Sun at sea......... 206 To find the Latitude by the Sun or Stars......... 207 To find the Latitude by the meridian Altitude of the Moon. 209 —211 CHAPTER II. A perfect Time-piece............... 213 Local Time-Rule to find it............... 216 Longitude by Chronometer.............. 217 CHAPTER III. Lunar Observations................. 220 Formula, for clearing the observed Distance from the effects of Parallax and Refraction............... 223 Examples for working Lunars........... 224-227 APPENDIX. Artifices to be resorted to in difficult circumstances.... 228 TABLES. INTRODUCTION. CHAPTER I. MENSURATION, SURVEYING, and NAVIGATION, are but branches of the same science, and should be regarded as the application of geometry and trigonometry, and in this light we shall present them to our readers. In this volume we shall not demonstrate geometrical truths unless we wish to present them in some new form, or unless the demonstration is not readily to be found in the proper places, in the elementary books. It is expected that all readers of a work of this kind, have previously made themselves more or less acquainted with Algebra and Geometry, and where this' is the case the reader will have no difficulty; and readers who are not thus prepared should be careful not to charge imaginary defects to the book: in no work of this kind would it be proper to demonstrate every elementary principle. These remarks apply only to the educational character of the book. Preparatory to a course of practical mathematics, it is proper to give such descriptions of the instruments to be used as will enable the operator to understand their use. But some of these instruments can never be understood from a book, it must be from the instrument itself; we might as well attempt to give a person an idea of color by the means of language, as to give a person a correct idea of the sextant and theodolite by a mere book description. It is true we can do something by drawings and descriptions, and that something we intend to do. To represent plane surfaces and tracts of land on paper, no other instruments are necessary than the scale, and dividers, and a pro(9) 10 SURVEYING. tractor to measure angles. In fact, every thing can be done with the scale and dividers only - other instruments, as the protractor, sector, and parallel rulers,only add to our convenience; at the same time they could be dispensed with. THE PLANE SCALE. The plane scale, or the plane diagonal scale of equal parts, as here represented, is the most common and useful of all the instruments used in drawing. It is also a ruler, and if wide and well made will serve as a square also, by which right angles may be drawn. The very appearance of this scale will show its construction, the side of the square a b may be of any length whatever, it is generally taken an inch, but this is not imperative. By means of the 10 parallel lines running along the length of the scale, and the 10 diagonal lines parallel to-each other in the square a b c d, we have 100 intersections in the square, by which we are enabled to find any and every hundredth part of the division of a b. For example, I wish to find 27 hundredths of the:line a b. I go to the division:2 on a 6, and then run up that diagonal line to the 7th parallel, and from that intersection to the line a d is 27 hundredths of a b. The distances a b, a g, g h, &c., may be taken to represent 1, 10, 100, or in fact any number we please. Suppose we take any one of the equal divisions a b, ag, &c., to represent 100, and then require 234. From ito ea.represents that distance. If the base a b were 10, from I to ewould be 23.4; if 1, then from I to e would be 2.34; and so on proportionally for any other change of base, or change of the unit. To transfer distances from the scale (as I e, p q, &c.) to paper, we require INTRODUCTION. 11 DIVIDE R.S. Dividers are nothing more than a delicate pair of compasses - two bars turning on a joint. They are too well known to require representation by a figure. They are also used for describing circles and parts of circles. THE PROTRACTOR. The following diagram accurately represents this instrument. It consists of a semicircle of brass ABC, divided into degrees. The degrees are numbered both ways, from A to B and from B to A. There is a small notch in the middle of AB, to indicate the center. To lay off an angle. Place the diameter AB on the line, so that the center shall fall on the angular point. Then at the degree required, at the edge of the semicircle make a point with a pin. Then remove the protractor and draw a line through the point so marked and the angular point; this line, with the given line, will make the required angle. The reader will observe a great similarity between this instrument and the circumferentor, which is described in a subsequent portion of this work. This instrument is designed merely to draw angles on paper, that to draw lines marking given angles, with other lines, in the field. 12 SURVEYING. In addition to this, both the protractor and circumferentor may be used in taking levels, and measuring angles of altitude, when no better instruments for such purposes are at hand. For instance, if a delicate plumb should be suspended from the center of the protractor, and the thread rest at the point C, while the instrument is held in a frame, then A and B would be as a level, and as many degrees as the plumb line rested from C so many degrees would be the inclination of A and B from a horizontal level. Levels and angles of altitudes were formerly taken in this way. With the instruments previously described, solve the following problems. The references are to Robinson's Geometry. Thus, (th. 15, b. 1, cor. 1,) indicates theorem 15, book 1, corrolary 1, where the demonstrations of the problem referred to will be found. PROBLEM 1 To bisect a given finite straight line. Let AB be the given line, and from its extremities, A and B, with any radius greater than the half of AB (Post. 3), describe arcs, cutting each other in n and m. Join n and m; and C, where it cuts AB, will be the middle of the line required. Proof, (th. 15, b, 1, cor. 1). PROBLEM 2. To bisect a given angle. Let ABC be the given angle. With any radius, from the center B, describe the are A C. From A and C, as centers, with a radius greater than the half of AC, describe arcs, intersecting in n; and join Bn, it will bisect the given angle. Proof, (th. 19, b. 1). INTRODUCTION. 13 PROBLEM 3..From a given point, in a given line, to draw a perpendicular to that line. Let AB be the given line, and C the given point. Take n and m equal distances on opposite sides of C; and from the points rn and n, as centers, with any radius greater than nIC or or in C, describe arcs cutting each other in S. Join SC, and it will be the per-: pendicular required. Proof, (th. 15, b. 1, cor. ). The following is another method, which is preferable, when the given point, C, is at or near the end of the line. Take any point, 0, which is manifestly one side of the perpendicular, and join 0 C; and with OC, as a radius, describe an arc, cutting AB in m and C. Join m O, and produce it to meet the arc, again, in n; man is then a diameter to the circle. Join Cn, and it will be the perpendicular required. Proof, (th. 9, b. 3). PROBLE M 4. From a given point without a line, to draw a perpendicular to t/hat line. Let AB be the given line, and C the given point. From 0, draw any oblique line, as Cn. Find the middle point of Cn by (problem 1), and from that point, as a center, describe a semicircle, having Cn as a diameter. From the point m, where this semicircle cuts AB, draw Cnm, and it will be the perpendicular required. Proof, (tlh. 9, b. 3). 14 SURVEYING. PROBLEM 5. At a given point in a line, to make an angle equal to another given angle. Let A be the given point in the line AB, and D CE the given angle. From C as a center, with any radius, CE, draw the arc ED. From A, as a center, with the radius AF= CE, describe an indefinite arc; and from F, as a center, with FG as a radius, equal to ED, describe an arc, cutting the other arc in G, and join AG; GAF will be the angle required. Proof, (th. 5, b. 3). PROBLEM 6. Fro& a given point, to draw a line parallel to a given line. Let A be the given point, and CB the given line. Draw AB, making an angle, ABC; and from the given point, A, in the line AB, draw the angle BAD-ABC, by the last problem. AD and CB make the same angle with AB; they are, therefore, parallel. (Definition of parallel lines). PROBLEM 7. To divide a given line into any number of equal parts. Let AB represent the given line, and let it be required to divide it into any number of equal parts, say five. From one end of the line A, draw AD, indefinite in both length and position. Take any convenient distance in the dividers, as Aa, and set it off on the line AD; thus making the parts Aa, ab, bc, &c., equal. Through the last point, e, draw EB, and through the points a, b, c, and d, draw parallels to eB (problem 6.); these parallels will divide the line as required Proof (th. 17, b. 2). INTRODUCTION. 15 PROBLEM 8. To find a third proportional to two given lines. Let AB and A C be any two lines. Place A B them at any angle, and join CB. On the A C greater line, AB, take AD=A C, and through D, draw DE parallel to BC; AE is the third proportional required. Proof, (th. 17, b. 2). PROBLEM 9. To find a fourth proportional to three given lines. Let AB, AC, AD, represent the A B three given lines. Place the first two A C together, at a point forming any angle,' A D as BAC, and join BC. On AB place AD, and from the point D, draw (problem 6) DE parallel to BC; AE will bt the fourth proportional required. Proof, (th. 17, b. 2). PROBLEM 10. To find the middle, or mean proportional, between two given lines. Place AB and B C in one right line, and, on AC, as a diameter, describe a semicircle (postulate 3), and from the' 1 point B, draw BD at right angles to A Ca (problem 3); BD is the mean proportional required. Proof, (scholium to th. 17, b. 3). 16 SURVEY I (;. PROBLEM 11. To find the center of a given circle. Draw any two chords in the given circle, as AB and CD; and from the middle point, n, of AB, draw a perpendicular to AB; and from the middle point, m, draw a perpendicular to CD; and where these two perpendiculars intersect will be the center of the circle. Proof, (th. 1, b. 3). PROBLEM 12. To draw a tangent to a given circle, from a given point, either in or without the circumference of the circle. When the given point is in the circumference, as A, draw AC the radius, and from the point A, draw AB perpendicular to A C; AB is the tangent required. Proof, (th. 4, b. 3). When A is without the circle, draw A C to the center of the circle; and on A C, as a diameter, describe a semicircle; and from the point B, where this semicircle intersects the given circle, draw AB, and it will be tangent to the circle. Proof, (th. 9, b. 3), and (th. 4, b. 3). PROBLEM 13. On a given line, to describe a segment of a circle, that shall contain an angle equal to a given angle. INTRODUCTION. 17 Let AB be the given line, and C the given angle. At the ends of the given line, make angles DAB, DBA, each equal to the given angle,6 C. Then draw AE, BE, perpendicularsto AD, BID; and from the center, E, with radius, EA or EB, describe a circle; then AFB will be the segment required, as any angle F, made in it, will be equal to the given angle,:C. Proof, (th 11. b. 3), and (th. 8, b. 3). PROBLEM 14. To cut a segmentfrom any given circle, that shall contain a given angle. Let C be the given angle. Take any point, as A, in the circumference, and from that point draw the tangent AB; and from the point A, in the line AB, make the angle BAD=C (problem 5), and A.ED is the segment required. Proof, (tk. 11, b. 3), and (th. 8, b. 3). PROBLEM 15. To construct an equilateral triangle on a given finite straight line. Let AB be the given line, and from one extremity, A, as a center, with a radius equal to AB, describe an arc. At the other extremity, B, with the same radius, describe another arc. From C, where these two arcs intersect, draw CA and CB; A BC will be the triangle required. The construction is a sufficient demonstration. Or, (ax. 1). 2 s8 SURVEYING. PROBLEM 16. To construct a triangle, having its three sides equal to three given lines, any two of which shall be greater than the third. Let AB, CD, and EF represent the three E F lines. Take any one of them, as AB, to be one C D side of the triangle. From A, as a center, with n a radius equal to CD, describe an arc; and from B, as a center, with a radius equal to EF, describe another arc, cutting the former in n. Join An and Bn, and AnB will be the iz required. Proof, (ax. 1). A B PROBLEM 17. To describe a square on a given line. Let AB be the given line, and from the extre- C D mities, A and B, draw AC and BD perpendicular to AB. (Problem 3.) From A, as a center, with AB as radius, strike an arc across the perpendicular at C; and from C, A B draw CD parallel to AB; A CDB is the square required. Proof, (th. 21, b. 1.) PROBLEM 18. To construct a rectangle, or a parallelogram, whose adjacent sides are equal to two given lines. Let AB and AC be the two given lines. A C From the extremities of one line, draw per- A B pendiculars to that line, as in the last problem; and from these perpendiculars, cut off portions equal to the other line; and by a parallel, complete the figure. When the figure is to be a parallelogram, with oblique angles, describe the angles by problem 5. Proof, (th. 21, b. 1). INTRODUCTION. 19 PROBLEM 19. To describe a rectangle that shall be equal to a given square, and have a side equal to a given line. Let AB be a side of the given square, and C D CD one side of the required rectangle. A B Find the third proportional, EF, to CD E F and AB (problem 8). Then we shall have, CD AB:: AB: EF Construct a rectangle with the two given lines, CD and EF (problem 18), and it will be equal to the given square, (th. 13, b. 2). PROBLEM 20. To construct a square that shall be equal to the diference of two given squares. Let A represent a side of the greater of two given squares, and B a side of the lesser square. On A, as a diameter, describe a semicircle, and from one extremity, m, as a center, with a radius equal to B, describe an are, n, and, from the point where it cuts the circumference, draw mn and np; np is the side of a square, which, when constructed, (problem 17), will be equal to the difference of the two given squares. Proof, (th. 9, b. 3, and 36, b. 1.) PROBLEM 21. To construct a square, that shall be to a given square, as a line, M, to a line, N. Place _I and N in a line, and on the sum describe a semicircle From the point where they join, draw a perpendicular to meet the circumference in A. Join Am and An, and produce them indefinitely. On Am or An, produced, take AB= to the side of the given square; and from B, draw B C parallel to nn; A C is a side of the required square. 20 SURVEYING. Besides the numerical scale of equal parts, we have scales of chords, sines, and tangents, which can be constructed corresponding to any radius. Such scales of course are not scales of equal parts. Such scales are constructed in the following manner. Take CA any radius, and describe a semicircle. Draw CD at right angles to AB, and draw a tangent line from A. Divide the arc AD into equal parts 10, 20, 30, &c., beginning at ), and subdivide them as much as required. Draw 10 10, -- 20 20, -30 30, &c., all parallel to CD. From Cto 10 on the line CA, is the sine of - 100. From C to 20 is the sine of 20~ &c. &c. The line 10 10 is the sine of 800, and CD( ox CA is the sine of 90~. The distance from A to D is the chord of 900~, from A to 10 is the chord of 800, and from A to 20 is the chord of 70~, and so on down. Thus we perceive that we can take off any sine or chord and lay it down on a ruler; and chords and sines thus laid off constitute the scale of chords, sines, &c. Lines drawn from C through any division of the arc, commencing at A to strike the tangent line, will mark off the tangent corresponding to that arc. Thus, if the angle A CH is 300, then the line AH placed on a scale, will represent the tangent of 30~ to the radius CA, and thus any other tangent can be laid down on the same scale. The scale of chords and sines, as well as the scale of equal parts, are to be found on the SECTOR. The sector is commonly made of ivory, and consists of two arms which open and turn round a joint at their common extremity. For some operations, particularly the projection of solar eclipses, the sector is a very useful instrument. INTRODUCTION. 21 The figure before us represents one side of a sector with the plane scale only upon it. More than one scale can be put on to a side, but we represent but one to avoid confusion. The scale must be alike on both arms -and it must commence exactly at the joint - hence when near the center the different scales crowd each other. The two arms of the sector always form two sides of a triangle, and by opening and closing them we vary the angle, yet the distance across from one arm to the other is always proportional to the sides of the triangle. The advantage of the sector will appear from the following problem. A map is before me, its scale is 2Q miles to an inch; I wish to find the distance in a right line between two points laid down on it. 1st. I take one inch in the dividers and open the sector, so that the distance between 20 and 20 on the two arms, shall just correspond to the measure in the dividers, that is, shall be one inch. Let the sector lie on the table thus opened. 2nd. Now take the distance you wish to measure. in the dividers; place one foot on one arm of the sector, and the other foot on the other aim; so that the feet of the dividers shall fall on the same. number on both arms of the sector. The number thus marked by the dividers will be the distance required. The distance between any other two points may be measured on the same map, without any computation whatever. For another illustration of the utility of the sector, let us suppose, that the sine of 20~ is required corresponding to a radius of 6 inches. 22 SURVEYING. Take 6 inches in the dividers, and open the sector so that the sine of 90~ from arm to arm shall be 6 inches. The sector being thus open, take the distance from 20 to 20, on the line of sines from arm to arm, in the dividers, and that is the distance required. GUNTE R' S SCALE. Gunter's scale is commonly two feet in length, containing the plane scale and the scale of sines, chords, and tangents" on one side of it, and the scale for the logarithms of numbers, sines, and tangents on the other. This scale is very ingenious, but it is not so much used nor considered so important as formerly. CHAPTER II. LOGARITHMS. ART 1. Logarithms are exponents. Thus, if a2 =9 and a3-27 Then a5 =243; by multiplying the two equations together term'by term. The exponent 2 of the first equation may be considered the logarithm of 9; the exponent 3 the logarithm of 27, and the exponent 5 the logarithm of the number 243. In these equations a=3 the base of the system. By the preceding operation it is obvious that adding the exponents 2 and 3, corresponds to multiplying the numbers 9 and 27. If we take the equation a5 =243, and divide it by a2 -9 member by member we shall have a — =a3 — 27. Hence adding exponents (logarithms) corresponds to the multiplica LOGARITHMS. 23 tion of their corresponding numbers and subtracting the exponents (logarithms) corresponds to the division of their numbers. It is this property of logarithms that gives them their utility and importance. ART. 2, The base of our common system of logarithms is 10, and in any equation in the form 10x=n, x is the logarithm of the number n whatever number n may represent. If n= 10, then the equation becomes 10x= 10. Whence x= 1 because 101 =10. Therefore in our common system of logarithms the logarithm of 10 must be 1. Now because 100=1 101 = 10 102 =100 103 = 1000 104 = 10000 &c. &c.; it is plain that the logarithm of 1 is O, of 10 is 1, of 100 is 2, &c., every power of 10 increasing the logarithm by 1. It is also obvious, that every number between 0 and 10 must have a fractional or decimal number for its logarithm, and every number between 10 and 100 must have one, and some decimal for its logarithm. In the equation 10-=3, x is the logarithm of 3, and if we multiply this by 10' = 10, member by member, we shall have 101 +X=30 Multiply this by 10 =10 and we have 102 1+x300. These results show that 3, 30, 300, have logarithms containing the same decimal number; x differs from each exponent only by whole numbers, and thus, generally; any number multiplied or divided by 10, or any power of 10, will have logarithms containing the same decimal part. ART. 3. For the general computation of logarithms we refer to algebra, and in a work like this we shall only attend to such portions of theory as to enable the student to use them understandingly and with as much practical facility as possible. 24 SURVEYING. Let it be observed that the logarithm of 10000 is 4,00000 of 1000 is 3,00000 of 100 is 2,00000 of 10 is 1,00000 of I is 0,00000 T is —1,00000,- 10-2 is-2,00000 For every division of the number by 10 we subtract 1 from its logarithm, and when the number comes down to 1, and its logarithm of course to 0, if we again divide by 10, making it J- or 10-1, we must subtract one from the logarithm, making it — 1. The decimal portion of a logarithm is always positive, but the index or whole number part of it, becomes minus when the value of the number is less than 1. ART. 4. The whole number belonging to any logarithm is called its index, a very appropriate term, because it indicates, it points out where to place the decimal point between whole numbers and decimals. The index, or (as some call it) the characteristic, is never put in the tables (except from 1 to 100), because we always know what it is. It is always one less than the number of digits in the whole number. This is obvious from Art. 3. Thus, the number 3754 has 3 for the index of its logarithm, because the number consists of 4 digits; that is, the logarithm is 3, and some decimal. The number 34.785 has 1 for the index of its logarithm, because the number is between 34 and 35, and 1 is the index for all numbers between 1 and 100. All numbers consisting of the same figures, whether integral, fractional, or mixed, have logarithms consisting of the same decimal part. (Art. 2.) The logarithms differ only in their indices. Thus, the number 7956. has 3.900695 for its log. the number 795.6 has 2.900695 " the number 79.56 has 1.900695 " the number 7.956 has 0.900695 " LOGARITHMS. 25 the number.7956 has — 1.900695 for its log. the number.07956 has -2.900695 " &c., &c. For every division by 10, we diminish the index by 1. When the index is minus it indicates a decimal number; but let the learner remember that the index only is minus; the decimal part is alwuy, positive. ART. 5, To take out the logarithm of any number from the tables, we only consider the digits; for the logarithms of 7956, or of 7.956, or of.007956, have the same decimal part; and when that decimal part is found we then consider the value of the number to prefix the index. To prefix the index to a decimal, count the decimal point 1, and each cipher 1, up to the first significant figure, and this is the negative index. For example, find the logarithm of the decimal.00085. To accomplish this we must look for the logarithm of the whole number 85, and we find its decimal part to be.929419; and now, to determine the index, we count one for the decimal point and three ciphers, making 4; hence, we have Num..00085 - log. -4.929419. The smaller the decimal, the greater the negative index; and when the decimal becomes 0, the logarithm becomes negatively infinite. ART. 6. The logarithm of any number consisting of four digits or less, can be taken out of the table directly and without the least difficulty. Thus, to find the logarithm of the number 3725, we find the number 372 at the side, and over the top we find 5, and opposite the former and under the latter we find.571126 for the decimal part of the logarithm. The 57, the first two decimal, is under 0, which is the same for the whole horizontal column. Hence, the logarithm of 3725 is 3.571126 of 37250 is 4.571126 of 3.725 is 0.571126 &c., &c, Find the logarithm of 1176. We find 117 at the side, and 6 at the top, and opposite the former and under the latter we find.407 3 26 SURVEYING. The point here demands a cipher, and is put in to arrest attention to make the operator look to the next horizontal line below for the first two decimals. Thus, we find.070407 for the decimal part of the logarithm required. Hence, the log. of 1176 is 3.070407 1. What is the log. of.001176? Ans. -3.070407 2. What is the log. of 13.81? Ans. 1.140194 3. What is the log. of 72.55? Ans. 1.860637 4. What is the log. of.6762? Ans. -1.830075 5. What is the logarithm of the number 834785? This number is so large that we cannot find it in the table, but we can find the numbers 8347 and 8348. The logarithms of these numbers are the same as the logarithms of the numbers 834700 and 834800, except the indices. 834700 log. 5.921530 834800 log. 5.921582 Difference, 100 52 Now, our proposed number, 834785, is between the two preceding numbers; and, of course, its logarithm lies between the two preceding logarithms; and, without further comment, we may proportion to it thus, 100: 85=52: 44.2 Or, 1.:.85=62: 44.2 To the logarithm 5.921530 Add 44 Hence, the logarithm of 834785 is 6.921574 the logarithm of 83.4785 is 1.921574 From this we draw the following rule to find the logarithm of any number consisting of more than four places of figures. RuLe. —Take out the logarithm of the four superior places directly from the table, and take the difference between this logarithm and the next greater logarithm in the table. Multiply this difference by the inferior places in the number as a decimal, and add the result to the logarithm corresponding to the superior places, the sum will be the logarithm required. Example. Find the log. of 357.32514. The four superior digits are 3673; the logarithm of these LOGARITHMS. 27 corresponds to the decimal,.553033, for its decimal part. The inferior digits, taken as a decimal, are.2514 122 6028 5028 2514 30.6708 This result shows that 30, or more nearly,31, should be added to the logarithm already found, thus giving.553064 for the decimal part of the logarithm 357.32514. Therefore, as three digits of the given number are whole numbers, the index must be 2, and the logarithm of 357.32514 is 2.553064 of 3573251.4 is 6.553064 of.035732514 is — 2.553064 The change between the place of the decimal point in a number, and the corresponding change of the index to its logarithm, should be strongly impressed on the mind of a learner. Example 2. What is the log. of 366.25636? Ans. 2.563785 3. What is the log. of 39.37079? Ans. 1.595174 4. What is the log. of 2.37681? Ans. 0.375812 ART. 7. We now give the converse of the last article; that is, we give the decimal part of a logarithm to find its corresponding number. Taking the decimal in Example 1, (Art. 6,).6553064, we demand its corresponding number.* The next less logarithm in the table, is.553033, corresponding to the figure 3573. The difference between this given logarithm and the one next less in the table, is 31; and the difference between two consecutive logarithms in this part of the table, is 122. Now divide 31 by 122, and write the quotient after the number 3573. * To take out a number from its logarithm, never enter the first part of the table between 1 and 100. Go to the main table, as it contains many more logarithms. 28 SURVEYING. That is, 122)31.(254 244 660 610 500 488 The figures, then, are 3573254, which corresponds to the decimal logarithm.553064; and the value of these figures will, of course, depend on the index to the logarithm. If this given logarithm contained an index, such index would point out how many of these figures must be taken for whole numbers, the others will be decimals; thus, if the index had been 4, the number would be 35732.54 If the given decimal had been.553063.67, which is the exact converse of example 1, then we should have found that number, 35732514; but we did not give that decimal logarithm, because the table contains only six decimal places. From this obvious operation we derive the following rule to find the number corresponding to a given logarithm. RULE.- If the given logarithm is not in the table, find the one next less, and take out the four figures corresponding; and if more than four figures are required, take the difference between the given logarithm and the next less in the table, and divide that difference by the difference of the two consecutive logarithms in the table, the one less, the other greater than the given logarithm; and the figures arising in the quotient, as many as may be required, must be annexed to the former figures taken from the table. EXAMPLES. 1. Given, the logarithm 3.743210, to find its corresponding number true to three places of decimals. Ans. 5536.182 2. Given, the logarithm 2.633356, to find its corresponding number true to two places of decimals. Ans. 429.89 3. Given, the logarithm -3.291742, to find its corresponding number. Ans..0019577 LOGARITHMS. 29 MULTIPLICATION. BY LOGARITHMS. ART. 8. If the principle first laid down in (ART. 1) is true, the sum of the exponents will be the exponent of the product of any number of factors. In other words, The sum of the logarithms of any number of factors will be the logarithm of the product of those factors. N. B. The logarithmic table corresponds to this principle, and we may see by the following EXAMPLES. The log. of 3 (taken from the table,) is 0.477121 The log. of 4 " " " " is 0.602060 Therefore the log. of 12 must be 1.079181 Given, the log. of 7 and the log. of 9, to find the logarithm of 63. Because 7X 9=63, therefore, To log. 7=0.845098 Add log. 9=0.954243 Sum 1.799341 By inspecting the table, we shall find this logarithm stands opposite 63, and by this process the logarithms of all the composite numbers have been found. In this we may consider that the logarithm pointed out the product 63. Hence we have the following rule for obtaining the product of any number of factors. RULE.- Find the logarithm of each factor, add those logarithms together and the sum will be the logarithm of the product. The number corresponding to this last logarithm taken from the table, will be the product itself. EXAMPLES. 1. To multiply 23.14 by 5.062. 2. To multiply 2.581926 by Numbers. Logs. 3.457291. 23.14 1.364363 Numbers. Logs. 5.062 0.704322 2.581926 0.411944 Product 117.1347 2.068685 3.457291 0.538736 Product. 8.92648 0.950680 30 SU RVEYING. 3. To mult. 3.902 and 597.16 4. To mult. 3.586, and 2.1046, and.0314728 all together. and 0.8372, and 0.0294 all Numbers. Logs. togrether. 3.902 0.591287 Numbers. Logs. 597.16 2.776091 3.586 0.554610.0314728 -2.497935 2.1046 0.323170 Prod. 73.3333 1.865313 0.8372 -1.922829 Here the-2 cancels the 2, and 0.0294 2.468347 the one to carry from the decimals Prod. 0.1057618 -1.268956 is set down. Here the 2 to carry cancels the -2, and there remains the -1 to set down. DIVISION BY LOGARITHMS. ART. 9, As division is the converse of multiplication we draw the following rule for division by use of logarithms. N. B. Addition and subtraction is to be understood in the algebraic sense. RULE. - From the logarithm of the dividend subtract the logarithm of the divisor, and the number corresponding to the remainder is the quotient required. EXAMPLES. 1. Divide 327.5 by 2207 log. 327.5 2.515211 log. 2207 3.342028 Quotient.14839 — 1.173183 2. Divide.054 by 1.75 log..054 -2.732394 loO. 1.75 0.243038 Quotient.030857 -2.489356 ART. 10. The preceding examples in multiplication and division were adduced only to show the nature of logarithms: had our object been results, the common arithmetical operations would hasve been more convenient for some of them; but there are cases that demand the use of logarithms, and such cases mostly occur in Involution and Evolution. LOGARITHMS, 31 RULE FOR INVOLUTION. —Take out the logarithm of the given number, and multiply it by the index of the proposed power. Find the number corresponding to the product, and it will be the power required. EXAMPLES. 1. What is the 2d power of 2. What is the cube of 1.72? 351? log. 1.72 0.235528 log. 351 2.545307 3 2 Ans. 5.0884 0.706584 Ans. 123201 5.090614 3. What is the 4th power of 4. What is the 17th power of.0916? 1.04? log..0916 — 2.961895 log. 1.04 0.017033 4 17 Ans..000070401 -5.847580 0.119231 Here 4 times the negative in- 0.17033 dex is -8, adding the 3 to carry Ans. 1.9476 0.289561 gives -5. N. B. This last example begins to disclose the utility of logarithms. 6. What is the 6th power of 6, What is the 21st power of 1.037? 2.02? log. 1.037 0.015779 log. 2.02 0.305351 6 21 Ans. 1.243+ 0.094674.305351 6.10702 Ans. 25684454.6 6.412371 EVOLUTION. ART. 11. Evolution is the converse of Involution; hence we have the following rule for the extraction of roots: Take the logarithm of the given number out of the table. Divide the logarithm, thus found, by the index of the required root; then the number corresponding is the root sought. 32 SURVEYING. EXAMPLES. 1. What is the cube root of 2. What is the cube root of 125? 200? log. 125 3)2.096910 log. 200 3)2.301030 Ans. 5 0.698970 Ans. 5.848+ 0.767010 3. What is the 4th root of 4. What is the 20th root of 751? 1.035? log. 751 4)2.875640 log. 1.035 20)0.014940 Ans. 5.235+ 0.718910 Ans. 1.001718 0.000747 5. What is the cube root of the decimal.00048 log..00048 -4.681241 To the inexperienced here would be a difficulty, as the index is negative, and the decimal part positive. How then shall we divide by 3? Add -2 and +-2 to the index; and this is, in effect, adding nothing; it merely changes the form of the index, thus, -6+2.681241 Now, we can divide by 3, and the quotient is — 2.893747. The corresponding number, or root, sought, is.07829+ Ans. REMARK.-In the preceding articles we have taught all the preliminary rules for the use of logarithms; " but there is a wisdom beyond rules," and he who does not arrive at it, attains only the burdens of knowledge without its benefits. Rules are necessary through the first rudiments of any science; but he who can instantly fall back on to first principles, and do the most advantageous thing at the most advantageous point of time, has a practical tact of the highest value. To awaken this faculty in the mind of the learner, we give what follows on the subject of logarithms. It may not be necessary for some, but to many it will be interesting and new ART. 12. Persons who possess both theoretical knowledge and practical skill, rarely, if ever, use the first part of the logarithmic table of numbers, except for exact numbers, especially if they pretend to any thing like accuracy. Such persons take some artifice to throw their logarithm into the last part of the table, where the variation of the logarithm is slower than in the first part of the table. To illustrate these remarks let us take into consideration the resulting logarithm to Example, 4, Art. 11. LOGARITHMS. 33 It would be troublesome, and, indeed, quite impossible, to take out the number corresponding to this logarithm 0.000747 if we simply apply the usual rules, and go directly to the table with the logarithm. From the given log. 0.000747 Subtract the log. of 1.01 0.00432i This log. corresponds to.9918 -1.996426 Subtracting the logarithm of 1.01 was equivalent to dividing the number by 1.01. We must, therefore, multiply.9918 by 1.01 to produce the number corresponding to the log. 0.000747. Thus, 0.9918 9918 1.001718 Subtracting the logarithm of 1.01 produced a logarithm having a large decimal part, and this was the object. We can, then, take it into the table, and find its number, to great accuracy, by mere inspection. We might have added the logarithm of 9, and thus produced a large decimal, and then have divided the corresponding number by 9, for the required result. Again: Suppose the logarithm of the number 101248, was required to as great exactness as our tables will allow. The operator who exercises no original thought, and depends only upon rules, will go directly to the table for the logarithm required; but he cannot find it there without some trouble to proportion to it; and even then his result will not be accurate, because the logarithms vary by no exact numerical ratio. Take the number and divide by some number that will give a large integer quotient. Thus, 102) 101248(992.6} 918 944 918 268 204 640 612 Now, we can find the logarithm of 992.6- very accurately, by 34 SURVEYING. inspection. Not regarding the fraction - it would be very accurate; but the practical man always makes a little correction for such fractions, wit]hout taking any proportion to do so, or using any formality about the matter. In this case, we perceive that 10 or 11, added to the last two decimal figures, will correct it; hence, log. 992.6- 2.996785 log. 102 2.008600 Therefore, log. 101248 is 5.005385 We give one more example. What number corresponds to the log. 2.111497 Add log. of 7 0.845098 Num. 904.88 log. 2.956595 Dividing this by 7, gives 129.27 for the number required. REMARK.-The foregoing comments and illustrations are sufficient for all practical problems, that can come before the surveyor, navigator, or engineer. The common table of logarithms, to six decimal places, extending from page two to twenty, of tables, is sufficiently accurate for all common problems; but there are cases in Astronomy, and in very delicate scientific investigations, where it is important to have the logarithms extend to a greater number of decimal places. Accordingly I have computed a table to 12 decimal places, corresponding to the consecutive whole numbers up to 110, and the prime numbers from thence to 1129. These logarithms, together with the auxiliary logarithms on page 71, are sufficient to find the logarithm of any number that can be proposed, and to find the number corresponding to any given logarithm containing ten decimal places. It is a general impression that a table of logarithms must be practically useless, unless it is voluminous and complete; and for constant practical use it should be so; but for occasional service, the tables here given are sufficient, and for educational purposes they are better than they would be if they were more full, for now they demand proper theoretical knowledge to use them with success. On the contrary, whoever cannot use them with success, must be deficient in theoretical knowledge, or wanting in practical tact to bring such knowledge into immediate use. To show the importance and practical utility of these tables, is the object of the following illustrations and examples. ART. 13. To make a table of logarithms anew, to contain any particular number of decimal places, the following formula taken from algebra, appears to be the most practical and convenient. The investigation of the formula belongs to the science of algebra, and not to a work like this. LOGARITHMS. 35 log (z+ ) —log z= 0.8685889638 2z- 1 -- 1 & 2z+ 3(2z+I)" 5(z+ 1)5 By this formula we perceive that the log. of (z 1l) becomes known when that of z is known; but the log. of z is known when z-= 1, 10, 100, 1000, &c. Then the formula will give the logarithms of 2, 11, 101, 1001, &c. After a commencement has been made and the logarithms of a few numbers obtained, the logarithms of others can be deduced from them - hence the formula is used for the prime numbers only. When z is large, over 100, the series converges very rapidly and only two terms need be used. When z is over 2000 only one term is required even for twelve decimal places. The auxiliary logarithms markedA,B,C page 71, were computed by this formula. For example, the log. of 1000 is 3,000000: make z=1000, then (z+1)=1001, and (2z+1)=2001. The formula now readily gives the log. of 1001; and the log. of 1.001 is the same, if we suppose the index or rather make the index 0. Having the log. of 1001, we find that of 1002, and thus we run through A. In the same manner we run through B and C. The greater the number the more readily can its log. be computed. That the learner may fully comprehend the application of the auxiliary logarithms A, B, and C, he must call to mind the following principle. ART. 14. Tile product of any number of factors consisting of one and a small fraction, is very nearly equal to one and the sum of those fractions. Thus the product of (1.0001) (1.00002) (1.000003) is very nearly equal to 1.000123. If this be true, we can immediately separate 1.000123 or any other similar quantity in the following factors. (1.0001) (1.00002) (1.000003) The number 1.00008 may be taken as the product of (1.00002) (1.00005) (1.00001) without any material error. This principle may be proved algebraically, thus: Let a, b, and c represent very small fractions, then the product of (1 +a) ( 1+b)= 1 a +b+ab. 36 SURVEYING. But a and 6 being very small fractions their product ab is still much less, and the material part of the whole product is 1+a+-b. Multiply this by (1 +c) making the same consideration, and we shall have 1+a+6+c for the essential value of (1+a) (1+b) (1+c). Try it by numbers, thus: multiply 1.0001 by 1.00004. 1.00004 1.0001 100004 1.00004 1.000140004 But the value of this is extremely near 1.00014, the sum of unity and the fractional parts of the factors. ART. 15. When the difference of two quantities of the same kind, is very small in relation to the quantities themselves, such a difference is called a differential. Thus, the difference between 1.000140004 and 1.00014 is.000000004, and it may be called the differential of 1.00014, and in reference to it may be omitted. The difference between 8 and 9 is 1, but in this case 1 cannot be considered the differential of 8, it is too large. But the difference between 8000 and 8001 is 1, and here 1 is sufficiently small to be considered as the differential of 8000. There is no exact line of demarkation where a difference may be taken for a differential, that depends on the nature of the case; hence the prejudice in a certain class of minds against the calculus. Now, if we take the logarithmic formula from Art. 13, and conceive z to be very large, then the difference between (z- -1) and z which is 1, may be considered as the differential between the two numbers; and in that case log. (z+ 1)-log. z, is the same as the differential of the logarithm of z. Making this supposition, the formula in Art. 13 becomes (dif.) log.z =0.8685889638 X (dif.) z (2z~1) We take only one term of the series, because the other terms are of no essential value, compared with the first; moreover, as z is very large, 2z+-1 is comparatively so little greater than 2z, that for all LOGARITHMS. 37 practical purposes it may be taken as 2z; this being admitted, the preceding equation reduces to (dif.) log. z- 0.4342944819 (dif.) z z The symbol (dif.) stands for the differential of the quantity. Observe that the decimal 0.4342944819 is the modulus of our system of logarithms. Now this equation put in words, is the following: The differential of a logarithm is equal to the modulus, into the differential of the number divided by the number. This equation also gives (dif.) z z (dif.) log. z 0.4342944819 Or in words, The diferential of a number is equal to the number into the dcifferential of the logarithm of the number divided by the modulus. The practical use of these principles will be shown in the following articles. ART. 16. When the diameter of a circle is 1, the circumference is 3.14159265359. Find the log. of this number, true to at least ten decimal places. When the logarithm is found its index will be 0. Now, consider the digits as composing one whole number, and then pay no attention to the index. Take the three superior digits 314. Its factors are 2.157. (Table commencing on page 67.) Log. 157 2.195899653409 Log. 2 0.301029995644 Log. 314 2.496929649053 Table B. 1.0005 log. 0.000217099966 Prod. 3141570 log..497146749019 Table C. 1.000007 log. 3040058 (7b) 2199099 314157 Prod. 314159199099- log..497149789077 38 SURVEYING. REMARK.-Let the learner take hold of the preceding problem with great deliberation, understand the reason of every part of the process - and make every necessary consideration, and then he will understand how to manage every problem of the like kind. Here then, we have the exact decimal part of the logarithm to these digits. If the value of that superior digit 3 is simply three, then the index to the log. is 0; if it is 30, 1, if 300, 2, &c. Knowing the value to be 3, we shall in the end, put the index at 0. We have obtained the exact logarithm of a certain number, but it is not the number required; it is a number however, very near the number required. Take their difference From 314159265359 Take 314159199099 66260 This difference, great as it may appear by itself, is so small in relation to 314159265359 that it may be taken as its dferential. But corresponding to this differential of the number, there is a differential for the logarithm, which is given by the equation in Art. 15. That is, (dif.) log. (0.4342944819)(66260) 314159265359 It would be a tedious operation to draw out the result of this expression arithmetically. We will, therefore, use the common table of logarithms, which will answer every purpose. We have the logarithm of the modulus as a constant quantity on page 71 of tables; and having the logarithm of the denominator, we have only to look for the logarithm of 66260, the operation stands thus, log. 66260 4.821251 log. m. — 1.637784 4.459035 log. z. 11.497150 Num. 0.000000091596 log. -8.961885 To 0. 497 149 789 077 Add 0. 000 000 091 596 Log. of 3.14159265359 - 0. 497 149 880 673 The factor 1.0005 was obvious enough from inspection; but the LOGARITHMS. 39 other factor 1.000007 is not obvious. The question then arises, how did we find it? Wanting a factor, which with the other factor 314157, would produce the given number, we represented it by x, then 314157x — 314159265359 By division x= 1000007 + The sidereal year consists of 365.2563744 mean solar days. What is the logarithm of this number? We know that the index must be 2, therefore pay no attention to the index during the operation. Take the three superior digits 365: its factors are 73 and 5. 73 log. 1. 863 322 860 120 5 log. 698 970 004 336 Prod. 365 log. 2. 562 292 864 456 Table B, 10007 log. 0.000 303 836 798 Prod. 3652555 log. 2. 562 596 701 254 (1.000002 log. 868 588 (2b) C( 1.0000003 log. 130 287 (3c) 1.00000009 log. 39 087 (9d) Prod. = given number nearly, log.=2. 562 597 739 216 We found these factors by taking the value of x out of the following equation: 3652555x-=3652563744 Whence, x= 1.00000239. And by Art. 14, x=(1.000002)(1.0000003)(1.00000009) In place of these last factors we might have taken the differential equation (Art. 15), and that would have been more direct and to the point. The factor method but approximates to the given number: the differential method comes directly to it, but it will not be so generally understood as the factor method. The differential equation applied in place of the last three factors, gives 2.562 597 740 854 for the required logarithm. But, if very great accuracy was required, the factor (1.000002) should be employed and then the differential equation. When the radius of a circle is 1, the natural sine of 7~ 30' is expressed by the decimal 0.1305261921 what is the logarithm 40 SURVEYIN G. of this number, true to nine places of decimals, consider the decimal a whole number? Take its four superior digits 1305: the factors of these are 15 and 87; therefore,.87 log. — 1. 939 519 252 619.15 log. -1. 176 091 259 059 Prod..1305 log. -1. 115 610 511 678 10002 log. 0. 000 086 850 213 13052610 log. -1. 115 697 361 891 Correction, 306 450 Num..1305261921 log. -1. 115 697 668 341 For the radius of our common tables add, 10 Hence, log. sine of 7 30'= 9. 115 697 668 341 In this manner the logarithmic sine of any other arc can be found when we have its natural sine. The correction was found by the differential equation, Art. 15, thus, (dif.) log. z (0.4342944819) (921) 1305261000 Having the logarithm of the modulus, and of the denominator, we can readily deduce the result by logarithms, using the common table to find the logarithm of 921. 921 log. 2.964260 m. log. -1.637784 Numerator, log. 2.602044 Denom. log. 9.115697 Num..000000306450 log. -7.486347 We give one more porblem of this kind. The mean distance between the Sun and Earth is 1; the greatest distance in 1800 was 1.01685317: what is the logarithm of this number? Take the whole as the whole number, 101685317, and pay no attention to the index during the process. The four superior digits are 1016; the factors of this number are 8 and 127: hence, LOGARITHMS. 41 8 log..903 089 986 992 127 log..103 803 720 956 Prod. 1016 log..006 893 707 948 Table B. 10008 log..000 347 233 698 Prod. 10168128 log..007 240 941 646 Table C. 100004 log. 17 371 430 10168128 40672512 Prod. 1016853472512 log..007 258313 076 From 1016853472512 Take 101685317 30.25 Taking the given quantity as a whole number, our last product exceeds it by 301, which is the differential of the number. The differential of the logarithm will therefore be m. 30.25 z By llog. og. m -1. 637 784 log. 30. 25 1. 480 725 1. 118509 log. z* 8. 007 257 log. 0.0000001292 -7. 111 252 From.007 258 313 076 Take.000 000 129 200 Log. sought 0. 007 258 183 876 From the foregoing problems, the reader will perceive that the logarithm of any number whatever, can be found by these tables - and to any degree of accuracy within ten decimal places. We are now prepared to take the converse problem; that is, Given a logarithm, to find its corresponding number. What number corresponds to the log. 4. 636 747 519 487? Carrying this log. to the table, we find it corresponds to a number a little greater than 43, but the index being 4, the number must be a little over 43000. Let this be one factor of the number sought, and the reason for the following operation must be obvious. * z=101685317, a number now considered as a whole number, its log. is taken approximately, and its index is 8. 4 42 SURVEYING. Given log. 4.636747519487 1st Factor, 43000 log. 4.633468455579 3279063908 2nd Factor, 1.007 table A. log. 3030465635 248598273 3rd Factor, 1.0005 table B. log. 217099966 31498307 4th Factor, 1.00007 table C. log. 30399546 1098761 5th Factor, 1.000002 log. 868588 (2b) 230173 6th Factor, 1.0000005 log. 217145 (5c) 13028 7th Factor, 1.00000003 log. 13029 (3d) The product of these factors is the number required, and that product can be obtained with great facility. Prod. of 1st and 2nd factors=43301 This, by the 3rd 10005 43301 21.6505 43322.65050000 Prod. of 4th, 5th, 6th, 7th 1.00007253 43322.6505 3.032585535 866453010 2166132525 1299679515 Num. required 43325.792691840765 2. What number corresponds to log. 2. 563 785 181 020? Taking this decimal log. to the table, we find that its place is between 36. and 37. and as the index is 2., it must be between 360. and 370., nearer the latter than the former. Suppose it near 366. It is not 367., for the table gives a log. of 367., greater than our given log. LOGARITHMS. 43 366=6 X 61 From given log. 2. 563 785181020 Sub. log. 61 1. 785329 835011 0. 778 455 346 009 log. 6 o. 778 151 250 384 304 095 625 3rd Factor 1.0007 303 836 798 258 827 The product of these three factors is 366.2562, which is very near the number required. When a logarithm is reduced below its sixth decimal place, what is left may be taken as a differential of the given logarithm. This differential will give a corresponding differential, to be applied to the number by using the equation in Art 15. That is (dif.) N N (dif.) log. modulus By the common table of log. Given log. 2. 563 785.0000002588 log. — 7. 412 964 -5. 976 749 log. m -1. 637 784 0.0002182 log. -4. 338 965 Add 366.2562 Number sought 366.2564182 In this manner, the number to any logarithm may be found. EXAMPLES. 1. What number corresponds to the log. 2.204923118054? Ans. 160.29616 2. What number corresponds to the log. 4.133409102? Ans. 13595.93 3. What number corresponds to the log. 3.278902074620? Ans. 1900.64967 14 SURVEYING. C HAP TE PR III. ELEMENTARY PRINCIPLES OF PLANE TRIGONOMETRY. TRIGONOMETRY in its literal and restricted sense, has for its object, the measure of triangles. When the triangles are on planes, it is plane trigonometry, and when the triangles are on, or conceived to be portions of a sphere, it is spherical trigonometry. In a more enlarged sense, however, this science is the application of the principles of geometry, and numerically connects one part of a magnitude with another, or numerically compares different magnitudes. As the sides and angles of triangles are quantities of different kinds, they cannot be compared with each other; but the relation may be discovered by means of other complete triangles, to which the triangle under investigation can be compared. Such other triangles are numerically expressed in Table II, and all of them are conceived to have one common point, the center of a circle, and as all possible angles can be formed by two straight lines drawn from the center of a circle, no angle of a triangle can exist whose measure cannot be found in the table of trigonometrical lines. The measure of an angle is the are of a circle, intercepted between the two lines which form the angle —the center of the arc always being at the point where the two lines meet. The are is measured by degrees, minutes, and seconds, there being 360 degrees to the whole circle, 60 minutes in one degree, and 60 seconds in one minute. Degrees, minutes, and seconds, are designated by o,',. Thus 27~ 14' 21", is read 27 degrees, 14 minutes, and 21 seconds. All circles contain the same number of degrees, but the greater the radii the greater is the absolute length of a degree; the circumference of a carriage wheel, the circumference of the earth, or the still greater and indefinite circumference of the heavens, have the same number of degrees; yet the same number of degrees in each and every circle is precisely the same angle in amount or measure. PLANE TRIGONOM ETRY. 45 As triangles do not contain circles, we can not measure triangles by circular arcs; we must measure them by other triangles, that is, by straight lines, drawn in and about a circle. Such straight lines are called trigonometrical lines, and take particular names, as described by the following DEFINITIONS. 1. The sine of an angle, or an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end. Thus, BF is the sine of the arc AB, and also of the arc BDE. BK' is the sine of the arc BD, it is also the cosine of the arc AB, and BF, is the cosine of the arc BD. N. B. The complement of an arc is what it wants of 90~; the supplement of an arc is what it what it wants of 180~. 2. The cosine of an arc is the perpendicular distance from the center of the circle to the sine of the arc, or it is the same in magnitude as the sine of the complement of the arc. Thus, CF, is the cosine of the arc AB; but CF=GKB, the sine of BD. 3. The tangent of an arc is a line touching the circle in one extremity of the arc, continued from thence, to meet a line drawn through the center and the other extremity. Thus, AH is the tangent to the arc AB, and DL is the tangent of the arc DB, or the cotangent of the arc AB. N. B. The co, is but a contraction of the word complement. 4. The secant of an arc, is a line drawn from the center of the circle to the extremity of its tangent. Thus, CH is the secant of the arc AB, or of its supplement BDE. 5. The cosecant of an arc, is the secant of the complement. Thus, CL, the secant of BD, is the cosecant of AB. 6. The versed sine of an arc is the difference between the cosine and the radius; that is, AF is the versed sine of the arc AB, and D K is the versed sine of the arc BD. For the sake of brevity these technical terms are contracted thus: for sine CAB, we write sin.AB, for cosine AB, we write cos.A.B, for tangent AB, we write tan.AB, &c. 46 SURVEYING. From the preceding definitions we deduce the following obvious consequences: 1st, That when the arc AB, becomes so small as to call it nothing, its sine tangent and versed sine are also nothing, and its secant and cosine are each equal to radius. 2d, The sine and versed sine of a quadrant are each equal to the radius; its cosine is zero, and its secant and tangent are infinite. 3d, The chord of an arc is twice the sine of half the arc. Thus, the chord BO7, is double of the sine BF. 4th, The sine and cosine of any arc form the two sides of a right angled triangle, which has a radius for its hypotenuse. Thus, CF, and FB, are the two sides of the right angled triangle CFB. Also, the radius and the tangent always form the two sides of a right angled triangle which has the secant of the arc for its hypotenuse. This we observe from the right angled triangle CAH. To express these relations analytically, we write sin.+cos. 2=R2 ( 1 ) B2+tan.2=sec.2 (2) From the two equiangular triangles CFB, CAH, we have C F: FB= CA: AH 1 sin. That is, cos.: sin=R: tan. tan.= (3) COS. Also,. CF: CB= CA: CH That is,. cos: =R: sec. cos. sec.=-2 (4) The two equiangular triangles CAH, CDL. give CA: AH=DL: DC That is,. R:tan.=cot: R tan. cot.=R2 (5) Also,. CF: PB=DL': DC That is,. cos.: sin.-cot: R cos. R=sin. cot. (6) By observing (4) and (5), we find that cos. sec.=tan. cot. (7) Or, cos.: tan.=-cot.: sec. The ratios between the various trigonometrical lines are always tl., same for the same arc, whatever be the length of the radius; anc. therefore, we may assume radius of anylength to suit our convenience; and the preceding equations will be more concise, and more PLANE TRIGONOMETRY. 47 readily applied, by making radius equal unity. This supposition being made, the preceding become sin.2+ cos.2= 1 ( 1 ) 1~ tan.2=sec.2 (2) sin. 1 tan.= —- (3) cos.=- (4) cos. sec. tan.=-7 (5) cos.=-sin. cot. (6) The center of the circle is considered the absolute zero point, and the different directions from this point are designated by the different signs + and -. On the right of C, toward A, is commonly marked plus (+), then the other direction, toward E, is necessarily minus (-). Above AE is called (+), below that line (-). If we conceive an arc to commence at A, and increase contin. uously around the whole circle in the direction of ABD, then the following table will show the mutations of the signs. sin. cos. tan. cot. sec. cosec. vers. ist quadrant. + + + + + + + 2d " + - -+ + 3d " - - + + -- + 4th " - + - - + - PROPOSITION 1. The chord of 600 and the tangent 450 are each equal to radius; the sine of 300 the versed sine of 60~ and the cosine of 60~ are each equal to half the radius. (The first truth is proved in problem 15, book 1). On C=, as radius, describe a quadrant; take AD=450, AB =60~, and AE=900~, then BE=30~. Join AB, CB, and draw Bn, perpendicular to CA. Draw Bm, parallel to AC. Make the angle CAH=900, and draw CDIH. In the A ABC, the angle AC-B=60~ by hypothesis; therefore, the sum of the other two angles is (180-60)=120~. But CB= CA, hence the angle CBA= the angle CAB, (th. 15 b. 1), and as the sum of the two is 120~, each one must be 60'; therefore, each of the angles of triangle ABC, is 60~ 48 SURVEYING. and the sides opposite to equal angles are equal; that is, AB, the chord of 600, is equal to CA, the radius. In the A CAH, the angle CAH is a right angle; and by hypothesis, A CH,. is half a right angle; therefore, AHC, is also half a right angle;; consequently, AH=A C, the tangent of 45~= the radius. By th. 15, book 1, cor. Cn —nA; that is, the cosine and versed sine of 60~ are each equal to the half of the radius. As Bn and EC are perpendicular to A C, they are parallel, and Bm is made parallel to O(n; therefore, Bon= Cn, or the sine 30~, is the half of radius. PROPOSITION 2. Given the sine and cosine of two arcs tofind the sine and cosine of thle sunz, and deference of the same arcs expressed by the sines and cosines of the separate arcs. Let G be the center of the circle, CD, the greater are which we shall designate by a, and DF, a less are, that we designate by b. Then by the definitions of sines and cosines, D O —sin.a; G O=cos.a; FI=sin.b; GI=cos.b. We are to find FMf, which is -sin.(a+b); GM]=cos.(a+6); EP=sin.(a-b); GPP=cos.(a —b). Because IN is parallel to D O, the two As GD O, GIYL, are equiangular and similar. Also, the zA FII, is similar to GI-V; for the angle FIG, is a right angle; so is HLVi, and, from these two equi!, take away the common angle NI~L, leaving the angle FIH= GIV. The angles at H and Xi, are right angles; therefore, the A FHEI, is equiangular, and similar to the A GIN, and, of course, to the / GD 0; and the side HI, is homologous to IT, and D 0. Again, as FI=IE, and IK, parallel to FJll, FH=IIKf, and HI=K E. By similar triangles we have aD: DO- GI: ILN. sin.a cos.b That is, R: sin.a=cos.b: IN, or In= —----- Also, GD: GO=-FI: FEH PLANE TRIGONOMETRY. 49 cos.a sin.b That is, R:cos'.a=sin.6: FH, or i - Also, GD: GfO= GI: GN~ cos.a cos.b That is,': cos.a=cos.b: GN, or = cos Also, GD: D O=FI: IH That is, B:sin.a=sin.b: Ii, or IH= B. By adding thie first and second of these equations, we have I1V+FHI F-= sin.(a+b) That is, s. in. (+-b)= sin.a cos.b+cos.a, sin.b By subtracting the second from the first, we have sin.a cos.b-cos.a sin.b sin. (a) —-= R By subtracting the fourth from the third, we have C-fEIH= GM-=cos.(a+6-) for the first member. cos.a cos.b-sin.a sin.b Hence,. cos.(a+b)= - By adding the third and fourth, we have UVN+IH= GVY+NP=P- =P=cos.(a —b) cos.a cos. b+sin.a sin.b Hence,. cos. (a-b) BR Collecting these four erpressions, and considering the radius unity, we have { sin.(a+b) =sin.a cos.b —+cos.a sin.6 7 sin.(a —b) =sin.a cos.b —-cos.a sin.6 8) cos.(a+b) =cos.a cos.b-sin.a sinmb (9) cos.(a-.b)=cos.a cos.6+sin.v sin.6 (10) Formula (A), accomplishes the objects of the proposition, and from these equations many useful and important deductions can be made. The following, aire the most essential': By adding (7) to (8), we have ( 11-; subtracting (8) from (7), gives (12). Also, (9)+(10) gives (13); (9) taken from (10) gives (1 4). rsina.(aatb))+sin. (a —b) =-sin.a cos b (11 sin.(a+b)-sin.(a-b)=2cosa sin.b (12) cos.(a+b)+cos.(a-b)=2cos.a cos.b 13) cos.(a-b) — cos.(a6b)=2sin. a sin.b (14) 5 50 SURVEYING. If we put a+b=A, and a-b=B, then (11) becomes (15), (12) becomes (16), 13 becomes (17), and (14) becomes (18). sin.A+sin.B=2sin.(+B) cos. A-) (15) sin.A-sinB=2cos. (~ ) sin. (A ) (16) cos.A+cos.AB=2cos.( L ) cos. A- ) (17) cos.B —cos.A=2sin. ( A+B sin. (-2n ) (18) If we divide (15) by (16), (observing. that -=tan. and COS. COS.os. =-cot.. o al. as we learn by equations (6) and (5) tiigonomesin, tan. try), we shall have sin.A+sin.B sin. 2 - COS. 2 ) ta. +*- - os t - - X — (19) sin.A-sin.B. ( B-) sin A-B ) t AWhence, sin.A+sin.B: sin. A-sinBa.n:tan.: ta or in words. The sum of the sines of any two arcs is to the dzference of the same sines, as the tangent of the half sum of the same arcs is to the tangent of half their difference. By operating in the same way with the different equations in formula (C), we find, sin.A+sin.B ta. ( )+B) (20) sin.A-sin.B,44-B cos.A+cos.B 2 (22) sin.A-sin.B -cot. A (21 os.B- cos.A 2 (23 sin.A-sin.B=tan. A-B (22) eos.A+cos-.B (D) sin.A-sin.B = A+cot. _!~B (23) cos.B —-cos.A,an. (A,B) co. Q+ PLANE TRIGONOMETRY 51 These equations are all true, whatever be the value of the arcs designated by A and B; we may therefore, assign any possible value to either of them, and if in equations (20), (21) and (24), we make B= 0, we shall have, I+C,.Atan2.2X- tiA (25) 1 -~ cosA=a =2 cot'~A sin.A Aot 1( =cot. —-o (26) 1-cos.A 2 tan.JA 1+cos.A _ot.4A 1 (27) 1-cos.A tan.- tan.2A ( If we now turn back to formula (~A), and divide equation (7) by (9), and (8) by (10), observing at the same time, that sin-=tan. COS. we shall have, sin a cos.b+cos.a sin.b tan.(a+-b) ta(a )cos.a cos.b-sin.a sin.b sin.a cos.b —cos.a sin.b cos.a cos.b+sin.a sin.b By dividing the numerators and denominators of the second members of these equations by (cos.a cos.b), we find, sin.a cos.b cos.a sin.b,ta.(+b)= cos.a cos.b cos.a cos.b tan.a+tan.b cos.a cos.b sin.sin.b i1-tan.atan.b (28) cos.a cos.b cos.a cos.b sin.a cos.b cos.a sin.b cos.a cos.b cos.a cos.6 tan.a-tan.b tan'(a —) -cos.a cos.b sin.a sin.b I —+tan.a tan.b (29) cos.a cos.b cos.a cos.b If in equation (11), formula (B), we make a —b, we shall have, sin.2a=2sin.a cos.a (30) Making the same hypothesis in equation (13), gives, cos.2a+ 1-=2cos2.a (31) The same hypothesis reduces equation (14), to 1 -cos.2a=2sin2.a (32) The same hypothesis reduces equation (28), to 2tan.a tan.a — =I ------ (33) 52 SURVEYING. The secants and cosecants of arcs are not given in our table, because they are very little used in practice; and if any particular secant is required, it can be determined by subtracting the cosine from 20; and the cosecant can be found by subtracting the sine from 20. PROPOSITION 3. In any right angled plane triangle, we may have the following proportions: 1st. As the hypotenuse is to either side, so is the radius to the sine of the angle opposite to that side. 2d. As one side is to the other side, so is the radius to the tangent of the angle adjacent to the first-mentioned side. 3d. As one side is to the hypotenuse, so is radius to the secant of the angle adjacent to that side. Let CAB represent any right angled triangle, right angled at A. AB and A C are called the sides of the A, and CB is called the hypotenuse. (Here, and in all cases hereafter, we shall represent the angles of a triangle by the large letters A, B, C, and the sides opposite to them, by the small letters a, b, c.) From either acute angle, as C, take any distance, as CD, greater or less than CB, and describe the arc DE. This arc measures the angle C. From D, draw DF parallel to BA; and from E, draw EG, also parallel to BA or 1DF. By the definitions of sines, tangents,: and secants, DF is the sine of the angle C; E& is the tangent, CG the secant, and CF the cosine. Now, by proportional triangles we have, CB: BA=CD: DF or, a: cwR: sin.C) CA: AB —CE: EG or, b: c-R: tan.C Q. E..D. CA: CB- CE-: C6 or, b: aR: sec.CJ Scholium. If the hypotenuse of a triangle is made radius, one side is the sine of the angle opposite to it, and the other side is the cosine of the same angle. This is obvious from the triangle CDF. PLANE TRIGONOMETRY. 53 PROPOSITION 4. In any triangle, the sines of the angles are to one another as the sides opposite to them. Let ABC be any triangle. From the points A and B, as centers, with any radius, describe the arcs measuring these angles, and draw pa, C/D, and an, perpendicular to AB. Then,.. pa==sin.A, mn=sin.B By the similar Ass, Apa and A CD, we have, R: sin.A=b: CD; or, B(CID)=b sin.A (1) By the similar As Bmn and B CD, we have, R: sin.B=a:- CD; or, R(CD)-=asin.B (2) By equating the second members of equations (1) and (2). b sin.A=a sin.B. Hence,. sin.A: sin.B —a:b.. Or,.. a: b=sin A: sin. B Scholium 1. When either angle is 90~, its sine is radius. Scholium 2. When CB is less than A C, and the angle B, acute, the triangle is represented by A CB. When the angle B becomes B', it is obtuse, and the triangle is A CB'; but the proportion is equally true with either triangle; for the angle CB'D= CBA, and the sine of CB'D is the same as the sine of AB' C. In practice we can determine which of these triangles is proposed by the side AB, being greater or less than AC; or, by the angle at the vertex C, being large as A CB, or small as A CB'. In the solitary case in which A C, CB, and the angle A, are given, and CB less than A C, we can determine both of the As A CB and A CB'; and then we surely have the right one. PROPOSITION 5. If from any angle of a triangle, a perpendicular be let fall on the opposite side, or base, the tangents of the segments of the angle areto one another as the segments of the base. 54 SURVEYING. Let ABC be the triangle. Let fall the perpendicular CD, on the side AB. Take any radius, as Cn, and describe the arc which measures the angle C. From n, draw qnp parallel to AB. Then it is-obvious that np is the tangent of the angle D CB, and ng is the tangent of the angle A CD. Now, by reason of the parallels AB and qp, we have, qn: np=AD: DB That is, tan.A CD: tan.D CB=AD: DB Q. E. D. PROPOSITION 6. If a perpendicular be letfallfrom any angle of a triangle to its opposite side or base, this base is to the sum of the other two sides, as the diference of the sides is to the difference of the segments of the base. (See figure to proposition 5.) Let AB be the base, and from C, as a center, with the shorter side as radius, describe the circle, cutting AB in G, A C in F, and produce AC to E. It is obvious that AE is the sum of the sides A C and CB, and AF is their difference. Also, AD is one segment of the base made by the perpendicular, and BD=DG is the other; therefore, the difference of the segments is A G. As A is a point without a circle, by theorem 18, book 3, we have, A.EX AF=ABX A G Hence,.. AB: AE=AF: AG Q. E. D. PROPOSITION 7. The sum of any two sides of a triangle, is to their diference, as the tangent of the half sum of the angles opposite to these sides, to the tangent of half their d~iference. Let ABC be any plane triangle. Then, by proposition 4, trigonometry, we have, CB: A C=sin. A: sin.B Hence, CB+A C: CB-A C=sin.A+sin.B: sin.A-sin.B (th. 9 b. 2) PLANE TRIGONOMETRY. 55 But, tan. ( +2 ) tan. ( - ) =sin.A+sin.B: sin.A —sin.B (eq. (1), trig.) Comparing the two latter proportions (th. 6, b. 2), we have, CB+AC: CB-AC= tan. ( + B tan. 2) Q.E.D. PROPOSITION 8. Given the three sides of any plane triangle, to find some relation which they must bear to the sines and cosines of the respective angles. Let ABC be the triangle, and let the perpendicular fall either upon, or without the base, as shown in the figures; and by recurring to theorem 38, book 1, we shall find CDa' —b2-c 2 CD 2a (1) Now, by proposition 3, trigonometry, we have, B: cos. Cb: CD CD5bcos. C Therefore,. b (2) Equating these two values of CD, and reducing, we have, cos. CR( a Yb -- (t) 2ab (n) In this expression we observe that the part of the numerator which has the minus sign, is the side opposite to the angle; and that the denominator is twice the rectangle of the sides adjacent to the angle. From these observations we at once draw the following expressions for the cosine A, and cosine B. Thus,.. cos..A= (b2- - -a ) cos.B= R(a2+c(2) ) SURVEY NG. As these expressions are not convenient for logarithmic computation, we modify them as follows: If we put 2a=A, in equation (31), we have, cos.A + 1 =2 cos.2 JA In the preceding expression (n), if we consider radius, unity, and add 1 to both members, we shall have, cos.A+ 1- =1+-b 2bc 2bc~-b2+-c2 —a2 Therefore, 2 cos.2 -A (b+c)2 —a2 2bc Qonsidering (b+c ) as one quantity, and observing that we have the difference of two sguares, therefore (b+c)2 —a2=(b+c+a)(b+c-a); but (b+c-a)=bIc+a-2a (b+ca)(b-Jr-c+a —2a) Hence, 2 cos?.2 A= 2bc (b+c+a) ( b+c+a_ a ) Or,. cos.2:A= - By putting 2 —- s, and extracting square root, the final result for radius unity, is cosj.A= s(s —) For any other radius we must write, cos. A= —-- b By inference, cos.jB=JR2s(s-b) Also,. cos.i=4 C- ( C) In every triangle, the sum of the three angles must equal 180~; and if one of the angles is small, the other two must be comparatively large; if two of them are small, the third one must be large. The greater angle is always opposite the greater side; hence, by merely inspecting the given sides, any person can decide at once which is the greater angle; and of the three preceding equations, that one should be taken which applies to the greater angle, whether that be the par. ticular angle required or not; because the equations bring out the PLANE TRIGONOMETRY. 57 cosines to the angles; and the cosines, to very small arcs vary so slowly, that it may be impossible to decide, with sufficient numerical accuracy to what particular are the cosine belongs. For instance, the cosine 9.999999, carried to the table, applies to several arcs; and, of course, we should not know which one to take; but this difficulty does not exist when the angle is large; therefore, compute the largest angle first, and then compute the other angles by proposition 4. But we can deduce an expression for the sine of any of the angles, as well as the cosine. It is done as follows: EQUATIONS FOR THE SINES OF THE ANGLES. Resuming equation (m), and considering radius, unity, we have, a2~b2-c2 cos. C - 2ab Subtracting each member of this equation from 1, gives I Cos. C= I- 2ab ) (1) Making 2a= C, in equation (32), then a=j C, And.. 1-cos. C=2 sin.eCG (2) Equating the right hand members of (1) and (2), 2 sin.2C- _ 2+c2 2ab -C —-(a —b)2 2ab (c+l-a)(c+a-b) 2ab (cba) (c+a-'b) Or,... sin.2C= ab cBut. -a c -+b-a a and c+-a-b c+a+ -b But, *-a and + 2 2 2 2 Put. a +=s, as before; then, sin.+ C= — (-a) (s —) By taking equation (p), and operating in the same manner, we have... sin.=J(s-a)(s-c) From (n).. sin.A =|( b) f 58 SURVEYING. The preceding results are for radius unity; for any other radius, we must multiply by the number of units in such radius. For the radius of the tables, we write R; and if we put it under the radical sign, we must write R2; hence, for the sines corresponding with our logarithmic table, we must write the equations thus,... sin.A R( — sin.jB= S~ac sin. C4=R(sa)(sb) A large angle should not be determined by these equations, for the same reason that a small angle should not be determined from an equation expressing the cosine. In practice, the equations for cosine are more generally used, because more easily applied. In the preceding pages we have gone over the whole ground of theoretical plane trigonometry, although several particulars might have been enlarged upon, and more equations in relation to the combinations of the trigonometrical lines, might have been given; but enough has been given to solve every possible case that can arise in the practical application of the science. By the application of equations (1), (31), and (32), the table of natural sines and cosines has been computed. The operation is as follows. The sine of 30~ is half radius; making the radius unity, equation (1) gives +-cos.2 300=-1 whence cos.2 30-=1 or cos. 30=~-1,J From (32) we have, sin.a= —/ 1-cos. 2a Making 2a=300, then sin. 15~-=(-,/3)= o-0.2588190 4 From (31) we have, cos.a=/-/1+cos. 2a 2 Making 2a=300 as before, cos.a=(~+{-3 )2=0.96592582 Having sine and cosine of 150 the second application of these equations will give the sine and cosine of the half of 15", and so on through as many bisections as we please. PLANE TRIGONOMETRY. 59 Being desirous of giving a full exposition of the formation of table II, we give the following geometrical demonstration of equation 30, by the help of the figure in the margin. Let the arc AD=2a Then DG=sin. 2a, CG=cos. 2a, DI =sin. a, AD=2 sin. a, Cl =cos. a, DB —2D 0 —2 cos. a. The angle DBA being at the circumference, is measured by half the are AD, or by a. Now, by applying proposition 4 to the triangle DBG, we have sin. DBG: 1) G=sin. 900: BD. The sin. DBG=sin. a, and sin. 900~1, the radius being unity; therefore, the preceding proportion becomes, sin. a: sin. 2a= 1: 2 cos. a. Whence 2 cos. a sin. a=sin. 2a. (Same as eq. 30.) PROBLEM. Given the sine and cosine of an arc, to find the sine and cosine of one half that arc. Designate the given arc by 2a, the radius by unity, and whatever be the value of a, equation (1) gives cos.2 a+sin.2a=1 (in) It is proved in proposition 1, that the sine of 30~ is half the radius: therefore, let 2a=300, then sin. 2a=0.5: and equation 30, just demonstrated, gives 2 cos. a sin. a=-0. 5. (n) Add (m) and (n), and extract the square root of both members. Then cos. a+sin. a=1.22474486 (o) Subtracting (n) from (m), and extracting square root, gives cos. a-sin. a=0.70710678 (p) By subtracting, and adding (p) and (o), and dividing by 2, we find sin. a —sin. 15 —0.25881904 cos. a= cos.1 5~ 0.96592582 60 SURVEYING. Now let 2a-15P. Then cos.2 a+sin.2 a= 1. and 2 cos. a sin. a=0.25881904 Operating as before, we find sin. a=sin. 70 30'=0.1305261921 cos. a —cos. 7~ 30'=-0.9914447879 Again, put 2a=70 30' then as before, cos.2 a - sin.2 a= 1 2 cos. a sin. a= —0. 1305261921 These equations give sin. a=sin. 30 45'-0.0654031291 cos. a=cos. 30 45'=0.9978589222 Thus we can bisect the arc as many times as we please. After five more bisections, we have sin. a=sin. 7' 1" 521"'= —0.0020453077 cos. a —=cos. 7' 1" 521"'=0.99999799 As the sines of all arcs under 10', may be considered as coinciding with the arc, and varying with it, we can now find the sine of 1' by proportion. Thus, 7' 1" 52i"': 1':: 0.0020453077: sin. 1 Or, 25312.5'": 3600:: Or, 10125: 1440:: 0.0020453077: sin. 1' Whence sin. 1'-0.0002908882 sin. 2'=0.0005817764 sin. 3'=0.0008726646 In formula (B) equation (11), we find sin. (a+-b) +sin. (a-6)=2 sin. a cos. b Now, if a=3' and b=1' sin. 4'+sin. 2'=2 sin. 3' cos. 1' We have already sin. 2' and sin. 3', and cos. 1' does not sensibly differ from unity, therefore sin. 4'-2 sin. 3' —sin. 2'=0.0011635528 sin. 5'=2 sin. 4' cos. 1 —sin. 3' &c,. &c. to 15' When the sine of any are is known, its cosine can be found by the following formula, which is, in substance, equation (1) trigonometry cos. a= /(1 +sin. a) (1 —sin. a) PLANE TRIGONOMETRY. 61 In formula (A) equation (7) we find that sin. (a+b)=sin. a cos. b+cos. a sin. b Now, if we make a=30~ and b-4' Then sin. a —0.6 cos. a=ja /3=0.8660254 sin. b=0.001 16355 cos. b.=0.999999323 Whence sin. (300 4')1=(0.5) (0.999999323)+(0.8660254) (0,00116355) =0.499999661 + 0.0010007620 =0.501007281 Equation (8) gives sin. (290~ 6')=0.498992041 When the sine and cosine of any arc are both known, the sine and cosine of the half or double of the arc can be determined by equation 30;- and thus, from equations (30), (7), (8), (11), and (1), the sines and cosines of all arcs can be determined. But these sines and cosines are expressed in natural numbers, to radius unity, hence they are called natural sines and natural cosines, and they are all decimals, except the sine of 900 and the cosine of 0~, each of which is unity. To form table II, we require logarithmic sines, and cosines, which are found by taking the logarithms of the natural sines and cosines, and increasing the indices by 10% to correspond to the radius of 10000000000. The radius of this table might have been greater or less, but custom has settled on this value:. To find the logarithmic sine of 1', we proceed thus, Nat, sin. 1'=0,0002908882 log. -4. 463 726 To which add 10. The log. sine of 1', therefore is 6. 463 726 Nat. sin. 3'=0.0008726646 log. -4. 950 847 Add 10. Log. sin. 3' therefore is 6. 940 847 Thus the logarithmic sine and cosine of all arcs are found. After the logarithmic sine and cosine of any arc have been found, the tangent and cotangent of the same arc can be found by equations (3) and (5), and the secants by (4); that is, R sin.a Rcos.a Rtan. a= —. cot. a -— _- sec. acos a sin. a cos. a 62 SU RVEYING. For example, the logarithmic sine of 6~ is 9.019235, and its cosine 9.997614. From these, find tan., cot., and secant. R sin. 19.019235 Cos. - - - - subtract 9.997614 Tan. is - - 9.021621 Rcos. 19.997614 Sin. - - - subtract 9.019235 Cotan. is 10.978379.R2 is - 20.000000 Cos. - - - subtract 9.997674 Secant is - 10.002326 Thus we find all the materials for TABLE II. This table contains logarithmic sines and tangents, and natural sines and cosines. We shall confine our explanations to the logarithmic sines and cosines. The sine of every degree and minute of the quadrant is given, directly, in the table, commencing at 0~ and extending to 45~, at the head of the table; and from 45~ to 90~, at the foot of the table, increasing backward. The same column that is marked sine at the top, is marked cosine at the bottom; and the reason for this is apparent to any one who has examined the definitions of sines. The difference of two consecutive logarithms is given, corresponding to ten seconds. Removing the decimal point one figure will give the difference for one second; and if we multiply this difference by any proposed number of seconds, we shall have a difference corresponding to that number of seconds, above the logarithm, corresponding to the preceding degree and minute. For example, find the sine of 190 17' 22". The sine of 190 17', taken directly from the table, is 9.618829 The difference for 10" is 60.2; for 1", is 6.02X22 133 Hence, 19~ 17' 22" sine is 9.518952 From this it will be perceived that there is no difficulty in obtaining the sin. or tan., cos. or cot., of any angle greater than 30'. PLANE TRIGONOMETRY. 63 Conversely. Given the logarithmic sine 9.982412, to find its corresponding arc. The sine next less in the table, is 9.982404, and gives the arc 730 48'. The difference between this and the given sine, is 8, and the difference for 1", is.61; therefore, the number of seconds corresponding to 8, must be discovered by dividing 8 by the decimal.61, which gives 13. Hence, the arc sought is 730 48' 13". These operations are too obvious to require a rule. When the arc is very small, such arcs as are sometimes required in astronomy, it is necessary to be very accurate; and for that reason we omitted the difference for seconds for all arcs under 30'. Assuming that the sines and tangents of arcs under 30' vary in the same proportion as the arcs themselves, we can find the sine or tangent of any very small arc to great accuracy, as follows: The sine of 1', as expressed in the table, is 6.463726 Divide this by 60; that is, subtract logarithm.. 1.778151 The logarithmic sine of 1", therefore, is. 4.685575 Now, for the sine of 17", add the logarithm of 17 1.230449 Logarithmic sine of 17", is..... 5.916024 In the same manner we may find the sine of any other small arc. For example, find the sine of 14' 214"; that is, 861"5 To logarithmic sine of 1", is,. 4.685575 Add logarithm of 861.5.... 2.935255 Logarithmic sine of 14' 214"... 7.620830 Without further preliminaries, we may now preceed to practical E X AMP L ES. 2. In a right angled triangle, ABC, given C the base, AB, 1214, and the angle A, 510 40' 30", to find the other parts. To find BC. A As radius.. 10.000000:tan.A 510 40' 30" 10.102119:: AB 1214. 3.084219: BC 1535.8. 3.186338 N. B. When the first term of a logarithmic proportion is radius, the resulting logarithm is found by adding the second and third logarithms, rejecting 10 in the index, which is dividing by the first term. In all cases we add the second and third logarithms together; which, in logarithms, is multiplying these terms together; and from that sum f64 SU tVEYING. we subtract the first logarithm, whatever it may be, which is dividing by the first term. To find A C. As sin. C, or cos. A 51~ 30' 40" - - 9.792477:.AX 1214. 3.084219:; Radius - 10.000000 ACI 157. - 3.291742 To find this resulting logarithm, we subtracted the first logarithm from the second, conceiving its index to be 13. Let ABCrepresent any plane triangle, right angled at B. 1. Given A C 73.26, and the angle A 490 12' o0"; required the other parts. Ans. The angle C 400 47' 40", BC 54.46, and AB 47,87. 2. Given AB 469.34, and the angle A 51~ 26' 17", to find the other parts. Ans. The angle C 380 33' 43", BC 588.5, and AC 752.9. 3. Given AB 493, and the angle C 200 14'; required the remaining parts. Ans. The angle A 690 46', BC 1338, and A C 1425. It is not necessary to give any more examples in right angled plane trigonometry, for every distance in the traverse table is but the hypotenuse of a right angled triangle, and its corresponding latitude and departure form the sides of the triangle. If any one should suspect an error in the traverse table, let him test it by computing the triangle anew. OBLIQUE ANGLED TRIGONOMETRY. Of the six parts of a triangle, three sides and three angles, three of them must be given and one of the given parts must be a side. The subject presents four cases. 1. When two sides are given, and an anole opposite one of them. 2. When two sides are given, and the included angle. 3. When one side and two angles are given. 4. When the three sides are given. The principles previously demonstrated are sufficient, and indeed ample, to give all solutions that can come under any one of these OBLIQUE ANGLED TRIGONOMETRY. 65 cases. The operator must use his own judgment in applying these principles. We give an example in each case, which, with the incidental examples, will be sufficient to fix the principles in the mind of the operator. EXAMPLE 1. In any plane triangle, given one side and the two adjacent angles, to find the other sides and angle. In the triangle ABC, given AB=376, the angle A=48~ 3', and the angle B=40~ 14', to find the other parts. As the sum of the three angles of every triangle is equal to 180~, the third angle C must be 180~-88~ 17'-=91~ 43'. INSTRUMENTALLY. Take 376 from the scale, by means of the dividers, and place it on paper; making one extremity of the line A, and the other extremity B. From A, by means of the protractor (or otherwise), make the angle A=480 3', and from B, make the angle B=40' 14'. The intersections of the lines AC, BC, will give the angle C, which being measured will be found to be a little more than a right angle. Take A C in the dividers, and apply it to the scale, and it will be found to be 243; and BC will be found to be 279.8, if the projection is accurately made; but no one should expect numerical accuracy from this mechanical method. N. B. Our figures in the book do not pretend to accuracy, they should be drawn on paper on a larger scale. BY L O GA I T HMS. To find A C. As sin. 91~ 43' - - - 9.999805: AB 376 - 2.575188:: sin. AB 400 14' - - - 9.810167 12.385355: AC243 - 2.385550 Observe, that the sine of 910 43' is the same as the cosine of 1~ 43'. 6 66 SURVEYING. To find BC. As sin. 91~ 43' - - - 9.999805: AB 376. 2.575188:: sin. A 48~ 3' - - 9.871414 12.446602: B C279.8 - - - - 2.446797 EXAMPLE 2, In a plane triangle, given two sides, and an angle opposite one of them, to determine the other parts. Let AD= 1751. feet, one of the given sides. The angle D=310 17' 19", and the side opposite, 1257.5. From these data, we are required to find the other side, and the other two angles. In this case we do not know whether A C or AE represents 1257.5, because A C=AE. If we take A C for the other given side, then D C is the other required side, and DA C is the vertical angle. If we take AE for the other given side, then DE is the required side, and DAE is the vertical angle; but in such cases we determine both triangles. I N STRUMENTALLY. Draw DE indefinitely- from the point D make the angle D=)31~ 17'. AD-1751., but call it 175.1, which -take from the scale. Place one foot of the dividers at D, the other foot will extend to A, thus finding the point A. Take 125.75 in the dividers, place one foot at A. as a center, and with the other strike an arc, cutting DE in C and E. Join A C, AE, and one or the other of the triangles A CD ADE, will be the triangle required. DC and DE applied to the scale, will give onetenth of the required side, and the angle E or D CA, measured, will be one of the required angles. We can also take one hundredth part of the sides, as well as the tenth; this will make no difference with the angles, the triangles thus formed will be similar. In that case AD= 17.51, and the side sought will be 23.64, which can be changed to 2364. OBLIQUE ANGLED TRIGONOMETRY. 67 BY L OGARITHMS, To find the angle E= C. (Prop. 4.) AsAC=AE=1257.5 log. 3.099508: 31~ 17' 19" sin. 9.715460;: AD 1751 log. 3.243286 12.958746 E= C: 460 18' sin. 9.859238 From 1800 take 46~ 18', and the remainder is the angle D CA 1330 42'. The angle DA C= A CE-D (th. 11, b. 1); that is, DA C=46~ 18'-31~ 17' 19"- 15~ 0' 41". The angles D and E, taken from 1800, give D)AE= 102~ 24' 41". To find D C. As sin. D 3io 17'19" log. 9.715460: AC 1257.5 log. 3.099508:: sin. DA C 15~ 0'41" log. 9.413317 12.512825 D: C626.86 2.797165 To find DE. As sin.D 31~ 17'19 9.715460: AC 1257.5 3.099508:: sin. 102~ 24' 41" 9.989730 13.089238: DE 2364.5 3.373778 N. B. To make the triangle possible, A C must not be less than AB, the sine of the angle D, when DA is made radius. EXAMPLE 3. In any plane triangle, given two sides and the included angle, to find the other parts. Let AD=)1751 (see last figure), )DE=2364.5, and the included angle D=41~ 17' 19". We are required to find DE, the angle DAE, and angle E. Observe that the angle E must be less than the angle DAE, because it is opposite a less side. 68 SURVEYING. INS T UME N TAL LY. Take 1)E=236.45 from the scale (as near as possible), and from D draw DA, making the given angle 410 17' 19". Take 175.1 from the scale, in the dividers, and with it mark off DA. Join AE; and ADE will be the triangle in question, and AE applied to the scale will give the tenth part of the side sought; and measuring the angle E with the protractor (or otherwise), will determine its value. BY LOGARITHMS. From 180~ Take D - - 31~ 17' 19" Sum of the other two angles =148~ 42' 41" (th. 11, b. 1) i sum - - - = 74~ 21' 20" By proposition 7, DE+DA: DE-DA_=tan. 74~ 21' 20": tan. i (DAE —E) That is, 4115.5: 613.5=tan. 740 21' 20"': (DAE —E) Tan. 740 21' 20" - - - 10.552778 6'13.5 - 2.787815 13.340593 4115.5 log. (sub.) 3.614423 i(DAE-E) tan. 280 1' 36" 9.726170 But the half sum and half difference of any two quantities are equal to the greater of the two; and the half sum, less the half difference, is equal the less. Therefore, to 740 21' 20" Add 28 1 36.DAE= 1020 22' 56" E= 46 19 44 To find AE. As sin. E46~ 19' 44" - - 9.859323: DA 1751 - - - - 3.243286: sin.D310 17' 19" - - 9.715460 12.958746: AE 1257.2 - - 3.099423 OBLIQUE ANGLED TRIGONOMETRY. 69 EXAMPLE 4. Given the three sides of a plane triangle to find the angles. Given A C=1751, CB=1257.5, AB=-2364.5 If we take the formula for cosines, we will compute the greatest angle, which is 0. INSTRUMENTALLY. Construct a triangle with the three given sides 236.45, 125.75, and 175.1, according to problem 16, chapter 1. The angles then measured will show their value. BY LOGARITHMS. R2 20.000000 s=2686.5 3.429187 s-c=322 2.507856 Numerator, log. 25.937043 a 1257.5 3.099508 b 1751. 3.243286 Denominator, log. 6.342794 6.342694 2)19.594249 i= 510~ 11' 10" cos. 9.797124 C= 102 22 20 The remaining angles may now be found by problem 4. We give the following examples for practical exercises: Let ABC represent any oblique angled triangle. 1. Given AB697, the angle A 810 30' 10", and the angle B 400 30' 44", to find the other parts. Ans. A C534, BC 813, and the angle C 57~ 59' 4". 2. If A C=720.8, the angle A=700 5' 22", and the angle B= 590 35' 36", required the other parts. Ans. AB 643.2, BC 785.8, and the angle C 500 19' 6". 3. Given B1C980.1, the angle A 7~ 26' 26", and the angle B 1060 2' 23", to find the other parts. Ans. AB 7284, A C 7613.3, and the angle C 66~ 51' 11". SURVEYING. SuwrvEna is the art of running definite lines on the surface of the earth, measuring them, and finding the contents of lands; and the subject necessarily includes the measure of surfaces generally. We shall therefore commence with mensuration. Mensuration'is the application of the principles of Geometry, to the measure of surfaces and solids, and when lands are measured it is a part of surveying. We shall be very brief on mensuration proper, because the rules are so simple and obvious. For the demonstration of the rules, we refer to (Legendre and Robinson's Geometry.) All surfaces are measured by the number of square units which they contain. The unit may be taken at pleasure; it may be an inch, foot, yard, rod, mile, &c., as convenience and propriety may dictate. The square unit is always the square or the linear unit. PROBLEM I. To find the area of a square, or a parallelogram. RULE. —Multiply the length by the perpendicular breadth, and the product will be the area. (Leg. b. IV, prop. V. Rob. book I, th. 29). 1. What is the area of a square whose sides are 6 feet 3 inches? Ans. 39-T- square feet. * 2. How many square feet are in a board that is 13"- feet long and 10 inches wide? Ans. 1 square feet. 3. A lot of land is 80 rods long, and 45 rods wide, how many square rods does it contain, and how many acres? Ans. 3600 rods, 22- acres. * NOTE.- Reductions from one measure to another have no reference to the rules here given. (70) MENSURATION. 71 4. A man bought a farm 198 rods long, and 150 rods wide, at $32 per acre; what did it come to? Ans. $5940. PROBLEM II. To find the area of a triangle, when the base and altitude are given. RULE.- Multiply one of these dimensions by half the other, and the product will be the area required. (Leg. book IV, p. VI. Rob. book I, th. 30). 1. How many yards in a triangle whose base is 148 feet, and perpendicular 45 feet? Ans. 370 yards. 2. What is the area of a triangle whose base is 18- feet and altitude 251 feet? Ans. 231w1 feet. PROBLEM III. Investigate and give a rule forfinding the area of a triangle when two sidces and their included angle are given. Let ABC be the triangles, AB, BC the given sides, and B the given angle. Represent the side opposite to the angle A, by a, opposite C, by c, and opposite B, by b. Now a and c are the given sides, and by problem II, the area is ~a(AD) (1) The trigometrical value of AD can be found from the right angled triangle ABD. Thus, sin. ADB: c:: sin. B: AD. That is, 1 c:: sin. B: AD. Whence AD=c sin. B. This value of AD substituted in (1) gives ~ac sin. B= area A (2). This expression is the area of the triangle, and from it we draw the following rule. RULE.- Take half the product of the two given sides and multiply it by the natural sine of the included angle, and the last product will be the area required. 72 SURVEYING. 1. One side of a triangle is 82 feet, another 90 feet, and their included angle is 27~ 31'. What is the area? Ans. 1749.4 square feet. 270 31' Nat. sine. - - - 46201 ac - - - - 3780 3996080 323407 138603 1749.39780 Ans. When we use logarithms we have the following rule: RULE. — To the logarithms of the two sides, add the log. sine of the included angle, and the sum rejecting 10, in the index, is the logarithm of twice the area of the triangle. 2. A certain triangle has one side 125.81, another equal 57.65, and their included angle 570 25', what is its area? Ans. 3055.7. 125.81 log. 2.099715 57.65 log. 1.760799 570 25' sine 9.925626 2 Area, 6111.4 log. 3.786140 3. How many square yards in a triangle, two sides of which are 25 and 21- feet, and their included angle 45~?'Ans. 20.8695. PROBLEM IV. Investigate and give a rule for finding the area of a triangle when the three sides are given. (See figure to problem III). Let A represent the area of any plane triangle, then by problem III A=' ac sin. B. (1) But sin. B=2 sin. ~.B cos. -B. (Eq. 30, trigonometry). Therefore, A-ac, sin. ~B cos. -}B. (2) Now in proposition 8, trigonometry, we find Sin. B /(s-a)( )s —c (3) ac and cos. j B= 18'(s5b) (4) ac PLANE TRIGONOMETRY. 73 The product of (3) into (4) is sin. I Bcos.!B= ls(ss 2 -2 or ac sin. Bcos. -B=-,,s(s —a)(s.-)(s-c). (6) By comparing (2) and (6) we perceive that A= —s( —.a) (s —b)(s —) Here s represents the half sum of a, b, and c, therefore, we have the following rule to find the area when the three sides ~are given. RULE. — Add the three sides together and take half the sum. From the half sum take each side separately, thus obtaining three remainders. fi7ltiply the said half sum and the three remainders together; the gquare root of this product is the ared required. i. Find the area of a triangle whose sides are 20, 30, and 40. Ans. 290.47. 2 sum =45, 1st Rem. =25, 2d —15,. 3d=5. 45,.25. 15.5=_'225.25. 1 — 15.5f/15=675(3.873)= 290.474. 2. How many acres in a triangle whose sides are severally 60, 50, and 40 rods? Ans. 61 nearly. 3. How many square yards are there: in a triangle:whose sides are 30, 40 and 50 feet? Ans. 66AL. 4. There is a triangular lot-of land containing' 8 acres, two of its sides are 64, and- 46 rods respectively; what is:the angle between these side, -and what is the length of the remaining side?:Ans..; The angle is 600 27', or is supplemenht 119' 33'. -The:: side is 57.37 rods, or 95.535 rods; the less angle corresponding iO dtie iesser side. In short, there are two triangles answering to the conditions, the one is ABE, the other ABC. They are equal because they are on the same base and between the same parallels. AE = 57.37, AC = 95.535. 7 74 SURVEYING. PROBLEM V. To find the area of a trapezoid. RULE.- Add the two parallel sides together, and take half the sum. lJldtiply this half um by the perpendicular distance between the sides. Or, The sum of the parallel sides multiplied by their distance asunder will give twice the area. (Leg. book IV, prop. VU. Rob. b. I, th. 31). Rxvaax.- The application of this problem is the most important of any in general surveying, as will appear in the sequel, and if the geometrical theorem is not familiar to the student he should again review it. Ex. 1. In a trapezoid, the parallel sides are 750 and 1225, and the perpendicular distance between them 1540 links: to find the area. 1225 750 1975X 770=152075 square links =15 acr. 33 perches. Ex. 2. How many square feet are contained in a plank, whose length is -2 feet six inches, the breadth at the greater end 15 inches, and at the less end 11 inches? Ans. 131 feet. Ex. 3. In measuring along one side AB of a quadrangular field, that side, and the two perpendiculars let fall on it from the two opposite comers, measured as below, required the content. AP- Io0 links AQ= 745 A4B 1110 CP= 352 DQ 595 Ans. 4 acres, 1 rood, 5.792 perches. - Here we perceive a trapezoid and two right angled triangles. N. B. A chain is 4 rods, and contains 100 links; 10 square chains make an acre. PROBLEM VI. Tofind the Area of any Trapezium. DvIDE the trapezium into two triangles by a diagonal; then find the areas of these triangles, and add them together. MENSURATION. 75 Or thus, let fall two perpendiculars ou1 the diagonal from the other two opposite angles; then add these two perpendiculars together, and multiply that sum by the diagonal, taking half the product for the area of the trapezium. Ex. 1. To find the area of the trapezium, whose diagonal is 42, and the two perpendiculars on it 16 and 18. Here 16+18- 34, its half is 17. Then 42X 17= 714 the area. Ex. 2. How many square yards of paving are in the trapezium, whose diagonal is 65 feet; and the two perpendiculars let fall on it 28 and 33} feet? Ans. 222A- yards. When the sides of a trapezium, and two of its opposite angles are given, the most convenient rule for finding its area is found in problem III. Conceive C(B joined, then the whole figure consists of two triangles and the whole area is found in the following expression (AB X AC X sin.A) + (C2D X D)B X sin. CDB.) EXAMPLE. In the quadrilateral A CDB we have AC 15.7, CD 20.4, DB 14.24, and BA 27.7 rods. The angle A 78~0 15' and the opposite angle CIDB 970 30'. What is the area enclosed? Ans. 356.65 square rods. PROBLEM VII. To find the area of an irregular figure bounded by any number of right lines. RULE. - Draw diagonals dividing the figure into triangles. Find the areas of the triangles so formed and add them together for the area of the whole. Let it be required to find the area of the adjoining figure of five sides. On the supposition. that AC=36.21 EC-39.11 Aa- 4.18 Bb-=4 and d-d-7.26 Ans. 296.129. 76 SURVEYIN G. PROBLEM VIII. To find the area of a long irregular figure like the one represented in the margin, it is necessary to divide it into trapezoids. Then find the area of each one of the trapezoids (by problem V.) and add them together for the whole area, If however the trapezoids have equal distances between their parallel sides we can take a more summary process, which we discover by the following investigation. The trapezoid AEFD=I (a+b) X AE. " " EG, eHF= (b+c)XEG.,, GIKH =I ( c + d) X "IBCK =- (d+-e) X IB. On the supposition that AE, E6S, GI, &c. are all equal to each other the sum of these is! a e X " +b+c+d+) X AE, which represents the area of the whole figure. From this we draw the following rule to find the area of a long and narrow figure bounded by a right line on one side, and a broken or curve line on the other, to which off sets are made at equidistant points along the right line. RtUL. —Add the intermediate breadths or of sets together, and the half sum of the extreme one: then multiply this sum by one of the equal parts of the right line, the product will be the area required, very nearly.* 1. The breadths of an irregular figure at five equidistant places, being 6.2, 5.4, 9.2, 3.1, 4.2, and the length of the base 60, what is the area? Mean of the Extremes 5.2 Sum of 5.4, 9.2, 3.1 17.7 Sum 22.9 One of the equal parts 1 5 114 5 229 Area=343.5 * In case DF, PF, &c. are right:lines we shall have the area exactly, if they are other than right lines the area will be nearly. MENSURATION. 77 2. The length of an irregular figure being 84, and the breadths at six equidistant places 17.4 20.6 14.2 16.5 20.1 24.4; what is the area? Ans. 1550.64. PROBLEM IX. To find the area of a circle, also any sector or segment of a circle. RULE 1. - The area of a circle is found by multiplying the radius by half the circumference. (Leg. book V, prop. 12. Rob. book V, th. 1.) RULE 2. Mfultiply the square of the diameter by the decimal..7854. When the radius of a circle is 1, the length of one degree on the circumference is 0.01745 and the whole circumference is 3.1416. The radius and the circumference increase and decrease by the same ratio, therefore the length of any are corresponding to any radius is easily computed. A sector of a circle is to the whole circle as the number of degrees it contains is to 360. The area of a segment of a circle as FAE,, may be found by first finding the sector FCE, and from it taking the area of the triangle F CE. This same triangle added to the greater sector will give the greater segment. These principles and rules are sufficient to solve the following examples which are given merely as educational Exercises. 1. What is the area of a circle whose diameter is 10? Ans. 78.54. 2. What is the area of a circle whose diameter is 20? Ans. 4 times 78.54. 3. What is the area of a circle whose circumference is 12? Ans. 11.4595. 4. How many square yards are in a circle whose diameter is 3, feet? Ans. 1.069. 5. Find the length of an are of 200, the radius being 9 feet. Ans. 3.141. 78 SURVEYING. 6. Find the length of an arc of 600, the radius being 18 feet. Ans. 18.846. 7. To find the length of an arc of 30 degrees, the radius being 9 feet. Ans. 4.7115. 8. To find the length of an arc of 120 10', or 120~, the radius being 10 feet. Ans. 2.1231. 9. What is the area of a circular sector whose arc is 18~0 and the diameter 3 feet? Ans. 0.35343. 10. To find the area of a sector, whose radius is 10, and arc 20? Ans. 100. 11. Required the area of a sector, whose radius is 25, and its arc containing 147~ 29'. Ans. 804.3986. 12. What is the area of the segment, whose height is 18, and diameter of the circle 50? Ans. 636.375. 13. Required the area of the segment whose chord is 16, the diameter being 20? Ans. 44.728. 14. What is the length of a chord which ctts off one-third of the area from a circle whose diameter is 289? Ans. 278.6716. 15. The radius of a certain circle is 10; what is the area of a segment whose chord is 12? Ans. 16.35. 16. What is the area of a segment whose height is 2 and chord 20? Ans. 26.88. 17. What is the area of a segment whose height is 5, the diameter of the circle being 8? Ans. 33.0486. PROBLEM X. To find the Area of an Ellipse. RuLE. — Multiply the two semi-axes together and their product by 3.1416. (See conic sections). 1. Required the area of an Ellipse whose two semi-axes are 25 and 20. Ans. 1570.8. 2. The two semi-axes of an Ellipse are 12 and 9, what is its area? Ans. 339.29. To find the area of any portion of a parabola we multiply the base by the perpendicular height, and takce two-thirds of the product for the area required. (See conic sections). MENSURATION OF SOLIDS. 79 Required the area of a parabola, the base being 20, and the altitude 30. Ans. 400. The surfaces of prisms, cylinders, pyramids, cones, &c., are found by the application of the preceding rules. From theorem 16, book VII, Geometry, we learn that The convex surface of a sphere is equal to the product of its diameter into its circumference. The surface of a segment is equal to the circumference of the sphere, multiplied into the thickness of the segment. In the same sphere, or in equal spheres, the surfaces of diferent segments are to each other as their altitudes. MENSURATION OF SOLIDS. BY the Mensuration of Solids are determined the spaces included by contiguous surfaces; and the sum of the measures of these including surfaces, is the whole surface or superficies of the body. The measures of a solid, is called its solidity, capacity, or content. Solids are measured by cubes, whose sides are inches, or feet, or yards, &c. And hence the solidity of a body is said to be so many cubic inches, feet, yards, &c., as will fill its capacity or space, or another of an equal magnitude. The least solid measure is the cubic inch, other cubes being taken from it according to the proportion in the following table, which is formed by cubing the linear proportions. Table of Cubic or Solid.Measures. 1728 cubic inches make 1 cubic foot 27 cubic feet 1 cubic yard 166- cubic yards 1 cubic pole 64000 cubic poles 1 cubic furlong 512 cubic furlongs 1 cubic mile. As the mensuration of solids has little to do with surveying or navigation, we shall leave this subject after simply stating the following truths, which are demonstrated in solid geometry. In fact, these truths may be called rules for practical operations. 80 SURVEYING. 1. The solidity of a cube, parallelopiped, prism, or cylinder, is found by multiplying the area of its base by the altitude. 2. The solidity of a pyramid or cone is found by multiplying the base by the altitude, and taking one-third of the product. 3. The solidity of the frustum of a pyramid or cone is found by calculating the solidity of the pyramid when complete, and subtracting from it the solidity of the part removed; or by multiplying the top and bottom diameters together and that product by the altitude, and then to this last product adding a sum produced by squaring the difference between the top and bottom diameters and multiplying it by one-third of the height. 4. Guaging is performed by considering a cask to be made up of two frustums of cones placed base to base, and applying the rules for the measurement of such solids. 5. The solidity of a sphere is two-thirds of the solidity of its circumscribing cylinder. CHAPTER I. MENSURATION OF LANDS. LANDS are; not only measured to find their areas, but their exact positions must be ascertained, the direction which each line makes with the meridians, or with the north and south lines on the earth. The boundaries of each tract of land are referred to that meridian which runs through or by the side of it. All meridian lines meet at the poles, therefore they are not parallel, (except at the equator,) but the poles are so far distant that no sensible error can arise from supposing them parallel, and all surveys are made on the supposition that the surface of the earth is a plane and the meridians parallel. When large surveys are made, like a county or a state, the spherical form of the earth should be taken into consideration. Meridian lines in surveys are usually determined by the magnetic needle, but the needle does not settle exactly north and south, gen MENSURATION OF LANDS. 81 erally speaking, and the direction which it does settle is called the magnetic meridian. Surveys are often made by the magnetic meridian as the true one, and this would answer every purpose, provided the difference between the magnetic and true meridians were every where and at all times the same, but this is not so. The magnetic meridian is variable, and for this reason it is very difficult to trace old lines, unless visible monuments are left, or unless the record refers to the true meridian. Lines are generally measured by a chain of 66 feet or 4 rods in length, containing 100 links, each link is therefore 7.92 inches. The area of land is estimated in acres and hundredths, formerly in acres, roods, and perches, but the modern method is more simple and convenient; we have a clearer conception of 35 hundreths of an acre than we have of 1 rood and 16 perches. JAn acre is eqtal to 10 square chains or 100,000 square links. We may note down the length of a line in chains and hundreths, or in links only, for it is nearly one and the same thing: thus, 12 chains and 38 links may be written 12.38, or 1238 links. The area of a field may be found by measuring with the chain only, and dividing it into rectangles and triangles, and computing each of them separately, according to the rules laid down in mensuration. The most common method for measuring a field for calculation, is, to take the length of all the sides of the field with the chain, and their bearings with the surveyor's compass. With these notes an accurate plan or plot of the field may be made on paper, and then its contents ascertained by cutting it into triangles and measuring their bases and perpendiculars with a scale and dividers. A very little instruction from a teacher will enable the student to practice this method with success; yet no instrumental measures p2retend to be numerically accurate, they are but app2roximately so. TO MEASU RE A LINE. Provide a chain and 10 small arrows or marking pins to fix one into the ground, as a mark, at the end of every chain; two persons take hold of the chain, one at each end of it; and all the 10 arrows 82 SURVEYING. are taken by one of them who goes foremost, and is called the leader; the other being called the follower, for distinction's sake. A picket, or station-staff being set up in the direction of the line to be measured, if there do not appear some marks naturally in that direction, they measure straight towards it, the leader fixing down an arrow at the end of every chain, which the follower always takes up, as he comes at it, till all the ten arrows are used. They are then all returned to the leader, to use over again. And thus the arrows are changed from the one to the other at every 10 chains' length, till the whole line is finished; then the number of changes of the arrows shows the number of tens, to which the follower adds the arrows he holds in his hand, and the number of links of another chain over to the mark or end of the line. So, if there have been 3 changes of the arrows, and the follower hold 6 arrows, and the end of the line cut off 45 links more, the whole length of the line is set down in links thus, 3645. In all these measures horizontal distances are required, and they are obtained, at least very nearly, by holding the chain in a horizontalposition, both on ascending and descending ground. If the declivity is too great to admit of measuring a whole chain at a time, take a part of it, and in all cases the proper position of the elevated extremity should be determined by a plumb line. The reason of these operations is obvious by the adjoining figure; we require the line AB, and not the line along the ground as A C. AB=ab+cd+fC. It is not only necessary to measure lines but we must also know their direction or the angles which they make with the meridian. This is commonly determined by means of the SURVEYOR' S COMPASS. The surveyor's compass consists of a horizontal circle to which are attached sight-vanes and a magnetic needle delicately balanced on its centre. When the compass is set, that is, standing in a free horizontal MENSURATION OF LANDS. 83 position and the needle free to move on the center, the needle will keep the magnetic meridian, and the circular plate may be turned under it to bring the sight-vanes to any line; - the needle will then point out the degree of inclination which the line makes with the meridian. It is important that this part of the subject be most clearly understood by the learner, we therefore give the following minute illustration of it. Let the reader now face the north with the book open before him, his right hand is then toward the east, and his left hand toward the west. The following figure represents the compass set to the magnetic meridian, that is the sight-vanes Vv and the needle, lie in the same direction. The degrees on the plate are numbered both ways from Nand S to E'and W. At the first view of this subject, it has surprised many to find WVfor west on the right hand toward the east, and E in the direction toward the west. The reason of this is explained by the next figure. Suppose we wished to find the direction of a line from the center of the compass to the object B. We set the compass, that is place it horizontal on its staff or tripod, the needle will take the same direction as in the first figure, parallel to the margin of the paper. The sight-vanes Vv are turned toward the object B which turns the whole plate, but the needle retains its position. 84 SURVEYING. We now read the degree pointed out by the north end of the needle, and we find it to be about 500 on the are between XN and E, showing that the course or direction from the center of the compass to B is North about 500 toward the East - a result obviously true. Turning the sightvanes toward the north west will bring the are between N and TV to the north point of the needle. For example, if it were required to run a line North 310 West, from a certain point, all we have to do is to set the compass over that point, level the plate, see that the needle is free to move on its pivot, and so turn the plate that the north end of the needle will settle at 31~ between NV and W, the range of the sight-vanes will then show the required line. Proceed in the same manner to find any other line. Care should be taken that no iron or steel comes near the compass while operating with it. To insure a correct position of the needle is the principal difficulty, but; if it settles with a free motion, describing iearly equal arcs, slowly decreasing on each side of a given point and finally rests at that point, it operates well, and may be relied upon. In whatever direction we run, the north point of the needle should always lie on some part of the north side of the plate, that is, nearer to N than to S and this can always be except when we run due east or west per compass. Lines should be tested by taking back sights or reverse bearings, which will be exactly in the opposite point of the compass, in case there is no local attraction to disturb the needle. If the line just MENSURATION OF LANDS. 85 run over does not correspond to the exact opposite point of the compass, it shows carelessness in running or some local attraction of the needle. We shall show how to overcome this last difficulty further on. Compasses are usually marked to half degrees, some of them are subdivided to one fourth degrees, but by the aid of a vernier scale we can theoretically read the arc to one minute of a degree. DESCRIPTION OF THE VERNIER. The vernier to a compass is on the outer edge of the graduated limb. It is a slip of metal made to fit the graduated limb of an instrument, and the equal divisions upon it are so made that n divisions on the vernier will cover n-t~1 divisions on the limb. The vernier of the compass is on the outside of the dial plate and it is firmly attached to the bar that holds the sight-vanes. The dial plate can be moved to and fro along it by means of a screw. The vernier is used when the needle points between two divisions on the limb: the dial plate is then gently moved by the screw until the needle points exactly to the preceding division on the limb, this being done, that division on the vernier which makes a right-line, that is, coincides with a division on the are is the number of small divisions (minutes) to be added to the division on the limb now pointed out by the needle. Practical and experienced men, never use the vernier of the compass, because they can read the compass without it to greater accuracy than they can really run a line. But as the vernier scale is of the greatest importance attached to several other instruments, which will be referred to in this work, we now make an effort to give the learner a clear comprehension of it. Let AB represent a portion of an are and ED the vernier whice is conceived to be attached to an index bar and made to revolve with it. In case 0 on the vernier makes a right line with 100 on the are as is represented in the figure, then the index marks 10~. But if 0 on the vernier is a little beyond 100 we then look along the vernier 86 SURVEYING. scale to see what division of it makes a right line with some division on the limb, suppose the division 8 on the vernier coincided with a division on the limb then the index would mark 10~ 8'. To understand the philosophy of this: Let x represent the value of a division on the vernier and n the number of them which cover (n —1) divisions on the limb, then nx=n- 1 1 n 1 n -— 2-2x n — 3 -3x &c. &c. to n, number, from which it appears that one division on the vernier beyond a division on the limb corresponds with the nth part of the unit of graduation, two divisions of the vernier above two divisions on the limb, correspond with 2nth division of the unit of graduation, &c. In our figure n=30, the graduation of the limb is to half degrees or 30 minutes, and this vernier measures minutes. Verniers on many instruments measure as small as ten seconds of are and on some very large instruments as low as four seconds. MENSURATION OF LANDS. 87 CHAPTER II. Having shown in the preceding chapter how to use a compass - to run lines, and to measure them; the next step is to keep a proper record of all the lines run, and compute the areas they enclose. A line traced on the ground, is called a course, the angle that it makes, with the meridian passing through the point of beginning, is called its bearing. A course written NV420 E, indicates that the line runs between the north and the east, and makes an angle of 42~ with the meridian; when between the north and west, we write N. W., putting the number of degrees and minutes between. Lines from the south point, are also written S. E. and S. W.; that is, bearings are reckoned from the north and south points, east and west, as the case may be. Hence, to make a record of a survey, all we have to do is to write the bearing and distance of each course, and if the last side runs to the point of beginning, it is a complete survey; otherwise it is not. Of course, no area can be attached to any un-enclosed space. To complete a partial survey, to enclose a space, to find an area, or to test the accuracy of a complete survey; the most satisfactory method of investigation, is that known as LATITUDE AND D:EPARTURE. Latitude is the distance of the end of a line north or south of its beginning, measured on a meridian, and it is called either northing or southing, according as the line runs north or south. Departure is the distance of the end of a line east or west of its beginning, measured perpendicular to a meridian, and it is called easting or westing, according as the line runs east or west. For example, suppose that we have the following bearings and distances, which enclose a space represented by AB CD. Bearings. Distances; AB N. 230 E. - - 17 BC Y. 830 E. - - 11 CD S. 140 E. - - 23 DA V: 770~ W. - - 2366 S URVEY ING. Let NS represent the meridian running through A, the most western point of the field; make the angle IAB —23~, and AB-17; then Ab is the latitude, and 1b is the departure, corresponding to the course AB. By means of the right angled triangle ABb, having the hypotenuse AB, and the angles, we can compute Ab, and Bb, 16.65, and 6.64, or we can turn to the traverse table, and under 23~ and opposite 17, we shall find the value of these lines at once; and this is the utility of having the traverse table. In the same manner we find Bm and m C, the latitude and departure corresponding to the bearing and distance of the line BC. We find Bm= 1.34, and mnC=10.92. Thus we go round the field, taking the latitude and departure of each side, and arrange the whole in a table as follows: Bearings. Dist. N. S. E. W. AB E. 23~.E 17 15,651 6,64 B C V. 830 E. 1 1,34 10,92 CD S. 140 E. 23 22,32 5,56 DA -. 770 W 23,66 5,33 23,05 ____ 22,32 22,32 23,12 23,05 When the several operations are performed with perfect accuracy, the sum of the northings will be equal to that of the southings, and the sum of the eastings to that of the westings. This necessarily follows from the circumstance of the surveyor's returning to the place from which he set out; and it affords a means of judging of the correctness of the work. But it is not to be expected that the measurements and calculations in ordinary surveying will strictly bear this test. If there is only a small difference, as in the above example, between the northings and southings, or between the eastings and westings, it may be imputed to slight imperfections in the measurements. Here the northings and southings agree, hut the eastings are a little greater than the westings; we will therefore decrease the MENSURATION OF LANDS. 89 eastings by half the error, and increase the westings by the same amount; the sums will then agree. We do this without any formal statement, but the operation is strictly that of proportion; the greater the line the greater the correction to be applied. When the errors are considerable, a re-survey should be made, and if the errors are still great, and in the same direction, there is reason to suspect that some local attraction disturbs the free action of the needle; and then, if the importance demands it, a survey can be taken without the compass, by methods we shall explain in some following chapter. We shall make use of this example and this figure to illustrate the TAKING OF ANGLES BY THE COMPAS S. For this, and for several other operations in practical mathematics, the learner must not expect a written rule: original principles are far more simple and reliable. We now require the angle ABC; conceive the AB to be produced, then the angle between BC and the produced part, is 830 less 23~, or 600. Now 600 taken from 180~ gives 1200 for the angle ABC. Again the line BC makes an angle with the meridian toward the north of 830~, therefore toward the south it must be 970 on the east side of it. The line BA makes an angle of 23~ with the meridian on the west side of it; therefore, the angle ABC = 97+-23 - 120, the same as before. To find the angle BCD, we add 83~ and 14~. Why? To find the angle CDA, we subtract 14~ from 770. Why? To find the angle DAB, we add 14~ to 770~. The sum of these 4 angles must equal 4 right angles. Suppose now that the surveyor runs the lines AB, BC6, CD, and then wishes TO CLOSE THE SURVEY. To close a survey is to run the last side so as to strike the first point, when we are not able to see it. To accomplish this, we sum up the latitude and departure as far 8 90 SURVEYING. as the point D, the result will show Dd and dA. Having then two sides of the right angled triangle, the angle dAD will be the bearing for dAD = nDA, because nD and NS are parallel. The side DA can also be computed, but it should be measured also, as a test to the accuracy of the whole survey. If, on running D)A, according to computation, we actually strike the point A, or very near to it, and there is little or no difference between actual measure and computation, then we may be sure that all the sides have been run correctly; but if on running DA, we do not strike A, or the distance does not correspond to computation, we may be sure of errors somewhere - either in want of skill or care in the operation, or the action of the compass has not been uniform at all the angular points. In case of material errors a re-survey should be made. In case that we have no means, of making computations in the field, we may take a course as near the true one as our judgment will permit, and run it. This line must bring us near the point of beginning, if it does not strike it, and when we get opposite to that point we must measure to it at right angles from the line run; then we shall have data to correct our course. Running a line thus by guess work, is called running a random line, from which the true line can be found as follows: Suppose that when we arrive at.D, we judge the course to the first point to be N. 750 W., and run that course, and after measuring 28.65 chains we find that we are passing the first point, which is 83 links in perpendicular distance toward the south; what course should have been taken? By the following investigation we draw out a rule that will apply to all such cases. Let AB represent a true course, AD a random line, and PDB its amount of deviation. Also let.DAE equal one degree, and take Ad= 1, then the deviation at d will be the natural sine of one degree, and may be taken from the. table of natural sines (which is.01745). By proportional triangle we have 1:.01745:: AD: DE; Whence DE= —0.01745(AD). Now DE is contained in PDB as often as 1~ is contained in the MENSURATION OF LANDS. 91 number of degrees in the angle DAB. Let x represent the number of degrees in DAB, then x (DB) 1.01745(Dj)) That is, x_(DB)(57.3) because 1 57.3. (AD) I 5 Hence, to correct a course, we have the following RULE.- -.Multiply the deviation by 57.3, and divide that product by the distance, aud the quotient will be the number of degrees and parts of a degree to add to, or subtract from, the random course. This rule, applied to the present example, gives.83(57.3) _2 23.65 Hence, the true course is 75~+2~=77~. Had the deviation of the random line been toward the south, we should have subtracted the correction; but for this the operator must rely on his judgment. We now come to the COMPUTATION OF AREAS. By inspecting the figure we perceive that c CDd is a trapezoid, from which, if we subtract the triangles ADd, ABb, and the trapezoid bBCc, the area of the field ABCD will be left. OBSERVATION.- To preserve uniformity of expression, and clearness and brevity in forming a rule, we shall call triangles, trapezoids, while discussing this subject. A trapezoid becomes a triangle, when its smallest parallel side is so small as to call it zero - conversely, then, a triangle is a trapezoid, whose smallest parallel side is zero. We observe that C is the most northern point of the field, and.D is the most southern. In traversing from C to D, from the north to the south, we pass along the oblique sides of trapezoids that we shall call south areas, and in traversing from D to A, B, and C', 92 SURVEYING. from the south to the north, we pass along the oblique sides of trapezoids, which we shall call north areas. Now it is obvious that if we subtract the surm of the north areas from the sum of thIe south areas, we shall have a remainder equal to the area of the field. We now require a systematic method of finding these areas, or the area of these several trapezoids. In the first place, we must have latitudes and meridian distances. Latitude and departure have already been defined and explained. MERIDIAN DISTANCES. Meridian distances are the distances of the angular points of the field from the meridian which runs through the most westerly point of the field; thus, bB, cC, dD, are meridian distances. Double meridian distances are the bases of triangles, or the sum of the parallel sides of the trapezoids. Thus bB is the double meridian distance of the side AB, or it is the double meridian distance of the middle point of the line AB. bB + cC is the double meridian distance of the line BC, or double the meridian distance of the middle point of BC. This double meridian distance (bB+cC) multiplied by bC, or Bn?, the latitude corresponding to BC, will give the double area of the trapezoid bB, Cc. We are now prepared to give the following summary or rule for finding the area of any field bounded by any number of right lines: RuzE. 1. Prepare a table headed as in the example, namely: Bearings, Distance, North, South, East, West, fMeridian distance, Double meridiin distance, North areas, South areas. 2. Begin at the most western point of thefield, and conceive a meridian to pass through that point. Find, by the traverse table or by trigonometry, the northings, southings, eastings, and westings of the several sides of thefield, and set them in the table opposite their respective stations, under their proper letters AV., S., E., or W. 3. For the first meridian distance take the departure of the first line; for the second, take the first meridian distance and add to it the departure of the second line, if the departure is east, or subtract if west, &c. MENSURATION OF LANDS. 93 4. Add each two adjacent meridian distances, and set their sum opposite the last of the two in the column of double meridian distances. 5. Multiplg each double meridian distance by the latitude to which it is opposite, and set the product in the column of N. areas, if the latitude is north, and in that of S. areas, if the latitude is south. 6. Subtract the sum of the N'. areas from that of the S. areas, and take half the remainder, which will be the area of the field in square chains. Dividing this by 10 gives the acres; and the roods and rods are found by multiplying the decimal parts by 4 and by 40. Bearings. Dis. N. S. E. W. M. D. D. M.D. N. areas. S. areas AB N 230 E 17 15.65 - 6.64 -6.63 6.63 103.76 (6.63) BC N 83~ Ell 1.34 10.92 17.53 24.16 32.37 (10.90)1 CD S 14~ E 23 22.32 5.56 23.08 40.61 9.06.42 (5.55) DA N 770 W 23.66 5.33 23.05 0.00 23.08 123.02 (23.08) 23.08 259.15 906.42 259.15 Diff., 647.27 Half, 323.63 Dividing by 10, 32.363 Hence the field contains 323 square chains and 63 hundredths, or thirty two acres and a little more than 36 hundredths of an acre. The numbers in parentheses, as (6.63), and all others in parentheses are the numbers corrected to make the eastings and westings agree, -the numbers above them are taken from the table. Before we give any more examples, it is proper to give some examples to show the practical utility of the TRAVERSE TABLE. This table is computed to every half degree, but if a course is between two courses in the table, the operator can use his judgment and take out the proper intermediate numbers. Those who are not satisfied with this method, can use the table of natural sines and cosines, as we shall subsequently explain. The distances are consecutive to 30, then 35, 40, &c., to 100. But a little thought in the operator will enable him to use the table for any distance whatever. 94 SURVEYING. The following examples will illustrate. 1. A course is N. 220 30' E. distance 62.43; what is the corresponding latitude and departure, as found in the table? We shall regard the distance as 6243'links, andseparate it into parts. Lat. Dep. Thus: 6000 5543 2296 240 221.7 91.8 3 2.77 1.15 6243 5767.47 2388.95 If we now return to chains and links, the latitude is 57.67 and the departure 23.89. We entered 240 in the table, as 24 chains, and took the numbers corresponding. 2. A course is N. 480 30' W., distance 187.61; what is the corresponding latitude and departure? Dis. Lat. Dep. 180.00 119.30 134.80 7.60 5.036 5.692 1 066 075 187.61 124.3426 139.4995 If we now take the distance as 187 chains and 61 links, the Lat. is 124 ch. 34 links, and the Dep. is 139 ch. and 50 links. 3. A course is S. 810 W., distance 76.87; what is the corresponding latitude and departure? Dis. Lat. Dep. 75.00 1173. 7408. 1.80 28.2 177.8 7 1.10 6.91 76.87 1202.3 7592.71 If 76 ch. 87 lin., Lat. 12 ch. 2 lin., Dep. 75 ch. 93 links. Thus we can find the latitude and departure for any distance corresponding to any degree and half degree. We can find it to any degree and minute of a degree by the table of natural sines and cosines. The common tables containing natural cosines and sines are nothing more than latitude and departure corresponding to unity of distance. MENSURATION OF LANDS. 95 Therefore, a double distance will correspond to a double distance in latitude and departure, a treble distance will give a treble amount of latitude and departure, and so on in proportion. Lat. = Nat. cosine. Dep. = Nat. sine. Hence: The natural cosine of any course taken as a decimal, multiplied by any distance, will give the latitude corresponding to that course and distance. Also, the natural sine taken as a decimal, multiplied by a given distance, will give the departure corresponding to that course and distance. N. B. Nat. sines and cosines are found in table II., pages 21-65 of tables. For common purposes, four places of decimals are sufficient. 1. The bearing of a certain line is N. 350 18' E.; distance 12 chains; what is the corresponding latitude and departure? Angle 350 18' N. cos..81614 N. sin..57786 Dis. (multiplier) 12 12 Diff. Lat= 9.79368 Dep. 6.93432 2. A certain line runs S. 4~ 50' E.; distance 74.40; what is the corresponding latitude and departure? Angle 40 50' N. cos..9964 N. sin..0842 Distance 74.4 74.4 39856 3368 39856 3368 69748 5894 Lat. 74.13216 Dep. 6.26448 3. A line makes an angle with the meridian of 750 41', at a distance of 89.75 chains; what is the latitude and departure? 750 47' cos..2456 sin..9694 Distance 89.75 89.75 Prod. Diff. Lat. 22.042 Dep. 87.001 4. A line bearing N. 70 40' W.; distance 31.20 chains; required the difference of latitude and departure. 7~ 40' cos..98106 sin..13341 Multiplier 31.2 31.2 Diff. Lat. 30.92 Dep. 4.16 96 SURVEYING. 5. A line running S. 80~ 10' E. distance 35.25 chains; what is the difference of latitude and departure? 80~ 10' cos..17078 sin..9853 Multiplier 35.25 35.25 Diff. Lat. 6.02 Dep. 34.72 In the last three examples, we have given only the results of the multiplications to two places of decimals; that is, to the nearest link, which is a degree of accuracy sufficient for all practical purposes. We are now prepared to estimate the areas of the following general surveys, given as EXAMPLES. 1. In May, 1845, the following measures of a field were taken. Beginning at the western-most point of the field; thence,-.N 20~ 30' E. 5 chains 83 links; thence S. 790 45'E. 10 chains 15 links; thence S. 27~ 30' W. 9 chains 45 links; thence 2iV. 63~ 15' tW. 8 chains 28 links; thence N. 15~ 30' W. 1 chain and 4 links, to the place of beginning; required the area. It is not absolutely necessary to make a plot or figure of -Ae field, but for the:ake of perspicuity, it is best to do so; yet no reliance is placed on the accuracy of the constructed figure. We perceive by the figure, that there are two south areas, bBCc, and c CDd; and three north areas, eEdD, AeE, and ABb. Let the reader observe that all the north areas are on the outside of the field. MENSURATION OF LANDS. 97 Bearings. Dis. N. S. E. W. M. D. D.M.D. N. areas. 8. area. A4 N.-20o30'E. 5.83 5.461 _ 2.04 2.04 2.04 11.1384 l BC S.790 45'E. 10.15 1.81 9.99 12.03 14.07 25.4667 CD S.27o30'W. 9.45 8.38 4.36 7.67 19.70 165.08601 DE N.63015'W. 80.28 7.95 29.6535 EA N.15030'W. 104 1.001 0.28 0.00 0.28 0.2300 10.19 10.19 12.03 12.031 41.0719 190.5527 41.0719 2)149.4808 10)74.7204 Area in acres, 7-472The operator can rely on the rule used in the last operation, whatever be the number of sides, or whatever be the shape of the figure, provided that the lines are right lines, from one angular point to another. In case of re-entering angles, like HITA, represented in the adjoiniRg figure, a portion of the figure AIr, is reckoned twice. But this is corrected by the subtractive space iih, which includes not only the exterior portion hE-1; but also the whole additive triangle Ali, belonging to the last side of the figure IA. In the following example are several such re-entering angles. 2. Find the area of a lot of land, of which the following are the field notes. Beginning at the south west comrer, the ancient land mark; thence 1. XN 270 15' E. distance 9.42 chains. 2. $. 800 00' E. " 1.15 " 3. S. 69 00' E. " 12.73 " 4. S. 150 45' W. " 5.00 "..V. 66~0 45'W W. 1.05 " 6. S. 31 00'. " 2.90 " 7. A. 70 45' W. 1" 8.92 " 8. S. 410 45'WV. t" 2.08 " 9. NV. 630 00' " 4.18 " la 98 SURVEYING. We can contract the operation in reference to the space it will occupy, by putting the difference of latitude in one column, and all the departures in another column. Marking all the northings by the sign +, and all the southings by the sign -. Also, all the eastings by the sign +, and all the westings by the sign -. This being understood, the work will appear as follows: Bearings. Dis. Lat. Dep. M. D. D.M.D. N. areas S. areas 1 N. 27015' E. 9.42 — 8.371 4.31 4.31 4.31 36.0747 2 S. 800 00' E. 1.15.-020. 1.13 5.44 9.75 1.9500 3 S. 690 00'. E. 12.73.-4.57 4-11.89 17.33 22.77 104.0589 4 S. 150~45' W. 5.00 -4.81 - 1.36 15.97 33.30 160.1730 5 N. 660 45' W. 1.05 +0.41_ 0.96115.01 30.98 12.7018 6 S. 310 00' W. 2.90 -2.49 - 1.49 13.52 28.53 71,0397 7 N. 700 45' W. 8.92 +2.941- 8.42 5.10 18.62 54.7428 8 S. 410 45' W. 2.08 -1.55- 1.38 3.72 8,82 13.6710 9 N. 630 00' W. 4.18 +1.90 - 3.72 0.00 3.72 7.0680 Sum 0'-00- 0.00 - 110.5873 350.8926 110.5873 2)240.3053 10)120.1526 12 acres, and a small fraction over. 12.015261 3. Having the following field notes, it is required to find the closing side and the area of the field. Bearngs..Dis. N. S. S. W. S. 75~0 W. 13.70 3.54 13.24 2jS. 200 30' W. 10.30 9.65 3.60 3 W. 16.20 16.20 4 N. 330~ 30' E. 35.30 29.51 19.49 51 N 760 E. 16.00 3.87 15.52 6 South 9.00 9.00 33.38 22.19 35.01 33.04 22.19 33.04 11.19 1.97 This result shows, that if we commence at the first station, and traverse round to the sixth, we shall then be 11.19 chains to the north of the place of beginning, and 1.97 chains east of it. This is sufficient data to compute the course and distance. To compute the area, however, it is not necessary to find either the course or the distance. We do know, however, by merely inspecting the traverse table, MENSURATION OF LANDS. 99 that the course totheplaceofbeginning, is south about 10~ 20' west, and distance near 12 chains. We are now prepared to compute the area, and as we wish to commence at the western-most point of the field, we shall begin at the 4th station, calling it the first: thus, l Bearings. Dis. Lat, Dep. I. D. D.M.D. N. area. S. area. 1 N. 330 30' E. 35.30 29.51 9.49 19.49 19.49 575.1499 _ 2 N. 76~ E. 16.00 + 3.87 +15.52 35.01 54.50 210.9150 3 South 9.00 9.00 0. 0 35.01 70.02 630.1800 4 S. W. -_ —11.19 1.97 33.04 68.05 761.4795 5 S. 75 W. 13.70 -- 3.54 -13.24 19.80 42.84 151.6536 6 S. 20~ 30' W. 10.30 - 9 65 - 3.60 16.20 36.00 347.4000 7 W. 16.20 0. 0 -16.20 0. 0 16.20 786.0649 1890.7131 786.0649 2)1104.6482 This result shows that the field contains 55 acres, and 10)552.3241 a little more than 23 hundredths of an acre. 55.23241 4. What is the area of a survey, of which the following are the field notes. Stations. Bearings. Distances. 1 S. 46~ 30' E. 80 rods. 2 S. 51~ 45' V. 34.16 3 West. 85.00 4 N. 56~ W. 110.40 5 N. 330 15'E. 75.20 6 S. 74~ 30' E. 123.80 Ans. 104.35 acres. 5. Required the contents and plot of a piece of land, of which the following are the field notes. Stations. Bearings. Distances. 1 S. 340 W. 3.95 ch. 2 S. 4.60 3 S. 36~0E. 8.14 4 N. 69~ E. 3.72 5 N. 250~ E. 6.24 6 NA. 16~ E. 3.50 7 V. 650~ W. 8.20 Ans. 10A. OR. 5P. 100 SURVEYING. 6. Required the contents and plot of a piece of land, from the following field notes. Stations. Bearings. Distances. I S. 40~ W. 70 rods. 2 N 450 E. 89 3 N.; 360~ E. 125 4 N. 54 6 S. 81~ E. 186 6 S. 8~ W. 137 7 W. 130 Ans, 207A. 3R. 33P. 7. Given the following bearings and distances of the several sides of a field, namely, 1. X., 680 E. 19 ch. 2. E. 60 S. 20 3. S. 170 W. 20 4. W. 20 5, N, 420 3' W. 16.10 to find the area, Ans. 64.9 acres. 8. Given the following bearings and distances, namely, I. N. 405~ E. 40 ch. 2. S. 300 W. 25 3. S. 5~ E. 36 4.'W 29.60. N. 20~ E. 31 to find the corrected difference of latitude and departure, and the area. N. B.-In this last example, as in most others, the northings and southings will not exactly balance; nor will the eastings and westings balance, This arises from inaccuracies in the data. In such cases ( if the errors are but trifling ) we balance off the errors. When a course is east or west, as the 4th in this example, some operators have expressed doubts, whether any correction should be applied to latitude in that course. We reply, that errors ldo really exist, and, therefore, we cannot MENSURATION OF LANDS. 101 say that the course marked west, in the example, was really west, or not; the probability is, that the course was not exactly due west, and it is therefore proper to put a correction in the latitude column, as shown in the following results: CORRECTED LATITUDES AND DEPARTURES. I N. S. E. W. 1. 28.30 28.30 2. 21.63 12.49 3. 35.84 3.16 4. 0.02 29.59 5. 29.15 10.62 57.47 57.47 42.08 42.08 On inspecting these latitudes and departures, we perceive station 5 is the most westerly point of the field, therefore to find the area, we will arrange these results in the following order: Lat. Dep. M. D. D. M. D. N. areas. S. areas. 5 "+29.15 +10.62 10.62 10.62 309.5730 1 +28.30 +28.30 38.92 49.54 1400.9820 2 -21.63 -12.49 26.43 65.35 1403.5205 3 -35.84 + 3.16 29.59 66.02 2007.7568 4 +00.02 -29.59 0. 0 29.59 0.5918 1711.1468 3410.2773 1711.1468 2)1699.1305 10)849.5652 Ans. areas, 84.956 9. What is the area of a survey of which the following are the field notes. From the place of beginning, N. 310 30' TW., distance 10 chains: thence N. 620 45' E., 9.25 chains: thence S. 36~ E., 7.60 chains: thence S. 450 30' W., 10.60 chains, to the place of beginning. Ans. 8-l5-40 acres. 10. Do the following bearings and distances enclose a space? If not, give an additional bearing and distance that will, then determine the area so enclosed. 102 SURVEYIN G. Stations. Bearings. Distances. 1 S. 40~ 30' E. 31.80 ch. 2 7. 540 00'E. 2.08 3 N. 29~ 15' E. 2.21 4 N. 28~ 45' E. 35.35 5 Y. 570 00' TW. 21.10 Ans. These bearings and distances do not enclose a space. A line run from the further extremity of the 5th to the first station will bear south 46~ 43' W., distance 31.21 chains, and the area thus enclosed will contain 92.9 acres. 11. Do the following bearings and distances enclose a space? If not, determine the additional line that will, and the area of the space so enclosed. Stations. Bearings. Distances. I,S. 850 00' W. 46.4 rods. 2 N. 530 30' W. 46.4 " *3 N. 36~ 30' E. 76.8 " 4 N. 220~ 00' E. 56.0 " 5 S. 760 30' E. 48.0 " Ans. These bearings and distances do not enclose a space. A line run from the last station to the first would bear S. 30~ 25' W., distance 128.6 rods. Area 56.86 acres nearly. The operation for the area is as follows: We commence at station 3, for reasons that have been several times explained. Station 6 is the one we supplied. Sta. I Lat. Dep. M. D. D. M. D. N. area. S. area. 3 + 61.73 +45.65 45.65 45.65 2817.9745 4 + 51.92 +20.98 66.63 112.28 5829.5776 5 -- 11.21 +46.67 113.30 179.93 2017.0153 *6 -125.18 -29.85 83.45 196.75 24629.1650 1 - 4.85 -46.15 37.30 120.75 585.6375 2 + 27.59 -37.30 0. 0 37.30 1029.1070 19676.6590 27231.8178 9676.6590 2) 17555.1588 160)8777.5794(54.86. *The stations marked with a * are those supplied. THE MERIDIAN LINE. 103 12. What is the area of a survey of which the following are the field notes. Stations. Bearings. Distances. I N. 750 00' E. 54.8 rods. 2 N. 20~ 30' E. 41.2 " 3 E. 64.8 " 4 S. 330 30' W. 141.2 " 5 S. 760 00' W. 64.0 " 6 N. 36.0 " 7 S. 840~ 00' W: 46.4 " 8 NV: 530 15' W. 46.4 " 9 N. 360 45' E. 76.8 " 10 NY. 220 30' E. 56.0 c" 11 S. 76~ 45' E. 48.0 " 12 S. 150 00' W; 43.4 " 13 S. 160~ 45' W. 40.5 " In this survey 4 is the most easterly and 8 the most westerly station. The area is equal to 1llA. 2R. 23P. It may vary a little, on the account of the way in which the balancing is done. CHAP TER III. ON THE MERIDIAN LINE AND THE VARIATION OF THE COMPASS. THE -meridian is an astronomical line, having no necessary connection with the magnetic needle. Meridians would be primary lines to which we would refer all surveys, if there were no such thing as magnetism, or a magnetic needle. It is only a coincidence, that the magnetic needle settles near the meridian, so near, that for a long time it was considered the meridian itself, but accurate observations have shown that the needle does not point rigorously north and south, but has a variation which is not the same at all times in the same place, therefore a line run by the compass is still unknown in respect to the meridian, 104 SURVEYING. unless we know the variation of the compass, that is, the declination of the needle. In the year 1657 the needle at London pointed due north, since that time its variation has been west, previous to that time the variation was east. In the Atlantic ocean, between Europe and the Uniteal States, the variation is from 12 to 18~ west. The needle seems to point to the region of greatest cold, which is in the northern part of America, and not the north pole, and if this be true, if the point of minimum heat changes its position, there will be a corresponding change in. the direction of the magnetic needle. The needle has a small annual and also a diurnal variation, corresponding to the temperature of the different seasons of the year, and of the different times of the day, but these variations are too small to trouble the common operations of surveying. Some of the variations of the compass are regular, others irregular; some amount to many degrees and require a long period of;ime, others are small in amount and require but short intervals to pass through all their changes. The daily variation consists of an oscillation eastward and westward of the mean position, and is different in different places. Generally the greatest oscillation eastward is between six and nine in the morning, and westward about one in the afternoon, gradually returning toward the east until eight P. M. At night it is stationary. On the subject of magnetism we know nothing, beyond facts drawn from observation, but there is no doubt that the earth is a great magnet, made so by the action of the sun, and the poles of this great magnet are near the poles of the equator.. Indeed, all observations made correspond to this hypothesis, for changes of the weather, clouds, and storms, all have an influence on the needle. These facts are sufficient to convince any reader that to survey correctly we must know the VARIATION OF THEZ COMPASS. As the true meridian is an astronomical line, we must find it by astronomical observations, and then by comparing the meridian of VARIATION OF THE COMPASS. 105 the compass with it, we shall have the variation of the compass. When the sun is on the equator, it rises due east, and sets directly in the west. Should we then observe the direction of its center, just as it was rising or setting, at the time it had no declination, and trace that line a short distance on the ground, we should then have a due east and west line. If from any point in that line we draw another line at right angles, we should then have the true meridian. If we now put the compass on this meridian, and make the sightvanes range with it, the needle will also range with it, if there is no variation, but if the north point of the needle is to the west of the sight-vane, the variation is westerly, if to the east, easterly, and the number of degrees and parts of a degree that the needle deviates from the direction of the sight-vanes shows the amount of the variation. But it is not to be supposed that any particular observer can be at the points and places, where the sun is either rising or setting just at the time the sun is on the equator. We must have a broader basis, and in fact by means of the latitude of the observer and the declination of the sun, any observer has the means of knowing the precise direction in which the sun will rise or set, any day in any year. Let us suppose that the sun on a certain day, observed from a certain place, must have arisen S. 81 E., but by the compass it was observed to rise S. 790 E., the variation of the compass was therefore 2~ west. These observations are called taking an azimuth. Azimuths are often taken at sea to determine the variation of the compass. On land, however, the horizon is rarely visible, and very few observations on sun rise or sun set can be made, besides there are other objections arising from atmospherical refraction; it is therefore best, most convenient, and more conducive to accuracy, to take the sun when up 10, 15 or 250 above the horizon, and observe its direction per compass, and compare the result to the computed bearing for the same moment, and if the two results agree the compass has no variation; if they disagree the amount of such disagreement is the amount of the variation of the compass. 106 SURVEYING. By means of spherical trigonometry the true bearing of the sun can be determined at any time, on the supposition that the observer knows his latitude, the declination of the sun, and its altitude; these three conditions furnish a triangle like PZS. The altitude subtracted from 900, gives ZS, the latitude from 900, gives ZP, and PS is found by adding or subtracting the sun's declination to 900, according as it is north or south. ZS is co-altitude, ZP is co-latitude, and PSis the sun's polar distance; the angle PZS is required, and it can be found by the following rule. 1. Add the three sides of the triangle together and take the half sum. From the half sum, subtract the sun's polar distance, thus finding the remainder. 2. Add the sin-complement of the co-altitude, the sin-complement of the co-latitude, the sine of the half sum, and the sine of the remainder. The sum of these four logarithms divided by 2, will be the cosine of half the azimuth angle. N. B. This rule is the application of equations on page 204 Robinson's Geometry. The sin-complement is the logarithmic sine of an arc, subtracted from 10. EXAMPLES. In latitude 390~ 6' 20" north, when the sun's declination was 120 3' 10" north, the true altitude of the sun's center was observed to be 30~ 10' 40", rising. What was the true bearing of the sun, or its azimuth? 900 ~ 90 90 Lat. 39 6 20 Alt. 30. 10. 40 Dec. 120 3' 10 co-Lat. 50 53 40 co-Alt. 59. 49. 20 PD 77. 56 50 VARIATION OF THE COMPASS. 107 P. D. 77~ 56. 50 co-Lat. 50. 53. 40 sin. com. 0.110146 co-Alt. 59. 49. 20 sin. comrn. 0.063295 2)188 39 60 iS. 94 19 55 sin. 9.997758 77 56 50 Rem. 16 23 5 sin. 9.450376 2)19.622575 49038'30" cosine 9.811287 Bearing, 990 17' 0 from the north, or 80~ 43' from the south. If at the time of taking the altitude of the sun, another observer had taken its bearing by the compass, and found it to be S. 80~ 43' E., then the compass would have no variation, and whatever it differed from that would be the amount of variation. If a line were run along the ground, direct toward the center of the sun, at the time the altitude was taken, and sufficiently marked, that would be a standing line of known direction; and if from any point in that line, we could draw another line, making an angle with it of 990 17' on the north, or 800 43' on the south, such a line definitely marked, would be a permanent meridian line, for all time to come; on which we could at any time place a compass, and observe its variation. Let AS be the line toward the sun, along the ground, AE a line due east, and Mm a true meridian line. The angle SAE must equal 90 17'. To make that angle, take AS, one chain or 100 links; from S, draw the line SE at right angles to AS, by means of a surveyor's cross.* From Stake SE, of such a value as will make SAE 90 17', which is determined by trigonometry; as follows, * Surveyor's cross is nothing more than a pair of sight vanes, set at right angles with each other, for the purpose of making right angles. 108 SURVEYING. As 100.: SE=Rad.: tan. 90 17' Whence SE=100. (tan. 90 17')_ 9.213405 B. 2 SE= 16.347 links. 1.213405 That is, from S measure off 16 and a little more than -I of a link, and there is the point E. A line drawn from A to E, is a due east and west line. If we put the surveyor's cross on this line, at any point as A, and range one branch of it along the line AE, the other branch will mark out the line Mm, a true meridian, if everything has been done to accuracy. In the afternoon, or some other day, another meridian may in like manner be drawn near this one, and if they are both true meridians, they will be parallel. If not parallel, other observations should be made until some two or three are obtained, that are parallel or very nearly so; and the mean direction then, may be regarded as the true meridian. A true meridian will always be a test line for a compass; and by placing any compass upon it, the declination* of the needle can be determined. Again. In the triangle PZS, if we compute the angle ZPS (as is done on page 211 Robinson's Geometry), we shall have the sun's distance from the meridian or the apparent time; then if we have a time piece that can be relied upon, for three or four hours, we can determine the time within a few seconds, when the sun will be on the meridian. A line at that time, run direct toward the center of the sun, will define the meridian. The objections to these methods are, 1. The sun is a large body, and its center cannot be exactly defined. 2. The sun changes position so rapidly that, unless we are in an observatory, where every thing is prepared and in order, it is difficult to get observation upon it. * Declination of the needle, in common language, is called the variation of the compass; and, as a general thing, we adhere to common language. VARIATION OF THE COMPASS. 109 3. The sun is so bright an object that it cannot be viewed without prepared glasses. 4. The majority of persons that have been, and probably will be practical surveyors, have not the instruments to take altitudes of the sun, and they are not and cannot be at home in astronomical obser. vations and computations. Some of these objections are deserving of little respect, and others can be partially removed. For instance, if the sun is too large, and too brilliant to be accurately and deliberately observed, we can take the planet Venus, Jupiter, or Saturn, and, by proper observations, determine their directions, during the twilight of evening, when we can see the planet distinctly, and at the same time that other objects are sufficiently distinct to run lines.* But the method most known and most in favor among practical men, is that of taking the direction of the north star. The north star is a star of the second magnitude (Polaris), whose right ascension, Jan. 1, 1851, was lh 5m 18s ( at present increasing at the rate of 17s 71 per annum ), and declination was then 88~ 30' 65", with an annual increase of 19"8, it is therefore, but 10 29' 5" from the pole, and it is called the pole star or north star because it is so near the pole. If the star were situated directly at the polar point, a line toward it would be the true meridian line, but being 1~ 29' 5" distant, the star apparently makes a circle round the pole in a siderial day, making two transits across the meridian, one above and the other below the pole, - a direction to it, at these times, would be a true meridian line. To find these times, subtract the right ascension of the sun from the right ascension of the star; increasing the latter by 24h, to render the subtraction possible, when necessary. * For example, in the year 1853, from the 25th of July to the 5th of August, the planet Jupiter will pass the meridian in the evening twilight. On the first of August, Jupiter will pass the meridian of New York, at 8h 11m 53s, and it will pass the meridian of Cincinnati, at 8h 11 39s, mean local time; and, of course, whoever is able to designate that time within a few seconds, and is also prepared to mark the direction of the planet, will have a true meridian line. The moon is not a good object for this purpose; it changes its place too rapidly. 110 SURVEYING. The difference will be the time of the upper transit, and 1 lh and 69 minutes from that time will be the time of the lower transit. The right ascension of the sun is to be found in the Nautical Almanacs, for every day in the year; and it is nearly the same, for the same day, in every year. For example. At what times will the north star make its transits over the meridian on the first day of July, 1853. H.I M. S. *- R. A.+24h - - - - 25 6 0 0. R. A.- - 6 41 16 18 24 44 This result shows that the upper transit will occur about 6h 24m, in the morning of the 2d of July. I say about, because I took the sun's right ascension for the morning of July 1, and from that time to 6, next morning, is 18 hours: and during this time the right ascension of the sun will increase full 3 minutes,- therefore the upper transit will take place Gh 21m in the'morning, and the previous lower transit 1 lh 59m previous, or at 6h 22m, evening. But neither of these transits will be visible, as they both occur in broad day light, from any place where the north star is ever distinctly visible. In summer, then, when most surveying is done, the meridian transits of the north star are not visible, nor is this important: for the transits are seldom used, by reason of two objections: 1. The star changes its direction most rapidly while passing the meridian. 2. Observers, generally, have not the means of knowing the time to sufficient accuracy.* To obviate these objections, observations may be taken on the star at its greatest elongations; for, about those points and for full * NOTE.- Very few persons consider that their clocks and watches, however good and valuable, do not give the exact time, but only approximations to the time. For any astronomical purpose, like the one under investigation, the character of the time piece should be well tested — its rate of motion known - and its errors established by astronomical observations. VARIATION OF THE COMPASS. 111 15 minutes before and after, the star does not scarcely change its direction; hence the observer has a sufficient interval to be deliberate, and he can be sufficiently exact as to time without any extra trouble. The following tables show the times of the greatest eastern and western elongations, which occur in the night season. These tables are not perpetual, but they will serve without correction for 20 years or more to come. EASTERN ELONGATIONS. Days. April. May. - June. July. August. Sept. H. M. H. M. H.. H. M_. H. M. H. M. 1 18 18 16 26 14 24 12 20 10 16 8 20 7 17 56 16 03 14 00 11 55 9 53 7 58 13 1734 1540 1335 11 31 930 736 19 17 12 15 17 13 10 1107 9 08 715 25 16 49 14 53 12 45 10 43 8 45 6 53 WESTERN ELONGATIONS. Days. Oct. Nov. Dec. Jan. Feb. i larch. H. lI. H. M. H. M. H. M. H. M. I. M. 18 18 16 22 14 19 12 02 9 50 8 01 7 17 56 1559 13 53 11 36 9 2617 38 13 17 34 15 35 13 27 11 10 9 02 7 16 19 17 12 15 10 13 00 10 44 8 39 6 54 25 16 49 14 45 12 34 10 18 8 16 633 It will be observed that these times are astronomical; the day commencing at noon, and 12h 40 means 40m after midnight, etc. Now, admitting that the direction of the star can be observed, the next step is to find how much that direction deviates from the meridian - and this is a problem in spherical trigonometry. A great circle passing through the zenith of the observer to the star, when the star is at one of its greatest elongations will touch the apparent small circle made by the apparent revolution of the star about the pole, and will therefore, with the star's polar distance, form a right angle - and we shall have a right angled spherical triangle, of which the observer's co-latitude is the hypotenuse, the star's polar distance one side, and the angle opposite to this side is the angle required. 112 SURVEIYING. EXAMPLE, What will be the bearing of the north star observed from latitude 420 NV. in the year 1860, when the star's polar distance will be 1~ 26' 12"? Ans. 10 56'. As cos. Lat. 42~ 0 - 9.871073 is to radius - -. - 10.000000 So is sin. 1~ 26' 12" - - 8.399183 To sin. 1~ 66' - - 8.628110 In this manner the following table was computed. The mean angle only is put down, being computed for the first of July in each year. A IMUT T T ABL IYears Lt. 0! Lat. 350 Lat. 40~ Lat. 450 I at. 0~ 0 Years Azimuth. Azimuth. Azimuth. Azimuth. Azimuth. 1852 10 42' 30" 10 48' 21" 1o 55' 52" 2~ 5' 32" 20 18' 6" 1854 1~ 41' 45" 10 47' 39" 10 55' 2" 20 4' 30" 20 17' 6" 1856 1~ 41' 2" 10~ 46' 49" 10~ 54' 12" 2~ 3' 44".2~ 16' 9" 1858 10~ 40' 27"1~ 046' 11"11~ 53' 30"2~ 03' 2" 20 15' 12" 1860 1~ 39' 43" 1~ 45' 24" 1~ 52' 32" 2o 2' 4' 2i 14' 16" 1862 10~ 38' 50" 10 44' 29" 10~ 51' 44"1~ 1' 2"1~ 13' 18" This table is given for those who may wish to use it, but we would recommend each observer to follow the example which precedes the table, and compute the azimuth corresponding to his latitude and time. THEI PRACTICAL DIFFICULTY. The north star is not brilliant, it cannot be seen until it is so dark that all minute terrestrial objects are totally invisible, it is therefore difficult to draw a line and accurately mark it. All these night operations are, at best, perplexing and inaccurate, yet, by the means of lights and artificers, lines can be drawn. If the observer have a theodolite and an assistant, there will be no difficulty. Let them be at the place from which they wish to take the observation in time, adjust the instrument and direct the telescope to the north star. Now sufficient light must be reflected into the telescope to enable the observer to see the cross hairs, and this may be done by the assistant holding a light before a stiff sheet of white TO SURVEY WITHOUT A COMPASS. 113 paper, so as to throw the reflected light from the paper into the telescope, or this may be done by means of a stand to hold both the light and the paper. When the vertical spider's line becomes visible, let the star be brought directly upon it, and if it is near the time of greatest elongation it will appear to remain so, for some time. But if the star has not reached it greatest elongation, it will move from the line more to the east, if the elongation is easterly, and more to the west, if westerly. The telescope must be continually directed to the star, by means of the tangent screw of the horizontal plate, but for some time the spider line and star will coincide without moving the screw, and then the star will depart from the line in the contrary direction to its former motion, but the telescope must no longer follow the star, its position will now show the direction to the star, when the star had its greatest elongation, and thus it should be left until morning. In the morning, carefully range and mark a line through the telescope. If we now make an angle with this line equal to the azimuth, by means of the theodolite, or by means of measuring a triangle as explained in the former part of this chapter, and mark this new line either to the right or left, as the case may require, we shall then have a permanent meridian line for all future use. By placing a compass on any well defined and true meridian we can determine its variation by simple observation. If we have not a theodolite, we can obtain a tolerably accurate direction to the north star by means of illuminated plumb lines suspended in vessels of water, so placed as to range to it. CHAPTER IV. TO SURVEY WITHOUT A COMPASS. THE inquiry is sometimes made, whether lands could be surveyed without a compass; we reply in the affirmative. The compass is only a convenience, and if it had never been discovered, it is probable 10 114 SURVEYING. that surveys would have been more accurately made. Too much reliance has been placed on the accuracy of the compass, and in consequence little attention has been paid to defining any-astronomical lines. Were it not for the compass, it is probable, that every countrytown, and even every large land holder, would have meridian lines well defined about his premises. Having a meridian line to start upon, we can find angles and define the position of lines very accurately by means of a CIRCUMFERENT OR. The circumferentor consists of a horizontal circular plate divided into 360 degrees, over which an index bar, or another circular plate, is made to revolve. This index bar carries sight-vanes or a telescope. The index bar or the revolving circular plate also carries a vernier scale, which will enable the operator to make an angle to one minute of a degree. The whole instrument is placed on a tripod, and by the aid of spirit levels attached to the lower plate, the horizontal position is attained with a sufficient degree of accuracy. The figure before us, represents the essential parts of a circumferentor. VNS is considered as the primative or meridian line, and AB is the index bar, which turns horizontally on the common center. Vernier scales are fitted into the index bar, and revolve over the graduated arc. At A and B are openings, to receive cross hairs or a telescope. When a vertical semicircle is made to revolve vertically through TO SURVEY WITHOUT A COMPASS. 115 the plane AB, and the diameter of that circle a telescope, then we have all the essentials of a theodolite. To most theodolites, a magnetic needle is attached, but the magnetic needle is, properly speaking, no part of the instrument. To show the manner of finding the direction between two given points, by means of the circumferentor, we propose the following problem. Mr. T. H. Jones wishes me to run the east line of his lot, in the town of A, and give the true bearing, the corners being known. In the public square of the town, about one and a quarter miles distant, a meridian line has been established. Let Mm be the established meridian in the public square, and GHthe direction of the line required. Place the circumferentor on the meridian line Mm, so that NLS of the instrument will coincide with it, the center of the instrument being at a in a road. The general direction of the road is a b, and the index bar AB is made to revolve over the plate, which is firmly fixed, until the index bar or sight vanes point out the line ab. The line is run by means of ranging objects; such as flag staffs, if the line is long; or if short, by sending on a flag, and stationing it at b. Now clasp the index bar on the plate, by the clamp screw under it (made for the purpose). Leave a flag at a; take the instrument to b, and there place it, so that the sight vanes will range back to a; then the position of NVS on the instrument will show a meridian line through that point. Here the road bends a little, unclamp the index (being careful that VS rigidly retains its position), and direct it to the general direction of the road be. Mark the point c, by an object as before, and mark some other point, so as to secure the line (the other point may or may not be b). Now clamp the index again, and remove the instrument to c. Place the instrument firmly as before, and make the index range along the line bc; the line 2,S of the instrument, 116 SURVEYING. will mark out a meridian line at the point c, and t1 us we can transfer the meridian line iMm to any other point whatever. Thus we may go to ajny point d, in the given line; no matter, theoretically speaking, how many angles we have made during the traverse. Placing the instrument at d, with its index to range along the last line, the line NS of the instrument gives the meridianMf'm' Now unclamp the instrument, and direct its index along the required line GH; the position of the index, on the graduated plate, will give the angle from the north, which, by means of the vernier, can be determined with great exactness. In this manner we may go to any point, and place a meridian there, and then run any required line whatever; therefore, we can survey any field, farm, or tract of land, without a compass, if we have a circumferentor, and a meridian line. It would not be safe to transfer meridians, as we have just done, Jbver any very great extent of country, for at every angle, small errors might be made, and the accumulation of many small errors may produce too great inaccuracies to be tolerated or overlooked. When using the magnetic needle, no errors accumulate, for every setting of the compass is primary, and independent of every other. Therefore, in case no compasses were in existence, primary meridians, astronomically established, would be necessary in every town; and it would be better to have several of them in the same town. From the foregoing illustrations, we perceive that surveying can be done, and well done, without a compass, yet the compass is an inestimable blessing to mankind; for it is the only index to direction over the wild waste of waters, when the heavens are obscured, and no mariner would dare brave the ocean without it. CHAP TER V. ORIGINAL AND SUBSEQUENT SURVEYS.DIFFICULTIES AND DUTIES OF A SURVE YOR. IN this country, lands were ceded to States, or sold to companies in large tracts, without any definite surveys; the boundaries SURVEYS AND SURVEYORS. 117 described, were mountains, rivers, or a certain number of miles along the shores of a lake, and then a certain number of miles back. The land companies hired surveyors from time to time, to survey off their lands, into lots of 100, 200, and 500 acres; and wherever these surveyors left monuments for the corner of lots, established the corners for all time to come, whether correctly placed or not. These surveys were very loose and inaccurate; it could not be otherwise, for a company of surveyors would frequently run 15 miles in a day; when to run a line accurately, and measure it,four miles is a good day's work. But, notwithstanding inaccuracies, these surveys are legal and cannot be changed; " thou shalt not move thy neighbor's ancient land mark," and it is right it should be so; for any attempt at correction, would create more trouble, confusion, and injustice, than it could remedy. Lots originally sold for 100 acres in the state of New York, generally contain from 101 to 106 acres, in consequence of the original surveyors having directions to have their lots hold out. Where the lots thus overrun in one portion of the tract, they fall short on another, for the surveyors were probably desirous to show to the company, that their grant actually contained as much land as was anticipated. The author surveyed one of these lots, that originally sold for 100 acres, and found that it contained but a little over 76 acres. Some of the companies had their grants laid off into townships 6 miles square, or 6 by 8 miles; then each township into four sections, each section divided off into lots, and the lots numbered, generally beginning at the south-west corner. The description of the lots in the deeds given, were very loose and indefinite, stating the township, section, and number of the lot, containing 100 acres, " be the same more or less," and in some lots it was more, and in other lots it was less. As we before remarked, any land mark to the corner of a lot laid down by these original surveyors, must remain; subsequent surveyors can straighten lines between point and point, and decide what the true courses are, and how many acres the lot contains. When a surveyor is called to survey any farm or estate that has 118 SURVEYING. been previously surveyed, he must find some corner as a place of commencing, and from thence run a random line, as near the true line as his judgment permits.;.and if he strikes another corner he has run the true course, if not, he corrects his course, as taught in chapter II. Thus, he must go round the field from corner to corner. He has a right to establish corners only where no corners are to be found, and no evidence can be obtained as to the existence and locality of a former land mark. It may be the case, that a surveyor is called to survey a lot where no corners are to be found. If a fence or line exists, which has been the undisputed boundary for a long time, that boundary cannot be changed, and the surveyor must establish a corner by ranging some other line to meet the first. Sometimes corners may be found to some neighboring lot, from which lines can be run, to establish a corner to the lot we wish to survey. Lines of lots in the same town, are generally parallel, and a surveyor who offers his services to the public, must make himself acquainted with the general directions of the lines of lots, over that section of country where his services are required. When a surveyor is called to divide a piece of land, he is then an original surveyor, and not liable to be embarrassed by old lines and old traditions, he has then only his mathematical problem before him. Owing to the inaccuracies of original surveys, and the impossibility of leaving proper land marks, in consequence of the great haste in which lands were originally surveyed; great confusion has followed, in some sections of our country, in respect to lines, and it has been no uncommon thing to have whole neighborhoods at variance, if not in law, in reference to the boundaries of their lands. In cases of this kind, one, and then another of the disaffected, have successively employed surveyors, and surveyors thus employed, are apt to act the part of.advocates, rather than arbitrators, and survey too much according to the direction of their employer; but all such efforts to settle difficulties, but aggravate them more and more. On the contrary, however, if the surveyor clearly understands his duties, and can rise above being a special advocate for any one of the parties concerned, hrecan do more than judges or juries to restore SURVEYS AND SURVEYOR'S. 119 harmony and peace. To illustrate these views, and possibly to give some valuable instruction to some readers, we give a history of a case of this kind, which occured in the year 1837, in the county of Ontario, in the State of New York. A tract of land consisting of about 670 acres, of an irregular shape, was divided on paper into five equal parts, and sold to five different individuals. The whole 670 acres was bounded by four lines, no two of them were equal, and neither of the angles was a right angle. The largest boundary line could not be directly measured on account of an impassable ravine; and the banks of this ravine was so thickly set with hemlocks, that it was impossible even to sight across. In consequence of the irregular shape of the whole, and the impossibility of directly measuring the principal boundary, they had never been able to agree on their division lines. Each one imagined that his neighbor was inclined to crowd upon him, and although permanent fences were desirable, none could be made until lines were agreed upon. They had employed several surveyors, but they had not been able to agree on their divisions. In this state of things a surveyor was called upon to go and make a division of this land, but the difficulties of so doing were carefully concealed from him. When he arrived on the ground, ready for operations, the whole neighborhood was present, and by unmistakable signs he soon learned that an unusual degree of interest was taken in the survey. He also found that the chief difficulty arose from not being able to measure the line CD. All the corners, A, B, C, and D, were established. The surveyor commenced at C to run a random line as near CD as possible. After going a few chains, he came to the bank of'the ravine at F, where it was impossible to pass or sight across. Driving a stake at F, he took a direction FPH along the bank of the ravine, carefully noting the angle, and measuring the line to H, a point where objects were clearly to be seen on the other side of the ravine. The surveyor then sent a man over with a flag, stationing his staff, first at K, 120 SURVEYING. then at L, carefully noting the direction of each, and being careful to have the angle KHL greater than 300. He then passed over and set the compass at K, took the direction of KL and measured it. Having now KL one side, and all the angles of the triangle RHKL, he computed HL. He now set the compass at L and took a definite direction Lm; this definite direction gave him the angle HLm, and he now had all the angles of the quadrilateral LmFiH, and two of its sides. Whence he computed the exact distance to m, to strike the line CF produced. He measured that distance and drove a stake at m, and computed mF. The company now ran through the random line, driving stakes at the end of every third chain, and the random line came out within a few feet of the established corner at D. The surveyor measured the perpendicular distance to D, and corrected the course by the rule in chapter II. He also computed how far each stake that had been placed on the random line must be moved to transfer it to the true line; this, the reader will perceive, was done by proportional triangles. At D he set the compass, and carefully noted the course and distance to A. He then returned to C, taking care not to pass along the line AB. At C he set the compass, and carefully noted the course and distance to B. He now computed the course and distance from B to A. The line lay in the open fields, over tolerably smooth ground, and it could be directly and accurately measured. The surveyor now called all the parties interested, including the sour and the belligerent, and told them that the distance from B to A was a certain number of chains and links, and that they would now measure it, and if they found it to correspond without any material error, they must then be convinced that he had obtained the true length of CD, and that he could then divide the land into five equal parts, as required. To this test they all cheerfully assented; the line was measured and corresponded to the computation within three links; all parties were satisfied, and thus ended a neighborhood quarrel of six years' standing. Previous surveyors commenced at the point D and run DA, AB, SURVEYS AND SURVEYORS. 121 and BC, and then computed CD. This was more direct, simple, and proper, than the method just described, but it left no test behind it, and it is vain to expect that the mass of men will receive theoretical computation as actual measurement. Here, and in most other cases that involve contention, the surveyor must not only convince himself that the survey is correctly made, but he must, if possible, show others that his conclusions are not only right, but cannot be wrong; hence judicious surveyors must often measure lines, where there is no mathematical necessity for so doing. The next duty of this surveyor was to divide the land into five equal parts. Each one had previously purchased his part, and he knew its locality, but not his exact boundary line on the division. As CD was not parallel to AB, and AD not exactly parallel to CB, to divide this mathematically exact was a problem of considerable difficulty, and this will be explained in the next chapter; but practically we need not apply all mathematical rigor, the surveyor can divide this more strictly conformable to justice without, than with the mathematical rigor. The persons who had the two most eastern lots, had the worthless part of the land in the ravine, and of course if any one had an excess of area it should be those. To find where or nearly where the divisions come, divide the line AB into five parts and suppose a one of those parts. Now BG is not quite long enough, because the field is a little narrower at this end than at the other; the surveyor took a distance BG a few links greater than one fifth of AB, and from that point run a line in a medium direction between B C and AD, and then computed its area, the result would show whether the area was too great or too small, and if it were within a very small fraction of the area required, the line is left as the true one, otherwise it is moved as the case requires. In the same manner the other division lines were run. UNITED STATES' LANDS. Soon after the organization of the present government, several of the States ceded to the United States large tracts of unoccupied land, and these, with other lands, since acquired by treaty and purchase, constitute what is called the public lands. 11 122 SURVEYING. Previous to 1802, there was no general plan for surveying the public lands, or in fact, no surveys were made, and when grants were made the titles often conflicted with each other, and in some cases different grants covered the same premises. In the year 1802, Colonel I. Mansfield, then Surveyor General of the north-western territory, adopted the following method: Through the middle, or about the middle of the tract to be surveyed, a meridian is to be run, called the principal meridian. At right angles to this, and near the middle of it, an east and west line is to be run, and called the principalparallel. Other meridians are to be run, six miles distant from the principal meridian, both east and west. Also, parallels of latitude are to be run, six miles from the principal parallel, both north and south. When this was done ( and it has been on all the public lands east of the Mississippi river ), the whole country is divided into squares, six miles on a side, called townships. Each township contains 36 square miles. Each square mile is called a section, and it contains 640 acres. Sections are divided into half sections, quarter sections, and eighths. But these divisions are only made on paper. When a person makes a purchase of a half or quarter section, it is supposed that he will find it himself, or employ a surveyor to mark it out. Townships which lie along a meridian, are called a range, and numbered to distinguish them from each other. Sections are regularly numbered in every township, and to designate any particular one, we say, section 13, in township number 4 north, in range 3 east. This shows that the third range of townships east of the principal meridian, in township No. 4 north of the principal parallel, is the township, and the thirteenth section of this township is the one sought. Not more than ten townships north or south of a principal parallel should be drawn, before a new principal parallel should be designated, and new measures made between meridians: because meridians tend toward the pole, and the north lines of townships DIVISION OF LANDS. 123 will be theoretically shorter than south lines, if the meridians are run by the compass. Where the public lands extend to rivers and lakes, there will be fractional townships along the shores. Where the locality of a particular number is found to be occupied by a lake or pond, the sale is void. CHAPTER VI. METHODS OF SURVEYING IRREGULAR FIGUJRES AND OF DIVIDING LANDS. FARMS and tracts of lands, wholly or partially bounded by water, as represented in the figure before us, are surveyed and - there areas determined by drawing right I lines within the tract as near the real - boundaries as possible, and from these right lines, at equal intervals, measuring the offsets to the real boundary. These off-sets form the parallel sides of trapezoids, and as they are all equally distant from each other, the computation of the areas they occupy will be very easy. A summary rule for finding the united area of all these trapezoids that are bounded by one line, is to be found in Prob. VIII, Mensuration. The area of the right lined figure ABCDEFG, is found as directed in Chapter IV, to which add the area of all the trapezoids, and we shall have the area of the whole, We have now investigated every possible case of computing areas, and we are now prepared to divide them. Commencing with the most simple case of the most simple figure, the triangle, or rather the figure that has the least number of sides. 124 SURVtYIVN G PROBLEM I. To divide a triangle into two parts, having a given ratio of m to n, CASE 1. By a line drawn from one angle to its opposite side. Let ABC represent the triangle; divide its base into two parts, corresponding to the given ratio, and let AD be one of the parts; then we shall have the following proportion, AD AB::: m+ft Whence, ADS=-~-(AB) and BD) -— (AB) Now the two parts are numerically known, and are to each other as e to n. Triangles, having the the same altitudes, are to one another as their bases. Therefore, ADC': CDB1: m: n as required. CASEI ~. By a line parallel to one of its sides, tet DE divide the triangle as required, and as similar triangles are to one another as the squares of their homologous sides, therefore (AB)2:(AAD)2: m+n: n Whence, AmDAB%/ m Which shows that if we have the numerical value of AB, and of tn and rn, we can find that of AD, and from 1) draw DE parallel to BC, and the triangle is divided as required. CASr 3. By a line parallel to a given line, or by a line running in a given direction. To make this case clear, we commence by giving a definite example: There is a triangular piece of land, from one of the angular points, A, one line runs N. 250~ I., distance 12 chains; another from the same point runs N. 420 E., distance 16 chains, It is required to divide this DIVISION OF LANDS. 125 triangle into two parts in the ratio 2 to 3, by a line running due east and west. Let ABC be the given triangle, and B' C' the required division line. It is required to find the numerical value of A C' or AB', to make the area AB' C' I of the area ABC. Let b represent the side of the triangle opposite B, and c the side opposite C. Let A C'=x. As A C' and, C'B' have definite directions, the angle A C'B' is given, also AB' C' is given. AC'B'-480, AB' C' =65~, BAC-670~. In the triangle AB' C' we have sin. 650: x:: sin. 480 AB sin. 48~ Whence AB':sin. 650 x (1) Now, by Prob. III, Mens., area ABC=_ Ibe sin. A. Also, c" " i" area AB'C' — ( sin. 650 x2 sin. A?.sin, 6 50%,J By the conditions of our problem, we have the following proportion sin. 48~\ bc si.'A 2sin. 6 50\ sin. A: 5: 2 sin. 48~ Or, b' sin. 65o 5 2 (2)5 We may here stop, and make the problem general. If B' C' is given in direction, the angles B' and C' will be given. We now require the division of the triangle ABC into two parts, in the ratio of m to n by a line opposite to the angle A, running in a given direction. Represent the sides of the given triangle adjacent the angle A by b and c, b extending from A to B, and c extending from A to C. Put x= the distance from A to the division line, on the side CA. Then, by the preceding proportion we have, 126 SURVEYING. sin. C' b6: sin. B:: m+n: m Whence, a= )( sin. B ) 2 Observe that x is opposite the angle B', the sine of which stands in the numerator of the second fraction. Had x represented AB', sin C' would have been the numerator. Drawing out the result for (2), we find that 5 sin 480 sin 650x=2bc=2.15.12 Or,, /72 sin 48 =9.446 chains. CASE 4. By a line that shall pass through a given point within the triangle. A point in a triangle cannot be given, unless the perpendicular distances from that point to the sides are given, and if these perpendicular distances are given, then we can readily find the three distances from the angular points of the triangle, and the angles which these lines make with the sides of the triangle are known. For instance, if the point P, in the triangle ABC, is known, PR and PT are known, and all the angles of the quadrilateral ATPR are known. These are sufficient data to compute the line AP, and the angles RAP and TAP. REMARK. - We may now require a triangle to be cut off, by a line running through P, which shall contain any definite portion of the triangle ABC, not involving an impossibility. For instance, if the point P is near the center of the triangle, it would not do to require us to cut off a tenth part of the triangle, or any smaller portion, for it would be impossible to do so. When it is required to cut off a very small portion of the whole triangle, the point P must be near one of the sides, or near one of the angular points. Sometimes the required quantity can be cut off from one angular point, sometimes from another, and sometimes from all three. Let us now require one-third of the triangle cut off, by a line passing through DIVISION OF LANDS. 127 P, taking the angular point A, and let EF be that line. We are to determine the value of AF. In the triangle ABC, the angles A, B, and C, are known, and the sides opposite to them, a, b and c, are also known. AP is known, and call it A. Put the angle EAP-p, PAF=q. Then, A-+-q. Put AF=x, AE=y. By Prob. III. Mens. area AB C=jbc sin. A Also " area AEF=jxy sin. A By the conditions of the problem, ]xy sin. A=Ibc sin. A Whence, 3xy=bc (1) The triangle AFE consists of two parts, APP, APE; therefore, jhx sin. q+ihy sin. p= jxy sin. A Or x sin. q+y sin. p nA (2) If we had required the nth part of the triangle ABC, in place of the 3rd part, equation (1) would have been nzy=bc. Making this supposition, to make the problem more general, we bcCbc have xy= b and y= b. By the aid of these last two equations, (2) becomes be sin. p.bc sin. A x sin. q+- - nx np bc sin. A bc sin. p Or, X2-., X- -. nh sm. q nsin. q (3) Whence, Jbc sin. A b2c2 sin.2A be sin. p 2nh sin.q \4 f2 h2 sin. q n sin. qj In case n is large, that is the part to be cut off small, the value of x may be imaginary, corresponding to the preceding remark. CASE 5. When the given point is on one side of the triangle. The two parts must be equal, or one of them will be less than half of the whole. We always compute the less part. Let P be the point in the line AB, P Q and PR perpendiculars to the other sides, are known; BC and A G are both known. Now, through the given point P, it is 128 SURVEYING. required to draw PD, so that the triangle BPJD, shall be the sth part of ABC. That is BC. A 0 2Q. BD= n PQ CASE. 6. When the given point is without the triangle. Let ABC be the given triangle, and P any given point without it. It is required to run a line from P, to cut off a given portion of the triangle ABC, or (which is the same thing) to divide the triangle into two parts having the ratio of m to t. Let PG be the line required. As P is a given point, AP is a line given in distance and position; therefore, the angle PAHis known. Solution. —Put the angle PAH=u, CAB=v; then PAG=(u+-v). Also put A G=x, AH=y, AP=a, and the area of the triangle AHGt=mc, mc being a known quantity. Now, (by Prob. III. Mens.) jxy sin. v=mc (1) Also " j.ax sin. (u+v)=area APG And " " C ay sin. u=area PA H Therefore, lax sin. (u+v)-.-ay sin. un=sc (2) Or, sin. (u+v)-y sin. =2M (3) From (1), we find y=: 2mc which value substituted in (3) gives xsin. v 2 me sin. u 2 me x sin. (u+v-) —_ -. x sin. V a 2 sac 2 mc sin. u Whence, z*sin. (u+v) —- 2 ~2mcsm.u a sin. v 2 me 2 me sin. u a sm. (u+v) sin. v sin. (u+v) TherefOlre for(+ e~\'Sm+ +m, 2 me sin. u ~ r ec~t a n. ('-~~'/~sin, v sin.(u+-,v) DIV ISION OF LANDS. 129 EXAMPLES. 1. In the triangle ABC, the side AB=23.645 chains, AC -17.51 chains, and BC- 12.575 chains. The given point P from the angle A, is distant 10 chains, at an angle of 400 from the line A C. It is required to draw a line from this given point P, through the triangle, so as to divide it into two equal parts. Whereabouts on AB -will P G intersect? The angle BA C310~ 17' 19"-=v. PAH=400~u. Therefore PAG=-710 17' IO" (u+v.) The area of the triangle ABC is 107.52 square chains. The part to be out off by: the triangle A/HG is therefore =53.76-=m. We must use the natural sines, or the logarithmic sines if we omit them in the index. me log. - 1.730464 a sin.(u+-v) log. - - 0.976406 mc _5.676 log. 0.754058 asin. (u+-v) 2 2 —12 g( -=32.22 log. 1.508116 a2 sin.2(u+v) 2mc log. - - 2.031494 sin. u log. - - -1.808067 2mc sin. u 1.839561 sin. v sin.(u+-v) - - -1.691866 2mc sin. u - sin. 702.56 log. 2.147695 sin. v sin. (u-v) The part of the formula under the radical is therefore (32.22+ 702.56) or 734.78 Whence x=5.676i14:/734.78-32.7829 or -21.43.'REMARK. - In geometry plus and minus generally indicate opposite directions, here the sign i means, sum and difference of the two numbers, the sum is 32.7829, the difference is 21.43; in fact there can be no such thing as minus a line The value of AG is 21.43; the other value is not admissible. 130 SURVEYING 2. We have a right angled triangle whose base is 48.87 chains, and perpendicular 54.46 chains. From a given point without it, we are required to run the center of a straight road, to leave one third of the triangle on one side and two thirds on the other. From the acute angle at the base, the distance to the given point is 20 chains, and the line to it makes an angle with the hypotenuse of 300~. REMARK. — If P is a given point, its distance and direction from one of the angular points must be given; and if the distance and direction from one of the angular points is given, the dis. tances and directions from all of them are virtually given; thus, if we have AP, AC, and the angle PAC, we have CP, and the angle ACP, and we may theorize on the triangles PCH, CHG' as well. as on APH and AHG. The area of the triangle AB C= 1304.94 square chains. One third of this is mc —= 434.97. BAC=v-490~ 12' 20". PA Cu=300. (u+v)=790~ 12' 20". AP=a=20. AH=y. A G =x. mc log. 2.638459 a= —20 log. - 1.301030 sin. (u+-v) log. -1.992238} 1.293268 mc - =22.142 log. 1.345191 a sin. (u+v) 2 m2 c- =490.34 log. 2.690382 a2 sin.2(u+v) 2mc log. - - 2.939489 sin. u - - - - -1.698970 2mc sin. u - - - - 2.638459 sin. v sin.(u-+v) - - - -1.871339 2mcsinu = —584.98 log. 2.767120 sin. v sin. (u+v) Whence, x=22.142: /V 1075.32=54.932 or -10.648. Here AG=54.932. AB=47.87. Hence BG=7.062. Having AP, A G, and the angle PAG, we can compute the angle APG. DIVISION OF LANDS. 131 PROBLEM II. To divide a triangle into THREE PARTS having the ratio of the three numbers m, n, p. CASE 1. By lines drawn from one angle of the triangle to the opposite side. Let ADE be the triangle and A the angle from which the lines are to be drawn. Divide DE the opposite side into parts in the ratio of m, n, and p, and from the points of division C and B, draw A C, AB, and the triangle is divided as required. Demonstration. - The areas of triangles are their bases multiplied into their altitudes, but here all the triangles have the same altitude; therefore multiplying the bases into that altitude gives the same proportional product, and the areas of the triangles are as m, n, p. CAsE 2. By lines parallel to one of the sides. Let ABC be the triangle. Divide its numerical area into three parts in the ratio of m, n, p. Conceive the problem solved and DF, EG, the division lines parallel to AB. We are to determine the numerical values of CD, and CE. CB is known, put it equal to a. Put CD=x. CE=y. Now as similar triangles are to one another as the squares of their homologous sides, therefore x2 a2: m: m+n+p. Whence, x=a,/ m m+n+p In the same manner, Y=a~/ m+n In this manner we might divide the triangle into any proposed number of parts, having given ratios. CASE 3. By lines drawn from a given point on one of the sides of the triangle. 132 SURVEYIN G. Let ABC be the given triangle, and P the given point on the side AB. It is required to draw lines from P, as PD and PE, dividing the triangle into three parts me, nc,pc, that is, assume the given numerical area to be (mc+-nc+pc), then the required parts will be minc, nc, and pc.* Put AD=x, then (by Prob. III, Mens.) 2mc - ax sin. A=m or x=_.c a sin. A 2pc By comparison, sin. B b sin. ]B When mc and pc are cut off, nc is left. Having a and x, the angle ADP is easily determined. In a similar manner we can divide the triangle into any proposed number of parts, by lines drawn from the given point P. CAsE 4. By lines drawn from a given point within the triangle. Let ABC be the given triangle, and P the given point within it. A variety of lines may be drawn from P, to divide the triangle into the parts required. Conceive PD, PF, and PE to make the requisite division. As P is a given point, AP, PB, and P C are known lines, and the angles DAP, PAXl are known angles. Take AD=b, any convenient assumed value. Take AE=x. Put A-P=a. Then, sab sin DAP= area - DAP Also, ~ ax sin PAE= area A PAE * Suppose we had a triangular piece of ground containing 320 square rods, and we wished to divide it into three parts in the ratio of 2, 3, and 5, what is the area of each of the parts? We decide it thus: m —2. --— 3. p-5. c is at present unknown, but, mc+ —nc+pc-320 That is, 0lc=320 or c=32. Whence, mc=64. nc=96. pc=160. The quantity c becomes known on dividing the area, and the parts separately mc, nc, and pc, are always known. DIVISION OP LANDS. 133 Conceive the triangle ABC, divided into three parts, in the ratio of m, n, p; and conceive ADPE to be one of these parts represented by mc. Then ab sin, DAP+ax sin, PAE.=2mc 2mc —ab sin, DAP Whence, x- a sin. PAE Had we taken b greater than we did, xwould have been less, and a variety of lines could be drawn as well as PD and PE, and the same area cut off, Having x, we have ElB as a known quantity, and by the two triangles, PEB and BPF, we determine y in precisely the manner as we found x: thus, we have two parts of the triangle mc and pc, and, consequently, the remainder DPEC corresponds to nc. CAsE 5. By lines drawn from a given point without the triangle. Let ABC be the triangle, and P the given point without it, Divide the numerical area of the triangle into the three required parts, me, nc, and pe, as in former cases. Draw PD, cutting off the portion nec, as in Case 6 of the last problem; then cut off the two portions ( mc+nc) by the line A G: and the portion pc will be left. Or, we may cut off pc, and the portion ne will be left, PROBLEM III. To divide a triangle into three parts, having the ratio of mt, n, and p, by three lines drawn from the three angular points to some point within. Divide any side, as A C, into three parts in the proportion of m, n, and p. Let Aa represent the portion corresponding to m, and Cc the part corresppnding to p. Through a, draw ab parallel to AB, 134 SURVEYING. and through c, draw cd parallel to CB, Where these two lines intersect is P, and the triangle ABC is divided into three triangles, APB, CPB, and AP C. Demonstration. — Any triangle having AB for its base, and its vertex in the line, ab will have the same ratio to the triangle ABC, as Aa has to A C, that is, as m to m+n+p. Also, the triangle CPB is to AB C, as Cc is to CA, that is, as p to m+n+p, for triangles on the same base are to one another as their altitudes. If these two triangles, APB and'CPB, are in due pro. portion, the third one, AP C, is in due proportion, of course. PROBLEM IV, To divide a quadrilateral into two parts, having any given ratio m to n. OAsE 1. By a line drawnifrom a given point in the perimeter. Let AB CD be the given quadrilateral, and P the given point in the side AB. it is required to determine the magnitude, and the direction of the line PG, which divides the figure into parts in the ratio of m to n. All the sides and angles of the quadrilateral are known, and its area is known. AB and D C are, or are not parallel; if they are parallel the figure is a trapezoid, then the method of finding G, in the opposite side, is easy and obvious. If AB and CD are not parallel, we can produce them and form a triangle in the one direction or the other; by this figure we form the triangle BCE, whose area we may represent by t. As BC, and the angles as B and C are all known, the triangle EB C is determined in all respects. As PB is known, PE is known, and designate EG by x. Let cm and cn designate the portions of the quadrilateral after it is divided, and let cm represent the part BPG C. Put PE=a. Now (by Problem III, Mensuration ), we have -ax sin. E==t+cm. DIVISION OF LANDS. 135 2t+-2cm Whence, X=a sin. E We now have the numerical value of x, from which we subtract EC, and we have CG, which being measured from C will give the point G, through which to draw the line from P, to divide the figure as required. CASE 2. By a line making a given angle with one of the sides. If the division line makes a given angle with one side, it must also make a known angle with the opposite side. Taking the last figure, conceiving PG6 to take a given direction across AB and CD, so as to cut off the area mc. As in the former case, let t represent the area of the triangle EB C, to this add mc, and we have the area of the triangle P GE. But in this case P is not a given point, and EP is not known. Put EG=x. Let P represent the given angle at P, and G the given angle at G. Now, by trigonometry, sin. P: x:: sin. G: EP sin. P (Prob. III. Mens.) Sin. E sin. G2t+mc 2sin.P Whence, X= /(2t+.2mc)sin. P sin. G? Sin. E From x we take EC, measure off the remainder along CD, to the point G, there making the given angle, and the figure will be divided as required. CASE 3. By a line drawn through a given point within the quadrilateral. Let ABCD be the quadrilateral as before, and P the given point within it; and as P is the given point, EP is a known line, I and the angles PEH, PEG are known. Let t equal the area of the triangle EBC, as before, and mc the area GUHCB. Put EH-x, and EG-Y. 136 SURVEYING. Now we have a problem precisely like Case 4, Problem I, of this chapter; therefore, further explanations would be superfluous. CAse 4. By a line drawn through a given point without the quadrilateral. Let AB CD be the quadrilateral as before, and P the given point without it. By producing the two sides AB CD, we form the triangle ADE. Let the area of the triangle B CE be represented by t, and the part GH CB by mc, then from a given point P, without a triangle, we are required to draw a line PH, to divide the triangle into two given parts, and this is Case 6, of Problem I of this chapter, which has been fully investigated. REMARK. — By extending the principles of these several cases we may divide a quadrilateral into three or more parts. PROBLEM V. To divide any polygon (regular or irregular) into tuwo parts having a given ratio, m to n, by a line drawn through a given point. CASE 1. When the given point is on one of the sides of the polygon. Let AB CDEF be the polygon, and (mc+nc) express its numerical area. Let P be the given point on the side AB. Let the surveyor run a random line as near to the line required as his judgment permits, and generally it will be best to run a line from P to one of the opposite angular points. In this figure, let PE represent such a random line, and let the surveyor compute the area of the figure PAFE, thus cut off, which axrea will be equal to, or greater, or less than one of the required parts. We will suppose it less; then subtract it from the required portion mc, and let the tiiangle PEG represent that known dilerence, which we shaH designate by t. PE is known, the angle PEG is known; and put ECG=:x Then, ~ PE x sin. P-EG=t Whence, X= PEsin. PEG DIVISION OF LANDS. 137 This determines the point G, and PG divides the polygon as required. CASE 2. When the given point is within the polygon. Let ABCDEF be the polygon as before, and (mc+nc) express its numerical area; also, let P be the given point within. Through P let the surveyor run the random line, RHPR, measuring from Hto P, and from P to K, let him also observe the angles that this line makes with the sides of the polygon AFand CD, and compute the area HAB CK, and note the difference between it and mc, the required portion of the polygon; call this difference d, a known quantity. Let hPk represent the true line through P, which divides the polygon as required; but this line diminishes the area HAB CK, by the triangle P-ik, and increases it by the triangle P1Hh. Therefore the diference of these triangles must equal d. The sine of the angle AHP, has the same numerical value as the sine of P11A, and the sine of the angle PKID has the same numerical value as the sine of the angle PKk. Put the angle AHP-u, and the angle PK.D=v; let the acute verticle angles at P be designated by the letter P. Let HP=a, PK-=b, hP=x, Pk=y In the triangle PhTl, we have sin. u::: sin. (u-P): a a sin. u Whence, =. - (1) sin. (U —P) Similarly, b sin.(2) sin. (v- P) The area of the triangle PhHR=iax sin. P Also, PkK-=by sin. P Whence, b sin. Py-a sin. Px=2d (3) By substituting the values of z and y, taken from (I) and (2), we have, 12 138 SURVEYING. b2sin. P sin. v a2sin. P sin. u sin. (v-P) sin. (u-P) Equation (4) contains only one unknown quantity P, the value of P, or the angle EPh can therefore be deduced. (2 ( sin. P sin. v )a sin. P sin. u / sin. v cos. P —~cos. v; sin. ]P - sin. u cos. P —cos. u. sin. P =2d. Dividing the numerator and denominator of the first fraction by (sin. P sin. v), and of the second fraction by (sin. P sin. u.), recollecting that cosine divided by sine gives cotangent. Thus we shall obtain b(, _ ( =2d () \cot. P —cot. v cot. P —cot. ul This last equation shows the surveyor that if he can make it convenient to run his random line from P, perpendicular to one of the sides, his equation will be less complex. For instance, if AIHP=900, its cotangent will be 0, and cot. u would then =0, and equation (5) would become. ~b2a -- -2d (6) cot. P — cot. v cot. P For the sake of convenience put cot. P=z, and cot. v=c. Then 62 a2 -— 2d Or, z2+ (a2 b C) a2c The numerical value of z will be the numerical value of cot. P; its logarithm taken, and 10 added to the index will be logarithmic cot. in our table. The same remarks will apply to cot. v or c. CAsE 3. When the given point is without the polygon. Let ABCDE be the polygon, and P the given point without it. From the last case we learn that the surveyor had better run his random line perpendicular to one of the sides, therefore let PHG be the random line, perpendicular to AB. DIVISION OF LANDS. 139 As before, compute the area, AEGH, subtract it from me, the difference is the difference between the triangles PGL and PHK. Draw PKL the line that divides the polygon as required. Put PH=a, PG =b, PK=x, PL=y, angle H=900, angle P GE =u, and the angle at P, designated by P. The triangle PHK=Iax sin. P e" PGL — by sin. P Whence, b sin. Py-a sin. P x-=2d. (1) Here (2) represents a similiar quantity as in the last case. In the triangle PHI, we have 1: x: cos. P a, or s (2) cos. P (2) In P L, sin. u: y: sin. (u-P): b. b sin. u (3) When the values of x and y are substituted in (1) we have sin. P sin. u sin. P 62 2 -2d sin. (u-P) cos. P- (4) I sin. P sin. u sin. P Or, b2 sin. u cos. P —os. u sin. P) 2d( b2 a2 Or, *cot. P-'cot. u cot. P-d (6) This equation is exactly similar to equation (6) of the last case, and it is reduced in the same manner. PROBLEM VI. To divide a polygon into three or more parts, having a given ratio, m, s, p, P, by lines passing through a given point. This problem admits of three cases. CAsE 1. When the given point is on one side of the polygon. Divide the numerical area of the whole into parts, mc, nc, pc, qc, corresponding to the given ratio. Unite these into two parts (mc+nc) and (pc+qc). 140 SURVEYING. From the given point P, draw PG, by Case i, Problem V, so as to divide the polygon into the two parts (mc +nc) and (pc+qc). We have now to divide the polygon PAEG into two parts, mc, nc, by the line PL, and the polygon PBCDG into two parts, pc and qc, by the line PH. CASE 2. When the given point is within the polygon. Let AB CDE be the given polygon and P the given point within. Draw hk through P, by Case 2, Problem V, so that the area AhkiDE shall equal mc, and the area hk CB shall equal (nc+pc), when the whole is required to be divided into three parts in the ratio of m, n, p. When the whole is to be divided into four parts, in the ratio of m, n, p, q, then draw hk, so that one portion shall be (mc+nc) and the other (pc+-c). Then we have P as a given point, in one side of the polygon, AhkDE, to divide it into two parts, in the ratio m to n, and P a given point on one side of the polygon, hkCB, to divide it into two parts, in the ratio of p to q, and this is done by Case 1, Problem V. CAsE 3. When the given point is without the polygon. Let AB CDEF be the given polygon, and P the given point without it. Divide the numeral area into the required proportional parts, me, nc, pc, &c., as many as required. From the point P draw the line PH, as directed in Case 3, Problem V, dividing the polygon into' two parts, mc and (nc+pc+&e.). Then ividafthe polygon, GHED CB, into two parts, one of which is nc,'and the other (pc+qc, &c.),' and. thus'we can proceed and cut off-one portion after another, as many as may be required. The application of the foregoing principles will meet any case that DIVISION OF LANDS. 141 can occur in the division of lands; and we now close this subject with the following practical EXAMPLES. 1. A triangular field, whose sides are 20, 18, and 16 chains, is to have a piece of 4 acres in content fenced of from it, by a right line drawn from the most obtuse angle to the opposite side. Required the length of the dividing line, and its distance from either extremity of the line on which it falls 9 Ans. Length of the dividing line, 13 chains, 95 links, if run nearest the side 16. Distance it strikes the base from the next most obtuse angle is 5.55 chains. 2. The three sides of a triangle are 5, 12, and 13. If two-thirds of this triangle be cut off by a line drawn parallel to the longest side, it is required to find the length of the dividing line, and the distance of its two extremitiesfrom the extremities of the longest side. Ans. Distance from the extremity on 5, is 5(/3 —V2); on the side of 12, it is 12(d3 —^J2); both divided by,/3. The divisionline is 134J/. 3. It is required to find the length and position of the shortestpossible line, which shall divide, into two equalparts, a triangle whose sides are 25, 24, and 7 respectively. REMARK. - It is obvious that the division line must cut the sides 25 and 24, and to make it the shortest line possible, the triangle cut off must be Isosceles. Ans. The division line makes an angle with the sides 25 and 24 of 810 52' 31", and its length is 4.896. 4. The sides of a triangle are 6, 8, and 10. It is required to cut of nine-sixteenths of it, by a line that shall pass through thle center of its inscribed circle. Ans. The division line cuts the side of 10, at the distance of 7.5 from the most acute angle, and on the side of 8, at the distance 6 from the most acute angle. 5. Two sides of a triangle, which include an angle of 700, are 14 and 17 respectively. It is required to divide it into three equal parts, by lines drawn parallel to its longest side. 142 SURVEYING. Ans. The first division line on the side 17, cuts that side at the distance7; the second division line17 /2. The side 14 is cut at 42 4V3 14and 14,2. f3 J3 6. Three sides of a triangle are 1751, 1257.5, and 2364.5. The most acute angle is 310 17' 19". This triangle is to be divided into three equal parts by lines drawn from the angular points to some point within. Required the lengths of these lines. Ans. The line from the most acute angle is 1322.42, and from the next most acute angle 1119 7. The legs of a right-angled triangle are 28 and 45. Requiredthe lengths of lines drawn from the middle of the hypotenuse, to divide it into fotr equal parts. Ans. A line drawn from the middle of the hypotenuse to the right angle, divides the triangle into two equal parts. 8. In the last example, suppose the given point on the hypotenuse at the distance of 13 from the most acute angle, whereabouts on the other sides will the division lines fall to divide the triangle into three equal parts? N. B. The sine of an acute angle to any right-angled triangle is equal to the side opposite that angle divided by the hypotenuse. Ans, Both division lines fall on the side 28, distance of the first from the acute angle 12-, or the second 24~x9. There is a farm containing 64 acres, commencing at its south westerly corner, the first course is 2Vorth 150 E., distance 12 chains; the second is N. 800 E. (distance lost), the third S. (distance lost), the fourth is N. 82~ W. (distance lost), to the place of beginning. It is required to determine the distances lost. OBSERVATION. - Extend the northern and southern boundary westward, and thus form a triangle on the west side of 12. Ans. The 2nd side is 35.816 ch. 3rd, 23.21 ch. 4th, 38.76 ch. The two following problems are from GUMMERe's Surveying, and are considered very difficult. DIVISION OF LANDS. 143 1. There is a piece of land bounded as follows: Beginning at the south-west corner; thence, 1. N. 14~ 00' W:, Distance 15.20 chains-a; 2. NX. 70~0 30' E., " 20.43 " b; 3. S. 60 00' E., " 22.79 " -c; 4. N. 860 30' W., " 18.00 " -=d. Within this lot there is a spring; the course to it from the second corner is S. 750 E., distance 7.90 chains. It is required to cut of ten acres from the west side of this lot, by a line running through the spring. Where will this line meet the fourth side, that is, how far from the first corner Z Ans. 4.6357 chains. First make a plot of the field. It is as here represented. Produce the sides b and d, the second and fourth, until they meet at G. Let S be the position of the spring, and join SG. We may or may not find the contents of the field*: it is not necessary for the location of the line LSH. It is necessary to find the area of the triangle ABG. Conceive a meridian line run through B; then we perceive that the angle ABG- 140+700 30'-84~ 30'. Conceive also a meridian line to be run through A, and then we perceive that the angle BA G=860 30' —140~-720 30'; whence A GB=230. With the angles and the side AB=15.20, we readily find AG=38.72, BG=37.10, and the area AGB=280.65 square chains. It is necessary to find the line GS and the angle B GS. From the given direction of the lines BC and BS, we find the angle GBS=-145~ 30'; and then from the triangle GBS, we find BGS-50 51' 30", and GS=43.83. Also we have the angle SGA-170 8' 30'. To the area of the triangle A GB=280.65, add 10 acres, or 100 square chains: then the area of the triangle GLH/must equal 380.65 * When we have four sides only, and all the angles, as in this field, the best method of finding the contents is by conceiving it to be two triangles. Thus in this case the area is represented by ~ ab sin. ABC+- dc sin. CDA. 144 SURVEYING. square chains; but GL and GH are both unknown. Put GL=y, GH=x: then we shall have the equation. xy sin. 230=2(380.65). (1) It is obvious that the sum of the two triangles L GS, SGH is equal to the triangle GLH. But GS=m=43.83, sin. 230~-P, sin. (17~ 8' 30")=Q, sin. (50 51' 30")-R, and 2(380.65) or 761.3=a: then we have Pxy=a, (I) and Rmy+ Qmx=a, (2) From (1), y a This value put in (2), gives PX aPRm+ Qmx=a; (3) whence, X2- -a - =- (4) We now find the numerical values of. and abylogarithms, mQ PQ as follows As our radius is unity, we diminish the indices of the logarithmic sines by 10. log. a, 2.881556 log. a, 2.881556 log. m, 1.641771 log. R,-1.008880 log. Q,-1.469437 1.790436 1.111208 1.111208 58.932 1.770348 log. P,-1.591878 log. Q, —1.469437 -1.061315 -1.061315 674.72 - - - - - 2.829121 Equation (4) now becomes x2-58.932x=-674.7; whence x=-G=43.366: from which subtract GA=38.72, and we have AHi the distance required =4.646, which differs from the given answer about one link of the chain. DIVISION OF LANDS. 145 LEMMA. Find the point in any trapezoid, through which any straight line which meets the parallel sides will divide the trapezoid into two equal parts. Let AB CD be the trapezoid. Bisect the parallel sides AB and CD in the points n and m. Join mn, and bisect mn in 0, and 0 is the point required. Any line meeting the parallel sides, and passing through 0, will divide the trapezoid into two equal trapezoids. It is obvious that the line mn divides the figure into two equal parts, because the sum of the parallel sides is the same in each. Now draw any other line through 0, as p Oq: the trapezoid pqBD=mnBD; because the triangle Onp-=q On, and one triangle is cut off and the other is put on at the same time. The triangles are equal, because m O= On, the angle pm O= Ong, and the opposite angles at 0 are equal: therefore pm=nqi; and whatever more than Cm is taken on one side, the equal quantity qn less than An is taken on the other side. Another method of finding the point 0, is to bisect A C in H, and draw HO parallel to AB or CD, and equal to one-fourth the sum of AB and CD. 2. There is a piece of land bounded as follows Beginning at the westernmost point of the field; thence, 1. V. 350 15' E., 23.00 chains; 2. N. 750 30' E., 30.50 " 3. S. 30 15' E., 46.49 " 4. V. 660 15' W., 49.64 It is required to divide this field into four equal parts, by two lines, one running parallel to the third side, the other cutting the first and third sides. Find the distance of the parallel line from the first corner measured on the fourth side, and the bearing of the other line. Ans. Distance to the parallel, 32.50 chains; Bearing of the other side, S. 790 48' E. 13 146 SU R VEYING. CHAPTER V I. TRIANGULAR SURVEYING: THE PLANE TABLE-ITS DESCRIPTION AND USES: MAPPING: MARINE SURVEYING. TRIANGULAR SURVEYING, as here understood, requires the actual measurement of only one line, and all other lines can be deduced from this by means of observed angles forming triangles, of which this measured line forms a base of the first triangle in the series or chain of triangles. Some of the other lines however should be measured after being computed, as a test to the accuracy or inaccuracy of the operations. Let AB represent a base line which must be very accurately measured, for any error on AB will cause a proportional error in every other line. If at A we measure the angles BA C, BAD, and at B we measure or observe the angles ABC, AB)D, we then have sufficient data to determine the points C and D, and the line CD. With equal facility that we determine the point C, we can determine the point E, or F, or G, or any other visible point. Thus we may determine all the sides and angles of the figure CEFGHD, or any visible part of it, by triangulating from the base AB. The lines forming the triangles are not drawn, except those to the points C and D); we omitted to draw others to avoid confusion. After any line, as FG, has been computed, it is well to measure it, and if the measurement corresponds with computation, or nearly so, we may have full confidence of the accuracy of the work as far as it has been carried. We may take CD as the base, and determine any visible number of points, as A, B, H, F, G, &c., trace any figure and determine its area, or show the relative positions and distances of objects from each other, such as buildings, monuments, trees, &c. THE PLANE TABLE. 147 But to make the computation, triangle after triangle, for the sake of making a map, would be very tedious, and to measure every side and angle would be as tedious, and to facilitate this kind of operation we may have an instrument called the PLANE TABLE. The plane table is exactly what the name indicates; it is a plane board table, about two feet long, and twenty inches wide, resting on a tripod, to which it is firmly screwed, yet capable of an easy motion onits center, having a ball and socket like a compass staff. Directly under the table is a brass plate, in which four milled screws are worked, for the purpose of adjusting the table, the screws pressing against the table. To level the table, a small detached spirit level may be used. The level being placed on the table over two of the screws, the screws are turned contrary ways, until the level is horizontal; after which it is placed over the other two screws, and made horizontal in the same manner. The table has a clamp screw, to hold it firmly during observations, and also a tangent screw, to turn it minutely and gently, after the manner of the theodolite. The upper side of the table is bordered by four brass plates, about an inch wide, and the center of the table is marked by a pin. About this center, and tangent to the corners of the table, conceive a circle to be described. Suppose the circumference of this circle to be divided into degrees and parts of a degree, and radii to be drawn through the center, and each point of division. The points in which these radii intersect the outer edge of the brass border, are marked by lines on the brass plates; these lines of course show degrees and parts of degrees; they are marked from right to left, from 0 to 1800 on both sides, but on some tables the numbers run all the way round, from 0 to 3600~. Near the two ends of the table are two grooves, into which are fitted brass plates, which are drawn down into their places by screws coming up from the under side. The object of these grooves and corresponding plates, is tQ hold down paper firmly and closely to the table. 148 SURVEYING. The paper before being put on, should be moistened to expand it, then carefully drawn over the table, and fastened down by the plates that fit into the grooves; on drying, it will fit closely to the table. A delicate fine edged ruler is used with the plane table, it has vertical sights; the hairs of the sights are in the same vertical plane as the edge of the ruler. A compass is sometimes attached to the table, to show the bearings of the lines; but the most practical mathematicians prefer each instrument by itself. The plane table may be used to advantage for three distinct objects. 1. For the measurement of horizontal angles. 2. For the determination of the shorter lines of a survey, both as to extent and position. 3. For the purpose of mapping down localities, harbors, watercourses, &c. 1. To measure a horizontal angle. Place the center of the table over the angular point, by means of a plumb. Level the table, then place the fine edge of the ruler against the small pin at the center; direct the sights to one object, and note the degree on the brass plate; then turn the ruler to the other object, and note the degree as before. The diference of the degrees thus noted, is the angle sought. If the ruler passed over O, in turning from one object to the other, subtract the larger angle from 1800, and to the remainder add the smaller angle, for the angle sought. 2. To determine lines in extent and position. Let CD be a base line; having the paper on the table, all dried and ready for use. Place the table over C, so that the point on the table, where we wish Cto be represented, shall fall directly over C; and place the position of the table so that CD shall take the desired direction on the table. .TtE PLANE TABLE. 149 Now level the instrument, and clamp it fast; it is then ready for use. Sight to the other end of the base line, and mark it along the fine edge of the ruler. In the same manner sight along the direction of CE, and mark that direction in a fine lead line, that can be easily rubbed out, the point E is somewhere in that line. Sight in the direction of F, and mark the line on the paper; F is somewhere in that line. In this manner sight to as many objects as desired, as G, H, B, A, &c. Now the base on the paper, may be as long, or as short as we please; suppose the real base on the ground to be 1200 feet; this may be represented on the table by 3, 4, 5, 10, or 12 inches, more or less; suppose we represent it by 6 inches, then one inch on the paper, will correspond with 200 feet on the ground (horizontally). Take CD six inches, and place a pin at D, remove the instrument to the other end of the base, and place ) of the table right over the end of the base, by the aid of a plumb, and give the table such a position as will cause CD on the table, to correspond with the direction of the base. Level the table and clamp it. Now, if CD on the table, does not exactly correspond with the direction of the base on the ground, make it correspond by means of the tangent screw. Now from D, by means of the ruler and its sight vanes, draw lines on the paper, in the direction of the points E, F, G, H, B, A, &c.; and where these lines intersect those from the other end of the base, to the same points, is the real localities of those points, in proportion to the base line. Lines drawn from point to point, where these lines intersect, as EF, FG, GH, &c., will determine the distances from point to point, at the rate of 200 feet to the inch. Lines drawn from the center of the table, parallel to FE and FG, will determine the angle EFG, in case the angle is required. After the points E, F, G, &c., have been determined, the light pencil lines to them, from the ends of the base, may be rubbed out, except those that we may wish to retain. Here we perceive the utility of the plane table; we have a multitude of results, as soon as the observations are made. 150 THE PLANE TABLE. The plane table will give us at once, the relative distances of buildings from the base, and from each other, and if we are careful and particular, we can obtain the magnitudes of the: buildings, as is obvious by the adjoining figure. This is most useful to an officer or a spy, who wishes as. exact knowledge of an enemy's locality as possible.' OrTrom' a distant place AB, we may examine and measure any objectsiwhatever, on the other side of a river, or give a correct delineation of the river itself. The plane table can be made very useful to civil engineers, for mapping the localities through which a canal or railroad: passes. Take, for example, a railroad line, ABCD.E, represented in the next figure. The lines being all measured and marked in distances of 100 or 200 feet, the bases are all ready where the line is straight. Set the table as at A, and draw lines to all objects that you wish to appear on the map, both to the right and to the left,- then move the instrument to B, drawing lines to the same object from a corresponding base on the paper, and also draw lines'to other objects further in advance on the line that may be seen from another base. The intersections of the lines from the extremeties of any base to to the same object will locate the object. When all the objects are thus located, both to the right and to the left, we pass on to new objects, taking care to keep at least two of the old objects in sight, to connect one new observation with those previously taken. We now commence a series'of observations from a new base, which base must take its proper relative position on the SURVEYING. 151 r a a pr a I a a a a a a a a a r I r r a II s a a a r a r I a r r O a a I a a a -e -- -~ —— a- — UI-)-~I~~LI31L8) 152 SURVEYING. paper, and if the paper on the table is not large enough, it must be taken off and new paper put -on, and two of the objects on the old paper must appear on the new, and then these two papers can be put together so that the objects which are on both papers will coincide, and then the two papers will be the same as one, and thus we may put any number of papers together and form as large a map as we please. If the different bases are not in the same direction, the objects which are on two different papers on being put together will show it, and several papers put together may. make a very inconvenient figure; but they must be put:tgether and then -a square sheet of tissue paper put over the whole, and the ma taken off. From the tissue paper the map can be put on any other paper. The engineers of Napoleon's army freqtently made maps of the localities they were about to ps; indeed it is a military principle never to go into an unknown locality, except in eases f absolute neaessity. Tbis. subject naturally eads as to MSAR:IN: s i rU xVZYI tN G. Marine Surveying is too extensive a subject to be fully investigated in any work like this. We shall only explain how to find shoals, rocks, and turning points in a ehannel, by ranging to objects on shore. In-trigonometricl surveying on shore, the obswerer is supposed to take his angles from the extremities of a bass line, but in trigonometria surveying on water, the observer can take his angles only from single points which may be connected together by distant base.lines on the-shore. Important ponts along the shore are determined by taking latitude and longitude, and intermediate' places, by regular land surveying. The localities of rocks and shoals are also determined by astronoleical observations, establishing latitude and longitude, in case no land is nm sight, or they are'far from the shore,; but in the vicinity of the-land, the determinatimn of a point is commonly effected by thxe tAree poit poulem. MARINE SURVEYING. 153 The three point problem is the determination of any point from observations taken at that point, on three other distant points where the distances of these three points from each other are known. It is immaterial how those points are situated, provided the three points and the observer are not in the same right line, the middle one may be nearest or most remote from the observer, or two of them may be in one right line with the observer, or all three may be in one right line, provided the observer be not in that line. The following example will illustrate the principle. Coming from sea, at the point D, I observed two headlands, A and B, and inland C, a steeple, which appeared between the headlands. I found, from a map, that the headlands were 5.35 miles from each other; that the distance from A to the steeple was 2.8 miles, and from B to the steeple 3.47 miles; and I found with a sextant, that the angle ADC was 120 15', and the angle BDC 15~ 30'. Required my distance from each of the headlands, and from the steeple. If the direction of AB is known, the direction of AC is equally well known. The case i, which the three objects, A, C, and B, are in one right line may require illustration. At the point A, make the angle BAE= the observed angle CDB, and at B, make the angle CBE= the observed angle AD C. Describe a circle about the triangle ABE, join E and C, and produce that line to the circumference in D, which is the point of observation. Join AD, BD. The angle ADB is the sum of the observed angles, and AEB added to it must make 180~. The Trigonometrical Analysis. — In the triangle ABE, we have the side AB and all the angles, AE and EB can therefore be computed. In the triangle AEC, we now have A C, AE, and the angle CAE, from which we can compute A CE, then we know A CD. Now in the triangle A C:D, we have A C and all the angles, whence we can find AD and CD. 154 SURVEYING. When the bearing of the base line on shore is known, as it generally is, and the bearings to its extremities, or even to one extremity, are taken, the triangle is known at once. A pilot guides a vessel in and out of a port by ranging lines on the shore, minutely or approximately, as the case may require. We will illustrate this by a figure. Let the deep shaded lines represent the shores, A a light house on a rocky promontory. B another prominent object on the opposite shore; the position of the arrow indicates the direction north and south. The faint dotted lines represent the boundaries of shoal water, over which it would not be prudent, if even_ possible, to conduct a vessel. The line a, b, d, E, the center of the ship channel. All the pilots know that a line from A to a, which is nearly east south east, runs safe to the open sea, after passing the shoal coast near n. Now suppose a pilot boards a ship coming in from sea, sufficiently far from the coast, he directs her sailing so as to bring the light house at A to bear west north-west. He then sails toward the light house until he finds the object B bearing due north of him. He then knows that the ship must be near a, the mouth of the channel. He continues the same course, and knows when he is about half way from a to b by the two objects, C and D, appearing in the same line. When C and D becomes fairly open, and C- nearly north, and B not quite north-west, he is then at the turning point b of the channel. His course is then north, a little to the west, until the ship is nearly in a line between the two objects A and B. From thence, west north-west takes the ship directly through the proper channel into the harbor. LEVELING. 155 C HA P T E R VIII. LEVELING. Two or more points are said to be on a level, when they are equally distant from the center of the earth, or when they are equally distant from a tranquil fluid, situated immediately below them. A level surface on the earth, is nearly spherical, and is not a plane; it is everywhere perpendicular to a plumb line. Any small portion of a true level surface, cannot be distinguished from a plane; and, therefore, when observations are taken in respect to level, within short distances of each other, the spherical form of the earth is disregarded, and the level treated as a plane. But when any considerable portion of surface is taken into account, the curvature of the earth's surface must be considered. The apparent level, at any point on the earth, is a tangent plane, touching the earth at that point only, and the true level is below this, and the distance below, depends on the distance from the tangent point. Let T be any point on the surface of the _ earth, at right angles to the plumb line from this point is the plane or apparent level ATB; but the true level, or the surface of standing water, would be the curved surface GTH. The distance AG, depends on the distance AT, and the radius of the earth CG or CT. From G, draw GD, at right angles to AC; then the two triangles ATC, A GD, are equiangular and similar, and give the proportion CT: TA:: DG: GA. Practically. — TA is a very short distance compared to CT, for TA, the distance within which we can take observations, is never more than two or three miles, while the distance CT is near 4000 miles; therefore, CT is nearly equal to CA, consequently, DA is nearly equal to DG, so near that we shall call it equal. Observe that TD= DG; hence DG- TA. 156 SURVEYING. Now, in the preceding proportion, in the place of DG, put its equal ~TA, and we shall have, CT: TA:: ~TA: GA. Whence, GATA= Also, ga — (1) 2CT 2CT That is, The square of the distance, divided by the diameter of the earth, is the distance between the apparent and the true level. We can arrive at the same result by the direct application of the 36th proposition of the third book of Euclid, or by the application of theorem 18, third book of Robinson's Geometry. Because A is a point without a circle, and AT touches the circle, we must have A GX (2 CG+A G)=AIT2 But 2 CG, which is the diameter of the earth, cannot be essentially or appreciably increased, by the addition of AG, which is at most, but a few feet, therefore, AG within the parentheses, may be suppressed without making any appreciable error. Then divide by 2 CG. AT3 Whence, AG- A', the same as before. 2 CG If we take one mile for the distance TA, the value of GA will be 18 —— 8.001 inches. By comparing equations (1) we perceive that, GA:ga:: TA':Taa That is, The corrections for apparent levels, are in proportion to the aquares ofthe distances. The correction for one mile is 8.001 inches; what is itfor 10 miles. Ans.'It is x inches; then we have the following proportion, 8.001:x:: 12: 102 x —-= 800.1 inches. We have seen above, that the correction for one mile or 80 chains distance, on an apparent level, is 8.001 inches, what is the correction for the distance of 20 chains? Let z=the correction sought, and the solution is thus, 8.001::: (80)2: (20)2:: 42 12 =0.500 inches. LEVELING. 157 In this manner, the following table was computed. Table showing the dferences in inches, between the true and apparent level, for distances between 1 and 100 chains. Ch's. In's. Oh's. In's. Ch's. In's. Ch's. In's. 1.001 26.846 51 3.255 76 7.221 2.005 27.911 62 3.380 - 77 i 7.412 3.011 28.981 653 3.511 78 7.605 4.020 29 I1.051 54 3.645 79 7.802 6.031 30 1.126 55 3.785 80 8.001 6.045 31 1.201 66 3.925 81 8.202 7.061 32 1.280 67 4.061 82 8.406 8.080 33 1.360 68 4.205 83 8.612 9.101 34 1.446 69 4.351 84 8.832 10.125 35 1.531 60 4.500 85 9.042 11.151 36 1.620 61 4.654 86 9.246 12.180 37 1.711 62 4.805 87 9.462 13.211 38 1.805 63 4.968 88 9.681 14.245 39 1.901 64 5.120 89 9.902 15.281 40 2.003 65 5.281 90 10.126 16.320 41 2.101 66 5.443 91 10.351 17.361 42 2.208 67 5.612 92 10.587 18.405 43 2.311 68 5.787 93 10.812 19.451 44 2.420 69 5.955 94 11.046 20.500 45 2.631a 70 6.125 95 11.233 21.552 46 2.646 71 6.302 96 11.521 22.605 47 2.761 72 6.480 97 11.763 23.661 48 2.880 73 6.662 98 12.017 24.720 49 3.004 74 6.846 99 12.246 25.781 50 3.125 75 7.032 1 00 1 2.502 This table is of little or no practical use, for levelers rarely take sight to a greater distance than 10 chains, and at that distance the correction is only one-eighth of an inch, and if they put the level midway between two stations, they annihilate the corrections altogether. Suppose a level to be placed at T, midway between A and B; the instrument will show them to be on the same level, as so they really are, for they are at equal distances from the center of the earth; but if the observations were taken in reference to A and a, the apparent level would not show equal distances from the center of the earth, and a correction must be applied, if the difference of distances is more than four or five chains. 158 SU RVEY ING. To comprehend the whole subject, we must now describe the modern SPIRIT LEVEL. The figure before us represents this useful instrument, apart from its tripod. Its principal'parts are the telescope AB, to which is attached the leveling tube CD; the telescope rests in a bed, which is supported by posts yy, called the y's; EE is a firm bar, supporting the y's. In S is a socket, which receives the central pivot of the tripod (which is not here represented). When the instrument is put upon its tripod, the tube S can be clasped on the outside, and held firmly by a clamp screw, it can then be moved horizontally, as minutely and readily as desired, by means of a tangent screw. The tripod contains a pair of brass plates, to the lower one the legs of the tripod are firmly attached, the other plate moves in all directions on its center, and is worked by four screws; these are called the leveling screws; these plates are purposely made small as a greater surety against bending: the four leveling screws are placed at the four quadrant points of the circle, and, with the center, form diameters at right angles. The eye-glass of the telescope is at A, the object glass at B. The screw V runs out the tube which holds the object glass, to adjust it to different distances. The telescope is fastened into the y's, by the loops r r, which are fastened by the pins p p. The telescope can be reversed in the y's, by taking out the pins p p; opening the loops r r; taking up the tube, turning it round, and again placing it in the y's; then A will LEVELING. 159) take theposition of B, and B of A. The necessity of this construction will appear when we describe the adjustment. At n n are two small screws that are attached to a ring inside of the tube; this ring holds a horizontal spider line; the object of the screw is to elevate and depress that spider line. At q q ( only one q can be seen in the figure ), are two screws that work another ring, which holds a perpendicular spider line, which can be moved to the right and left by the screws q q. The two spider lines show a perpendicular and horizontal cross at the focus of the telescope. Before using the instrument it must be adjusted. The adjustment consists: 1st. In making the center of the eye-glass and the intersection of the spider's lines coincide with the axis of the telescope, and this line is called the line of collimation. 2nd. In making the axis of the attached level, C.D, parallel to the line of collimation, in respect to elevation. 3rd. In~ making the attached level lie exactly in the same direction as the line of collimation. To make the first adjustment, the telescope is made to revolve in the y's. To make the second adjustment, there is a screw a, which serves to elevate and depress the end of the leveling tube at C. To make the third adjustment, there is a side screw b, which drives the end of the tube D to the right and left, as the case may require. First Adjustment.- Plant the tripod, place the instrument upon it, and direct the telescope to some well defined and distant object. Draw out the eye-glass at A, until the spider's lines are distinctly seen, then run out the object glass by the screw V to its proper focus, when the object and the spider's lines will be distinct. Now note the precise point covered by the horizontal spider's line. Having done this, revolve the telescope in the y's half round, when the attached level will be on the upper side. See if in this position the horizontal spider's line appears above or below the same object. If this line should appear exactly in the same point of the object 160 SURVEYING. as before, this spider's line is already in adjustment, but if it ap. pears above or below, bring it half way to the same point by means of the screws n n, loosening the one and tightening the other. Carry back the telescope to its original position, and repeat the observation, and continue to repeat it until the telescope will revolve half round without causing the horizontal line to rise or fall. This will show that the horizontal line is a diameter of the circle of revolution, and not a chord of it. Make the same adjustment in respect to the vertical hair, and the line of collimation is then adjusted. Second Adjustment.- That is, to make the tube CD horizontally parallel to the line of collimation. Place the instrument properly on its tripod, and bring the horizontal bar EE directly over two of the leveling screws; turn these screws until the bubble d rests in the center of the tube. Now CD is on a level, but we are not able to say that the line of sight through the telescope, that is the line of collimation, is on a level also. To test this, take out the pins p p, open the loops r r, and take out the telescope with its attached level, and turn it end for end, put it back in its bed, and put the loops over and pin them down. If the bubble now rests in the middle, no adjustment is required; if not, the bubble will run to the elevated end. In that case the bubble must be brought half way back by the leveling screws, and the other half by the screw a. Repeat the operation until the bubble will settle in the middle of the tube after reversing the telescope. Third Adjustment.- The second adjustment being completed, revolve the telescope in the y's, and if the bubble continues in the middle, the axis of the telescope and the axis of the tube CD lie in the same direction, or in the same vertical plane; and if they be not in the same vertical plane the bubble will run to one end or the other; in that case the side screw b will remedy the defect. The three adjustments are now made approximately, no one of them can be made perfectly while the instrument is greatly out of adjustment in relation to the others; therefore commence anew.Bring the bar EE over two of the leveling screws, and level the instrument; then turn it over the two other screws and level it in that direction also. Now, if we can turn the instrument quite round LEVELING. 161 without removing the bubble from the center it is in pretty good adjustment, but if otherwise, as is to be expected, make all these adjustments over again; they can now be made with much less difficulty. It is important that a level should be in as perfect adjustment as possible, but perfection in all respects is almost, yea, quite impossible. Yet, with a level considerably out of adjustment, we can obtain the relative elevation of any two points, provided we can. set the level midway between them. To illustrate this, suppose the instrument placed at D), midway between two perpendicular rods Aa Bb. Let ab represent the true horizontal line, but suppose that the instrument is so imperfect, or out of adjustment, that when the leveling tube CD is horizontal, the telescope would point out the rising line DA, and the rise would be Aa. On turning the instrument round and sighting to B, the rise must be the same as in the opposite direction: for the distance is the same, therefore A and B are as truly on a level with each other as a and b. By this problem, practical men complete the second adjustment of the instrument. They make the three adjustments as just explained, as accurately as possible. They then measure, very carefully the distance between two stations, as E and F, and set the instrument exactly midway between them as represented in the last figure. They then level the instrument (that is the tube CD) ), and find the difference of the levels between E and F ( two pegs driven into the ground). INow, suppose AE measures on the rod, 4.752 feet. And BF "' " - - 6.327 feet. Then E is above F - 1.575 feet. 14 162 SURVEYING. They now bring the level near to one of the stations as E, and level it very accurately, and sight to the rod AE. Now, suppose the target stands at - - - 5.137 feet. To this add 1.575 feet. 6.712 The rod man now goes to the station F, puts his target on the rod exactly at 6.712, and the telescope is turned upon it, and the horizontal spider's line ought to just coincide with the target, and will, if the instrument is in perfect adjustment; if it is not, the error is taken out by the screws n n. If the error was but slight, as in such cases it always is with good instruments, the adjustment is as complete as it can be made. With the level there must be A ROD. The rod is commonly ten feet long, and divided into tenths and hundredths, some have also a vernier scale which in effect subdivides to thousandths. The target slides up and down the rod, and carries the vernier on the back of the rod; the target has equal alternate portions painted black and white for contrast. A party to take the necessary levels on the line of a railroad or canal, after the stations are measured off, should consist of a leveler, and assistant leveler, a rod man, and an axe man. The leveler and assistant leveler both keep book, and sometimes the rod man also. If there is no assistant leveler, the rod man will have an abundance of time to keep book, and under such circumcumstances always does so. Under all circumstances, two persons keep book, to have a check on each other and guard against mistakes. In the field the aim is to put the level midway between the two stations, but they are not particular about it if the instrument is in good adjustment; they rather take the most advantageous spot to sight from. When the ground admits of it, that is, sufficiently level, two or three intermediate stations are observed, as well as the extreme back and fore stations. The extreme back and fore stations are called changing stations; at these stations pegs are driven into the ground by the axe man for the rod man to plant his rod, so as to secure LEVELING. 163 the same point for both the fore and back sight. At the intermediate stations they have no pegs, and are not particular in any respect, for all errors cancel each other. The common railroad chain is 100 feet, divided into 100 links; each link is therefore onefoot. Levels are commonly taken at intervals of 200 feet, oftener if the ground is very uneven, but a station is considered 200 feet, and the number of the station multiplied by 200 gives the number of feet from the commencement. The field book is kept thus: B. S. means back sight, F. S. fore sight, A. ascent, D. descent, T. total elevation above a common base. N. B.- When the back sight is less than the fore sight, the ground is descending, and the converse. Sta. B. S. 5F. S. A. D. Total. 100 0 4.32 7.21 2.89 97.11 1 5.52 8.17 2.65 94.46 2 *9.18 6.27 2.91 99.37 3 6.27 6.12 0.15 97.52 4 6.12 3.76* 2.36 99.88 5 9.81 11.62 1.81 98.07 6 8.47 9.02 0.55 97.52 7 2.64 8.91 6.27 91.25 8 1.07 7.38 6.31 84.94 9 4.29 5.32 1.03 83.91 10 5.32 4.85 0.47 84.38 11 4.85 3.17 1.68 86.06 12 8.22 1.53 7.31 93.37 Thus we go through the whole line. We commenced with the constant 100, but this is arbitrary; the object of taking a constant is to avoid the minus sign, that is, getting below our ruled paper in making a profile of the vertical section. Where the line is to be generally ascending, we assume a small constant, where generally descending a large one; taking care in all cases to have it so large as not to run it out. At each and every section we know exactly how much we are above or below the constant base, and the exact ascent or descent from any one station to any other. The following diagram represents a vertical section of the ground 164 SURV E Y ING. where these levels were taken, with the exception of tile exaggeration of the roughness caused by the difference of scales for the base and perpendicular lines. From one station to another is 200 feet; we have made 10 feet occupy more space in the perpendicular direction than 400 feet does in a horizontal direction. We do this to show more clearly where any particular grade will enter the ground, and how much it is necessary to cut or fill at any particular point. The zigzag line from 100 of altitude to 86, represents the surface of ground, and suppose that we wished to reduce it to a regular grade so as to remove __ as little earth as possible. By the mere exercise of judgment, 0 9 7 6 5 2 1 0 we conclude to run the grade between 0 station and 11, from 98 to 92 ( but the propriety of this conclusion would depend on the contour of the ground on each aide of these stations ). The direct lilec a. b drawn, shows that the grade runs out of the ground at station 1, we must fill in about 21 feet at station 2, the grade runs into the ground again at about 80 feet before we come to station 3. At section 4 the cutting, must be a little over 2 feet, at station 5 a little over 5 feet, at 7, 2 feet, and runs out of the ground midway between stations 7 and 8. At stations 9 and 10 we must fill in about 8 feet, and so on; the depth of cutting or filling is obvious at every station. If the contour of the ground beyond 11 was generally level or descending, we would change the grade at station 7 and render it more descending, so as to make less filling up at stations 9, 10, and 11. In the adoption of grades for a railroad, an engineer has great scope to exercise his judgment. In the representation here made, ab appears like a steep grade, but it is not; it could scarcely appear on the ground other than a level, for the difference is only 6 feet in 2200 feet of distance, which is at the rate of 14~-4 feet to the mile. An engineer can freely vary his grade, while it keeps under 18 or 20 feet to the mile; but they submit to a great deal of cutting and filling, before they establish a grade over 40 feet to the mile. LEVELING. 165 CONTOUR OF GROUND. Contour of ground is shown on maps, by marlking where parallel planes run out on the surface. We shall give only the general principle. Let A2 be the top of a hill, whose contour we wish to delineate; measure any convenient line as AB, up or down the hill, and by the level or theodolite, ascertain the relative elevations of a, 6, c, d, &c., as many planes as we wish to represent. At a, place the level or theodolite, and level it ready for observation; measure the height of the instrument, and put the target on the rod at that height. Send the rod-man and axe-man round the hill, on the same level as the instrument.. Let the rod-man set the rod; the leveler will sight to it through the telescope, and if the target is below the level, he will motion the rod-man up the hill, if too high, down the hill; at length he will get the same level, and there:the axe-man will drive a stake.- In the same manner we will establish another stake further on; and thus proceed from point to point. To get round the hill, it may be necessary to move the instrument several times. The plane thus established, is represented by the curve am. In the same manner, by placing the instrument at b, we can establish the next plane bn. Then the next, and the next, as many as we please. Where the hill is more steep, two of these parallel planes will be nearer together in the figure; where less steep, they will appear at a greater distance asunder, and this, with the proper shading, will give a true representation of the ground. 166 SURVEYING. ELEVATIONS DETERMINED BY ATMOSPHERIC PRESSURE, AS INDICATED BY THE BAROMETER. The higher we ascend above the level of the sea, the less is the atmospheric pressure (other circumstances being the same), and therefore we can determine the ascent, provided we can accurately measure the pressure, and know the law of its decrease. As this work is designed to be educational as well as practical, we shall here make an effort to explain the philosophy of the problem, in such a manner, as toforce it upon the comprehension of the learner. The pressure of the atmosphere at any place, is measured by the height of a column of mercury it sustains in the barometer tube. It is found by experiment, that air is compressible, A and the amount of compression is always in pro- B portion to the amount of the compressing force. C Now, suppose the atmosphere to be divided into - an indefinite number of strata, of the same thickness, and so small that the density of each stratum may be considered as uniform. Commence at-an indefinite distance above the surface of the earth, as at A, and let iw represent the weight of the whole column of atmosphere resting on A. Let the small and indefinite distances between AB, BCM, CD, &c., be equal to each other, and we shall call them units of some unknown magnitude. The weight of the column of atmosphere supposed to rest on B, is greater than w, by some indefinite part of w, say the nth part. Then the weight on B, must be expressed by(+)o ( ). In the same manner, the weight or pressure resting on C, must be the weight above B, increased by its nth part; that is, it must be (nl)1 (nal)w, which by additionis( w. In the same manner, we find that the weight resting on D, must be(n+ X )3tw, and so on. For the sake of perspicuity, we recapitulate. n3 LEVELING. 167 The pressure on A is w. Units from A 0 D " on B is (l)w I 1 " " on C is (n+l)a w" " 2 n2 on D is (n1)3 w s 3 n3 cc" " on E is (n1) w " " 4 n 4 " onF is (n l)5sW w " " 5 n5 &c.&c. &c. &c. Here we observe the series which represents the pressure of atmosphere, at the different points A,B, C,&c., is a series in geometrical progression, and it corresponds with another series in arithmetical progression; therefore, by the nature of logarithms, the numbers in the arithmetical series, may be taken as the logarithms of the numbers in the geometrical series. But this system of logarithms, may not be hyperbolic nor tabular, indeed it is neither; the base of this system is as yet unknown, but our investigations will soon lead to its discovery. Now, let the number of units from A to S (the surface of the sea), or to the lower of two stations, be represented by x, then the expression for the pressure of the air would be n >I xw, but this n is neither more nor less than the weight of the column of mercury in the barometer, which is sustained by this pressure. By calling this b, and designating the logarithms of this unknown system by L', we shall have L'b =x (1) Taking y to represent the number of units from A to V, and b, to represent the pressure of the air at that point, we shall have L'b, =y (2) Subtracting (2) from (1), we shall have L'b-L'b, =x —y= S V This is, a certain peculiar logarithm of the barometer column at the lower station, diminished by the logarithm of the barometer at the 168 SURVEYING. upper station will give the difference of levels between the two stations. But still all is indefinite and unknown, because we know nothing of these logarithms. In algebra, we learn that the logarithms of one system can be converted into another by multiplying them by a constant multiplier called the modulus of the system, therefore, Assume Z to be the modulus or constant that will convert common logarithms into these peculiar logarithms. Then, Z(log. b-log. b,)=S V (3) Here, log. b denotes the common logarithms of the barometer column. Equation (3) is general, and determines nothing until we know SVin some particular case. Taking S V some known elevation, and observing the altitude of the barometer column at both stations, and then equation (3) will give Z once for all. Putting h to represent the known elevation, and we have, in general, h Z=~ (4) Z log. b-log. b, Example.- At the bottom and top of a tower, whose height was 200 feet, the mercury stood in the barometer as follows. At the bottom, - 29.96 inches =b At the top, - 29.74 inches =6b, the temperature of the air being 490 of Fahrenheit's thermometer. 200 200 Whenc, Z= 200 - 200 =62500 nearly. Whence, -log. 29.96- log. 29. 74-0.003201 "But this multiplier is constant only when the mean temperature of the air at the two stations is the same; and for a lower temperature the multiplier is less, and for a higher it is greater. A correction, however, may be applied for any deviation from an assumed temperature, by increasing or diminishing ( according as the temperature is higher or lower) the approximate height by its 449th part for each degree of Fahrenheit's thermometer. We can moreover change the multiplier to a more convenient one by assuming such a temperature as shall reduce this number to 60000 instead of 62500. Now 62500 exceeds 60000 by its 25th part; and, since 1~ causes a change of one 449th part, the proportion 4_. 1~'** I_ * 17.9 4 4 ()''*2 5s' * LEVELING. 169 gives 180 nearly for the reduction to be made in the temperature of the air at the time of the above observations, in order to change the constant multiplier from 62500 to 60000, or to 10000, by calling the height fathoms instead of feet. Thus, instead of the thermometer standing at 490, we may suppose it to stand at 49~-18~ or 310; and then,we take 10000 as the multiplier, and apply a correction additive for the 180 excess of temperature." The same observations, for example, being given, to find the height of the tower. 29.96 - - log. - - 1.47654 29.74 - - log. - - - 1.47334 Diff. of log. - - 0.00320 Multiplier - - - 10000 Product - - - 32 Then the height of the tower is 32 fathoms, or 32X 6= 192 feet, on the supposition that the temperature of the air was 310 in place of 49~. But it being 490, we must increase 192 by its 4-9 part for each degree above 310, that is, by —' or -o nearly of its approximate height, which gives nearly 8 feet to add to 192, making 200 feet for the height of the tower. The same method is applicable to other cases whatever be the temperature of the air at the two stations, provided it be the same or nearly the same at both stations, or provided we take the mean temperature of the two stations. We can find the difference of levels of two stations to considerable accuracy by the following RULE. - 1st. Take the dfference of the logarithms of the two baro. meter columns, and remove the decimal point four places to the riglt. This is the approximate dfference of levels in fathoms. 2d. Add 4T~ of the approximate height for each degree of temperature above 310, and SUBTRACT the same for each degree below 31~; the result cannot be far from the truth. EXAMPLES. 1. The barometer at the base of a mountain stood at 29.47 inches, and taking it to the top, it fell to 28.93 inches. The mean temperature of the air was 510. What was the height of the mountain in feet? Ans. 603.34 feet. 15 170 SURVEYING. 29.47 log. - - 1.469380 28.93 log. - - 1.461348 0.008032 Approximate height in fathoms, 80.32. Correciton.- Add ~2 of 80.32 to itself, that is, add 3.~7. Height, in fathoms, - - 83.89 Multiply by - - 6 Height in feet, - - 503.34 The average height of the barometer, at the level of the sea, is 30.09 inches; and now if we know the average height of the barometer at any other place, and the average temperature, it is equivalent to knowing the elevation of the latter place above the level of the sea. For example, the mean height of the barometer at Albany Academy is 29.96, and the mean temperature is 490~. How high is the academy above tide water? Ans. 117.3 feet. 30.09 log. 1.478422 490 29.96 log. 1.476542 31 0.001880 1 8 Approximate height 18.80 fathoms. Add A_-d or Wa 75 19.55X 6:=:117.3 feet. 2. The average height of the barometer at Amenia Seminary in Duchess, Co., New York, is stated in the Regents' report at 29.81 inches: average temperature 490. What is the height of that point above tide water? Ans. 253.32 feet. 3. The mean height of the barometer at Pompey Academy, Onondagua Co., New York, is 28.13, corrected and reduced to 320 Fahr. What then is the elevation of that locality? Ans. 1755 feet. Others have made it 1745 feet. 4. From various observations on the summit of Mount Washington, in New Hampshire, the mean height of the barometer there is 24.20, mean temperature at the times of observation was about 650 Fali. LEVELING. 171 Now admitting that the mean temperature at the surface of the sea, in the same latitude must have been 750, which would make the mean temperature between the two stations 700, what then is the height of Mount Washington above the sea? Ans. 6170 feet, nearly. By some observations the elevation is estimated at 6496 feet, by others at 6290 feet. 6. Lieutenant* Fremont, in his narrative of the exploring expedition across the Rocky Mountains, page 45, under date of July 13th, 1842, states his latitude at 410 8' 31", longitude 104~ 39' 37", height of the barometer 24.86 inches, attached thermometer 68~; what was his elevation above the sea? AAns. 6389.2 feet. REMARK.-The author states his elevation at 5449. feet. As he does not state the temperature of the air by the detached thermometer, we know not what correction he made. These solitary barometrical observations are more or less valuable, according to the settled or unsettled state of the weather. A person of experience and good judgment in such matters, like Fremont, would not of course record the inapplicable observations. 6. Lieut. Fremont, in his journal, page 104, under date of August 15th, 1842, when, as he supposed, on the highest point of the Rocky Mountains, observed the barometer to stand at 18.29 inches, and the attached thermometer at 440t; what was the elevation above the level of the sea? Ans. 13522 feet. Fremont estimates the elevation at 13570 feet. This shows that he estimated the mean temperature above 500, and no doubt a similar cause made the difference in the result of the previous example. 7. On page 140, of Fremont's Journal, under date of July 12th, 1843, he says; {" The evening was tolerably clear, with a temperature at sunset, of 630. Elevation of the camp, 7300 feet." Taking the mean temperature of the two stations, the sea and his + Lieutenant was his proper title at this time. t If the sea were at the base of the mountain, the temperature at the lower station would no doubt be as high as 600. Making this supposition, the mean temperature of the two stations would be 500. We therefore take 500 as the mean temperature. 172 SURVEYING. place of observation, at 67~, what must have been the height of his barometer? Ans. 23.21 inches. Represent the approximate elevation by y, then y+36Y=7300 Or, y=6738.14 449 Divide y by 6, which gives 1126.35. Divide this by 10000. Then, let x represent the altitude of the barometer column. Whence, 1.478422-log. x=0.112635 Therefore, log. x=1.365787 In the preceding examples we could only be general and approximate, we had only the observations at one station referred to general observations at the other; but our results cannot be far from the truth. When the difference of temperatures at the two stations is considerable, the result must be affected by it. When the upper station is the coldest, which is generally the case, the mercurial column will be shorter than it otherwise would be, and consequently indicate too great a height. If the temperature of the upper station is taken for the temperature of the lower, the mercurial column at the lower station would not be high enough, and the deduced result would be too small, as is the case in example 5. The contraction of mercury being about one 10000th part for each degree of cold, or, 0.0025 in a column of 25 in., it would require 40 difference of temperature, to produce an effect amounting to one division on the scale of a common barometer, where the graduation is to hundredths of an inch. This correction is combined with the former in the following equation, in which t t' represents the temperature of the air at the two stations; t at the lower station, q and q' the temperature of the mercury, as indicated by the attached thermometer. The fraction 0.00223, is equal to 7~V nearly; h=the height sought, b and b, represent the observed height of the mercurial column. h= 10000 1 +0.00223 (tt —31 log. b,(l+O (q.-') LEVELING. 173 Beside the corrections previously considered, regard is sometimes had to the effect of the variation of gravity in different latitudes, and at different elevations above the earth's surface. The latter, however, is too small to require any notice in an elementary work. The former may be found by multiplying the approximate height by 0.0028371 Xcos. 2 lat. It is additive, when the latitude is less than 45~0, and subtractive when greater. Or it may be taken from the following table. Latitude. Correction. 00 - - + — of the app. height. 50 - q- - 100 - - + 150 - - 4 0 200 - - 250 - _ + I300'' _ 350 - T -0T+ l' 40 - + 2 03 450!- _50 - - -}- 2-030 550 - I - 1 0 600, - 650 -' 54 8 70~- - - r0a 750 - -l 850 3.8 900~ - 352 Given, the pressure of the atmosphere at the bottom of a mountain, equal to 29.68 in. of mercury, and that at its summit, equal to 25.28 in., the mean temperature being 500, to find the elevation. Ans. 727.2 fathoms, or 4363.2 feet. The following observations being taken at the foot and summit of a mountain, namely, at the foot, bar. 29.862 attach. therm. 780 detach. therm. 710 at the summit, " 26.137 C 630 " 550 to find the elevation. Ans. 618.9 fathoms, or 3713.4 feet. 174 SURVEYING. It is required to find the height of a mountain in latitude 30~, the observations with the barometer and thermometer being as follows; namely, at the foot, bar. 29.40 attach. therm. 500 detach. therm.* 430 at the summit, " 25.19 " 460 390 Ans. 683.27 fathoms, or 4099.62 feet. If we assume any temperature, for instance 450, and the height of the barometer at the level of the sea, at 30.09 inches; we can compute the elevation of the point, where it would be 29.99, 29.89, 29.79, 29.69, &c., inches; and thus we might form a table, showing the elevations that would correspond to any assumed height of the barometer at that temperature. It will be found, that the first fall of -L of an inch will correspond to about 88 feet in elevation, but every subsequent tenth, will require a greater and greater elevation..* The attached thermometer measures the temperature of the mercury in the barometer, and, the detached thermometer, that of the surrounding air. NAVIGATION. CHAPTER I. INTRODUCTION. NAVIGATION is the art of conducting a ship from one place to another. In most works this art is mixed up with seamanship and elementary science. In this work, navigation will stand by itself - alone; and we shall presume that the reader is properly prepared in elementary science., This being the ease, it will not be necessary to take up time and space in giving definitions of latitude, longitude, meridian, horizon, &c., &c., the previous indispensible knowledge of geography necessarily gives a knowledge of all these terms. Navigation, rightly understood, requires an accurate knowledge of the geography of the seas - the winds and currents that here and there prevail, and also a good general knowledge of astronomy. Running a line in surveying and running a course at sea, are mathematically the same thing, except that the latter is on a larger scale than the former, without its accuracy, and it is for a different object. In surveying we take no account of the magnitude and figure of the earth. In navigation we are compelled to do so, unless the limits of the operation be very small. There are two methods of finding the position of a ship, 1st. By tracing her courses and distances, as in a survey. This is called dead reckoning. 2d. By deducing latitude and longitude from observations on the heavenly bodies. This is called nautical astronomy. (175) 176 NAVIGATION. No one expects accuracy from dead reckoning, and as a general thing it is only used from day to day, between observations; or to keep the approximate run during cloudy weather or until observations can again give a new starting point. Some navigators keep a continuous dead reckoning from port' to port, which enables them to judge of drifts, currents, and unknown causes of error. The earth is so near a sphere that for the practical purposes of navigation it is taken as precisely so. Its magnitude corresponds to 69j English miles to one degree of the circumference, but in the early days of navigation 60 miles were supposed to be about a degree, which for the sake of convenience is still retained. The sixtieth part of a degree is called a nautical mile, and it is, of course, larger than an English mile. In an English mile there are - - - 5280 feet. In a nautical mile there are - - - 6079 feet. The rate which a ship sails is measured by a line running off of a reel, called the log line. The log is nothing more than a piece of thin board in the form of a sector, of about six inches radius, the circular part is loaded with lead to make it stand perpendicular in the water. The line is so attached to it that the flat side of the log is kept toward the ship, that the resistance of the water against the face of the log may prevent it, as much as possible, from being dragged after the ship by the weight of the line or the friction of the reel. The time which is usually occupied in determining a ship's rate is half a minute, and the experiment for the purpose is generally made at the end of every hour, but in common merchantmen at the end of every second hour. As the time of operating is half a minute, or the hundred and twentieth part of an hour, if the line were divided into 120ths of a nautical mile, whatever number of those parts a ship might run in a half minute she would, at the same rate of sailing, run exactly a like number of miles in an hour. The 120th part of a mile is by seamen called a knot, and the knot is generally subdivided into smaller parts, calledfathoms. Sometimes (and it is the most convenient method of division ) the knot is divided into ten parts, more frequently perhaps into eight; but in either case the subdivision is called afathom. INTRODUCTION. 177 We shall consider a fathom the tenth of a knot, and as the nautical mile is 6079 feet, the 120th part of it is 50.66, the length of a knot on the line, and a little over 5 feet is the length of a fathom. The operation of ascertaining the rate of sailing is called by seamen heaving the log. At the End of an hour the loaded chip, or log, is thrown over the stern into the sea; a quantity.of the line, called the stray line, is allowed to run off, then the glass is turned and the number of knots that runs off the reel during the half minute is the rate of the ship's motion. The log is then hauled in and the same operation is repeated at the end of the next hour. The officer of the watch, who has been on deck during the hour, will mark on the slate or board, called the log board, the number of miles and parts of a mile which the ship has sailed during the last hour, according to the best of his judgment; the log was thrown only to help Iake up that judgment, for the rate at the time the log was thrown may have been considerably more or less than the average motion during the hour. The course of a ship is marked by the mariner's compass. The mariner's compass differs from the surveyor's compass only by its construction, that is, the magnetic needle is the motive power in both. In consequence of the motion of a ship at sea, the mariner's compass is suspended in a double box, moving on a double axis, one at right angles with the other, the whole balanced by a central weight which keeps the compass card nearly steady and horizontal, whatever be the motion of the vessel. The card is attached to the needle, and is moved by the needle. The card is divided into 32 equal parts, called points, and to read over these points in order, is called by seamen, boxing the compass, and to know the north star and box the compass is too often the amount of the common sailor's knowledge of navigation. 178 NAVIGATION. The figure before us re. presents the card of the m a finer' s compass. The four quadrant points are marked by a single letter as N. for north, E. for east. The midway pointsbetween these by two letters, as N. E. for northeast, N. W. for north-west, &c. One point either way from any one of these eight points is marked by the word by. Thus, N. by E. is one point from the north toward the east, and it is read north by east; S. E. by S. indicates one point from the south-east more to the south, and it is read south-east by south; W. by N. means west one point toward the north. To box the compass we begin at any point, as north, and mention every point in order all the way round, thus: North; north by east; north north-east; north-east by north; northeast; north-east by east; east north-east; east by north; east, &c. A point of the compass is 110 15', which is subdivided into four equal parts. Mariners never take into account a less angle than a quarter point, in running a course. When the mariner sets his course, he makes allowance for the variation of the needle, and his magnetic courses he reduces to true courses, by the following RuiE. - Make a representation of the compass card on paper, and draw a line through the compass course. N1ow, conceive the compass card turned equal to the variation to the eastward, if variation is west, and VICE VERSA. INTRODUCTION. 179 The line will now pass over the true course. In the following examples, the true courses are required. Answer. Compass Course. Variation. True Course. 1. S. S. E.E. 2 W. S. E. bE. 2. E. N. 3E. E. S. E. S. 3.N. W. b W. 31 E. N.b W. W. 4. W. S. W. S. 4 W. S.bW.j W. 5. S. S. 4W. W1 S.:.W. 6.1V. 5 E. lYVE. bE. 7. E.bS. 2 E. S. E. E 8. S. 60~s. 180W. S. 78~E. 9. 1.24 W. 36 E...12E. 10. S. 16 W. 40 E.. 566 W. L E EWAY. The angle included between the direction of the fore and aft line of a ship, and that in which she moves through the water, is called the leeway. When the wind is on the right hand side of a ship, she is said to be on the starboard tack; and when on the left hand side, she is said to be on the larboard tack; and when she sails as near the wind as she will lie, she is said to be close hauled. Few large vessels will lie within less than six points to the wind, though small ones will sometimes lie within about five points, or even less; but, under such circumstances, the real course of a ship is seldom precisely in the direction of her head; for a considerable portion of the force of the wind is then exerted in driving her to leeward, and hence her course through the water, is in general found to be leeward of that on which she is steered by the compass. Therefore, to determine the point toward which a ship is actually moving, the leeway must be allowed from the wind, or toward the right of her apparent course, when she is on the larboard tack; but toward the left, when she is on the starboard tack. It is only when a ship crowds to the wind, that leeway is made. It is seldom that two ships on the same course make precisely the same leeway; and it not unfrequently happens, that the same ship makes a different leeway on each tack. It is the duty of the officer 180 NAVIGATION. of the watch, to exercise his best skill in determining, or estimating, how much this deviation from the apparent course amounts to; and in the dark, the chief reliance must be placed on the judgment of the experienced mariner. C HAPTER I. PLANE SAILING. IN plane sailing, the earth is considered as a plane, the meridians as parallel straight lines, and the parallels of latitude as lines cutting the meridians at right angles. And though it is not strictly correct to consider any part of the earth's surface as a plane, yet when the operations to be performed are confined within the distance of a few miles, no material error will arise, from considering them as performed on a plane surface. And, as we have already seen, in all questions where the nautical distance, diference of latitude, departure, and course, are the objects of consideration, the results will be the same, whether the lines are considered as curves drawn on the surface of the globe, or as equal straight lines drawn on a plane. In all maps, and charts, and constructions, when it is not otherwise stated, it is customary to consider the top of the page as pointing toward the north, the lower part as the south, the right side as the east, and the left as the west. The meridians therefore, in any construction, will be represented by vertical lines, and the parallels of latitude, by horizontal ones. Hence, in constructing a figure for the solution of any case in plane sailing, the difference of latitude will be represented by a vertical line, the departure by a horizontal one, and the distance by the hypotenusal line, which forms, with the difference of latitude and departure, a right-angled triangle; and the course will be the angle included between the difference of latitude and distance. With this understanding, the solution of any case that can arise from varying the data in plane sailing will present no difficulty. PLANE SAILING. 181 EXAMPLES. If a ship sail from Cape St. Vincent, Portugal (Lat. 370 2' 54" north), S. W. ~ S. 148 miles; required her latitude in, and the departure which she has made? By Construction. —Draw the vertical line AB, to represent the meridian; from the point A, make the angle BA C=3~ points, the given course; and from a scale of equal parts, take A C= 148 miles, the given distance; from C on AB, draw the perpendicular CB, then AB will be the difference of latitude, and BC the required departure, and measured on the scale from which A C was taken, AB will be found 114.4, and BC 93.9. Lat. left - - - 370 2' 54" NY. Diff. lat. - 1 54 24 N. Lat. in - - - - 35~0 8' 30" iN. Dep. 93.9 2. If a ship sail from Oporto Bar, in Lat. 410 9' north, N. W. j W. 315 miles; required her departure and the latitude arrived at? Ans. Dep. 233.4 miles W.; Lat. 440 41' N. 3. If a ship sail from lat. 550 1' N., S. E. by S., till her departure is 45 miles; required the distance she has sailed and her latitude? Ans. Dep. 81 miles; Lat. 530 54' N.. 4. A ship from lat. 36~ 12' NV., sails south-westward, until she arrives at lat. 350 1' NV., having made 76 miles departure; required her course and distance. Ans. Course S. 460 57' W.; Distance 104 miles. 5. If a ship sail from Halifax, in lat. 440 44' NV;, S. E..E., until her departure is 128 miles; required her latitude and distance sailed. Ans. Lat. 420 51' V., and dis. sailed 165.6 miles. 6. A ship leaving Charleston light, in latitude 320 43' 30" north, sails N. eastward 128 miles, and is then, by observation, found 93 miles north of the light; required her course, latitude, and departure. Ans. Lat. 330 22' 30" N.; Course N. 72~ 16' E.; Dep. 122 miles. 7. A ship from Cape St. Roque, Brazil, in latitude 5~0 10' south, sails N. E. j V., 7 miles an hour, from 3 P. M. until 10 A. M.; required her distance, departure, and latitude in. Ans. Dis. 133. miles; Dep. 84.4 miles; Lat. in, 30 27' south. 182 NAVIGATION. 8. A ship from latitude 410 2' NV., sails NV. TV. 4 W. 5~ miles an hour, for 21 days; required her distance, departure, and latitude arrived at. Ans. Dis. 330 miles; Dep. 169.7 miles; Lat. 45~0 45' N. Similar examples, might be given without end, but these are sufficient, for they only involve the principles of the solution of a plane right-angled triangle. In the preceding examples it will be observed that we traced latitude from latitude, and the distances east and west we called departure- not diference of longitude. It now remains to determine difference of longitudefrom departure, On the equator, 60 miles of departure are equal to one degree of longitude, and the further we are from the equator, north or south, that is, the greater the latitude we are in, the same departure will cover more longitude. To discover the law for changing departure to difference of longitude, we adduce the following figure. Let C7 be the center of the earth, P the pole, PC a portion of the earth's axis. The plane P GCB is the plane of one meridian, and PCA the plane of another meridian. AB is a portion of the equator between the two meridians. A CB is a sector in the plane of the equator, and DEH and FGQ are sectors similar to A CB. Observe that DE, FrP, &c., are parallels of latitude, that is, they are parallel to AB, the plane of the equator. The magnitude of DE is called departure, and it corresponds to the difference of longitude AB. Also, FG( is departure corresponding to the same difference of longitude AB. The difference of longitude is always greater than any corresponding departure, that is, AB is obviously greater than any otherparallel distance between the same two meridians. PLANE SAILING. 183 Because the two sectors A CB DEH are similar, we have the proportion. ACp: r H:: AB: DE (1) Observe that A C is the radius of the sphere, DH is the sine of the arc PD, or the cosine of DA, which is the cosine of the latitude of the point D. Therefore the preceding proportion becomes, rad.: cos. lat.:: dif. Ion.: dep. Or, cos. lat.: rad.:: dep.: dif. ion. In words, The cosine of the latitude is to the radius so is the departure to the difference of longitude. These words are indelibly engraved on the memory of every navigator, and they embrace all the rules that can be made for changing departure into longitude, or longitude into departure. When a ship sails east or west, the distance sailed is called departure, and is reduced to longitude by the preceding proportion. This is called parallel sailing. EXAMPLES. I. What difference of longitude corresponds to 47 miles departure in the latitude of 370 23'? Ans. 59.15 miles. Let x-= the difference of longitude required. 47X rad. Then cos. 370 53': rad.:: 47: x cos. 370 23' By log. 47 R - - - 11.672098 Cos. 37~ 23 - - - 9.900144 Diff. Lon. 59.15 - - - 1.771954 2. How many miles, or how much departure corresponds to a degree in longitude on the parallel of 42~ of latitude? Ans. 44.59 miles. Here the longitude of one degree is given. 60 cos. 420 R.: cos. 420~:: 60: x= _. By log. 60 - - 1.778151 cos. 420 - - - - 9.871073 44.59 - 1.649224 184 NAVIGATION. N. B. — In this manner the length of a degree in longitude corresponding to each degree of latitude has been found and put in a table. 3. A ship sails east from Cape Race, 212 miles; required her longitude. The latitude of the cape is 460 40' N., longitude 530 3' 15" west.. Ans. lon. 470 54' west. 4. Two places in lat. 50~ 12' differ in longitude 34~ 48'; required their distance asunder in miles. Ans. 1336. 5. How far must a ship sail V. from the Cape of Good Hope, that her course to Jamestown, St. Helena, may be due north? Ans. 1193 miles. Cape lat. 340 29' Jamestown lat. 153 55' S. Cape lon. 180 23' E. IJamestown on. 50 43' 30" W. 6. How far must a ship sail E. from Cape Horn to reach the meridian of the Cape of Good Hope? The latitude of Cape Horn being 550 58' 30" S., Ion. 670 21' W., and the latitude and longitude of the Cape of Good Hope being as stated in the last example. Ans. 2878 miles. 7. In what latitude will the difference of longitude be three times its corresponding departure? In other words in what latitude will FG be one-third of AB? ( See last figure.) Ans. lat. 700 31' 44". 8. Take the 2d example in plane sailing ( page 181 ), the departure made, as stated in the answer, is 233.4 miles. What is the corresponding difference of longitude? Ans. 5~ 18' 24". This inquiry now arises. To what latitude does this departure correspond? Is it to the latitude left, 41~ 9', or to the latitude arrived at, 440 41'? Or does it correspond to the mean latitude between the two? If we suppose the departure corresponds to lat. 41~ 9', then the difference of longitude by the preceding rule must be 50 10', and if the departure corresponds to lat. 440 41', then the difference of longitude is 50 28'; the mean of these is 5~ 19', and if we take the departure to correspond with the mean latitude 42~ 55', then the difference of longitude would be 50 18' 24". In the examples under Plane Sailing we have supposed the earth a plane, and the course a ship sails a straight line, but neither supposition is strictly true. PLANE SAILING. 185 Meridians are not paradle with each other, and therefore when a ship sails by the compass, and cuts all the meridians at the same angle, the line that the ship sails will not be a right line: it will be a curve line peculiar to itself, called a rhumb line. For the sake of illustration, let us suppose that in the annexed figure, P is the north pole, KQ the equator, or a great circle, every part of which is a quadrant distance from P; PiK, PL, P., &e., great circles passing through P, and of course cutting the equator at right angles; Al, bB, R S, &c., arcs of smaller circles parallel to the equator, and therefore cutting the meridians at right angles; AE a curve cutting every meridian which it meets, as PK, PL, PM2, &c., at the same angle. Then PK, PL, &c., produced till they meet at the opposite pole, are called meridians; A4, bB, RS, &c., continued round the globe, are called parallels of latitude; AE is called the rhumb line, passing through A and E; the length of AE is called the nautical distance from A to E; and the angle BAb, or any of its equals, cBC, d CD, &c., is called the course from A to E. If a ship sail from A to E, EF will be her meridian distance; but if she sail from E to A, AI4will be her meridian distance. If AB, BC, CD, &c., be conceived to be equal, and indefinitely small, and their number indefinitely great; then the triangles A.Bb, Bc C, &c., may be considered as indefinitely small right-angled plane triangles. And as the angles BAb, CBc, &c., are equal, and the right angles A6B, Bc C, &c., are equal, the remaining angles ABb, BCc, &c., are equal; and as the sides AB, BC, &c., are also equal, these elementary triangles ABb, BCc, CDd, &c., will be all identical triangles; therefore AE will be the same multiple of AB, that the sum of Ab, Bc, Cd, &c., is of Ab; and that the sum of Bb, Cc, Dd, &c., is of Bb. It is obvious that Ab+Bc+ Cd+&c.=AF the whole difference of latitude. And, Bb+ Cc+Dd+Ee+&c. =the whole departure. 16 186 NAVIGATION. But this departure is neither EF nor Az, it is greater than iF and less than AI; because bB is less than its corresponding portion on AI and greater than its corresponding portion on FE. The same may be said of cC, Dd, &c. Therefore the departure on any course corresponds to neither of the extreme latitudes, but to some mean between the two, and it is so near the arithmetical mean, that the arithmetical mean is taken as the true. Therefore in all those examples in Plane Sailing, on page 181, we can take the departures and find the corresponding differences of longitude, provided we take the middle latitude and consider the departure run on that parallel. This method of connecting the change in longitude with a ship's change of place is called MIDDLE LATITUDE SAILING. But in reality there is no such thing as middle latitude sailing; the cosine of the middle latitude is compared to the radius, as the ratio between the departure made and the corresponding difference of longitude, but the departure made may be made on one course or on several courses. When a ship sails on several courses before the run is summed up, the summing up and finding the result in one course and distance is called working a traverse, and sailing from one point to another by several courses is called TRAVERSE SAILING. With adverse winds or crooked channels, vessels are obliged to run a traverse. Going round a survey and keeping an account of our course and distance from the starting point is working a traverse, and the operation is the same on sea or land, except on land we aim at coming round to the same point again, but on sea we wish to make some other point. With this explanation it is obvious that we must make a table as in a survey, and compute the course and distance from the starting point, and this is called the course and distance made good. To work a traverse we use the traverse table of course; that table is made to every half degree, and the column in the table nearest to the course is sufficiently exact. TRAVERSE SAILING. 187 The following table gives the degree and parts of a degree corresponding to every point and quarter point of the compass. Points. Deg. Points. Deg. 41- 20 48' 45" 4-4 470 48' 45" i ~. 50 37' 30" 44 500 37' 30" &~ 80 26' 15" 44 530 26' 15" 1. 110 15' 5 560 15' 1, 14~ 3' 45" 65 590 3' 45" 14 \16~ 62' 30" 54 610 52' 30" 1i 190 41! 15" 5 640 41' 15" 2 220 30' 6 670 30' i2 260 18' 45" 64 700 18' 45" 24 280 7' 30' 64 730 7' 30" 24 30~ 56' 15" 64 750 56' 15' 3 330 45' 7 78~ 45' 34 360 33' 45" 74 810 33' 45" h34 390 22' 30" 74 84~ 22' 30" 31 420 11' 15" 74 870 11' 15" 4 450 0' 8 900 0' In-works exclusively designed for practical navigation, the traverse table is adapted exactly to the points and quarter points of the compass, but the table in this work is sufficient for the purpose. The use of this table is to find the degree corresponding to any given course, thus: NV by E., N. by W., S. by E., S. by W., each correspond to 1 point or 11~ 15'. In using our traverse table for 1 point we should take a mean result between 110 and 110 30', which mean result can be taken by the eye. Again S.E. by E. 4 E. is 54 points, or S. 590 E. nearly, and so on for any other course that may be named. The student is now fully prepared to work the following examples in traverse sailing. E XAMPLES. 1. A ship from Cape Clear, Ireland, in lat. 51C 25' 1VY and longitude 90 29' WT., sails as follows: S.S. E. ~ E. 16 miles, E. S. E. 23 miles, S. W. by W. T W. 36 miles, W. i N. 12 miles, and S. E. by E. 1 E. 41 miles. 188 NAVIGATION. Required her course and distance made good, the departure, latitude and longitude of the ship. By Construction.- Take A for the place sailed from, and draw the vertical line NIASC, to represent the meridian. About A, as a center, with the chord of 60~, describe a circle, cutting NVC in Nand S; then Nand S will represent the north and south points of the compass. Take 21 points from the line of chords,* and apply it from S to a, join Aa, and on it take AD= 16 from a line of equal parts. Then D will be the place of the ship at the end of the first course. From S, set off Sb-6 points from the line of chords; join A b, and through D draw DE, parallel to Ab, and make it equal to 23 from the same scale of equal parts that AD was taken from. Then _ will be the place of the ship, at the end of the second course. Make Sc=51 points, Vd 7~ points, and Se 54 points, taken from the line of chords. Through E draw EF, parallel to Ac, and make it equal to 36, from the scale of equal parts; through F draw FP6 parallel to Ad, and make it equal to 12, from the scale of equal parts; through (G draw GB parallel to Ae, and make it equal to 21, from the scale of equal parts. Demit BC a perpendicular, on the meridian IVC, and join AB; then B will be the place of the ship, AB her distance from the place which she left, A C her difference of latitude, BC her departure, and BA C the course which she has made on the whole. Now, AB, A C, and B C being measured on the scale of equal parts from which the distances were taken, we have AB= 62.7, A C= 59.6, and B C=19.6 miles. And the arc included by A C and BC, if measured on the line of chords, gives about 180 for the measure of the course BA C. * Two and i points is 25~ 18' 45", that is, take the chord of 250 18' in the dividers, and set it off from s to a, and so on, for other angles. TRAVERSE SAILING. 189 T RAVE B 15E TA B L. Oourse. Pointe. Dis. Diff. Lat. D)ep. i_____ N. E W.I S. E. 1E. 16 14.5 6.8 E. S. E. 6 23 8.8 21.3 S.- W. by W:. IW. 5j 36 17.9 31.8 W.4j N. 74 12 1.8 11.9 S. E. by E. E. 54 41 21.1 35.2 ~ 1.8 614.63.3 4.7' Result 59.6 119.6 Lat. left - 510~ 2' Z. diff. lat. - - - 00 S. Lat. in - 50 $25 N. Mid. lat. 0' 55' To find the course and distance, by trigonometry, As dif. lat. 59.6 miles 1.775246: radius 90~ 10.000000:: dep. 19.6 miles 1.292256: tan. course 18~ 12' 9.517010 As si. course 180 12' 9.484621: dep. 19.6 miles 1.292256: Radius 10.000000: Dis 62,75 miles 1.797635 To find the dif. of longitude. As cos. 50 5i5' 9.799651: Radius 10.000000:: dep. 19.6 1.292256:: diff. on. 31.09 miles 1.492605 Longitude left 90 29' west diff. lon. 31 east Lon. in 8~ 58' west Thus, we have found the course 180 12'; Distance 62.75 miles; diff. longitude 31' E.; lat. in 50.25 N.; lon. 8~ 58' W. If these be the distances run in a day, from noon to noon again, then the preceding operation is called working a day's work; other. wise it is called working a traverse, as we have meantioned before, 1910 NAVIGATION. But this is not the seaman's way of working a day's work, he does it all by inspection, in the traverse table. For example, taking the result of the traverse 59.6 south, and 19.6 east, which shows that the resulting course is between the south and the east, and with these numbers he enters the traverse table, and finds, as near as possible, 59.6 and 19.6, standing as latitude and departure; and they are found nearly under the angle of 18~, and opposite the distance 63. nearly. Hence, he takes his course as S. 18~ E., and dis. 63. To find the difference of longitude, he takes the middle latitude as a course, and the departure as difference of latitude, then the distance in the table is difference of longitude. In this instance, we take 51~ as a course, and in the difference of latitude column we find 19.5, and the distance opposite to it is 31., which we take for difference of longitude. The reason for this is as follows: For the longitude we have, cos. mid. L: R:: dep.: diff. Ion. (1) In the construction of the traverse table, we have, cos. course: R.:: diff. lat.: dist. (2) Now, in proportion (2), if we take the middle latitude for a course, and the dep. for difference of latitude; it necessarily follows, that the last term of proportion (2) must be diff. of longitude; for proportion (2) would then be transformed into proportion (1). 2. A ship sails from. Cape Clear, as follows; S. by W: 23 miles; Wr. S. tW. 40 miles; S. W;. W. 18 miles; WV. I N. 28 miles; S. by E. 12 miles; S. S. E. I E. 16 miles. Required the course and distance made good, and the latitude and longitude arrived at. Ans. Course S. 450 47' W.; dis. 102.4 miles. Latitude of ship 500 14' N.; Lcn. 110 25' W. 3. A ship at noon, on a certain day, was in lat. 410 12' NV, and longitude 370 21' TV, she then sailed as follows: S. W. by W. 21 m.; S. WV.I S. 31 m.; W. S. W. i S. 16 m.; S. E. 18 m.; S. W.V W. 14, and WV. j Y.. 30 miles. Required her course, distance, latitude, and longitude. Ans. course S. 62~0 49' W.; dis. 11.1.7; lat;L40~ 5' N.; ion. 390 18' W. SAILING IN CURRENTS. I91 4. Last noon we were in latitude 28~ 46' south, and longitude 32~ 20' west; since then we have sailed by the log: S. W.4J W. 62 m.; S.by W. 16 m.; W;. S.40m.; S.IW.I.W. 29 m.; S. by E. 30 m.; and S. 4 E. 14 miles. Required the direct course and distance, and our present latitude and longitude. Ans. Course S. 430 14' W.; Dis. 158 m.; lat. 30~ 41' S.; lon. 340 24' W. 5. A ship from Toulon, lat. 430 7' N., lon. 5~ 56, E., sailed S. S. W.48 m.; S.by E.34m.; S. W. W. 26 m.; and E. 17 miles. Required her course and distance to Port Mahon, Lat. 390 52' N., and longitude 40 18 30" east. Ans. Lat. of ship 41~ 32' N.; Ion. 5~ 37' east. Course to Port M. S. 310 W. nearly, and distance 117.5 miles. 6. On leaving the Cape of Good Hope, for St. Helena, we took our departure from Cape Town, bearing S. E. by S. 12 miles, after running N. WI. 36 m., and N. W. by W. 140 miles. Required our latitude and longitude, and the course and distance made. N. B. Lat. of Cape Town 33~ 56' S. Lon. 18~ 23' E. Lat. of St. Helena 156 55' S. Lon. 5~ 43' 30" WI Ans. Lat. 32~ 3' S.; Lon. 18~ 25' E.; course N. 520 41' W.; dis. 187 miles. SAILING IN CURRENTS. If a ship at B, sailing in the direction BA, were in a current which would carry her from B to C, in the same time that in still water she would sail from B to A, then, by the joint action of the current and the wind, she would in the same time, describe the diagonal BD of the parallelogram AB CD. For her being carried by the current in a direction parallel to BC, would neither alter the force of the wind, nor the position of the ship,northe sails, with respect to it; the wind would therefore continue to propel the ship in a direction parallel to AB, in the manner as if the current had no existence. Hence, as she would be swept to the line CD, by the independent action of the current, in the same time that she reached the line AD, by the independent action of the wind on her sails, she would be found at D, the point of intersection of the lines AD and CD, having moved along the diagonal BD. 192 NAVIGATION. Now the log heaved from the ship in the ordinary way, can give no imitation of a current; for the line withdrawn from the reel is only the measure of what the ship sails from the log; and, consequently, as the log itself, as well as the ship, will move with the current, the distance shown by the log in a current, is merely what it would have been if the ship had been in still water. If the ship sail in the direction of the current, the whole effect of the current will be to increase the distance; but if she sail against the current, the difference between the rate of sailing given by the log and drift of the current, will be the distance which the ship actually goes; and she will move forward, if her rate of sailing be greater than the drift of the current, but otherwise, her motion will be retrograde, or she will be carried backwards, in the direction of the current. Problems relating to the oblique action of a current upon a ship, may be resolved by the solution of an oblique-angled plane triangle, such as ABD, in the preceding figure, where if AB represent the distance which a ship would sail in still water, and 4D the drift of the current in the same time, BD will be the actual distance sailed, and ABD the change in the course produced by the current. A great variety of problems might be proposed relative to currents, but the chief ones of any practical importance, are the following: 1. To determine a ship's actual course and distance in a current, when her course and distance by the compass and the log, and the setting and drift of the current, are given.. To find the course to be steered through a known current, the required course in still water, and the ship's rate of sailing, being known. 3. To find the setting and drift of a current, from a ship's actual place, compared with that deduced from the compass and the log. The first of these cases may be conveniently resolved, by considering the ship as having performed a traverse, the setting and drift of the current being taken as a separate course and distance. EXAMPLES. 1. If a ship sail T: 28 miles in a current, which in the same time carries her N.V. W. 8 miles, required her true course and distance. SAILING IN CURRENTS. 193 N. B. Conceive the current to be one course and distance, and with the other courses find the course and distance made good. Thus, by the traverse table: Diff. Lat Dep. Course. Dis. N. S. E. W. W. 28 28 N. N. W. 8 7.39 2.06 7.39 31.06 As 7.39: rad.:: 31.06: tan. 76~ 36', the course, cos. 76~ 31': B: 31.06: 31.93, the distance. 2. If a ship sail E. 7 miles an hour by the log, in a current setting E. N. E. 2.5 miles per hour; required her true course, and hourly rate of sailing. Ans. Course NY. 84~ 8' E., and rate 9.358 per hour. 3. A ship has made by the reckoning N. WV. 20 miles, but by observation it is found, that, owing to a current, she has actually gone NV. NV: E. 28 miles. Required the setting and drift of the current in the time which the ship has been running. Ans. Setting N. 64~ 48' E., and drift 14.1 miles. 4. A ship's course to her port is W. N. W., and she is running by the log 8 miles an hour, but meeting with a current setting W. i S. 4 miles an hour, what course must she steer in the current that her true course may be W. NT. W? Ans. Course N., 440 39' W]. 5. In a tide running \. W. b W. 3 miles an hour, I wished to weather a point of land, which bore N. E. 14 miles. What course must I steer so as to clear the point, the ship sailing 7 miles an hour by the log, and what time shall I be in reaching the point? Ans. Course Nf. 69~ 51' E., and time 2 hours 25 minutes. 6. From a ship in a current, steering. W. S. Wt 6 miles an hour by the log; a rock: was seen, at —6 in the evening, bearing S.& W:.. S. 20 miles. The ship was lost on the rock at I11 P. M. Rlequired the setting and drift of the current. Ans. Setting S. 7-50 10' E., and drift 3.11 miles per. hour. 17 194 NAVIGATION. C HA P T E R III. MERCATOR'S CHART AND MERCATOR'S SAILING. IX representing any small portion of the earth's surface, it is sufficiently accurate to represent the meridians as parallel; but if the portion of the earth is considerable, the representation will not be true unless the meridians are curved. If we make a chart and draw all the meridians parallel with each other, the length of a degree of longitude in all places, except on the equator, will be greater on the chart than' its true distance, but the true bearing of one place from another will be preserved, provided we increase the degrees of latitude in the same ratio as the degrees of longitude are increased. Gerrard Mercator, a Fleming, in 1556, published a chart which seemed to embrace this idea, but he did not show its construction, nor were his degrees in their true proportion; but from this came the name of Mercator's Chart. A Mr. Wright, an Englishman, in 1599, it is said, published the true sea chart, constructed on the following principles. 1. The distance between two meridians at the equator, is8 to their distance in any parallel of latitude, as the radius is to the cosine of that latitude. 2. any part of a parallel of latitude, is to a like part of the meridian, as the radius is to the secant of that parallel. We shall make an effort to illustrate these principles by the following figure. Conceive the equator to be extended both ways parallel to the earth's axis, thus forming a cylinder, whose circumference is just equal to the circumference of the earth. MERCATOR'S CHART. 195 Let Qq be the plane of the equator, Pp the earth's axis; conceive a globe enclosed in the cylinder, HLMX. Suppose there is an island on the earth at a, that island is projected on the cylinder at A. The surface of the earth at b is projected at B. Conceive this paper cylinder cut by a line at right angles to the equator and rolled out, it will then be a true representation of Mercator's chart. The scale on the globe at a is, to the scale on the chart at A, as Ca to CA, that is, as radius to the secant of the latitude at a. The scale on the chart at A is, to the scale on the chart at B, as CA is to CB, that is, the scale on the chart increases as the secants of the latitudes increase. The poles of the earth, and places very near the poles, can never be represented on this chart. The meridian distance of a degree on the globe, as at a, is 60 miles, on the chart at A it is 60, into A C the secant of the latitude, calling Ca unity. If we commence at the equator at Q, and take one mile for unity. Then, Mer. pts. of 1'= nat. sec. 1 Mer. pts. of 2'= nat. sec. 1+ nat. sec. 2 Mer. pts. of 3'= nat. sec. 1'+ nat. sec. 2'+ nat.: sec. 3 Mer. pts. of 4'= nat. sec. 1'-+ nat. sec. 2'+ nat. sec. 3' + nat. sec. 4', &c., &c. In this manner the table of meridional parts was originally constructed. It is Table IV of this work. The following figures represent any problem than can arise in Mercator's sailing. AC represents the true difference of latitude. AD represents the meridional difference of latitude, which is always taken from the table. CB represents the departure. DE the difference of longitude. 196 NAVIGATION. AB represents the distance. A, the angle at A, represents the course. Three of thesesix quantities must be given to solve a problem. Observe that the difference of longitude DE is always greater than the departure CB, as it ought to be. E XAMPLES. 1. A ship from Cape Finisterre, in lat. 42~ 66' N., and longitude 80 16' V., sailed S. W. i W. till her difference of longitude is 134 miles; required the distance sailed and the latitude in. By logarithms. As radius - - - - 10.000000 diff. ion. 134 miles - - - - 2.127105:: cot. course 44 points - - - 9.957295:mer. diff. lat. 121.5 miles - - 2.084400 Lat. Cape Finisterre 420 56' V. Mer. parts - 2858 Mer. diff. - 121 Lat. 410 27' IV., corresponding to - - - 2737 in table As cosine course - 9.827085: proper diff. lat. 89 miles - 1.949390::radius 10.000000:dis. 132.5 miles - 2.122305 2. A ship from lat. 40~ 41' N., ion. 160~37' WT., sails in the N. E. quarter till she arrives in lat. 430 57' N., and has made 248 miles departure; required her course, distance, and longitude in. Ans. course N. 51~ 41' E., dis. 316 miles, and ion. in 110 W. 3. How far must a ship sail gN. E. i E. from lat. 440 12' NT;, Ion. 230~,- to reach the parallel of 470 N., and what from that point will be the bearing and distance of UJshant, which is in lat. 480 28'.V. and lon. s~ 3' W.? Ans. She must sail'262, miles, and her course and distance to Ushant will then be NV. 800 32'., and dis. 535 miles. 4. A ship from the Cape of Good Hope steers E. i S. 446 miles; required her place, and her course, and distance to Kerguelen's Land, in lat. 480 41' S., and ion. 690east. MECATOR'S SAILNIG. 197 Ans. lat. in 350 13' S., ion. in 27~ 21' E., course S. 660 25' E., and distance 2018 miles. 5. By observation, a ship was found to be in lat. 410 50' S., ion. 680 14 E. She then sailed N. E. 140m, and E j S. 76m; required her place, and her course, and distance to the island of St. Paul, which is in lat. 380 42' S., and in lon. 770 18' E. Ans. lat 40~ 18' S., ion. 720 2', course NV. 680 35' E, and dis. 263 miles, nearly. CELESTIAL OBSERVATIONS. CHAPTER I. WE now come to the more scientific and essential parts of navigation, the determination of latitude and longitude by celestial observations. We shall at present confine ourselves to latitude, first calling to mind the following necessary definitions and explanations: 1. MERIDIAN. - The meridian of any place is the north and south line passing through that place, and it may be conceived to run along the ground or pass in the same direction in the heavens, through the point vertically over the place. The line on the earth is the terrestrial meridian, the line in the heavens is called the celestial meridian; they are both in one plane with the center of the earth. 2. EQUATOR. - The equator is that circle around the earth over which the sun seems to pass when the days and nights are equal all over the earth. 3. LATITUDE. - The latitude of any place is the meridian distance of that place from the equator, measured by degrees and parts of a degree of arc. 4. LONGITUDE.- The longitude of any place is the inclination of the plane of its meridian, with the plane of some other definite meridian from which reckoning is made. This inclination is measured on the equator by degrees, minutes, and seconds of arc, and it is either east or west.* * The first meridian to reckon from may be arbitrarily chosen, and different nations have taken different meridians for the commencement of longitude, but custom and long association have pretty firmly fixed the meridian of Greenwich (England) as the first meridian for all who use the English language. (198) NAVIGATION. 199 5. DECLrNAToN. -The declination of a heavenly body is its meridian distance from the equator north or south. 6. POLAR DISTANCE. - The polar distance of a body is its declination added to, or subtracted from 90~. If both added and subtracted, we shall have the meridian distances from each pole. The distance from the north pole, is called north polar distance, and from the south pole, south polar distance. The two polar distances must of course make 1800. 7. ZEiT-. - Zenith is the point in the heavens directly overhead. 8. HoaRzos. — The horizon is either apparent or real, or as commonly expressed, sensible or rationral. The sensible horizon is a plane conceived to touch the earth at any point at which an observer is situated. The rational horizon is a plane parallel to the sensible one, passing through the center of the earth. The zenith is the pole to the horizon. 9. GREAT CIRCLES. - A great circle in the heavens is any circle whose plane passes through the center of the earth. All great circles which pass through the zenith are perpendicular to the horizon, and such circles are called vertical circles, azinuth circles, or circles of alitude. 10. AZIUa.TH -- The angle which the meridian makes with that vertical circle which passes through any object is said to be the azimuth of that object. Henee, azimuths may be reckoned from the north or south points of the horizon. 11. ALTITUDE. - The altitude of any object is its angular distance from the horizon, measured on a vertical circle.* Altitudes are very frequently measured at sea, several times in a day in fair weather; but altitudes observed from the surface of the earth, or above it, require several corrections before the true altitudes can be deduced from them. We do not pretend to give all the definitions of the sphere, but we suppose the reader is already acquainted with them, from his knowledge of Geography and Astronomy. 200 CELESTIAL OBSERVATIONS. These corrections are for semi-diameter, dip, refraction, andparallax. The correction for semi-diameter is obvious. At sea, the visible horizon (from which all observed altitudes are taken) is where the sea and sky apparently meet, and when the eye of the observer is above the water, this visible horizon is below the sensible horizon, and the amount of the depression is called the dip of the horizon. Its correction is always subtractive, and its amount is to be found in Table VI. Refraction is to be found in Table V. It is always subtractive, and for the reason, see some treatise on natural philosophy. Parallax is always additive. Conceive two lines drawn to a heavenly body; one from an observer at the circumference of the earth and the other from the center of the earth, the inclination of these two lines is parallax, and when the body is in the horizon its parallax is greatest, and it is then called horizontal parallax. Parallax always tends to depress the object, but the parallax of any celestial object, except that of the moon, is so small, that we shall pay attention to lunar parallax only, but this is so important to navigation that we shall give it a full explanation. The moon's horizontal parallax is given in the Nautical Almanac for every noon and midnight of Greenwich time, and from the horizontal parallax we must deduce the parallax corresponding to any other altitude. Let A C be the radius of the earth, A the position of an observer, Z his zenith, and suppose H to be the moon in the horizon; then the angle AHC* is the moon's horizontal parallax, and the angle AhC is the parallax corresponding to the apparent altitude hAH. Draw Am parallel to Ch, then mAHwould be the true altitude. * From this figure we draw the following definition for horizontal parallax. The horizontal parallax of any body is the angle under which the semi-diameter of the earth would appear as seen from that body. Of course then, when the body is at a great distance its horizontal parallax must be small, hence the sun and the remote planets have very little parallax, and the fixed stars none at all. NAVIGATION. 201 Let CH and CA be each represented by R. Putp= the horizontal parallax, and x= the parallax in altitude, or the angle mAh or Ah C. Now in the triangle A CH, right-angled at A, we have 1: sin.p::.R: AC. In the triangle A CA/ we have sin. CAh: sin. x:: 1: AC. By comparing these two proportions, we perceive that 1: sin. p:: sin. CA: sin. x Whence, sin.'x= sin.p. sin. A Ch But sin. CAh= cos. h/IA, for the sine of any arc greater than 900 is equal to the cosine of the excess over 90~, hence, sin. x= sin.p cos. hAH The lunar horizontal parallax is rarely over a degree, commonly less, and the sine of a degree does not materially differ from the arc itself, hence, the preceding equation becomes the following, without any essential error. That-is, x=p cos. altitude. Or, in words, the parallax in altitude is equal to the horizontal parallax multiplied into the cosine of the apparent altitude (radius being imity). EXAMPLES. 1. The apparentaltitude of the moon's center after being corrected for dip and refraction was 310 25'; and its horizontal parallax at that time, taken from a nautical almanac, was 57' 37"; what was the correction for parallax, and what was the true altitude as seen from the center of the earth? p=57' 37"=3457" log. - - 3.538699 310 25' cos. - - 9.931152 x=49' 10"=2950 log. - - 3.469851 Ans. Cor. for parallax 49' 10" True altitude 32~ 13' 10" 2. The apparent altitude of the moon's center on a certain occasion was 42~ 17'; and its horizontal parallax at the same time was 68' 12"; what was the parallax in altitude, and what was the moon's true altitude? Ans. Parallax in alt. 43' 4" True alt. 430 O' 4" 202 CELESTIAL OBSERVATIONS. No other examples of this kind are necessary, as they will incidentally occur in several places further on. It now remains to describe the instrument used for taking angles at sea. We, therefore, give the following illustrations on the QUADRANT AND SEXTANT. The quadrant and sextant are essentially the same instrument, and the following is an explanation of the principle on which they are constructed. Let AB C be a section of a reflecting surface, FB a ray of light falling upon it, and reflected again in the direction BE, and BD a perpendicular at the point of i'mpact; then it is a well known optical fact, that the angles PB C and EBA are equal, and that FB, DB, and EB are in the same plane. Again, if A C were a reflecting surface, and a ray of light, SB, from any celestial object S, were reflected to an eye at E, the image of the object would appear at S' on the other side of the plane, the angles SBA and ABS', as well as EB C, being equal; and if EB bear no sensible proportion to the distance of S, the angles SES' and SBS' may be considered as equal; for their difference, BSE, will be of no sensible magnitude. Before we proceed to the direct description of the sextant, it is necessary to give the following important LEMMA. if the exterior angle of a triangle be bisected, and also one of the interior opposite angles, and the bisecting lines produced until they meet, the angle so formed will be half the other interior opposite angle. Let ABC be the triangle, and bisect the exterior angle ACD by the line CE, and the angle B by the line BE. The angle X will be half the angle A. Let each of the angles A CE, ECD, be designated by x ( as rep THE PLANE TABLE. 203 resented in the figure ), and each of the equal parts of the angle B by y. Let A represent the angle A, and X the angle E. Now as the sum of the three angles of every plane triangle is equal to 180~; therefore, in the the triangle ABC, we have +2y+ C=1800 (1) Also, in the triangle EB C, we have E+Y+-:C+x=18o0 (2) Subtracting (2) from (1) gives us A —E+y —x=O (3) Whence, A=E+(x — y) (4) But because x is the exterior angle of the triangle ECB x=E+y (see Elementry Geometry.) Or, (x-y)=E This value of (x —y) substituted in (4) gives A A=2E, or E=2 Q. E. D. Another Demonstration.- The angle x being the half of A CD is equal to A+2y 2 The angle x is also equal to E+y, because it is the exterior angle to the triangle EBC. Therefore, by equality, E+y-A+2y Whence,. E. D.2 Whence, L=~A Q. E. D. 204 CELESTIAL OBSERVATIONS. We are now prepared to show the construction of the sextant and quadrant. The instrument represented by.the annexed'cut is a quadrant or a sextant, according as the arc con-: tains 900 or 120~, but each actual degree of arc is graduated to 20, and the space that covers 90~ is really but 450, and so on. The reason why a half degree is counted and marked as a whole one, we are about to explain. AB C is a firm plane sector, commonly made of metal or ebony; AJ is a revolving index bar, turning on the center A, to which is attached a vernier scale, revolving over the graduated arc. The graduation commences at B. At A is a small plane mirror, perpendicular to the plane of the sector, it is attached to the index bar and revolves with it. This is called the index mirror or index glass. At H is another small mirror, half silvered and the other half transparent. This is called the horizon glass; it might be called the image glass. The horizon glass must be perpendicular to the instrument, and parallel to AB. Now conceive a ray of light coming from an object S, striking the mirror A, the index and mirror being turned so as to throw the reflecting ray into the mirror H, this mirror agains reflects it toward E, and an eye anywhere in the line DHwill see the image of the object behind the mirror H. Conceive the ray of light from S to pass right through the mirror at A, to meet the line HE; then, it is obvious that the angle SED measures the angle between the object S and its image D. Now, in the triangle AEH, by a little inspection, it will be found that HL bisects the exterior angle, and AJ, the index, bisects one of NAVIGATION. 205 the interior opposite angles; therefore, by the preceding lemma, the angle HLA is half the angle at E, but as AB and the mirror H are parallel, the angle HLA is equal JAB. It is obvious that JAB is measured by the arc BJ, or it measures the angle at E, if half degrees on B C are counted as whole ones, which was to be shown. A tube, and sometimes a small telescope, is attached to the bar AB, and placed in the direction of the line EH. This is called the line of sight. THE ADJUSTMENT OF THE INSTRUMENT. When this instrument is in adjustment, the two mirrors are perpendicular to the plane of the sector, and are parallel to each other when 0 on the vernier coincides with 0 on the arc. We therefore inquire: First Is the index mirror perpendicular to the plane of the instrument. The following experiment decides the question. Put the index on about the middle of the arch, and look into the index mirror, and you will see part of the arch reflected, and the same part direct; and if the arch appears perfect, the mirror is in adjustment; but if the arch appears broken, the mirror is not in adjustment, and must be put so by a screw behind it, adapted to this purpose. Second, Are the mirrors parallel when the index is at 0? Place the index at 0, and clamp it fast; then look at some welldefined, distant object, like an even portion of the distant horizon, and see part of it in the mirror of the horizon glass, and the other part through the transparent part of the glass; and, if the whole has a natural appearance, the same as without the instrument, the mirrors are parallel; but, if the object appears broken and distorted, the mirrors are not parallel, and must be made so, by means of the lever and screws attached to the horizon glass. Third, Is the horizon glass perpendicular to the plane of the instrument? The former adjustments being made, place the index at 0, and clamp it; look at some smooth line of the distant horizon, while holding the instrument perpendicular; a continued unbroken line will be seen in both parts of the horizon glass; and if, on turning the instrument from the perpendicular, the horizontal line continues 206 CELESTIAL OBSERVATIONS. unbroken, the horizon glass is in full adjustment; but, if a break in the line is observed, the glass is not perpendicular to the plane of the instrument, and must be made so, by the screw adapted to that purpose. After an instrument has been examined according to these directions, it may be considered as in an approximate adjustment — a reexamination will render it more perfect - and, finally, we may find its index error as follows:-measure the sun's diameter both on and off the arch —that is, both ways from 0, and if it measures the same, there is no index error; but if there is a difference, half that difference will be the index error, additive, if the greater measure is off the arch, subtractive, if on the arch. To measure the altitude of the sun at sea. Turn down the proper screen or screens, to defend the eye. Put the index at 0, having it loose, look directly at the sun through the tube, and you will see its image in the silvered part of the horizon glass. Now move the index, and the image will drop; drop it to the horizon, and clamp the index. Let the instrument slightly vibrate each side of the perpendicular, on the line of sight as a center, and the image of the sun will apparently sweep along the horizon in a circle. While thus sweeping, move the tangent screw, * so that the lower limb of the sun will just touch the horizon, without going below it. The reading of the index will be the altitude corresponding to that instant, provided there be no index error. To measure the angular distance between two bodies as the sun and moon, or the moon and a star. The most brilliant of the two objects is always reflected to the other. Loosen the index, place it at 0, and direct the line of sight to the brighter object, and catch a view of its image in the silvered part of the horizon glass. Turn the plane of the instrument into the plane between the two objects; now move the index, keeping the eye on the image, and * The screens, adjusting screws, clamp screw, and tangent screw, are not given in our description of the instrument, it is not necessary to describe them; should we attempt it, there is danger that the spirit and clearness of the description would be lost in the multitude of words. NAVIGATION. 207 bring it along to the other object; bring them as near as possible, then gently clamp the index. Hold up the instrument again, in the plane between the two objects, and view one object through the transparent part of the horizon glass; and when the instrument is in the right position, the image of the other object will appear also in the same field of view, and then with the tangent screw, make the limb of the reflected object just touch the other, as it moves past it to and fro, by the gentle motion of the instrument. When the observer is satisfied that he has got the measure as near as he'can, he cries out, mark, and his assistants mark the time by the watch, and the altitudes of the objects are also marked for the same time, if required, and observers are present to take them. The first experiments in the use of this instrument, other than measuring a simple altitude, are generally failures, but a little practice will establish dexterity, skill, and confidence. We are now prepared to give examples for finding latitude. Let it be remembered, that latitude is the observer's zenith distance from the equator, and the nautical almanac gives the distance of all the heavenly bodies from the equator, under the name of Declination. We can therefore observe our zenith distance from any celestial object, and then apply its declination, and we shall have our zenith distance from the equator, which is the latitude. EXAMPLES. 1. On a certain day, the meridian* altitude of the sun's lower limb was observed to be 310 44', bearing south. At that time its declination was 70 25' 8" south, semi-diameter 16' 9", index error +-2' 12", height of the eye 17 feet. What was the latitude? Ans. 50~ 48' north. * To obtain the meridian altitude of the sun, the observer commences observations before noon, while the sun is still rising; driving the index forward as fast as the image appears to rise, and there will come a time, a few minutes in succession, in which the image appears to rest on the horizon, neither rises nor falls, but at length the image will fall; then the observer knows that noon has passed, and the greatest apparent altitude will be shown by reading the index. 208 NAVIGATION. Semi-diameter +16' 9" NV.A. Alt. ob. 31~ 44 Index error + 2.12 Correction 12' 46" Refraction - 1.31 Table. Alt. ()'s center 31 56 46 Dip - 4.04 Table. 900 Sum +- 12'46" ('s zenith dis. 580 3' 14"Dec. south 70 25' 8" Latitude north 50~ 48' 6" In this example, if the meridian altitude had been observed in the north, in place of the south, what then would have been the observer's latitude? Ans. 65~ 28' 22" south. We may note the following RULE. - Subtract the corrected- altitude from 90~. Then if the observer and the object are both on the same side of the equator, add the declination, but if on different sides, subtract the declination, and the sum or difference will be the latitude of the observer. Find the latitude from each of the following meridian observations: Object. Alt. ob. Direc. S.D. Height. Declination. Latitude. I SLun L. L. 450 27' South 16' 15" 20 feet 170 19' 31" S. 270 2' 51" N. 2 1" L.L. 81o 43' South 15'47"114 " -22013' 7"N. 300 18'10" N. 3 Jupiter 730 17' South 17' 24o10'13" S. 7022'52" S 4 Saturn 82~ 12' North 17 " 120 9' 6"N. 40 16' 54" N. 5 Sirius 750 5' North 18 " 16~31' S. 31030' 26" S. 6 Sun U. L. 40042' North 16'17" 16 " 23022' S. 730 1'19" S. 7 Sun L. L. 87029' South 16'17" 16 " 22~ 9' S. 190 50' 12" S. 8 Sun L. L.15045' South 16' 0" 16 " 4043' N. 69~23' 16" N In this table L. L. indicates lower limb, U. L. upper limb, S. D. semidiameter, N. north, S. south, Direc. direction. In these examples, the instrument is supposed to have no index error. Night observations at sea are of little value, for it is very seldom that the horizon can be defined, unless it is in bright moon-light, in the tropical climates. For this reason, very few navigators attempt to find the latitude, by observations on the planets and fixed stars. Occasionally, however, when one of the bright planets, or a conspicuous fixed star, comes to the meridian in the morning or evening, twilight observations can be made on them, and the latitude deduced. Some navigators apply a summary correction to the sun's lower limb, for semi-diameter, dip, and refraction, such as is comprised in the following table. CELESTIAL OBSERVATIONS. 209 Correction to be added to the Observed Altitude of the Sun's Lower Limb, to find the True Altitude. ~ O] Height of the Eye above the Sea in Feet. 6 8 10 12 14 16 18 20 22 24 26 2i8 30 32 34 5 3.8 3.5 3.1 2.8 2.5 2.3 2.1 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.5 6 5.3 4.9 4.6 4.3 4.0 3.7 3.5 3.3 3.0 2.8 2.6 f2.4 2.2 2.1 1.9 7 6.4 6.0 5.7 5.4 5.1 4.8 4.6 4.4 4.1 3.9 3.7 3.5 3.3 3.2 3.0 8 7.2 6.8 6.5 6.2 5.9 5.7 5.4 5.3 5.0 4.8 4.6 4.4 4.2 4.0 3.9 9 7.9 7.5 7.2 6.9 6.6 6.4 6.1 5.9 5.7 5.5 5.3 5.1 4.9 4.7 4.5 10 8.5 8.1 7.8 7.5 7.2 6.9 6.7 6.5 6.2 6.0 5.8 5.6 5.4 5.3 5.1 11 8.9 8.6 8.2 7.9 7.6 7.4 7.2 6.9 6.7 6.5 6.3 6.1 5.9 5.7 5.6 12 9.3 9.0 8.7 8.3 8.01 7.8 7.6 7.3 7.1 6.9 6.7 6.5 6.3 6.2 6.0 14 9.9 9.6 9.2 8.9 8.7 8.4 8.2 7.9 7.7 7.5 7.3 7.1 6.9 6.8 6.6 16 10.4 10.1 9.7 9.4 9.1 8.9 8.7 8.4 8.2 8.0 7.8 7.0 7.4 7.2 7.1 18 10.8 10.4 10.1 9.8 9.5 9.3 9.0 8.8 8.6 8.4 8.2 8.0 7.8 7.6 7.5 20 11.1 10.7 10.4 10.1 9.8 9.6 9.3 9.1 8.9 8.7 8.5 8.2 8.1 7.9 7.7 22 11.4 11.0 10.7 10.4 10.1 9 8 9.6 9.4 9.1 8.9 8.7 8.5 8.3 8.2 8.0 26 11.7 11.4 11.0 10.7 10.5 10.2 10.0 9.7 9.5 9.3 9.1 8.9 8.7 8.6 8.4 30 12.0 11.7 11.3 11.0 10.8 10.5i10.5 10.3 10.0 9.8 9.6 9.4 9.2 9.0 8.9 8.7 35 12.3 11.9 11.6 11.3 11.0 10.7 10.6 10.3 10.1 9.9 9.7 9.4 9.2 9.2 9.0 40 12.5 12.2 11.8 11.5 11.3 11.0 10.8 10.5 10.3 10.1 9.9 9.7 9.5 9.4 9.2 45 12.7 12.4 12.0 11.7 11.5 11.2 11.0 10.7 10.5 10.21 0.1 9.8 9.7 9.6 9.4 50 12.8 12.5 12.2 11.9 11.6 11.3 11.1 10.9 10.6 10.4 10.3 10 0 9.8 9.7 9.5 55 13.0.12.6 12.3 12.0 11.7 11.5 11.2 11.0 10.7 1 0.5 10.3 10.1 9.9 9.8 9.6 60 13.1 12.7 12.4 12.1 11.8 11.6 11.3 11.1 10.9 10.6 10.4 10.2 10.1 9.9 9.7 65 13.2 12.8 12.5 12.2 11.9 11.7 11.4 11.2 11.0 10.7 10.5 10.3 10.1 10.0 9.8 70 13.3 12.9 12.6 12.3 12.0 11.8 11.5 11.3 11.0 10.8 10.6 10.4 10.2 10.1 9.9 75 13.4 13.1 12.7 12.4 12.1 11.9 11.7 11.4 11.2 11.0 10.8 10.6 10.4 10.2 10.1 80 13.6 13.2 12.91 12.6 12.3 12.0 11.8 11.6 11.3 11.1 10.9 10.7110.5 10.4 10.2 Monthly Jan. Feb. Mar. April, May, J une, Correction 40 3 +-0'.2 I 0'.1 0'0 -0'.1 0'.2 for Sun's July, Aug. Sept. Oct. 1 Nov. Dec. Semi-diam. -0.3 -0.2 -0'1 - 0'.2 +0'.3 practical navigator is disposed to give. Moreover, such like accuracy is not required in practical navigation. To know the latitude within a mile is all the ship master requires; and this can be done in a very summary manner, by observing the moon's meridian alititude and using, the following tables, according as the lowers or upper limb of the moon is observed. o The bright limb, is the one observed, whether it be the upper or lower. 18 210 NAVIGATION. These tables makebut one correction for semi-diameter, parallax and refraction TABLE I. CORRECTIONS to be added to the OBSERVED ALTITUDE of the Moon's lower limb. Part 1st. HORIZONTAL PARALLAX. (I's 53' 54' 55' 56/ 57' 58' 59' 60' 61' Alt. O- o7 / - o -- - -/ - / o / 0o /o 6 0.59 1. 0 1. 1. 3 1. 4 1.5 1. 6 1. 8 1.9 8 1. 0 1. 2 1. 3 1. 4 1. 6 1. 7 1. 8 1. 9 1.11 10 1.1.3 1.4 1.5. 7. 8 1. 9 1.10 1.12 15 1. 2 1. 3 1. 5 1.6 1. 7 1. 9 1.10 1.11 1.12 20 1. 2 1. 3 1. 4 1. 5 1. 6 1. 8 1. 9 1.10 1.11 25 1. 0.. 3 1.4 1. 5 1. 6 1. 7 1. 9 1.10 30 0.59 1. 0 1. 1 1. 2 1. 3 1. 4 1. 5 1. 7 1. 8 35 0.57 0.59 0.59 1. 0 1.. 2 1. 3 1. 4 1. 5 40 0.55 0.55 0.56 0.57 0.58 0.59 1. 0 1. 1 1. 2 45 0.51 0.52.5i3 0.54 0.55 0.56 0.57 0.58 0.59 50 0.48 0.49 0,50 0.51 0.51 0.52 0.53 0.54 0.55 55 0.44 0.45 0.46 0.47 0.48 0.49 0.49 0.50 0.51 60 0.40 0.41 0.42 0.43 0.44 0.44 0.45 0.46 0.47 65 0.36 0.37 0.38 0.39 0.39 0.40 0.40 0.41 0.42 70 0.33 0.33 0.34 0.34 0.35 0.36 0.36 0.37 0.37 75 0.28 0.28 0.29 0.29 0.30 0.30 0.31 0.31 0.32 80 0.24 0.24 0.24 0.25 0.25 0.25 0 26 0.26 0.27 85 0.19 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.22 TABLE II. CORRECTIoNS to be applied to OBSERVED ALTITUDE of the Moon's upper limb Part 2nd. HORIZONTAL PARALLAX. ('s 53' 54 55t 56' 57' 58' 59' 60' 61' Alt. o6 7 / O- 7 - o o o Io o E O 10 +0.33 +0.33 +0.34 +0.34 +0.34 +0.36 +0.37 ~-0.37 +-0.38 15 0.33 0.33 0.35 0.35 0.36 0.37 0.37 0.39 0.39 20 5.32 0.33 0.34 0.35 0.35 0.36 0.37 0.37 0.38 26 0.30 0.32 0.32 0.33 0.33 0.34 0.35 0.35 0.36 30 0.29 0.30 0.31 0.31 0.32 0.32 0.33 0.34 0.34 36 0.26 0.26 0.27 0.27 0.28 0.28 0.29 0.29 0.30 40 0.24 0.25 0.26 0.26 0.27 0.27 0.28 0.29 0.29 46 0.19 0.22 0.22 0.22 0.23 0.24 0.24 0.24 0.26 50 0.17 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.21 55 0.14 0.15 0.16 0.16 0.16 0.16 0.17 0.17 0.17 60 0.10 0.11 0.12 0.12 0.12 0.12 0.13 0.13 0.13 65 0. 6 0. 7 0. 7 0. 8 0. 8 0. 8 0. 8 0. 8 0. 9 70 0. 3 0. 3 0. 3 0. 3 0. 3 0. 3 0. 3 0. 3 0. 4 75 -0. 1 -0. 1 -0. 1 -0. I -0. 1I 0. 2 -0. 2 — 0. 2 -0. 2 80 0. 6 0. 6 0. 6 0. 6 0. 6 0. 6 -0. 6 0. 7 0. 7 85 0.10 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.12 Height of the eye, 4ft. 1 9ft. 16ft. 25ft. 36ft. Dip of the Horizon, — 2 1 -3' -4' — 5' — 6' EXAMPLES. 1. In longitude about 450 west, on the 5th of January, 1852, at about 1 lh. in the evening, I observed the altitude of the moon's lower limb as she passed the meridian, and found it to be 68~ 12' from the south, height of the eye 16 feet. What was my latitude? CELESTIAL OBSERVATIONS. 211 On the 5th of Jan., at 11h. evening, Ion. 45 west, corresponds to 2 hours after midnight at Greenwich. From the Nautical Almanac, we find, that, At midnight of the 5th, the moon's horizontal parallax was - 57' 40" At noon of the 6th,. 58' 1' Therefore, by proportion, the horizontal parallax at the time of observation, must have been 57' 43". Moon's declination at midnight of the 5th, (N. Almanac), 210 47' 53" N. noon of the 6th,. 22~ 16' 55" N Variation in 12.hours, 29' 2" Therefore, the variation for 2 hours, was not far from - 4' 50" Hence the dec. at the time of observation was, - - 22~ 52' 43" N. We enter table 1, and under the parallax, and opposite to the altitude as near as we can find them, we perceive that 37' must be about the correction for the altitude. Whence, Observed alt. L. L 68~ 12' Correction, + 37 68 49 Dip. always sub. - 4 68 45 90 Zenith dis. D 21 15 )'s dec. 22 53 Lat. in 440 8' North. Find the true altitude of the moon's center, in each of the following examples. L. L. means lower limb; U. L. upper limb. Height of Ans. Observed Alt. H. P. the eye. True Alt. 1. D L. L. 530 23' 58' 14" 14 feet 540 10' nearly. 2. D L. L. 480 58 60' 27" 19 " 490 48' " 3. D U. L. 570 11' 54' 30" 20 " 57~ 19' 4. ) L. L. 630~ 38' 55' 29" 12 " 640 14' " 5. ) U. L. 200 3' 54' 14" 16 " 20~ 32' 6. ) L. L. 160 2' 59' 38" 23 " 170 12' When the weather makes it doubtful whether meridian observations can be obtained, navigator's resort to double altitudes, or to the altitudes of two objects taken at the same time. We shall only show 212 NAVIGATION. the principle on which this method is founded; it is the application of spherical trigonometry. Let Pp be the earth's axis, Qq the equator. Suppose the sun to be the object, and let its position be S and T at two different times. The elapsed time measures the angle SPT. In the triangle PT7S, we have the two sides PT, PS, and the included angle, from which we compute the side TS, and the angle ~TSP. Subtracting the altitudes Sm and Tn from 900, we have ZS, and ZT, then we have all the sides of the triangle ZTS, from which we compute the angle TSZ. Subtracting this angle from TSP, gives us the angle ZSP. Now, in the triangle. ZSP, we have the two sides ZS, SP, and their included angle, from which we compute PZ the complement of the latitude. If the ship sails, during the interval between the observations, a correction will be required for the first altitude, and such corrections are found by the traverse table; a nautical mile in the direction of the sun, corresponds to one minute of a degree, to be applied to the altitude. When the proper correction is made, the result is equivalent to having both altitudes taken at the last station, and the deduced latitude is the latitude of that station. CHAPTER II. LONGITUDE. LONGITUDE, from celestial observations, is measured by time. A place 150 west of another, will have noon one hour of absolute time later; if 300 west, the local time, noon will be two hours later, &c., &c.; 15~ corresponding to an hour in time. Therefore, if we have any way of determining the times at two places, corresponding to the same absolute instant, the difference of such times will LONGITUDE. 213 correspond to the difference of longitude between the two places at the rate of 150 to an hour, or 4 minutes to a degree. A perfect time piece will keep the time at any particular meridian, and by carrying thatperfect time piece with us, by it we can see the time at that particular meridian; and then if we can find the time at the place where we are, the comparison of these two times will give the difference of longitude, that is, the difference between our longitude and that of the-particular meridian, to which the time piece refers. For instance, a gentleman leaves Boston; his watch is a perfect time pie:e, aid it;:: set to Bosl~ time, he travels west on the railroads, his watch all the while shows Boston time; when it is twelve o'clock by his watch:t:is' really so i" Boston, but not so at the place where he is. The sun has arrived at the meridian of Boston, but not yet at the meridian of Albany, or Buffalo, or Detroit; and when the gentleman arrives at any of these places, or any intermediate place, the local time, compared with the time in Boston, will give the longitude of that locality. from Boston, counting one degree for every four minutes in the difference of time. Unfortunately, however, there is no such thing as a perfect time piece, but some do approximate toward perfection. Such ones, made with the greatest care and solely for accuracy in rate of motion, are called chronometers; they are supposed to keep time within certain known limits, and in the place of perfect time keepers, they are used at sea for finding longitude. Chronometers show the time at the distant place, it then remains to find the time at ship, and this is done most accurately by spherical trigonometry, as will soon appear. The sun's altitude is greatest just at apparent noon, but by observations we cannot define just the moment when that takes place; hence meridian observations, valuable as they are for latitudes, are worth nothing for time, when time is to be settled to anything like accuracy. The best position of the sun (or any other celestial object) for an observation to find local time, is when it is nearly east or west, and its altitude more than ten degrees. In such circumstances, an observer can find the local time 214 NAVIGATION. within 5 or 6 seconds, by taking an altitude of the sun, provided he at the same time knows his latitude and the sun's polar distance. The operation is a beautiful application of spherical trignometry, and it is illustrated by the following figure. Let Z be the zenith of the observer, P the pole, S the position of the sun, and PS the sun's polar cdistance.* When S comes on to the meridian, it is then apparent noon; and the angle ZPS of the triangle ZPS measures the interval from apparent noon, at the rate of four minutes to one degree. The side PS is the polar distance, the side ZS is the co-altitude, and the side PZ is the colatitude. Now, in every treatise on spherical trigonometry, it is demonstrated as a fundamental principle, that The cosine of any angle, of a spherical triangle, is equal to the cosine of its opposite side, diminished by the rectangle of the cosines of the adjacent sides, divided by the rectangle of the sines of the adjacent sides. cos. ZS-cos. PZ cos. PS That is, cos.P'= sin. PZ sin. PS Now, in place of cos. ZS, we take its equal, sin. ST, or the sine of the altitude, and in place of cos. PZ, we take its equal, the sine of the latitude. In short, let A= the altitude, L= the latitude, and D- the polar distance. sin. A-sin. L cos. D.Then cos. P= cos. L sin. D *When the observer is in the northern hemisphere, the polar distance is counted from the north pole; when in the southern hemisphere, from the south Dole. LONGITUDE. 215 From a general equation, in plane trigonometry, we have 2 sin.2 ~ P=l-cos. P Substituting the value of cos. P, in this last equation, we have sin. A-sin. L cos. D 2 sin.2 P- 1 cos. L sin. D (cos. L sin. D+sin. L cos D) —sin. A cos. L sin. D By comparing the quantity in parentheses with eq. (7), plane trigonometry, we perceive that sin. (L+-D) —sin. A cos. L sin. ) Considering (L+ D) to be a single are, and then applying equation (18), plane trigonometry, and dividing by 2, we shall have sin. os. (L — Asin. (L+D —A sin2 iP- 2) - cos. L sin D. But L+D-A _ L+D+A A, and now if we put 2 2 S L+D- +A we shall have cos. Ssin. (S-A) sins. i P= cos. L sin. ) Or, sin. i PA/ cos. S sin. (S-A)'V cos. L sin. D This is the final result when radius is unity, when it is R times greater, then the sin. i P will be R times greater, and if R represents the radius of our tables, to correspond with these tables we must multiply the second member by R, and if we put it under the radical sign, we must multiply by R2; in short we shall have, Sin... P=V( R) (sn ) cos. S sin. (S-A) Sin. ~p=Jcos- L- sin. R The right hand member of this equation, shows four distinct logarithms; thus, is the cosine of the latitude subtracted from cos. L 10, which we shall call cosine complement. 216 NAVIGATION. This equation furnishes the following rule for finding apparent local time, when the sun's altitude, its polar distance, and the lati-,tude of the observer, are given. The altitude must be observed, the latitude must be known, and the Nautical Almanac will furnish the polar distance. RuLE. —1. Add together the altitude, latitude, andpolar distance; take the half sum, and from the said half sum subtract the altitude, thus finding the remainder. 2. The logarithms. Find the cosine complement of the latitude, the sine complement of the polar distance, the cosine of the half sum, and the sine of the remainder. 3. Add these four logarithms together, and divide by 2, the logarithm thus found, is the sine of half the polar angle, or half the sun's meridian distance. 4. Take out the arc corresponding to this sine, and divide its double by 15 (as in compound division in arithmetic), and the quotient will be the hours, minutes, and seconds from apparent noon; and if the sun is east of the meridian, the hours, minutes, and seconds, must be subtracted from 12 hours, for the corresponding time of day. The time shown by a chronometer or a perfect clock, or rather graduation of clocks, is to mean and not to apparent time, and to convert apparent into mean time, the equation* of time is given in the Nautical Almanac for the noon of every day at Greenwich. The amount of it, reduced or modified to correspond to the time of observation, can be applied to apparent time, and the mean time of taking the observation will be determined. The difference between this time and the mean time at Greenwich, as determined by the chronometer, will be the longitude. The longitude will be west, if the time at Greenwich is latest in the day, otherwise it will be east. If the observer is on land, without a sea horizon, and uses a reflecting instrument, he must have an artificial horizon. A proper artificial horizon, is a small dish of mercury, with a glass roof to put over it, to keep the mercury from being agitated by the wind. In place of the mercury, a plate of molasses will answer. In still calm weather any clear pool of water is a good artificial horizon. In either of these, the reflected image of the object appears as much below the horizon as it is above it, and to measure the altitude, * For the theory of equation of time, see works on astronomy. LONGITUDE. 217 the image reflected by the mirror of the instrument must be carried to the image in the artificial horizon; half of the angle shown by the index will be the apparent altitude. In using an artificial horizon there is no dip, other corrections are to be applied according to circumstances. EXAMPLES UNDER THE PRECE,DING RULE. 1. Being at sea,' May 20th, 1823, in latitude 430 30' N., and in longitude about 200 west, I observed the altitude of the sun's lower limb, and found it to be 320 4' rising, when an assistant marked the time per watch, at 7h. 43m. A. M.; height of the eye 16 feet. What was the true mean time? Just before the observation, the watch was compared with the chronometer in the cabin*, and found to be 1 hours 21 minutes, and 12 seconds slow of chronometer. On the 8th of May, the chronometer was 3m. 7s. fast of Greenwich time, and gaining ls.6 daily. What was the longitude? S.D., - - 15'49". o. Dip., - -..356 Watch, - - - 43 0 Ref.,.-.1 30 Diff., - - - -12112 Correction, - - - 10 23 Face of.ehron. at ob., - 9 4 12 Observation, 3- - 4* Error 3m, 7s., increase of error Oet.io cente, - -3 - 342d Alt. center, - 14 23 in 12 days 19s., whole error, - 3 26 Greenwich time, - - 9 0 46 At noon on the 20th of May, 1823, the sun's declination, by the N. A., was 190 52' 18" north, increasing at the rate of 30".'6 per' hour, and the time of taking the observation was 3 hours before noon' at Greefiwich; therefore, the declination must have been 193 50' 47" At. Altitude, 320 i4' 23" Lat., 43 30 cos. corn..139435 P. D., 70 9 13 sin. com..026603 2)145 53 36 S. 72 56 48 caosno 9.467253 32 14 23 (S-A) 40 42 25 sine 9.814363 2)19.447657 310 58' 8" sin. 9.723828 * Chronometers should never be, and by careful persons, never are, taken out of their places during a voyage. 19 218 N AVIGATION. 310 58' 8" 2 630 56' 16"= 4h. 15m. 458. 12 Apparent time, - - 7 44 15 A.M. Equation of time N. A. - 3 51 Mean time at ship, - - 7 40 24 Watch, - - - 7 43 Watch too fast, - - 2m. 368. Time at Greenwich per ohron., 9h. Om. 468s. Time at ship per observation, 7 40 24 Diff., -.... 1 20 22=200 5' west Ion. 2. August 10th, 1824, in latitude 540 12' north, at 2h 33m per watch, height of the eye 18 feet, I observed the altitude of the sun's upper limb 160 50' falling. My chronometer was 2h 20m 37s fast of the watch; and on the 7th, the same month, the chronometer was 40m 29.4s fast of Greenwich time, gaining 7T-5- seconds daily. What was the error of the watch, and the longitude per chronometer? Prepartion. Time per watch - 5h. 33m. Os. P.M. Diff. per chron. - - 2 20 37 Face of chronometer - 7 53 37 P. M. Chron. fast (whole error) -40 52 Greenwich, mean time, 7 12 45 P. M. On the 10 of August, 1824, the sun's declination at noon, Greenwich time, was 150 32' 14" north, decreasing at the rate of 45" per hour, as given in the Nautical Almanac. The decrease for 7' hours must be 5' 24"; whence, the declination at the time of observation, 150 26' 50" N., and the polar distance 740 33' 10". Observed altitude - 160 50' 00" Equation of time, per N. A., Aug. Semi-diameter, N. A. - -15 48 10, 1824, was - - +5m. 2s. Dip and Ref. - - 7 20 Hourly decrease AsJ. -2 True-alt. center - 160 37' 52" Equation at ob. - - 5m. Os. We now leave the problem to be worked through by the pupil, giving only the answer. Ans. Watch slow of local mean time, 3m 27s. Longitude by chronometer, 240 4' 30" west. 3. When it was 6h Om 21s, P. M., mean time, at Greenwich, by my chronometer, I observed the altitude of the sun's lower limb to be 300 17', in the afternoon of January 12th, 1852. At noon our LONGT ITUDE. 219 latitude, by a meridian observation, was 210 47' north, and since that time we have made 11 miles of southing, by the log. The dip was 4', and semi-diameter 16' 17". What was the longitude by chronometer? Sun's Declination Jan. 12,'52, at noon, G. T. - 210 44' 10" south. Hourly decrease, per N. A., 25", giving - — 2' 30" Declination at the time of observation - -210 41"40" south Equation of time at noon, Greenwich - — +8m. 25s. Hourly increase -A, of a second, making - - 6s. nearly Equation at time of observation (to add) - -8m. 31s. Were we sure that pupils would have access to nautical almanacs, we would give neither declination nor equation of time. Ans. Lon. 450 39' west. 4. On the 16th of January, 1852, when my chronometer showed Ilth 27m 41s, A. M., for the mean time at Greenwich, I observed the aititude of the sun's lower limb and found it 320 21' rising, height of the eye 16 feet, latitude 00 41' south. What was the longitude by chronometer? By the N. A., the sun's declination at that time was 21~ 2' 36' south, and the equation of time 9m. 53s. additive. Ans. Lon. 46~ 39' west. N. B. —Time at any place, is but the difference between the right ascension of the meridian and the right ascension of the sun; and to find the time from these two elements, we always subtract the right ascension of the sun from the right ascension of the meridian, increasing the latter by 24 hours, to render subtraction possible, when necessary. The right ascensions of the stars are given, and the right ascension of the sun is given, in the Nautical Almanac, for the noon of every day in the year, Greenwich time. Now, if we can find the meridian distance of any known star, by observation, we can establish the right ascension of the meridian, and, consequently, the local time. Hence, we can find longitude by comparing the chronometer with the altitudes of the stars, as the following example will illustrate. 5. If on the 8th of March, 1852, when my chronometer showed the Greenwich time to be 7h 22m 3s, P. M., I found by observation, 220 NAVIGATION that the true altitude of Sirius was 370 52' west of the meridian. My latitude was 32~ 28' south. What was the time at ship, and what "was my longitude; the elements for computation being as follows? 1. Right ascension of the star - - 6h. 38m. 38s. 2. Declination of the star 16~ 31' south 3. Right ascension of the sun - 23h. 17m. 25s. By means of the triangle we find, The meridian distance of the star - 3h. 40m. 58s. To which add *'s R. A., because * is west 6 38 38 Right ascension of the meridian - 10 19 36 Add - 24 34 19 36 Subtract the R. A. of the sun - - 23 17 25 Diff. is apparent time at ship - 11 2 11 P.M. Equation of time, add - - 10 48 Mean time at ship - - - - 11 12 59 Time at Greenwich. 7 22 3 Longitude in time - - - - 3 50 56 =570 44' east N. B.- When the chronometer remains in the same place for a week or more, its rate can be determined by comparing it with the observed altitudes of the sun, taken from day to day. In different climates the same chronometer will have different rates, and on returning to its original station it will frequently resume its original rate. For azimuths, and variations of the compass, see page 106. CHAPTER III. LUNAR OBSERVATIONS. A GOOD and well-tried chronometer is a valuable and reliable instrument for finding the longitude at sea, during short runs; but still it is but an instrument, and is not one of the reliable works of LUNAR OBSERVATIONS. 221 nature. Near the end of a long voyage, the best of chronometers very frequently give false longitude, and in such cases, good navigators always resort to lunar observations, which from the hands of a good observer, can be relied upon to within 10 or 12 minutes of a degree, and they usually come within 5 or 6 miles, and sometimes even more exact, but that is accidental and unfrequent. To comprehend the theory of lunars, we must call to mind the fact that the moon moves through the heavens, apparently among the stars, at the rate of more than 130 in a day, and any angular distance it may have from the sun or any star corresponds to some moment of Greenwich time. About three days before and after the change of the moon, she is too near the sun to be visible, but at all other times, her distance from the sun, some of the larger planets, and certain bright fixed stars, called lunar stars,* which lie near her path, are computed and put down in the nautical almanac, for every third hour of mean Greenwich time commencing at noon. For any particular day, the distances are given to such objects only, east and west of her, as will be convenient to measure with the common instruments. The distances put down in the nautical almanac, are such as would be seen if viewed from the center of the earth; but observers are always on the surface of the earth, and the distances thence observed, must always be reduced to equivalent distances seen from the center, and this reduction is called working a lunar, which is generally the highest scientific ambition of the young navigator. t The true distance between the sun and moon, or between a star and the moon, can be deduced from the apparent distance by the application of spherical trigonometry. The moon is never seen by an observer in its true place, unless the observer is in a line between the center of the earth and the moon, that is, unless the moon is in the zenith of the observer; in all other * There are nine lunar stars, Arietis, Aldebaran, Pollax, Regulus, Spica, Antares, Aquile, Fomalhaut, and Pegasi. t Many navigators, both old and young, direct all their efforts to knowing how to do, without attempting to comprehend the reasons for so doing; and this the world calls practical, - a complete perversion of the term. On the other hand, some men of the schools spend their energies in metaphysical nothings, splitting hairs in logic, and calling it scientific; this is equally a perversion. 222 NAVIGATION. positions, the moon is depressed by parallax, and appears nearer to those stars that are below her, and further from those stars that are above her, than would appear from the center of the earth. Therefore, the apparent altitudes of the two objects, must be taken at the same time that their distance asunder is measured. The altitudes must be corrected for parallax and refraction, thus obtaining the true altitudes. The annexed figure is a general representation of the triangles pertaining to a lunar observation. Let Z be the zenith of an observer, S' the apparent place of the sun or star, and S its true place. Also, let m' be the apparent place of the moon, and m its true place as seen from the center of the earth. Here are two distinct triangles, ZS'm', and ZSm. The apparent altitudes subtracted from 900, give ZS' and Zm', and S'm' is the apparent distance; with these three sides, the angle Z can be found. Correcting the altitudes, and subtracting them from 900, will give the sides ZS and Zm; these two sides, and their included angle at Z, will give the side Sm, which is the true distance. The definite true distance must have a definite Greenwich time, which can be readily found; and this, compared with the local time deduced from an altitude of the sun, will of course give the longitude. We shall now make a formula to clear the distance. Let S'=the apparent altitude of the sun or star, and S= —the true altitude. Also, Let m'=the apparent altitude of the moon, and mn =the true altitude. Observe that the letters with the accent, indicate apparent, and without the accent, the true altitudes. Put d to represent the apparent distance, and x to represent the true distance. Bear in mind, that the sine of an altitude is the same as the cosine of its zenith distance, and conversely, the sine of a zenith distance is the same thing as the cosine of the corresponding altitude. Now, by the fundamental equation of spherical trigonometry noted in the last chapter, we have LUNAR OBSERVATIONS. 223 cos z- cos. d -sin. S' sin.m' Also co cos. x-sin. S sin. onm. Cos. S' Cos. S' COS. S COS. m Whence C9OS. d-sin. S' sin. m' cos. x-sin. x S in. in. m Cos. S' Cos. m' Cos. S cos. m By adding unity to each member we have 1cos. d-sin. S' sin. m' 1 +cos. x —sin. S sin. m Cos. S' Cos. m' cos. S cos. m (cos. S' cos. m'-sin. S' sin. m')-+-cos. d (cos. S cos. m —sin. S sin. m)+cos. x. COs. 5' Cos. m' Cos. S Cos. m By observing equation 9, plane trigonometry, we perceive that the preceding equation reduces to cos. (S'+m')+cos. d cos. (S+m)-J cos. x COs. S' Cos. m' cos. S cos. m Whence cos. x=(cos. (S'+m')-+-cos. d) cO S os. m_cor (S+m). COs. S' Cos. mn' It is here important to notice that the moon's horizontal parallax given in the Nautical Almanac, is the equatorial horizontal parallax; that is, it corresponds to the greatest radius of the earth. The diameter of the earth through any other latitude is less, and of course the corresponding parallax is less. We therefore give the following table for the reduction of the equatorial horizontal parallax, to the horizontal parallax of any other latitude; it is computed on the supposition that the equatorial diameter is to its polar as 230 to 229. For example if the horizontal parallax in the Nautical Almanac is 55' —, in the latitude of 400 the reduction would be 6", and the parallax reduced would be 54' 54", and if the parallax from the Nautical Almanac were 60' the reduction would be6" 6, and reduced would be 59' 53".4. The semi-diameter of the moon given in the Nautical Almanac is her horizontal semi-diameter, but when she is in the zenith she is nearer to us by the whole radius of the earth, about one-sixtieth part of her whole distance, consequently she must appear under a larger and larger angle as she rises from the horizon, and this is called the augmentation of the semi-diameter. We give the reduction for the parallax; and the augmentation for the semi-diameter in the following tables: 224 NAVIGATION. Red. of ('s Eq. heor. parallax. Augmentation of the La Eq. par. I Eq. par. Moon's semi-diam. itude. 55' 60' Ap. Alt. Aug. 200 0".9 1" 60 2" 25 2.8 3 12 3 30 3.7 4 18 5 35 4.6 5 24 6 40 6.0 6.6 30 8 45 7.3 8 36 9 50 8 6 9.4 42 11 55 10.1 11 48 12 60 11 12 64 13 65 11.813 60 14 70 12.8 14 66 15 75 13.9 15 72 16 80 14.6 16 90 16 We now give an example showing all the details of finding the longitude by a lunar observation. EXAMPLE. Suppose that on the 25th of January, 1852, between three and four o'clock in the afternoon, local time, the observed distance between the nearest limbs of the sun and moon was 500 3' 20", the altitude of the sun's lower limb was 200 1', and of the moon's lower limb 480 57', height of the eye 16 feet. The latitude corrected for the run from noon was 340 12' N., and the supposed longitude about 650 west. What was the longitude? (the Nautical Almanac being at hand.) Preparation. Supposed time at ship, 3 15 P. M. Supposed longitude 65 4 20 Supposed time at Greenwich, 7 25 P. M. On the 25th at noon the N. A. gives the E)'s S. D. at 14' 47", and at midnight at 14' 45" 7; therefore at the time of observation we take it at 14' 46", by simple inspection. In the same summary manner we take the Equatorial horizontal parallax at 54' 12". E)'s semi-diameter, - - 14' 46" 1's Eq. hor. par. - 54' 12" Aug for Alt. - 12 Red. for lat. - 4 E)'s true S.D. - - - 14' 58" Reduced hor. par. - - 54' 8" Observed distance, - 500 3' 20" Alt. E's L L 480 57' Sun's S. D. - - 16 16 )'s S. D. 14' 58" Moon's S. D. - - 14 58 Dip - 3 56 Apparent central dis. 50~ 34' 34"=-d )'s app. alt. 490 8' 2"==m.' LUNAR OBSERVATIONS. 225 Alt. ('s lower L 200 1' N. B. To find the moon's parallax in Semi-diameter, +16' 16" altitude see problems on page 201. Dip, --- 3' 56" )'s app. Alt. 490 8' ocs. 9.815778 ('s app. alt. 20~ 13' 12"=S' 54' 8"=3488' log. 3.542576 Refraction, - 2' 34" 36' 2"=2282 3.358354 O's true alt. 20~ 10' 38" S. I)'s app. alt. 490 8' 2" Parallax in alt. 36' 2 Refraction, -49 True alt. 490 43' 15" —m (S'+-m')=690 21' 14" (S+m)=690 53' 53". We are now prepared to apply the equation to compute the true distance. The equation requires the use of natural sines and cosines. cos. x=(cos. (S'+m')~+cos. d) S cos. cos. (S+m) (S'-+m')=690~ 21' 14" N. cos..35259 d — 500 34' 34" N. cos.*.63449.98708 log. -1.994350 S=20~ 10' 38" log. cos. 9.972496 m~=490 43' 15" log. cos. 9.810578 S'-200 13' 12" cos. com. 0.017626 m'-490 8' 2" cos. com. 0.184228 Num..97561 log. -1.989278 sum less 20.t N. cos. ( S-m)=690 53' 53" —.34369 True distance, 50~ 48' 29" cos.63192 In the Nautical Almanac, we find that at 6 P. M. mean Greenwich time, on said day, the true distance between the sun and moon was 49~ 59' 26", and at 9 P. M., the distance was 51~ 20' 49", showing a change of 10 21' 23" in three hours of time. But the change * When d is greater than 900 its cosine becomes minus, and its numerical value is then the natural sine of the excess over 900. Thus if d were 1050, its cosine would be numerically equal to the sine of 150, and must then be subtracted from the cosine of the sum of apparent altitudes. The result (cos. x) would then be the sine of the excess over 90~. t Less 20 because the table of natural sines is to radius unity, and we used cos. S and cos. m to the radius of 10, making two tens to take away. 226 NAVIGATION. from 49~ 59' 26" to 50~ 48' 29" is 49' 3"; and now on the supposition that the change is in proportion to the time ( and it is very nearly ), we have the following analogy 1~ 21' 23": 49' 3": 3h.'t Or, 4883: 2943:: 3: h1. 48m. 29s. That is, the time that this observation was taken 1h 48m 29s after 6 at Greenwich. Or, 7h 48m 29s mean Greenwich time. With the true altitude of the sun 200 10' 38", the latitude 340 12', and the polar distance 1090 0' 48", we find the apparent time at ship 3h 10m 5s, to which we would add the equation of time, 12m 34, making the mean time 3h 22m 39s. From the Greenwich time 7h. 48m. 29s. Sub. time at ship - 3 22 39 Giving lon. in time 4 25 50=660 27' 38" W. West, because the time at Greenwich was later in the day. If a lunar is taken with a star, or with the sun, when the sun is not in a proper position to depend upon its altitude for local time, the time must be noted by a watch, and the difference between the watch and true time made known, by a previous or subsequent observation on the sun, or some star which is nearly east or west of the observer. The most material part of working a lunar is that of clearing the distance. We, therefore, give the following examples, without the little incidental details. We show the working of one in which the distance is greater than 90~. The apparent distance between the center of the sun and moon on a certain occasion, was 980 12'; the apparent altitude of the sun's center was 740 10', and of the moon's 200 37'; the moon's horizontal parallax at the same time was 57' 12". What was the true distance? Ans. 97~ 34' 27" Horizontal par. 57' 12"=3432 log. 3.532547 ) Alt. 20~ 37' cos. 9.971256 Parallax in alt. 530 31'=3211 log. 3.586803 LUNAR OBSERVATIONS. 227 )'s app. alt. 200 37' ( app. alt. 740 10' 0" Refraction -2 31 Refraction 16 Parallax - 53 31 9, Paralla s true alt. 740 9' 44" D's True alt. 210~ 28' (S't m')=94~ 47' (S+m)=950 37' 44" 940 47' Nat. cos.-0.08339 d=980 12' Nat. cos.-O0.14263 -0.22602 log. — 1.356150* S -74~ 9' 44" cos. - - 9.436021 m- =21~ 28' 00" cos. - - 9.968777 S'=740 10' 00" cos. complement 0.364092 mn'=200 37' 00" cos. complement 0.028744 -0.23007 log. -1.361784 cos. (S+m)=95~ 37' 66"+0.09826t True dis. 97~ 34' 27" N. cos.- 0.13181 EXA MPLES FOR PRACTICE. No. Ap. alt. of sun Moon's ap. Apparent cen- Moon's True or a fixed star. altitude. tral distance. hor. par. distances. 1 0 86 3 39 18 46 45 0 53 61 46 425 2 29 47 5722 27 35 0 60 3 28 8 24 3 (031 14 28 7 14 21 30 53 29 14 9 24 4 0 60 5 6312 51 321 5830 50 41 15 5 * 34 28 10 42 49 18 38 61 11 48 45 39 6 0 8 26 19 24 120 18 46 57 14 120 1 46 7 *43 27 40 9 18 21 35 60 20 18 8 12 8 53 13 57 32 60 13 49 60 52 59 48 12 9 ( 72 26 18 30 81 2 28 60 58 80 9 33 10 0 60 33 9 26 70 36 16 59 57 69 49 12 * The factor 0.22602 being minus renders the product minus. t The cos. ( S+m ) in the equation is minus, but where the value of ( S-m ) is more than 900, the minus necessarily becomes plus. APPENDIX. IT is comparatively an easy matter, to conduct a survey, or navigate a vessel, when there are no important difficulties to be overcome; but the true test of knowledge or skill in any pursuit, is to be found only in real adversity. The mariner who successfully manages his ship, when every thing is provided, when all is in order, and the weather favorable, is deserving of little credit; but let the ship become disabled, and the storm terrific, and then there is scope for the exercise of every necessary acquirement, and its kindred talent. So it is with the man of science; when every instrument is at hand, and all in order, it requires little skill, and but common knowledge, to make observations and experiments; but when we reverse the case; the tact, knowledge, and ingenuity of the man, may oft times more or less overcome the difficulties. For instance, suppose it were necessary to find the altitude of the sun, for the purpose of finding the latitude of the place (on shore), or for the purpose of finding the time; and we had no sextant or quadrant, and in fact, no instrument to measure angles. It could be done approximately as follows: Let a plumb line be suspended in water; have a knot in the line, and let the knot be at a known distance above the water. The knot will cast a shadow on the water; measure the distance of this shadow from the plumb line. The knot and its shadow, with the plumb line and water, will form a right angled triangle, and the angle at the base, computed by plane trigonometry, will be the altitude of the sun's upper limb, and this altitude may be used for any purpose, the same as if it were measured by a sextant, but the accuracy is not to be depended upon, for the want of delicacy in the instrument. A person on shore having a good watch, and knowing his latitude, can regulate his watch, or at least determine its rate and error for a short period of time. Then, if he have a nautical almanac, the common tables of logarithms, and a knowledge of spherical trigonometry, and a (228) APPENDIX. 229 corresponding knowledge of astronomy, he can find the longitude by a lunar observation, without a sextant, as follows: By the means of his watch and a plumb line, he will be able to range off an approximate meridian line. He will then observe the transits of stars, and of the moon across that meridian, taking those stars which are at that time near the moon's meridian, some to the east and some to the west of the moon, and some more north, and others more south than the moon. He will note the difference in time, between the transit of each star and the moon, across his approximate meridian, and by a combination, or rather comparison of these observations, he will be able to determine the moon's right ascension very nearly. By the moon's right ascension, and the aid of the nautical almanact he can find the Greenwich time. The Greenwich time, compared with the local time, will give the longitude. When we can find the moon in a vertical plane with any two fixed stars, and it be at the time the moon changes her declination very slowly, so that we can depend upon a declination taken from the nautical almanac for the supposed time, we can then determine the moon's right ascension, and from thence the longitude as before, whether we are on land or sea. Ship-wrecked mariners, and travelers similarly situated, have frequently resorted to these artifices to obtain their approximate localities. We have frequently remarked in the course of this work, that the best position of a celestial object, at the time of taking its altitude, for the purpose of more exactly defining the time, is when the object is nearly east or west; we now propose to show this conclusively, and therefore give the following INVESTIGATION. To find under what circumstances, in a given latitude, a small mistake in observing or correcting the altitude of a celestial object, will produce the smallest error in the time computed from it. Let Z be the zenith, P the pole, r the supposed place, and m the true place of the object. Let ms be a parallel of altitude, join the points m and r, and letpq be the arc of the equator contained between the meridians Pm and Pr. Then as Pm and Pr are equal, mr may be considered as a small portion of a parallel of declination rs will be the error in 230 APPENDIX. altitude, and pq the measure of the required error in time. And as the sides of the triangle msr will necessarily be small, that triangle may be considered as a rectilinear one, right angled at s; and because the angle Prm is also a right angle, the angles smr and PrZ, being each the complement of mrs, are equal to each other We now have, rs: mr': sin. smr (ZrP): rad. (1) Also, mr pq:: cos. qr: rad. (2) Multiplying these two proportions together, omiting the common factor mr, gives, rs: pq:: cos. qr sin. (ZrP): (rad.)s (a) But, sin. rP or cos. qr: sin. rZP ~ sin. ZP: sin. (ZrP) (4) Whence, cos. qr sin. ZrP=sin. rZP sin. ZP (5) The first member of equation (5), is the same as the third term in proportion (3); therefore, proportion (3) may be changed to the following, rs:pq: sin. rZP sin. ZP: (rad,)2 Whence, pq= (rs (rad.)) 1 Now, as the quantities in parentheses are supposed to be constant the value of pq, the error in time must vary as. varies; and it is sin. rZP obvious that pq will be least, when sin. rZP is greatest, that is, when rZP=900, or the object due east or west. LOGARITHMS OF NUMBERS FROM 1 TO 10000. N. Log. N. Log. N. Log. N. Log. 1 0 000000 26 1 414973 51 1 707570 76 1 880814 2 0 301030 27 1 431364 62 1 716003 77 1 886491 3 0 477121 28 1 447158 53 1 724276 78 1 892095 4 0 602060 29 1 462398 54 1 732394 79 1 897627 5 0 698970 30 1 477121 55 1 740363 80 1 903090 6 0 778151 31 1 491362 56 1 748188 81 1 908485 7 0 845098 32 1 505150 57 1 755875 82 1 913814 8 0 903090 33 1 518614 68 1 763428 83 1 919078 9 0 954243 34 1 531479 59 1 770862 84 1 924279 10 1 000000 35 1 544068 60 1 778151 85 1 929419 11 1 041393 36 1 556303 61 1 785330 86 1 934498 12 1 079181 37 1 568202 62 1 792392 87 1 939519 13 1 113943 38 1 579784 63 1 799341 88 1 944483 14 1 146128 39 1 591065 64 1 806180 89 1 949390 15 1 176091 40 1 602060 65 1 812913 90 1 954243 16 1 204120 41 1 612784 66 1 819544 91 1 959041 17 1 230449 42 1 623249 67 1 826075 92 1 963788 18 1 255273 43 1 633468 68 1 832509 93 1 968483 19 1 278754 44 1 643453 69 1 838849 94 1 973128 20 1 301030 45 1 653213 70 1 845098 95 1 977724 21 1 322219 46 1 662578 71 1 851258 96 1 982271 22 1 342423 47 1 672098 72 1 857333 97 1 986772 23 1 361728 48 1 681241 73 1 863323 98 1 991226 24 1 380211 49 1 690196 74 1 869232 99 1 995635 25 1 397940 50 1 698970 75 1 875061 100 2 000000 N. B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are now introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the corresponding natural numbers in the first column stands in the next lower line, and its annexed first two figures of the Logarithms in the second column. LOGARITHMS OF NUMBERS. 3 N. 0 1 2 3 4 5 6 7 8 9 100 000000 0434 0868 1301 1734 2166 2598 3029 3461 3891 101 4321 4750 5181 5609 6038 6466 6894 7321 7748 8174 102 8600 9026 9451 9876.300.724 1147 1570 1993 2415 103 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 104 7033 7461 7868 8284 8700 9116 9632 9947.361.775 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 107 9384 9789.195.600 1004 1408 1812 2216 2619 3021 108 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 109 7426 7825 8223 8620 9017 9414 9811.207.602.998 110 041393 1787 2182 2576 2969 3362 3755 4148 4640 4932 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 112 9218 9606 9993.380.766 1153 1538 1924 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 7666 8046 8426 8805 9185 9563 9942.320 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 117 8186 8557 8928 9298 9668..38.407.776 1145 1514 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 119 5547 5912 6276 6640 7004 7368 7731 8094 8467 8819 120 9181 9543 9904.266.626.987 1347 1707 2067 2426 121 082785 3144 3603 3861 4219 4576 4934 5291 5647 6004 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 95652 123 9905.258.611.963 1315 1667 2018 2370 2721 3071 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 1026 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 128 7210 7549 7888 8227 8565 8903 9241 9579 9916.253 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 0245 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 134 7105 7429 7763 8076 8399 8722 9045 9368 9690..12 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 136 3639 3858 4177 4496 4814 5133 5461 5769 6086 6403 137 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 138 9879.194.508.822 1136 1450 1763 2076 2389 2702 139 143015 3327 3630 3951 4263 4574 4885 5196 5507 5818 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 141 9219 9527 9835.142.449.756 1063 1370 1676 1982 142 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 143 6336 5640 5943 6246 6549 6852 7154 7457 7759 8061 144 8362 8664 8965 9266 9567 9868.168.469.769 1068 145 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 147 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 149 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 20 4 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 151 8977 9264 9552 9839.126.413.699.985 1272 1568 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 154 7521 7803 8084 8366 8647 8928 9209 9490 9771..51 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 158 8657.8932 9206 9481 9755..29.303.577.850 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 160 4120 4391 4663 4934 5204 5475 5746 6016 6286 6556 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 162 9515 9783..51.319.586.853 1121 1388 1654 1921 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938.193 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 172 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 173 8046 8297 8548 8799 9049 9299 9550 9800..50.300 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 175 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 9443 9687 9932.176 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 179 2853 3096 3338 3680 3822 4064 4306 4548 4790 6031 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 186 9513 9746 9980.213.446.679.912 1144 1377 1609 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 190 8754 8982 9211 9439 9667 9895.123.351.578.806 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473'8696 8920 9143 9366 9589 9812 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197 4466 4687 4907 5127 5347 6567 5787 6007 6226 6446 198 6665 6884 7104 7323 7642 7761 7979 8198 8416 8635 199 8853 9071 9289 9507 9725 9943.161.378.695.813 OF NUMBERS 5 N. 0 1 2 3 4 1 5 1 6 7 8 9 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 202 5351 5566 5781 5996 6211 6425 6639 6854 70S8 7282 203 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 204 9630 9843..56.268.481.693.906 1118 1330 1542 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 6760 207 6970 6180 6390 6599 6809 7018 7227 7436 7646 7854 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 211 4282 4488 4694 4899 5105 5310 6516 5721 5926 6131 212 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 213 8380 8583 8787 8991 9194 9398 9601 9805..8.211 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 8656 8855 9054 9253 9451 9650 9849..47.246 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 5766 5962 6157 222 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 223 8305 8500 8694 8889 9083 9278 9472 9666 9860..54 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 229 9835'..25.215.404.593.783.972 1161 1350 1539 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 231 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 232 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 234 9216 9401 9587 9772 9958.143.328.513.698.883 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 237 4748 4932 5115 5298 5481 6664 5846 6029 6212 6394 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 1..30 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 2017 2197 2377 2557 2737 2917 3097 3277 3466 3636 242 3815 3995 4174 4353 4533 4712 4891 5070 6249 5428 243 5606 5785 5964 6142 6321 6499 6677 6856 7034 I 7212 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 18989 245 9166 9343 9520 9698 9876..51.228.405.582.759 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 248 4452 4627 4802 4977 5152 65326 5501 6676 5850 6025 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 17766 =~~~ ~ ~ 75_76 6 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 250 397940 8114 8287 8461 8634 8808 8981 9154 9328 9501 261 9674 9847..20.192.365.538.711.883 1056 1228 262 401401 1573 1745 1917 2089 2261 2433 2606 2777 2949 253 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 254 4834 5005 5176 6346 6617 5688 5858 6029 6199 6370 255 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 257 9933.102.271.440.609.777.946 1114 1283 1451 258 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 259 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 261 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 262 8301 6467 8633 8798 8964 9129 9295 9460 9625 9791 263 9956.121.286.451.616.781.945 1110 1275 1439 264 421604 1788 1933 2097 2261 2426 2590 2754 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 267 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 269 9752 9914..75.236.398.559.720.881 1042 1203 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 271 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 273 6163 6322 6481 6640 6800 6957 7116 7275 7433 7592 274 7761 7909 8067 8226 8384 8542 8701 8859 9017 9175 275 9333 9491 9648 9806 9964.122.279.437.594.752 276 440909 1066 1224 1381 1538 1695 1852 2009 2166'323 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 278 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 279 5604 5760 5915 6071 6226 6382 6537 6692. 6848 7003 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 281 8706 8861 9015 9170 9324 9478 9633 9787 9941..95 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 284 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 235 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 288 9392 9543 9694 9845 9995.146.296.447.597.748 289 460898 1048 1198 1348 1499 1649 1799 1948 2098 2248 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 291 3893 4042 4191 4340 4490 4639 4788 4936 6085 5234 292 5383 5532 5680 5829 5977 6126 6274 6423 6671 6719 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 294 8347 8495 8643 8790 8938 9085 9233 9380 9627 9675 295 9822 9969.116.263.410.557.704.851.998 1145 296 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 297 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 2 i99 71 5,l 6 5962 6107 6252 6397 6542 6687 6832 6976 OF NUMBERS. 7 N. 0 1 2 3 4 5 6 7 8 9 300 477121 7266 7411 7555 7700 7844 7989 8133 8278 8422 301 8566 8711 88655 8999 9143 9287 9481 9575 9719 9863 302 480007 0151 0294 0438 0582 0725 0869 1012 1156 1299 303 1443 1586 1729 1872 2016 2169 2302 2445 2588 2731 304 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 305 4300 4442 4585 4727 4869 5011 6153 5296 5437 5579 306 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 307 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 308 8551 8692 8833 8974 9114 9255 9396 9637 9667 9818 309 9959..99.239.380.520.661.801.941 1081 1222 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 311 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 312 4155 4294 4433 4572 4711 4850 4989 5128 6267 5406 313 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 314 6930 7068 7206 7344 7483 7621 7769 7897 8035 8173 315 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 316 9687 9824 9962..99.236.374.511.648.785.922 317 501059 1196 1333 1470 1607 1744 1880 2017 2164 2291 318 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 319 3791 3927 4063 4199 4335 4471 4607 4743 1878 5014 320 5150 5286 5421 5557 5693 5828 5964 6099 6234 6370 321 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 322 7856 7991 8126 8260 8395 8530 8664 8799 8934 9008 323 9203 9337 9471 9606 9740 9874...9.143.277.411 324 610545 0679 0813 0947 1081 1216 1349 1482 1616 1750 325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4414 327 4548 4681 4813 4946 6079 5211 6344 5476 6609 5741 328 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 329 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 331 9828 9959..90.221.363.484.615.745.876 1007 332 521138 1269 1400 1530 1661 1792 1922 2053 2183 2314 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 335 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 337 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 338 8917 9045 9174 9302 9430 95569 9687 9815 9943..72 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1361 340 1479 1607 1734 1862 1960 2117 2245 2372 2500 2627 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 343 5294 5421 5547 5674 5800 6927 6053 6180 6306 6432 344 6558 6685 6811 6937 7060 7189 7316 7441 7567 7693 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9578 9703 9829 9954..79.204 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1464 348 1579 1704 1829 1953 2078 2203 2327 2462 2576 2701 349 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 8 LOGARITHMS N. 0 1 2 3 4 5 6 7 8 9 850 644068 4192 4316 4440 4564 4688 4812 4936 5060 5183 351 5307 5431 55555 678 5805 5925 6049 6172 6296 6419 352 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 353 7775 7898 8021 8144 8267 8389 8512 8635 8768 8881 354 9003 9126 9249 9371 9494 9616 9739 9861 9984.196 355 550228 0351 0473 0595 0717 0840 0962 1084 1206 1328 856 1460 1572 1694 1816 1938 2060 2181 2303 2425 2647 357 2668 2790 2911 3033 3155 8276 3393 3519 3640 3762 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 359 5094 5215 5346 5467 5578 5699 5820 5940 6061 6182 360 6303 6423 6544 6664 6785 6905 7026 7146 7267 7387 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 863 9907..26.146.266.385.504.624.743.863.982 364 661101 1'21 1340 1459 1578 1698 1817 1936 2055 2173 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 366 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 367 4666 4784 4903 6021 5139 5257 5376 5494 6612 5730 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 369 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 370 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 371 9374 9491 9608 9725 9882 9959..76.193.309.426 372 570543 0660 0776 0893 1010 1126 1243 1369 1476 1592 373 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 374 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 375 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 376 6188 5303 5419 5534 5650 5765 5880 5996 6111 6226 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 378 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 380 9784 9898..12.126.241.355.469.583.697.811 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 382 2063 2177 2291 2404 26518 2631 2745 2858 2972 3085 383 3199 3312 3426 3639 3662 3765 3879 3992 4105 4218 384 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 385 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7699 887 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9834 389 9950..61.173.284.396.507.619.730.842.953 390 591065 1176 1287 1399 1510 1621 1732 1843 1955 2066 391 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 392 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 393 4393 46503 4614 4724 4834 4945 5055 5165 5276 5386 394 5496 6606 5717 5827 6937 6047 6157 6267 6377 6487 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 396 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 397 8791 8900 9009 9119 9228 9337 9446 9556 9666 9774 398 9883 9992.101.210.319.428.537.646.755.864 899 600973 1082 1191 1299 1408 1517 1625 1734 1843 1951 OF NUMBERS. 9 N. 0 1 2 3 4 6 7 8 9 400 602060 2169 2277 2386 2494 2603 2711 2819 2928 3036 401 3144 3253 3361 3469 3573 3686 3794 3902 4010 4118 402 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 406 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 407 9594 9701 9808 9914..21.128.234.341.447.554 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 416 9293 9198 9302 9406 9511 9615 9719 9824 9928..32 417 620136 0140 0344 0448 0552 0656 0760 0864 0068 1072 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 419 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 420 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 422 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 426 9410 9512 9613 9715 9817 9919..21.123.224.326 427 630428 0530 0631 0733 0835 0936 1038 1139 1241 1342 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 432 5484 5584 5685 5785 5886 5986 6087 6187 6287 6988 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 436 9486 9586 9686 9785 9885 9984..84.183.283.382 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 442 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 445 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 446 9335 9432 9530 9627 9724 9821 9919..16.113.210 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 449 2246 2343 2440 2530 2633 2730 2826 2923 3019 3116 10 LOGARITHMS N. 0 1 2 3 4 6 6 7 8 9 450 653213 3309 3405 3502 3598 3695 3791 3888 3984 4080 451 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 452 5138 5235 5331 5427 5526 5619 5715 5810 5906 6002 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 455 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 456 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 457 9916..11.106.201.296.391,486.581.676.771 458 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 459 1813 1907 2002 2096 2191 2286 2380 2475 2669 2663 460 2758 2852 2947 3041 3135 3230 3324 3418 3512 3607 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 462 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 463 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 465 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 466 8386 8479 8572 8665 8759 8852 8945 9038 9131 9324 467 9317 9410 9503 9596 9689 9782 9875 9967..60.153 468 670241 0339 0431 0524 0617 0710 0802 0895 0988 1080 469 1173 1265 1368 1451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 471 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 472 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 473 4861 4953 6045 5137 5228 5320 5412 5503 5595 5687 474 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 475 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 477 8518 8609 8700 8791 8882 8972 9064 9155 9246 9337 478 9428 9519 9610 9700 9791 9882 9973..63.154.245 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 481 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 483 o4i7 4037 4127 4217 4307 4396 4486 4576 41666 4756 484 4864 4935 5025 5114 5204 5294 5383 5473 5563 5652 485 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 486 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 487 7629 7618 7707 7796 7886 7975 8064 8153 8242 8331 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 489 9309 9398 9486 9575 9664 9753 9841 9930..19.107 490 690196 0285 0373 0362 0550 0639 0728 0816 0905 0993 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 492 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 493 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 495 4605 4693 4781 4868 4956 5044 5131 5210 6307 5394 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 497 6356 5444 6531 6618 6706 6793 6880 6968 7055 7142 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883 OF NUMBERS. 11 N. 0 t 2 3 4 6 6 7 8 9 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 601 9838 9924..11..98.184.271.358.444.531.617 502 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 603 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 504 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 505 3291 3377 3463 3549 3635 3721 3807 3895 3979 4065 606 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 507 5008 5094 5179 5265 5350 5436 55622 5607 5693 5778 508 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 509 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 510 7570 7655 7740 7826 7910 7996 8081 8166 8251 8336 611 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 512 9270 9355 9440 9524 9609 9694 9779 9863 9948..33 513 710117 0202 0287 0371 0466 0540 0625 0710 0794 0879 514 0963 1048 1132 1217 1301 1386 1470 1554 1639 1723 515. 1807 1892 1976 2080 2144 2229 2313 2397 2481 2566 516 2650 2734 2818 2902 2986 3070 3154 3238 3326 3407 517 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 518 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 519 5167 5251 5335 5418 5502 55686 6669 5753 5836 5920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 521 6838 6921 7004 7088 7171 7254 7338 7421 7504 7687 522 7671 7764 7837 7920 8003 8086 8169 8263 8336 8419 523 8502 8586 8668 8751 8834 8917 9000 9083 9165 9248 624 9331 9414 9497 9580 9663 9745 9828 9911 9994..77 525 720159 0242 0325 0407 0490 0573 0655 0738 0821 0903 526 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 527 1811 1893.976 2058 2140 2222 2305 2387 2469 2552 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 529 3466 3538 3620 3702 3784 3866 3948 4030 4112 4194 630 4276 4358 4440 4522 4604 4685 4767 4849 4931 5013 531 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 534 7541 7623 7704 7786 7866 7948 8029 8110 8191 8273 535 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 636 9165 9246 9327 9403 9489 9670 9651 9732 9813 9893 537 9974..55.136.217.298.378.459.440.621.702 638 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 539 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 540 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 541 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 542 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 643 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 544 5599 5679 5759 5838 6918 5998 6078 6157 6237 6317 545 6397 6476 6556 6636 6715 6795 6874 6954 7034 7113 546 7193 7272 17352 7431 7511 7590 7670 7749 7829 7908 547 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 548 8781 8860 i8939 9018 9097 9177 9256 9335 9414 9493 549 9572 9651 / 9731 9810 9889 9968..47.126.205.284'21 12 LI Oa Gh C THMS N. 0 I 2 13 4 56 6 7 8 9 560 740863 0442 06211 0560 0678 0r157 0836 0916 0994' 1073 651 11-5 1230 1 309 1388 1467 1546 1624 1703 1782 1860 652 IM39 2018 2096 2175 2254'2332 2411 2489 2568 2646 553 W725' 28904 2882 2961 3089 3118 3196 3276 3369 3431 554 35 0 35568 3661 374' 3823 3902 3980 4058 41,36 4215 655 4293 467 444, 4628 4608 4684'4762 i 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1440' 1514 591; 16587 1661 1734 1B08 1881 19556 2028 210.2 2175 2248 692 2322 29395 468 3542 2615i' 2688.2762 2835, 2908 2981, 693 3055 31t28 320i 3274, 3348 3421V 3494- 3567 3640, 3716 594 3786. 3860'3933 4006 40T9 4152'4225 4298 4871. 4444 695 4517: 4690 4663' 4136, 4809 4882 i:4965, 5028' 5100' 6173 596 5246 6319 6392 6465 6538 5610 65683 6756 5829 5902 697 5074 6047:6120' 61031 6265 6338 6411 6483 6566'6629 698 6701:6774!6846 69109 6992 7064 7137 7209 7T282 7354 699 7427.,7499;7572 7644 7717 7789 17862 7934 8006 8079 OF NUMBERS. 13 N. 0 1 2 3 4 5 6 7 8 9 600 778161 8224 8296 8368 8441 8513 8585 8658 8730 8802 601 8874 8947 9%19 9091 9163 9236 9308 9380 9452 9624 602 9596 6669 9741 9813 9885 9957..29.101.173.245 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 604 1037 1109 ii81 1263 1324 1396 1468 1640 1612 1684 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 240i 606 2473 2544 2616 2688 2759 283i 2902 2974 3046 3117 607 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 609 4617 4689 4760 4831 4902 4974 B0>45 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 613 7460 7531 7602 7673 7744 78i5 7885 7956 8027 8098 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 616 9581 9651 9722 9792 9863 9933..,4..74.144.215 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 619 1691 1761 i831 i901 1971 2041 2111 2181 2252 2322 620 2392 2462 2532 2602 2672 2742 2812 2882 2952 3022 621 3092 3162 3231 3301 3371 3441 8511 3581 3651 3721 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 44i8 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 624 5185 5254 5324 5393 5468 5532 5602 6672 6741 6811 625 5880 5949 6019 6088 6158 6227 6297 6366 6436 65'05 626 6574 6644 6713 6782 6852 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1317 1382 1448 663 1514 1579 1645 i710 1i775 1841 1906 1972 2037 2103 664 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 665 2822 2887 2952 3018 3083 3148 3213 3279 3344'3409 666 3474 3539 3605 3670 3735 3800 3865 3930 3996 406i 667 4126 4191 4256 4321 4386 4451 4516 4581 4646 471i 668 4776 4841 4906 4971 6036 6101 5166 5231 5296 6361 669 5426 5491 5556 6621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 672 7369 7434 7499 7663 7628 7692 7757 7821 7886 7951 673 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 675 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 676 9947..11.,75.139,204.268,332.396.460.525 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 678 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 681 3147 3211 3275 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4974 5036 OF NUMBERS. 15 N. 0 1 2 3 4 5 6 7 8 9 700 845098 5160 5222 5284 5346 5408 5470 5532 5594 5656 701 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 703 6955 1017 7079 7141 7202 7264 7326 7388 7449 7511 704 7573 7634 7676 7758 7819 7831 7943 8004 8066 8128 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 709 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 713 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 717 5519 5580 5640 5701 5761 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6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 737 7467 7526 7685 7644 7703 7762 7821 7880 7939 7998 738 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 741 9818 9877 9935 9994..53.111.170.228.287.345 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 743 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 744 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 745 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 747 3321 8379 3437 3495 3553 3611 3669 3727 3785 3844 748 3902 3960 4018 4076' 4134 4192 4250 4308 4360 4424 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 16 L O G A R I T H M S N. 0 1 2 3 4 5 6 7 8 9 750 875061 5119 5177 R235 5293 5351 5409 5466 5524 5582 751 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 753 6795 6853 6910 6968 7026 7083 7141 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7085 7138 7190 7243 7295 7348 7409 7453 827 7506 7558 7611: 7663 7716 7768 7820 7873 7925 7978,28 8030 8083 8185 8188 8240 8293 8345 8397 8410 8502 829 8555 8607 8669 8712 8764 8816 8869 8921 8973 9026 830 9078 91&3.0 9183 i 9235 9287 9340 9392; 9444 9496 4 831 9601 9653, 9706 9768 9810 9862 9914 9967..19..71 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 833 0545 0697 0749 0801 0853 0906 0958 1010 1062 1114 834 1166 1218 1270 1322 13'74 1426 1478'1530 1582 1634 835 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 836 2206 2258 2310 2362 2414 2466 2618 267.0 2622 2674 837 2725 2777 2829 -2881 2933 2985 3037 3089 3140 3192 838 3244 3296 3348 3399 3451 35903 3555 3607 3658 3710 839 3762 3814 38656 3917 3969 4021 4072' 4124 4147 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 841 4796 484848 4899 4951 5003 5054 5106 5157 5209.5261 842 5312 5364 5415 5467 5518 5570 6621 5673 5725 5776 843 5828 5874 5931 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6764 6805 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 846 737.0 7422 7473 7524 7576 7627 7678 7730 7783 7832 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 9.8908 8959 9010 9061 9112 9163 9216 9266 9317 9368 18 LOGARITHMS N. 0 1 2 3 4 6 6 7 8 9 850 929419 9473 9521 9572 9623 9674 9725 9776 9827 9879 851 9930 9981..32..83.134.185.236.287.338.389 852- 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 853 0949 1000 1051 1102 11563 1204 1264 1305 1356 1407 854 1458 1609 1560 1610 1661 1712 1763 1814 1865 1915 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2524 2576 2626 2677 2727 2778 2829 2879 2930 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 869 3993 4044 4094 4145 4195 4246 4269 4347 4397 4448 860 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 861 5003 5054 5104 6164 5205 5255 5306 5356 5406 5467 862 5507 5558 5608 5658 6709 5759 5809 6860 5910 5960 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 864 6614 6664 6614 6665 6715 6765 6815 6865 6916 6966 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 866 7618 7668 7618 7668 7718 7769 7819 7869 7919 7969 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 868 8520 8570 8620 8670 8720 8770 8820 8870 8919 8970 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9519 9569 9616 9669 9719 9769 9819 9869 9918 9968 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 874 1511 1661 1611 1660 1710 1760 1809 1859 1909 1958 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 876 2504 2564 2603 2653 2702 2762 2801 2851 2901 2950 877 3000 3049 3099 3148 3198 3247.3297 3346 3396 3445 878 3495 3544 3693 3643 3692 3742 3791 3841 3890 3939 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 881 4976 5025 6074 5124 5173 5222 5272 5321 5370 5419 882 5469 5518 5567 5616 5665 5715 5764 5813 5862 6912 883 5961 6010 6069 6108 6157 6207 6256 6305 6354 6403 884 6462 6501 6551 6600 6649 6698 6747 6796 6845 6894 885 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 886 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8365 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 891 9878 9926 9975..24..73.121.170.219.267.316 892 950365 0414 0462 0511 0560 0608 0657 0706 0754 0803 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 894 1338 1386 1436 1483 1532 1580 1629 1677 1726 1775 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 896 2308 2356 2405 2453 2502 25560 2599 2647 5696 2744 897 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 898 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 899 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 OF NUMBERS. 19 N. 0 1 2 3 4 5 6 7 8 9 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 901 4725 4773 4821 4869 4918 4966 6014 6062 6110 5158 902 6207 5255 5303 5351 6399 5447 5495 5543 5592 5640 903 5688 6736 5784 6832 5880 5928 5976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6457 6605 6553 6601 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7665 7703 7751 7799 7847 7894 7942 7990 8038 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 909 8564 8612 8669 8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 912 9995..42..90.138.185.233.280.328.376.423 913 960471 0518 0566 0613 0661 0709 07566 0804 0851 0899 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 915 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 916 1896 1943 1990 2038 2085 2132 2180 2227 2275 2322 917 2369 2417 2464 2511 25569 2606 2653 2701 2748 2795 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 919 3316 3363 3410 3467 3504 3552 3599 3646 3693 3741 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 922 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 924 6672 5719 5766 5813 5860 6907 5954 6001 6048 6095 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 926 6611 6658 6705 6762 6799 6845 6892 6939 6986 7033 927 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 8016 8062 8109 8166 8203 8249 8296 8343 8a90 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 931 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 9882 9928 9975..21..68.114.161.207.254.300 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 937 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 3543 941 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 5064 5110 5156 5202 5248 5294 6340 5386 945 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6533 6579 6926 6671 6717 6763 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 949 7266 7312 7368 7403 7449 7495 7541 7586 7632 7678 20 LOGARITHMS N. 0 1 2 3 14 5 6 7 8 9 950 977724 7769 7815 7861 7906 7952 7998 8043 8089 8135 961 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9603 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9968 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 958 1366 1411 1466 1501 1547 1592 1637 1683 1728 1773 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 962 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 963 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 964 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 966 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 967 6426 5471 5516 5561 6606 5651 5699 5741 5786 5830 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 969 6324 6369 6413 6468 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 973 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 9005 9049 9093 9138 9183 9227 9272 9316 9361 9405 976 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 977 9895 9939 9983..28..72.117.161.206.250.294 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 979 0783 0827 0871 0916 0960 1004 1049 1093 1)37 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 983 2654 2698 2642 2686 2730 2774 2819 2863 2907 2951 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 988 4757 4801 4845 4886 4933 4977 5021 5065 5108 5152 989 6196 5240 6284 6328 5372 5416 5460 5504 5547 5591 990 5635 5679 5723 5767 5811 5854 5898 5942 5986 6030 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 997 8695 8739 8792 8826 8869 8913 8956 9000 9043 9087 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 TABLE II. Log. Sines and Tangents. (00) Natural Sines. 21 —' Sine. D. 10" Cosine.'D. 101" Tang. D.10" Colang. N.sine. N. cos. 0 0.000000 10.000000 0.0000000 Infinite. 00000 100000 60 1 6.463726 000000 6.463726 13.536274 00029 100000 59 2 764756 000000 764756 235244 00058 100000 58 3 940847 000000 940847 059153 00087 100000 57 4 7.065786 000000 7.065786 12.934214 00116 100000 56 5 162696 000000 162696 837304 00145 100000 55 6 241877 9.999999 241878 758122 00175 100000 54 7 308824 999999 308825 691175 00204 100000 53 8 366816 999999 366817 633183 00233 100000 52 9 417968 999999 417970 582030 00262 100000 51 10 463725 999998 463727 536273 00291 100000 50 11 7.505118 9.999998 7.505120 12.494880 00320 99999 49 12 542906 999997 542909 457091 00349 99999 48 13 577668 999997 577672 422328 00378 99999 47 14 609853 999996 609857 390143 00407 99999 46 15 639816 999996 639820 360180 00436 99999 45 16 667845 999995 667849 332151 00465 99999 44 17 694173 999995 694179 305821 00495 99999 43 18 718997 999994 719003 280997 00524 99999 42 19 742477 999993 742484 257516 00553 99998 41 20 764754 999993 764761 235239 00582 99998 40 21 7.785943 9.999992 7.785951 12.214049 00611 99998 39 22 806146 999991 806155 193845 00640 99998 38 23 825451 999990 826460 174540 00669 99998 37 24 843934 999989 843944 156056 00698 99998 36 25 851663 999988 861674 138326 00727 99997 35 26 878695 999988 878708 121292 00756 99997 34 27 895085 999987 895099 104901 00785 99997 33 28 910879 999986 910894 089106 00814 99997 32 29 926119 999985 926134 073866 00844 99996 31 30 940842 999983 940858 050142 00873 99996 30 31 7 955082 2298.999982 02 7.951002298 12.044900 00902 99996 29 32 968870 2297 999981 0.2 968889 227 031111 00931 99996 28 33 982233 227161 999980 2 982253 2161 017747 00960 99996 27 34 995198 209861 999979 0:2 995219 2098 004781 00989 99995 26 35 8.007787 2039 999977 02 8.007809 203911.992191 01018 99995 25 36 02002 103983 999976 0 2 020045 21983 979955 01047 99995 24 37 03~~~~~~~~1919 37 031919 1930 99997 02 031945 1 30 968055 01076 99994 23 38 043501 880 999973 2 043527 18 956473 01105 99994 22 39 054781 1880 999972 0 054809 183 945191 01134 99994 291 40 065776 1832 999971 02 065806 18 934194 01164 99993 20 144 11.923469 01193 99993 19 41 8.076500 17744 9.999969 0 2 8.076531 1744 11.923469 01193 99993 19 42 086965 7 999968 02 086997 170 913003 012229 99993 18 4 09131703 9996603 930 43 097183 1664 999966 002 097217 1 902783 01251 99992 17 44 107167 1664 999964 0:2 107202 1667 892797 01280 99992 16 45 116926 1626 999963 O3 116963 1621 883037 01309 99991 15 1591 99991 0~3 12610591 833 46 126471 1557 999961 0 126610 17 873490 01338 99991 14 47 135810 154 999959 0.3 135851 4 864149 01367 99991 13 14931524 2495 386414 48 144953 42 999958 0.3 144996 85004 01396 99990 12 49 153907 1 42 999956 0.3 153952 143 846048 01425 99990 11 50 162681 1462 9999540.3 162727 1463 837273 01454 99989 10 51 8 171280 1435 9.999952 0.3 8.171328 1434 11.898672 01483 99989 9 52 19131405'9.999952 0.3 7 132 406 ~ 88609720539 99 52 179713 1379 999950 0.3 179763 7 820237 01513 99989 8 53 187985 1353 999948.3 188036 133 811964 101542 99988 7 535; 0 1353 54 196102 138 999946 0.3 196156 138 803844 01671 99988 6 132 999946 1328 834 55 204070. 999944 204126 795874 01600 99987 5 55 200 04 56 211895 134 999942 211953 13 788047 01629 99987 4 1281 0.4 1~~~~281 57 219581 2819 999940.4 219641 12 780359 01658 99986 3 58 227134 1259 999938 0.4 227195 1259 772805 101687 99986 2,,~. 99938 137[238 77205 59 2345o.7 1216 999936 0.4 234621 12 765379!01716 99985 1 60 241855 26 999934 0.4 241921 27 758079) 01745 99985 0 _ Co~sin~e. Sinle. ColanZ. Tanlar. I N. cos.N. sine- I 89 Degrees. -_.. 22 Log. Sines and Tangents. (1~) Natural Sines. TABLE Il. " Sine. D.10o" Cosine. D.10" Tang. D.1O" Colall. IN. sine. N. cos. 0 8.241855 1196 9.999934 8.241921 1197 11.758079 01742 9998560 1 249033 1177 999932 04 249102 1177 750898. 01774 99984 591 2 256094 1158 999929.4 256165 1158 743835 0180399984 58 3 263042 1140 999927 04 263115 1140 736885 0183299983 57 4 269881 1122 999925 04 269956 1122 730044 01862 99983 56 6 276614 1105 999922 0.4 276691 1105 723309 01891 99982 655 6 283243 1088 999920 04 283323 1089 716677 01920 99982 54 7 289773 1072 999918 04 289856 073 710144 01949 99981 53 8 296207 1056 999915 0.4 296292 1057 703708 01978 99980 52 9 302546 1041 999913 04 302634 1042 697366 02007199980 51 10 308794 1027 999910 0:4 308884 1027 691116 02036 99979 50 11 8.314954 1012 9.999907 0 4 8.315046 1013 11.684954 02065 99979 49 12 321027 998 999905 0.4 321122 9 678878 02094 99978 48 13 327016 985 999902 04 327114 985 672886 02123 99977 47 14 332924 971 999899 05 333025 972 666975 02152 99977 46 15 338753 9 999897 333856 661144 02181 99976 45 16 344504 946 999894 05 344610 946 655390 02211 99976 44 17 350181 934 999891. 850289 649711 02240 99975 43 18 355783 922 999888' 355895 922 644105 02269 99974 42 19 361315 910 999885 0. 361430 911 638570 02298 99974 41 20 366777 899 999882 0.5 366895 899 633105 02327 99973 40 21 8.372171 888 9.999879 0. 8.372292 888 11.627708 02356 99972 39 22 377499 877 999876 0o 377622 879 622378 02385 99972 38 23 382762 867 999873 os 382889 867 617111 02414 99971 37 24 387962 856 999870 0'5 388092 857 611908 02443 99970 36 25 893101 846 999867 Os 393234 847 606766 02472199969 35 26 398179 837 999864 5 3983165 837 601685 02501 99969 34 27 403199 827 999861 o5 403338 828 596662 02530 99968 33 28 408161 818 999858 0 408304 818 691696 0256099967 32 29 413068 809 999854 0.5 413213 809 586787 02589 99966 31 30 417919 800 909851 0.6 418068 800 681932 02618 99966130 31 8.422717 791 9.999848 0.6 8.422869 791 11.577131 02647 99965 29 32 427462 999844 0 427618 78 572382 02676 99964 28 33 432156 999841 06 432315 774 567685 02705 99963 27 34 436800 999838 6 436962 766 663038 02734 99963 26 35 441394 758 999834 0 6 441560 758 558440 02763 99962 25 36 445941 750 999831 06 446110 750 553890 02792 99961 24 37 460440 742 999827 0.6 450613 743 549387 02821 99960 23 38 454893 5 999823 6 455070 736 544930 02850 99959 22 39 459301 727 999820 06 459481 728 540519 02879 99959 21 40 463665 720 999816 6 463849 536151 02908 99968 20 41 8.467985 712 9.999812 0.6 8.468172 713 11.631828 02938 99957 19 42 472263 706 999809 06 472454 707 527546 0296799956 18 43 476498 699 999805 0 6 476693 700 523307 02996 99965 17 44 480693 692 999801 6 480892 693 19108 0302599954 16 45 484848 686 999797 485050 514950 03054 99953 15 46 488963 679 999793 0.7 489170 680 510830 03083 99952 14 47 493040 673 999790 0.7 493250 674 506750 03112 999529 13 48 497078 66 999786 7 49793 668 602707 03141 99951 12 49 501080 661 999782 7 01298 661 498702 03170 99950 11 50 605045 65 999778 07 6.05267 655 494733 03199 99949 10 51 8.508974 649.99977'.7.09200 650 11.490800 03228 99948 9 52 512867 643 999769 0 13098 486902 0325799947 8 53 616726 643 999765 07 516961 638 483039 03286 99946 7 64 520551'1 632 999761 07 520790 633 479210 03316 99945 6 55 524343 999757 7 524586 627 475414 03045 99944 5 56 628102 1 999753 628349 471651 03374 99943 4 57 531828 621 999748 7 532080 467920 0340399942 3 68 35523, 616 999744 0.7 G35779 611 464221 03432 99941 2 59 539186 611 999740 07 639447 661 460563 03461 99940 1 5 5 6605 0.7 6 60 542819: 999735 543084 456916 03490 99939 0 Cosile. Sine. Cotang. Tang. N. cos. N.sine. 88 Degrees. TABLE II. Log. Sines and Tangents. (20) Natural Sines. 23 Sine. D. 10" Cosine. ID. 10"1'T'ang. D. 10"1 Cotanlg. IIN. sine.N. eos. 0 8.542819 600 9.999735 7 8.543084 602 11.456916 03490 9939 60 1 546422 6 999731 546691 6 453309 03519 99938 69 2 549995 69 999726 0.7 550268 691 449732 03548 99937 58 3 553539 565 999722 553817 587 446183 03577 99936 57 4 557054 999717 557336 442664 1 03606 99935 66 5 560540 5 999713 0.8 560828 58 439172 03635 99934 55 6 563999 576 999708 08 664291 7 435709 03664 99933 54 7 67431 572 999704 5.8 67727 8 432273 03693 99932 53 8 570836 663 999699 0.8 571137 664 428863 03723 99931 52 9 5674214 5 999694 0.8 574520 6 42480 03752 99930 51 10 577466 999689 0. 577877 422123 03781 99929 50 11 8.580892 9.999685 8.581208! 11.418792 03810 99927 49 12 584193 550 999680 0.8 584514 415486 03839 99926 48 1 546 0.8 5 547 13 587469 99967 587795 412205 03868 99925 47 14 9072 42 0.8 91051 408949 03897 9 9924 46 15 593948 384 999665 0.8 94283 63 405717 03926 99923 45 16 597162 999660.8 697492 402508 03955 99922 44 17 600332 530 999665 0.8 600677 531 399323 03984 99921 43 18 603489 526 999650 0.8 603839 527 396161 04013 99919 42 19 606623 522 999645 0.8 606978 523 393022 0404' 99918 41 20 609734 519 999640.8 610094 519 389906 04071 99917 40 218612823 515 9.99630 8.613189516 il.386811 04100 99916 39 22 615891 999629 0.9 616262', 383738 03129 99915 38 23 618937 508 999324 0.9 619313 380687 0415999913 37 24 621962 504 999619 0.9 622343!Oo 377657 04188 99912 36 25 624965 501 999614.9 626352 501 374648 04217 99911 35 26 627948 497 999608 499 628340 498 371660 04246199910 34 62 7 0.9 8 40 27 630911 999603.9 631308 68692 4275 33 28 633854 490 0.9 634256 491 365744 04304 99907 32 29 636776 48 999592.9 637184 8 362816 04333 99906 31 30 639680 481 99986 640093 48 359907 04362 99905 30 31 8.642663982 482 11.37018 04391 99904 29 32 645428 9995.9 645853 478 354147 04420 99902 28.33 648274 474 999570 0.9 648704 47 351296 04449 99901 27 34 651102 471 99964 0. 9 651537 472 348463 04478 99900 26 35 653911 46 99958 0.9 654352 46 345648 04507 99898 26 36 656702 46 999553 1 657149 466 342851 0453699897 24 37 659475 462 999547 1.0 669928 463 340072 04565 99896 23 37 659475 999547 6569928 38 662230 456 999541 1 662689 460 337311 04594 99894 22 39 664968 4 999535.0 66433 334567 04623 99893 21 40 667689 53 9995291 0 668160 4 331840 04653 99892 20 41 8.670393 9 9524.670870 11.329130 04682 99890 1(9 42 673080 999518 0 673563 326437 04711 99889 18 44 326437 04 43 675751 442 999512 0 676239 323761 04740 99888 17 44 678405 4 999506 1 678900 321100 04769 9886 16 4 681043 437 999500 1 681544 448 318456 04798 99885 15 46 683665 434 999493 10 684172 435 315828 04827 99883 14 47 686272 43 999487 686784 313216' 04856 99882 13 667 432 10 433048 9988112 48 688863 999481 689381 43 310619 04886 99881 12 42 9994811 430 34998911 49 691438 99947.0 691963 308037 4Q27 1428 305047 0494399878 10 50 693998 99469 694529 305471 04943 99878 10 51 8;696543 424 9.999463 8.697081 11.302919 04972 99876 9 52 609073 422 999456 1 699617 423 300383 05001 99875 8 63 701589 417 999450 1 702139 428 297861 0503099873 7 54 704090 417 999443 1.1 704246 41 295354 0505999872 6 65 706677 1 999437 1. 707140 415 292860 05088 99870 56 709049 41 999431.1 709618 413 290382 05117 99869 4 67 711507 410 999424 1.1 702083411 287917 0514699867 3 58 713952 407 999418 1.1 714534 408 285465 05175 99866 2 59 716383 405 999411 1.1 716972 406 283028 0520599864 1 60 718800 999404 719396 40 80604 05234 99863 0 Cosine. Si'e. Cotang. T'alg i- N. co-. N.sine. 87 Degrees. 24 Log. Silles and Tangents. (30) Natural Sines. TABLE 11. Sine. D. l" Cosine. o. -]-5, Tang. D. 1ol Cotang. 1(N.sine. N. cos; 0 8.718800 9.999404 8.719396 402 11.280604 05234 99863 60 i 721204 39 999398.i 721806 526399861 59 2 723595 396 999391 1. 724204 3 275796 05292 99860 58 3 725972 396 999384 1.1 726588 395 273412 05321 99858 57 4 728337 394 999378 1.1 728959 9 271041 05350 99857 56 5 730688 392 999371 i.1 731317 393 268683 05379199855 55 6 733027 390 999364 1 * 733663 391 266337 05408 99854 54 7 735354 88 999357 1; 2 735996 389 264004 05437 99852 53 8 737667 386 999350 12 738317 385 261683 05466 99851 52 9 739969 382 999343 1 2 740626 389 259374 05495 99849 51 10 742259 382 999336 12 742922 381 257078 05524 99847 50 11 8.744536 380 9.999329 1.2 8;745207 38 254798 055539984 49 12 746802 378 999322 747479 3 252521 05582 99844 48 13 749055 376 999315 1.2 749740 377 250260 05611 99842 47 14. 751297 374 999308 1.2 75198,9 376 248011 05640 99841 46 15 753528 372 999301 1.2 754227 371 245773 05669 99839 45 i6 755747 370 999294 1.2 756463 369 243547 05698 99838 44 17 757955 368 999286 1.2 758668 3 241332 06727 99836 43 18 760151 366 999279 1.2 760872 367 28918 0575699834 42 19 762337 964 999272 1.2 763065 365 236935 05785 99833 41 20 764511 362 999265 1.2 765246 364 234764 05814 99831 40 21 8.766675 861 9;999257 12 8.767417 362 i;232583 05844 99829 39 22 768828 8 999250 i2 769578 360 230422 05878 99827 38 23 770970 o 999242 771727 3 228273 05902 99826 37 24 773101 855 999235 773866 356 226134 05931 99824 36 25 775223,.3 999227.S 775995 35 224005 05960 99822 35 26 777333 352 999220 13 778114 3 221886 05989 99821 34 27 779434 350 999212 1.3 780222 351 219778 06018 99819 33 28 781524 848 999205 1,3 782320 350 217680 06047 99817 32 29 783605 347 999197 784408 348 215592 06076 99815 31 30 785675 345 999189 13 786486 346 213514 06105 99813 30 31 8,78776 843 9.999181 8.78855 4 4 11.211446 06134{99812 29 32 789787 842 999174 3 790613 343 20987 0616399810 28 33 791828 340 999166 1 3 792662 341 207388 06192 99808 27 34 793859 33 999158 79471 340 205299 06221 99806 26 35 795881 337 999150 i 796731 38 203269 06250 99804 25 36 797894 33 999i42 13 798752 337 201248 0627999803 24 37 799897 334 999134;3 800763 3 4 99237 06308 99801 23 38 801892 332 999126 802765 197235 0633799799 22 39 803876 331 999i18 1; 804858 331 195242 00366 99797 21 40 805852 328 999110 1i 3 806742 29 93258 06395 99795 20 41 8 807819 328 9. 999102 i8 8808717 32 ii 1912831 d6424 99793 19 42'809777 326 999094 1.3 810683 38 189317 06453 99792 18 43 811726 325 999086 1:4 8i2641 326 187359 06482 997908 17 44 813667 323 999077 144 8i4589 823 185411'0651199788 16 45 815599 322 999069 1.4 816529 323 183471 0654099786 15 46 817522 3120 999061 1:4 818461 322 181539 065 69 99784 14 47 819436 319 999053 14 820384 320 179616 06598}99782 13 48 821343 9990440 4 822298 319 177702 06627 99780 12 31-6 1 4 318 {. - 49 823240 316 999036 1.4 824205 318 175795 06656 99778 { 1 50 825130 315 999027 1{4 826103 6 173897 06685 99776 1i 51 8 827011 313 9.999019 1.4 8827992 315 11.172008 d6714 99774 9 52 828884 312 999010 1.4 8129874 314 170126 06743 9977 8 53 830749 311 999002 1'4 831748 312 1682521 06773 99770 7 54 832607 309 998993 1s4 833613 166387 06802 99768 6 55 834456 308 998984 1 4 835471 308 164529 06831 99766 5 56 836297 307 998976 144 837321 308 162679 06860 99764 4 57 838130 306 998967 14, 889163 30 160837 06889 99762 3 58 839956 304 998958 1.5 840998 304 i59002 06918 99760') 303 15 304 59 841774 302 998950 1.5 842825 303 157175 106947 99758 1 60 843585 302 998941 1.5 844644 155356 06976 9976.C Osifl('. | Fine. Cotang. = i Ta-n. r N.e (co I5.6.sie 8( I)egrees.I ,. --... TABLE II. Log. Sines and Tangents. (40) Natural Sines. 25 I _ Sine. D. 10", Cosine. D. 10" Tang. D. 10" Cotang. iN. sine. N. eos. O 8.843585 a0o 9.998941 1.5;844644 11.155356 06976 06 9756 60 1 845357 299 998932 846455 1 153545 07005 99754 59 2 847183 298 998923: 5 848260 2 151740 0703499752 8 3 848971 298 998914. 850)57 99 149943 07063 99750 57 4 850751 2957 998905 1.5 851846 29 148154 07092 99748 56 s 852525 295 998896 1:5 853628 297 146372 07121 99746 55 6 854291 294 998887 1 856 8403 296 144597 0715099744 54 7 856049 292 998878 1. 857171 14289 07179 99749 53 8 8Ei7801 291 998869 1:5 858932 29 14068, 017208 99740 52 9 859546 90 9988601:1 860686 292 13314 07237 9738 6 10 861283 288 998861 1:' 862433 290! 37567 07266 99736 50 8863 287 98841 1. 8.864173 289 11.135827 0729599734 49 287 1 3 4 09 0391 48 12 864738 286 998832 1: 5 865906 28 134094 0732499731 48 13 866465 285 998823 1 6 867632 287 132368 07353 99729 47 14 868169 284 998813.6 869351 130649 07382999727 46 15 869868 283 9988014 1 871064,285 128936 074i 1199725 45 16 871565 282 998795 1.6 872770 283 127230 07440 997238 44 17 873255 281 9987816 874469 282 125531 0746999721 43 18 874938 27981 998776 1.6 76162 28 23838.07498 99719 42 19 876615 279 998766 877849 281 2151 0752799716 41 20 878285 279 998757 1.6 879529 280 120471 07556 99714 40 277 1.6 8 7999 *' 21 8.879949 27 9.998747 1 6 8.881202 278 11.i18798 0758599712 39 22 881607 275 998738 882869 117131 0276i499710 38 275 1.6 277 23 883258 274 998728 1 6 884530 276 116470 [07643 99708 37 24 8849( 273 998718 16 886185;275 113816 07672199705 36 25 886542 272 998708 1. 6 887833 274 112167 0770 199703 35 26 888174 271 998699 1 6 889476 27 110524 07730 99701 34 27 889801 270 998689 1:6 891112 279 108888 0779 99699 33 28 891421 269 998679 1 6 892742 272 107258 07788 99696 34 29 893035'268 998669 1 7 894366 270 105634 07817199694 31 30 894643 267 99869 7 895984 269 10416 0784699692 30 31 8.896246266 9.998649 7.897596 26 11102404 07875 99689 29 266 2'8 ~'' 32 897842 265 998639 17 899203 127 00797 07904 9987 28 33 899432 64 998629:. 900803 2 099197 07933199685 27 34 901017 263 998619 1 7 902398 266 097602 07962 99683 26 35 902596 262 998609 1 7 903987 264 096013 07991 99680 25 36 90'416 262 1.7 2091 36 904169 251 998599 1 7 905570 263 094430 0802099678 24 2 9961 1.7 269 37 905736 260 998589 1 907147 262 092853i 0804999676 23 38 907297 259 998578 1.7 908719 26 091281 080,78 99673 22 39 908853 58 998568 17 910285 26 0891715 08107 9967121 40 910404 25 998558 17 911846 2 6881541 08136 99668 20 41 8.911949 257 9.998548 1.7 8.913401 258 11.086599 1 08165 99666 19 42 913488 256 998537 1*7 914961 257 085049 0819499664 18 256 1.7 256 17 8 43 915022 255 998527 I;7 916495 256 083505 08223 961 17 44 916550 254 998516 1:8 918084 256 081966 08252 99691 16 45 918073 2 99806 9195 25 080482 0828 199657 15 46 919591 252 998495 1, 921096 254 078904 108310 9965'4 14 47 921103 251 998486 1-8 922619 253 077381 08339 99652 13 48 922610 251 998474 1.8 994136 252 075864 0836899649 12 49 924112 249 998464 1,8 i!25649 2521 074361, 08397 99647 11 50 925609 249 998453 1 8 927156 50 072844 08426 99644 10 51 8.97100 9248 9.998442 1.18 8.9 8658 249 1.071342 08455'99642 9 62 928587 247 998431 1 8 930155 249 069845' 0848499639 8 53 930068 998421 1. 931647 249 068353 0851;3 99637 7 54 931544 24 998410 18 933134 2 066866 08542 99635 6 655 933015 94 99839 1'8 934616 24 065384 108571 99632 5 56 934481 998399 66 19344 A24 A 98388 1 8 936093 245 63907 08600 99630 4 57 935942 243 998377 8 93765 06243 08629 99627 3 243 1.8 244:68 937398 22423 998366., 9890392 060968 0865 99625 2 242 1 8 244 59 938850: 998355' 940494 059506 08687 9622 1 60 940296 998344 941952 058048 08716 9619 0 Cosine. Sine. Tg Cotang. N. co.sine. 85 Degtees, Log. Sines and Tangents. (50) Natlral Sinies. TABLE II. t Sine. D. 10' Cosine. D. 10" Tang. D. 10" Cotang. N. sine. N. cos. o -8.940296 240 9.998344 1-9 8.941952 242 11.058048 08716 99619 60 1 941738 239 998333 1.9 943404 241 056596 08745 99617 69 2 943174 239 998322 1.9 944852 240 055148 08774 99614 68'3 944606 238 998311 19 946295 240 053705 08803 99612 57 4 946034 237 998300 1J' 947734 2 4 052266 108831 99609; 66 5 947456 236 998289 19 949168 238 050832 i108860 99607 [55 6 948874 235 998277 1.9 050597 237 049403 08889 99604 54 7 950287 235 998266 1.9 952021 237 047979 08918 99602 53 8 951696 234 99825 1.9 953441 23 0465659 08947 99599 52 9 953100 233 998248 1'9 954856 235 045144 08976 99596 51 10 954499 232 998232 19 956267 234 043733 09005 99594 50i 11 8.955894 232 9.998220 1,9 8.957674 234 11.042326 09034 99591 49 12 957284 2231 998209 1'9 959075 233 040925 09063 99588 48 13 958670 230 998197 1 960473 23 039527 09092 99586 47 14 960052 20.998186 19 961866 232 038134 09121199583 46 15 961429 229 998174 19 963255 231 0367456 09150199580 45 16 962801 228 998163 1.9 964639 231 035361 09179 99578 44 17 964170 227 998151 1'9 966019 230 033981 09208 99575 43 18 965534 227 098139 20 967394 229 032606 09237 99572 42 19 966893 226 998128 20 968766 -229 031234 09266 99570 41 20 968249 998116 20 970133 227 029867 09295199567 40 21 8.969600 224 9.998104 2' 8.971496 22 t11.028504 09324199564 39 22 970947 224 998092 2 0 972855 226 027145 09353 99562 38 23 972289 223 998080 2.0 974209 226 025791 09382199569 37 24 973628 222 998068 20 J 975560 225 024440 09411 99556 36 25 974962 222 998056 2|0 976906 224 023094 09440199553 35 26 976293 22 998044 2 0 978248 2 021752 09469199551 34 7 97 21 2. 6 223 J27 977619 221 998032 20 979586 222 020414 09498199548 33 28 978941 220 998020.0 980921 222 019079 09527199545 32 2 9 20 22.10 222 29 980259 219 998008. 982251 221 017749 09556 99542 31 30 981573 218 997996 2.0 983577 22 016423 10958599540 30 31 8.982883 8 9.997984 2 8.984899 2 11.015101 09614199537 29 32 984189 217 997972 2:0 986217 219 013783 09642199534 28 33 985491 2 9 27959 987532 219 012468 109671j99531 27 34 986789 916 997947 6.20 988842 218 011158 1 09700199528 26 35 988083 21 997935 990149 918 009851 1,097291995261 25 36 989374 214 97922 21'991451 216 008549 067581995231 24 37 990660 214 997910 21 992750 216 007250 09787J995201 23 38 991943 213 997897 2.1 994045 215 005955 098161995171 22 39 993222 22 9978835 99337 2 004663li0984519951421 40 994497 2 997872 996624 003376 09874!995111 20 41 8.995768 211 9.997860 2 1 8.997908 214 11.002092 09903199508 19 42 997036 211 997847. 1 999188 M 000812 109932t99506 18 43 998299 210 997835 2.1 9.000465 213 10.9995351109961199503 17 44 999560 2 997822 2.1 001738 12 998262 09990 99500 16 45.000816 09 997809 003007 211 9969931 10019199497 15 46 002069o 997797 004272 2 957281 1004899494 14 208 2.1 210 47, 003318 08 997784 2' 005534 21 994466 1 10077 99491 13 48 004563 207 997771 2.1 006792 209 9932081 10106[99488 12 49. 005805 20 997758 1 008047 991953 10135[99485 11 650 007044 206 997745 2.1 009298 208 99070211101641994821 10 51 [.008278 205 9.997732 9"1 9.010546 21 101989454 10192 99479 9 52 009510 205 997719 1 011790 98810 1022199476 8 2055 2,1 010207 63 010737J. 997706 013031 686969 I10250 99473 7 6541 011962 997693 0142680 985732 1110279199470 6 I 3 89 206 55J 013182 203 997680 22.2 015502 0205 984498 10308]99467 5 56 04400 2 097667 22 016732 4 983268 10337 99464 4 57! 015613 202 997654 22 017959 204 983041 110366]99461 3 568 916824 201 997641 2.2 019183 0 980817 10395199458 2 "9 201 2 2 203 9 59 018031 201 997628 2.2 020403 203 979597 10424l994551 1 60 019235 997614 021620 978380 20 1045319942 | Cosine. Sine. I Cntang ng. _ _;ln!.; i N. os.!N.sioe. 81 DIgrees. ___ TABLE II. Log. Sines and Tangents. (6c) Natural Sines. 27 I Sine. D. 10j" Cosine. D. 10"1 Tang. iD.10'1 Cotang. N. sine.lN. cos. 0 9.019235 00 9.997614 292 )9.021620 2 10.9783801 1045399452 60 1 020435 1 997601 22 022834 20 977166 10482 99449 9 2 021632 199 997588 2-2 024044 201 975956 10511 99446 58 3 022825 198 997674 2 2 ()25251 20 974749 10540 99443 57 4 024016 198 997561 2.2 026455 200 973545 10569 99440 56 5 025203 97 997547 2.2 027655 199 972345 10597 99437 55 6 026386 197 997534 3 028852 199 971148 10626 99434 54 7 027567 196 997620 23 030046 198 969954 10655 99431 53 8 028744 196 997507 2.3 031237 198 968763 I 10684,99428 52 9 029918 19 997493 2 032425 197 967575 1071399424 51 10 031089 195 997480 2-3 033609 197 966391 10742 99421 50 11 9.032257 194 9.997466 2.3 9.034791 196 10.965209 10771 99418 49 12 033421 194 997452 2 3 035969 196 964031 10800 99415 48 13 034582 193 997439 2 3 037144 195 9628561 10829 99412 47 14 035741 192 997425 2. 038316 196 961684 1/0858 99409 46 15 036896 192 997411 2 3 039485 194 960515 10887 99406 45 16 038048 1 997397 23 040651 194 959349 10916 99402 44 17 039197 191 997383 2:3 041813 193 958187 10945 99399 43 18 040342 190 997369 2 3 042973 193 957027 110973 99396 142 19 041485 190 997355. 044130 190 955870 11002 99393 41 20 042625 189 997341 2 3 045284 192 954716 11031 99390 40 21 9.043762 189 9.997327 2-4 9.046434 191 10.953566 1106099386 39 22 044895 180 997313 2 4 047582 191 952418 11089 99383. 38 23 046026 188 997299 2.4 048727 190 951273 11118 99380 37 24 047164 187 997285 2.4 049869 190 950131 11147 99377 36 25 048279 187 997271 2 *4 051008 189 948992 11176 99374 35 26 049400 186 997257.4 052144 189 947856 11205 99370 34 27 050519 186 997242 24 053277 188 946723 11234 99367 33 28 051635 185 97228 24 054407 188 945593 11263 99364 32 29 052749 185 997214 24 055535 187 944465 11291 99360131 30 053859 184 997199 2:4 056659 187 943341 11320 99357 30 31 9.054966 184 9,997186 2.4 9.057781 186 10.942219 11349 99354 29 3-2 056071 184 997170.4 058900 186 941100 11378 99351 28 33 057172 183 997156 24 060016 185 939984 11407 99347 27 34 058271 183 997141 2:4 061130 185 938870 11436 99344 26 35 059367 182 997127 2.4 062240 185 937760 11465 99341 25 36 060460 182 997112 2 4 063348 184 936652 11494 99337 24 37 061551 181 997098.4 064453 184 935547 11523 99334 23 38 062639 181 997083 25 065556 183 934444 11552 99331 22 39 063724 180 997068 2.5 066655 183 933345 11580 99327 21 40 064806 180 997053 2.5 067752 182 932248 1160999324 20 41 9.065885 179 9.997039 25 9.068846 182 60.931154 11638199320119 42 066962 179 997024 25 069038 181 930062 11667 99317 18 43 068036 179 997009 25 071027 181 928973 111696 99314 17 44 069107 178 996994 5 072113 181 927887 11725 99310 16 45 070176 178 996979 073197 180 926803 11754 99307 15 46 071242 177 996964 074278 180 925722 11783 99303 14 47 07,2306 177 996949 5 076356 179 924644 i 11812 99300 13 48 073366 176 996934 2 076432 179 923568 1184099297 12 49 074424 176 996919 2 5 077605 178 922495 1186999293 11 50 075480 5 996904. 078576 178 921424 11898i99290 10 9076633 175 9.996889 2.5 9.079644 178 10.9203561111927'99286 9 512 077583 175 996874 80710 177 919290 11956199283 8 53 078631 174 99685 8 2 081773 177 918227 11985199279 7 54 079676 1f4 996843 2-'5 082833 176 917167 12014 99276 6 56 080719 17 9968287 17 083891 176 916109 12043 99272 5 56 081759 3 996812 2' 084947 175 915053 1207199269 4 57 082797 12 996797 26 086000 175 914000 12100 99265 3 58 083832 172 996782 6 087050 175 912950 112129 99262 2 59 084864 172 996766 2.6 088098 174 911902 112158 99258 1 60 085894 996751 2 6 089144 910856 112187992551 0 Cosine. Sine. Cotang. Tang. N. cos.iN.sine. 83 Degrees. 9,2 28 Log. Sines and Tangents. (70) Natural Sines. TABLE II. __ Sine..' Cosine. D.10" a. 1 l otang. N.sine. N. cos. 0 9.085894 171 9.996751 2.6 9.089144 174 10.910856 12187 9925 60 1 086922 171 996735.6 090187 173 909813 12216 99251 59 2 087947 170 996720 2. 091228 173 908772 12245 99248 58 3 088970 170 996704 26 092266 173 9077341 12274 9944 57 4 089990 170 996688 26 093302 172 9066981 12302 99240 56 5 091008 169 996673.6 094336 172 905664 12331:99237 55 6 092024 169 996657 2.6 095367 171 904633 12360199233 54 7 093037 168 996641 2. 096395 171 903605 12389 99230 53 8 094047 168 996625 2.6 097422 171 902578 1241899226 52 9 095056 168 996610 2.6 098446 170 901554 12447199222 51 10 096062 167 996594 2.6 099468 170 900532 12476 99219 50 11 9.09706o 167 9.996578 27 9.100487 169 10.899513 12504199215 49 12 098066 166 996562 27 101504 169 898496 12533199211 48 13 099065 166 996546 2.7 102519 169 897481 12662 99208 47 14 100062 166 996530 2.7 103532 168 896468 12591 99204 46 15 101056 165 996514 2.7 104542 168 895458 12620 99200 45 16 102048 16 996498 27 105550 894450 12649 99197 44 17 103037 164 996482 2 7 106556 167 893444!12678 99193 43 18 104025 164 996465 2.7 107559 167 892441 12706 99189 42 19 105010 164 996449 2 7 108560 166 891440 12735 99186 41 20 105992 163 996433 2' 7 109659 166 890441 12764 99182 40 21 9.106973 163 9.996417 27 9.110556 166 10.889444 12793 99178 39 22 107951 163 996400.7 111551 165 888449 12822 99175 38 23 108927 162 996384 27 112543 165 887457 12851 99171 37 24 109901 162 996368'27 113533 165 886467 1288099167 36 25 110873 162 996351 2. 114521 164 885479 12908 99163 35 26 111842 1 996335 2.7 115507 164 884493 12937 99160 34 27 112809 16 996318 116491 883509 12966 99156 33 28 113774 161 996302 27 117472 164 882528 12995 99152 32 29 114737 160 996285: 2.8 118452 163 881548 13024 99148 31 30 115698 160 996269 28 119429 162 880571 13053 99144 30 31 9.116656 159 9.996252 2.8 9.120404 162 10.879596 13081 99141 29 32 117613 159 996235 2.8 121377 162 878623 1311099137 28 33 118567 159 996219 28 122348 161 877652 13139199133 27 34 119519 158 996202 2:8 123317 161 876683 13168 99129 26 35 120469 1 8 996185 28 124284 875716 13197 99126 26 36 121417 18 996168 2. 125249 160 874751 13226 99122 24 37 122362 11 9961518'.8 160 37 122362 15 996151 28 126211 160 873789 13254,99118 23 38 123306 7 996134 28 127172 16 872828 1328399114 22 39 124248 167 996117 2.8 128130 1 871870. 13312 99110 21 40 125187 157 996100 2.8 129087 169 870913 13341 99106 20 41 9.126125 1 6 9.996083 28 9.130041 19 10.869959 13370 99102 19 42 127060 156 996066 2 9 130994 158 869006 13399 99098 18 43 127993 156 996049 29 131944 158 868056 13427 99094 117 44 128925 165 996032 2.9 132893 158 867107 13456 99091 16 45 129854 164 996015 2.9 133839 157 866161 13485 99087 15 46 130781 154 995998 2.9 134784 157 866216 13514199083 14 47 131706 154 995980 2-9 135726 157 864274 13543 99079 13 48 132630 m1 995963 29 136667 156 863333 13572 99075 112 49 133551 153 995946 2-9 137605 156 862395 13600 99071 11 50 134470 163 995928 29 138542 156 861458 13629 99067 10 51 9.135387 162 9.996911 929 9.139476 155 10.860524 13658 99063 9 52 136303 162 995894 29 140409 15 859591 13687 99059 8 63 137216 152 995876 2-9 141340 858660 113716 99055 7 54 138128 152 995859 29 142269 154 857731 13744 99051 6 55 139037 151 995841 2.9 143196 154 856804 13773 99047 5 56 139944 151 995823 29 144121 154 855879 13802 99043 4 57 140850 151 996806 29 145044 153 854956 13831 99039 3 58 141754 150 995788 29 145966 153 854034 13860 99035 2 59 142655 150 995771 29 146885153 853115 j 13889 99031 1 150 2.9 9 7 163478 60 143555 995753 147803 852197 13917 99027 0 Cosine. Sine. ne. Cotang. Tang. IN. cosN.sine. 82 Degrees..........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE II. Log. Sines and Tangents. (80) Natural Sines. 29 Sine. D. 10" Cosine. D. 10" Tang. D.10"! Cotang. N. sie. N. cos. 0O9.143555 10 9.995753 9.147803 153 10.852197 13917 99027 60 1 144453 149 995735 30. 148718 152 851282 13946 99023 59 2 145349 995717, 149632 152 850368 13975 99019 58 149 3.0 152 3 146243 149 995699 3:0 150544 152 849456 14004 99015 57 149 319 0 4 147136 995681 151454 848546 14033199011 56 148 3.0 151 f 148026 14 995664 0 152363 151 847637 14061 99006 55 6 148915 148 995646 153269 1 846731 14090 99002 54 7 149802148 995628 3* 151 845826 14119198998 53 I 147 995628 154174 150 8 150686 147 995610 3:0 155077 150 844923 14148 98994 52 197 116 150 9 1.51569 14 995591 3.0 155978 150 844022'14177 98990 151 10 152451 7 995573 3.0 156877 1 843123 1420598986 50 1,[7 8 1 9 50 11 9.153330 14 9 995555 3 9.157775 10.842225 114234 98982 49 12 1 8 146 9955 30 158671 149 841329 114263198978 48 13 155083 13 I 13 155083 146 9966519 3.0 159565 149 840435 14292 98973 47 14 155957 15 99501 160457 839543 1l 432098969 46 15 156830 145 995482 3.1 161347 148 838653 14349 98965 45 16 157700 145 995464 3.1 162236 148 837764 14378 98961 44 17 158569 44 995446 3.1 163123 148 836877 14407 98957 43 184 31594 18 159435 1 995427 3.1 164008 147 835992 14436 98953 42 19 160301 144 995409 3.1 164892 147 835108 14464 98948 41 20 161164 144 995390 3.1 165774 147 834226 14493 98944 40 21 9.162025 143 9 995372 3.1 9.166654 1 10.833346! 14522 98940 39 22 162885 1 995353 9 167532 1 832468 14551 98936 38 23 163743 143 95334 3.1 168409 831591 14580 98931 37 24 164600 143 995316 3.1 169284 146 830716 14608 98927 36 25 165454 142 995297 1 170157 89843 i 14637 98923 35 142 995278 3 26 166307 14 995278 3.1 171029 146 828971 14666I98919 34 27 167159. 142 995260 3.1 171899 145 828101 14695 98914 133 28 168008 141 995241 3.2 172767 827233 14723 98910 32 29 168895622 I 29 168856 141 995222 3.2 173634 144 826366 14752 98906 131 30 169702 141 99520 3.2 174499 144 825501 14781 98902 30 31 9 170547 140 995184 3 2 9175362 14 110.824638 14810 98897 29 32 171389 140 995165 3.2 176224 143 823776 1483898893 28 33 172230 140 995146 2 177084 143 822916 14867 98889 27 34 173070 140 995127 2 17794 3 8220581 14896 98884 26 35 173908 140 995108 3.2 178799 14 8212011 14925 98880 25 139 995089 3.2 178799 142 36 174744 139 995089 3.2 179655 142 820345 114964 98876 24 37 175578 139 995070 3.2 180508 142 819492 i 14982 98871 23 38 176411 -13 995001' * 181360 81864011 15011 98867 22 39 177242 138 99503.2 3.2 182211 141 817789 15040 98863 21 40 178072 138 995013 3.2 183059 141 8169411506998858 20 41 9.178900 138 29994993 32 9.183907 141 10.816093 15097149884 19 42 179726 137 994974 3 2 184752 141 815248 1112698849 18 43 180551 994965 185597 814403 1515598845 17 137 32' 88 140 44 181374 137 994935 3 186439 140 813561 15184 98841 16 137 9.9491 45 182196 137 994916 3.3. 18728 140 812720 1521298836 1 46 183016 136 994896 3.3 188120 811880 115241 98832 14 47 183834 994877 188958 811042 16270 98827 13 48 184651 136 994857 3 3 189794 139 8102061 15299 98823 12 49 185466 136 994838 3 190629 139 8093711 15327 98818 11 50 186 6 38, 191462 139 50 186280 994818 191462 8085381 1535698814 10 139 3.3 139 1 9.187092 13,5 9.994798 3 9.19294 138 10.807706 15385 988091 9 52 187903 135 994779 3.3 193124 38 806876 15414 988051 8 63 188712 13 994759 33 1939 83 1 806047 15442 98800 7 64 189519 994739 194780 138 805220 15471 98796 6 55 190325 134 994719 3 195606 804394 15500 98791 5:994719 150 13 56 191130 134 994700'33 196430 137 803570 15529 98787 4 57 191933 134 994680 3 197253 17 802747 15557 98782 3 58 192734 134 994660 3.3 198074 13 801926 15586 98778 2 59 193534 133 994640 3 198894 137 801106 15615 98773 1 60 194332 133 994620 3' 199713 136 800287 115643 98769 0 - CFosine. I. ISine. Cotang. Tang. I N. eos.lN.sine 81 Degrees......~~~~~~~~~~~.... 30 Log. Sines and Tangents. (9o) Na:tural Sines. TABLE II. Sine(. D. 10" Cosine. D. 10" Tang. D. 10'" (oting.. sine. N. cos. 0 9.194332 133 9.994620 9.199713 136 0.8002871 1564398769 60 1 195129 133 994600 3.3 200529 136 799471 15672 98764 59 2 195925 133 994580 3'3 201345 136 798655 15701 98760 58 3 196719 132 994560 3.4 202159 135 797841 15730198755 57 3 19719 0 202971 4 197511 132 994540 3.4 202971 135 797029 15758 98751 56 5 198302 132 994519 3'4 203782 135 796218 15787 98746 55 6 19302 9 9 203 6 199091 1312 994499 3 24 904592 135 795408 15816 98741 54 7 199879 131 994479 3.4 205400 134 794600 15845 98737 53 8 200666 994459 206207 134 793793 15873 98732 52 9 201451 131 994438 334 207013 134 792987 15902 98728 51 10 202234 130 994418 3.4 207817 134 792183 15931 98723 50 11 9.203017 130 9.994397. 9.208619 133 10.791381 15959 98718 49 12 203797 130 994377 3.4 209420 133 790580 15988 98714 48 13 204577 130 994357 3-4 210220 133 789780 16017198709 47 14 205354 129 994336 3.4 211018 133 788982 16046 98704 46 15 206131 129 994316 3.4 211815 133 7881805 16074 98700 45 16 206906 129 994295 212611 132 787389 16103 98695 44 17 207679 129 994274 3. 213405 132 786595 16132998690 43 18 208452 128 994254 3.6 214198 132 785802 16160 98686 42 19 209222 128 994233 3 214989 132 735011 16189 98681 41 20 209992 128 994212 3- 215780 131 784220 16218198676 40 21 9.210760 128 9.994191. 9.216568 131 10.783432 16246 98671 39 22 211526 127 994171 3. 21736 131 782644 16275198667 38 23 212291 127 994150 3'6 218142 131 781858 16304 98662 37 24 213055 127 994129 3.5 218926 130 781074 16333198657 36 25 213818 127 994108 3 5 219710 130 780290 16361 98652 35 26 214579 127 994087 3 220492 130 779508 16390 98648 34 27 215338 126 994066 32 221272 130 778728 16419 98643 33 28 216097 126 994045 5 222052 130 777948 16447 98638 32 29 216854 126 994024 3. 222830 129 7771701 16476 98633 31 30 217609 126 994003 3 223606 129 776394 16505 98629 30 31 9.218363 125 9.993981 5 9.224382 129 10.775618 16533 98624'29 32 219116 125 993960 3/5 225156 129 7748441 1656298619 28 33 219868 125 993939 3.5 225929 129 774071 16591 98614 927 34 220618 125 993918 3'6 226700 128 773300 1662098609 26 35 221367 125 993896 36 227471 128 772529 1664898604 125 36 222115 124 993875 3.6 228239 128 771761 16677198600 24 37 222861 124 993854 3.6 229007 128 770993 16706J98595 23 38 223606 124 993832 3'6 229773 127 770227 16734 98590 22 39 224349 1 993811 36 230539 127 769461 11676398585 121 40 225092 124 993789 3.6 231302 127 768698 1116792 98580 I 20 41 9.225833 123 9.993768 6 9.232065[ 127 10.767935 116820198575 19 42 226573 123 993746 3.6 232826 127 767174 16849 98570 18 43 227311 123 993725 3.6 233586 126 7664141 167898565 17 44 228048 123 993703 3:6 234345 126 765655 16906898561 16 45 228784 123 993681 3.6 235103 126 764897 1693598556 15 46 229618 122 993660 3.6 235859 126 764141 16964398551 14 47 230252 122 993638 3'6 236614 126 763386 1699298546 13 48 230984 122 993616 3.6 237368 125 762632 17021198541 12 49 231714 122 993594 3. 238120 125 7618801 17050|98536 11 50 232444 122 993572 3.7 238872 125 761128 17078!98531 10 51 9.233172 121 9.993550 3.7 9.239622 125 10.760378 17107198526 9 52 233899 101 994528 240371 125 759629 17136 98521' 8 53 234625 21 993506 3.7 241118 1 ]24 758882 17164198516 7 54 235349 121 993484 241865 124 758135 17193J98511 6 55 236073 120 993462 2.7 9 42610 124 757 90 17222198506 6 56 236795 120 993440 243354 1 4 756646 17260198501 4 120 9341. 124 57 23751620 993418 3.7 244097 1124 755903 17279198496 3 58 238235 12 993396 244839 123 755161 17308 98491 2 59 238953 120 993374 245579 123 754421 17336J98486 1 60 239870 11 993351 246319 753681 17365 98481 0 Cosine. Sine. Cotang. Tang. N. cos. N.ine 80 Degrees. TABLE II. Log. Sines and Tangents. (10~) Natural Sines. *1 Sine. D. 10" Cosine. D.1O0" Tang. D. 10" Cotang4..ine.1N. cos. 0 9.239670 9.993351 37 9.246319 123 10.763681 17365 98481 60 1 240386 1 9 993329'7 247057 123 752943 17393 98476 59 2411011119 993329 247 7 123 2 241101 119 993307 8 7 247794 123 752206 17422198471 58 3 241814 119 993285 7 248530 122 751470 17451198466 657 4 242526 118 993262 3.7 249264 122 750736 17479198461 56 6 243237 118 993240 249998 1'2 750002 17508 98455 55 6 243947 118 993217 3.8 250730 192 749270 17537 98450 54 ]7 244656 1 993195 3 8 251461 122 748539 17565 98445 53 8 245363 118 993172 3 8 252191 122 747809 17594198440 52 9 246069 111 993149 3. 252920 12 747080 17623 98435 51 10 2677 17 993127 53648 121 746352 1765198430 50 1 11 9.247478 11 9.993104 3.8 9.254374121 10.745626 17680 98425 49 12 2481811 17 993081 3.8 255100 121 744900 17708 98420 48 13 248883 11 30 2558 24 13 248883 117 993039 3.8 255824 121 744176 17737 98414 47 14 524983 11 993036 3 8 256547 120 743453 17766 98409 46 15 250282 116 993013 257269 120 742731 17794 98404 45 16 250980 116 992990 3.8 257990 120 742010 17823 98399 44 17 2516777 16 992967 3.8 258710 741290 17852 98394 43 1i 252373 116 99296' 120 18 252373 116 992944 3 8 259429 120 740571 17880198389 42 19 253067 116 992921 3.8 260146 120 739854 17909 98383 41 20 253761 116 992898 3.8 260863 119 739137 17937198378 40 21 9.254453 115 9.992875 3.8 9.261578 119 10.738422 17966 98373 39 22 255144 115 992852 3'8 262292 119 737708 17995198368 38 23 255834 115 992829. 263005 119 736995 18023 98362 37 24 256523 115 992806 2 63717 1819 736283 18052198357 36 115 3.9 118 25 257211 114 992783 264428 118 735572 18081198352 35 26 257898 114 992759 3 265138 118 734862 18109198347 84 27 2585831 992736 265847 118 734153 18138 98341 33'28 259268 114 992713 3.9 266555 118 733445 18166 98336 32 29 259951 114 992690 3'9 267261 118 732739 18195 98331 31 30 260633 113 992666 3.9 267967 17 732033 18224198325 30 31 9.261314 113 9.992643.9.268671 1 10.731329 18252198320 29 32 261994 113 992619 269375 117 730625 18281 98315 128 33 262673 113 992596 3. 270077 1 729923 18309 98310 27 34 263351 113 992572 3.9 270779 117 7292211 18338 98304 26 35 264027 113 992549 3'9 271479 116 728521 i8367 98299 25 36 264703 12 992525 3' 2721781 727822 18395 98294 24 37 265377 12 9925012 2 72876 116 727124 18424198288 23 113 3 9 116 38 266051 112 992478 4.' 273573 116 726427 18452 98283 22 3 9 266723 1 1 992454 4.0 2 74269 116 726731 1848198277 2 21 ll~ 40 116 40 267395 - 112 992430 4'- 274964 116 725036 18509198272 20 112 40 116 1 41 9.268065 11 9.992406 4 9.275658 10.724342/ 18538 98267 19 42 268734 l11 992382 4. 276351 115 723649 18567198261 118 43 26940211 I 992359 4'0 277043 11i 722957 18595 98256 17 44 270069 l l 992335 4' 0 277734 115 722266]18624198250 16 45 2707351 l 992311 4.1 278424 115 721576I 18652 98245 16 46 271400 1il 992287 4.0 279113 115 720887 18681198240 14 47 272064 11 9922963, 4.0 279801 1 4 720199 18710198234 13 48 272726 110 992239 4'0 280488 114 719512 18738198229 12 49 273388 110 992214 4'2 281174 11 718826 18767198223 11 50 274049 110 992190 4 | 281858 114 718142 18795 98218 10 51 9, 274708 1109.992166 4 0 9.282542 114 10.717458 18824198212 9 52 2743670 110 992142 4'0 283225 114 716775 18852198207 8 53 276024 1 992117 4'1 83907 716093 18881198201 7 109 41 113 54 276681 - 109 992093 4' 284588 113 715412 18910198196 6 65 2773371 109 992069 4.1 285268 113 714732 18938198190 5 56 277991 109 992044 41 285947 113 714053 18967198185 4 57 278644 1 09 992020 41 286624 113 713376 I 18995198179 3 58 279297 109 991996 4.1 287301 113 712699 19024 98174 2 59 279948 18 991971 1 287977 112 712023 19052 98168 1 60 280599 991947 288652 711348 19081198163 0 Cosine. I I Sine. Cotang. Tang. N. cos. N.ine 79 Degrees. 32 Log. Sines and Tangents. (110) Natural Sines. TABLE II.'T Sine. DiSS" C' osine. D. Tang o;. l, Cotaung. IN. sine. N. cos. 09.280599 108 9.991947 41 9.288652 112 10.711348 19081 98163 60 1 281248 991922 289326 710674 19109 98167 59 2 281897 108 991897 4.1 289999 112 710001 19138 98152 58 3 282544 108 991873 4 290671 709329 19167 98146 57 4 283190 991848 2913421 708658 19195 98140 56 18 23 292013 112 6 283836 107 991823 4.1 292013 707987 19224 98135 55 6 284480 991799 4.1 292682 70i318 19252 98129 54 7 285124 991774 293360 706650 19281 98124 53 8 28576(6 991749 4.2 294017 705983 19309 98118 59 9 286408 991724 294684 111 705316 19338 98112 51 10 287048 991699 295349 704651 19366 98107 50 107 4.2 6 13 1 11 9.287687 9.991674 4. 296013 10.703987 19395198101 49 12 288326 991649 296677 703323 19423 98096 48 13 288964 991624 4.2 297339 110 702661 19452 98090 47 14 289600 108 991599 4.2 298001 110 701999 19481 98084 46 15 290236 991574 298662 710 01338 19509 98079 45 16 290870 106 991549 299322 700678 19538 98073 44 17 291604 10 991524 299980 700020 19566 98067 43 18 292187 105 991498 4.2 300638 109 699362 19595 98061 42 19 292768 991473 301295 1069 98706 19623 98056 41 20 293399 105 991448 4.2 301951 109 698049 19652 98050 40 21 9.294029 05 9.991422 4.2 9.302607 109 10-697393 19680 98044 39 22 294658 991397 303261 696739 19709 98039 38 I 105 9 4.2 109 23 2956286 4991372 303914 109 696086 19737 98033 37 24 295913 104 991346 4.3 304567 109 696433 19766 98027 36 25 296539 991321 305218 694782 19794 98021 35 26 297164 104 991295 305869 108 694131 19823 98016 34 27 297788 104 991270 306519 108 693481 19851 98010 33 28 298412 104 991244 307168 692832 19880 98004 32 29 299034 991218 307815 692185 19908 97998 31 104 4.3 3084 108 691 30 299655 991193 3 308463 691537 19937 97992 30 31 9.300276 03 9,991167 9.309109 07 10.690891 19965 97987 29 32 300895 103 991141 309754 690246 19994 97981 28 33 301614 103 991115 310398 689602 20022 97975 27 34 302132 103 991090 311042 688958 20051 97969 26 356 302748 103 991064 311685 107 688315 20079 97963 25 36 303364 10 991038 4,3 312327 687673 20108 97968 24 37 303979 991012 312967 687033 20136 97952 23 38 304593 102 990986 313608 10 86392 20165 97946 22 39 305207 102 990960 4 314247 685753 20193 97940 21 102 4.3 108 40! 305819 102 990934 314885 685115 20222 97934 20 41 9.306430 102 9.990908 9.315523 1 10,684477 20250 97928 19 42 307041 102 990882 4 316169 108 683841 20279 97922 18 43 307650 11 990855 316795 683205 20307 97916 17 44 308259 990829 317430 106 682570 20336 97910 16 45 308867 101 990803 318064 1 681936 20364 97905 15 46 309474 101 990777 318697 681303 20393197899 14 47 310080 101 990750 319329 105 680671 20421197893 13 48 310685 101 990724 4 319961 105 680039 20450197887 12 49 311289 101 990697 320592 105 679408 204781971 11 50 311893 100 990671 321222 106 678778 20507978765 10 51 9.312495 100 9.990644 9.321861 10.67814911 20535 97869 9 E2 313097 100 990618 322479 104 677521 20563 97863 8 53 313698 990591 323106 676894 20592197857 7 54 314297 990565 323733 676267 20620197851 6 55 314897 100 990538 324368 676642 20649 97845 5 56 316495 100 990511 324983 104 675017 20677 97839 4 57 316092 990485 325607 104 674393 20706 97833 3 58 316689 990458 4. 326231 673769 j20734.97827 2 59 317284 99 990431 326853 673147 20763197821 1 60 317879 990404 327476 104 672525 20791197815 0 Cosine. -- I Sine. Cotang. Tang. N. cos. Nl.ne. 78 Degrees. TABLE II. Log. Sines and Tangents. (120) Natural Sines. 33' Sine. D.10" Cosine..1' Tang. Dlul Cotang. N. sine. cos. 0 9.317879 09.990404 4 6 9.327474 103 10.672526 20791 97815 60 1 318473 98 990378 4-5 328095 103 671905 20820!97809 59 2 319066 990351. 328715 103 671285 20848'97803 58 3 319658 98.7 990324 5 329334 670666 2087797797 57 4 320249 98.6 990297 4.5 32995334 3 670047 2090597791 56 s 320840 983 990270 330570 1 669430 20933 97784 54 98.3 4.5 103 6 321430 9. 990243 4. 331187 03 668813 2096297778 54 7 322019 98.2 990215 4. 331803 103 668197 20990 97772 53 8 322607 980 990188 332418 102 667682 21019197766 62 9 32319497 990161 45 333033 102 66967 21047197760 51 10 323780 97:6 990134 4 333646 66635 2107697754 50 11 9.324366 9.990107 6 9.334259 10.665741 21104!97748 49 12 324950 990079 4. 334871 102 665129 2113297742 48 13 326534 990052 46 335482 664518 21161197736 47 97.2 4.6 102' 1 4 326117 97.0 990025 4:6 336093 1 663907 21189197729 46 15 326700 96 9 989997 4 6 336702 10 663298 21218517723 45 16 327281 8 989970 46 337311 1 662689 21246197717 44 17 327862 96. 989942 4'6 337919 101 662081 21275 197711 43 18 328442 96.6 989915 4.6 338627 01 661473 21303197705 42 19 329021 9' 989887 4.6 339133 101 660867 21331 97698 41 96.4 4.6 101 20 329599 989860 339739 660261 21360297692 40 21 9.330176 96.2 9.989832 46 9.340344 01 10.659656 21388197686 39 22 330753 96.1 989804 46 34094801 659052 21417697680 38 23 331329 96.0 989777 46 341552 101 658448 21445197673 37 24 331903 95.8 989749 4.6 34155 100 657845 21474197667 36 25 332478 9 989721 7 342757 100 657243 21502197661 35 28 334196 5.3 989637 4.7 34458 100 65442 21687 97642 32 96 47. 101 29 334766 95.2 989609 47 34517 100 654843 21616 97636 31 30 335337 95*0989582 4.7 346765 100 664246 21644 97630 30 31 9.335906 94.9 9.989553 7 9.346353 00 10.653647 2167297623 29 32 336475 94.8 989525 4.7 346949 99 653051 21701 97617 28 333337043 94.6 989497 34745 652455 21729 97611 27 945 4 3481417 987604 34 337610 94. 989469 4.7 34814191 651859 21758 97604 26 5 338176 94 989441 348735 99. 651265 21786 9798 25 36 338742 94 989413 7 349329 99.0 60671 21814 97592 24 37 339306 941 989384 349922 98.8 650078 21843 9785 23 38 339871 4. 989356 47 350514 98.7 649486 21871 97679 22 39 340434 93 989328 351106 98.6 648894 2189997573 21 40 34996 3 989300 351697 98.5 648303 21928 97566 20 41 9.341558 9.989271 7.352287 98.3 10.647713 21956 9760 19 93.6 4.7 - 98.2 42 342119 93.4 989243 7 362876 1 647124 21985 9753 18 43 342679 932 989214 47 353465 980 646535 12201397547 17 44 343239 3. 989186 47 35405 97 645947 122041197641 16 45 343797 3.0 989157 4'7 354640 97 645360 22070497534 15 46 3443596 9.9 989128 48 355227 976 644773 22098197628 14 47 344912 927 989100 48 355813 976 644187 22126 97521 13 48 345469 926 989071 48 356398 974 643602 2215597515 12 49 346024 9 989042 4"8 356982 97 643018 2218397508 11 50 346679 989014 357566 971 642434 2221297502 10 51.347134 92.1 9.988985 4.8 9.358149 97010.641861 22240 97496 9 52 347687 92. 988956 48 358731 96 641269 1226897489 8 92.1 4.8 96.9tl 3 348240 9 988927 4. 359313 96 640687112229797483 7 54 348792 92. 988898 4.8 39893 96-7 6401071 2232597476 6 55 349343 91.9 988869 4.8 360474 966 639526 22353197470 5 - 56 34 3 988840 4 8 361053 96 6 68948 2238297463 4 57 330443 91.6 988811 4-9 361632 96 638368122410 3 58 350992 91.4 988782 49 362210 96 637790'2243897450 69 351540 91. 988753 4 9 362787 961 637213 22467197444 1 60 352088 91.3 988724 363364 636636 22495 97437 0 Cosine. Sine. Cotang. Tang.! N. cos.N.sine. I 77 Degrees. 44! 3429l. _.[ 816._. {345l,, ~ 497 1 34 Log. Sines and Tangents. (130) Natural Sines. TABLE II. Sine. D. 10' Cosine. D. 10' Tang. D. 10/'1 Cotang. N.sine N. cos. 9.352088 911 9.988724 4 9 9.363364 960 10.636636 22495 97437 60 1 352635 91 0 988695' 363940 96 636060 22523 97430 69 2 353181 9.0 988666 4.9 364515 95.8 635485 22552 97424 58 3 353726 90.8 988636 4 365090 95.8 634910 22580 97417 57 4 354271 90.7 988607 4.9 365664 95.7 634336 22608 97411 56 354815 90.5 988578 4.9 366237 95. 633763 2263797404 6 355358 9 4 988548 4'9 366810 95.4 633190 22665 97398 54 7 355901 904 988519 49 367382 95.3 632618!a693 97391 53 8 356443 902 988489 49 3679 95.2 632047 22722 97384 52 9 356984 90.1 988460 4.9 368524 95. 631476 22750 97378 51 10 357524 01 988430 4 9 369094 95.0 630906 22778 97371 50 11 9.358064 89.8 9.988401 9 9.369663 94. 10.630337 2807 97365 49 12 358603 89.7 988371 4.9 370232 94. 629768 22835 97368 48 13 359141 896 988342 49 370799 94.6 629201 22863 7351 47 14 359678 89.' 988312 5-0 371367 94.5 628633 22892 97345 15 360215 89. 988282 s0 371933 94.4 628067 22920 97338 45 16 360752 89/ 2 988252' 0 372499 94.3 627501 22948 97331 44 17 361287 89. 988223' 0 378064 94.2 626936 2297797325 43 18 361822 890 988193' 373629 94.1 626371 2300597318 42 19 362356 88.9 988163 50 374193 94.0 626807 23033 97311 41 20 36889 988133 374766 93.9 625244 23062 97304 40 21 9.363422 88.8 59988103 0 9.375319 93.8 10.624681 2309097298 39 9.863422 8:7 9.988103' 9.376319 22 363954 88.6 988073 6 0 375881 93. 64119 23118 9791 38 23 364485 88'4 988043 376442 93.4 623558 2146 97284 37 24 365016 88.3 988013!'0 377003 93. 622997 23176 97781 36 25 366546 88'2 987983 5'0 377563 93.3 622437 2320397271 35 26 366075 88'1 987953'0 378122 93.2 621878 23231 97264 34 27 [366604 88. 987922 0 378681 93.1 6213191 23260 9725733 28 367131 87.9 987892 5.0 379239 92.9 620761 23288 97251 32 29 367659 877 987862 [.0 379797 92.8 620203 123316 97244 31 30 368185 876 987832 5'1 380354 92.7 619646 213345 97237 30 31' 9.368711 87' 19.987801 5'1 9.380910 92.7 10.619090 23373 97230 29 32 369236 87'4 987771 5.1 381466 92.6 618534 923401 97223 28 33 369761 873 987740 51 382020 926 617980 1 23429 97217 127 34 370286 87'2 987710 6 1 382/575 92. 6174251 2345897210 26 35 370808 8721 987679 5.1 383129 92.3 616871 123486197203 26 36 371330 *870 987649./1 383682 92.2 616318 23514 97196 24 37 371852 8769 987618 6 1 384234 93.1 615766 123542 97189 23 38 372373 86'7 987588 6' 384786 9109 615214 23671 97182 39 372894 8676 987557. 386337 91.9 614663 23599 97176 121 40 373414 865' 987626.1 385888 91 614112 112362797169 20 41 9.3 73)83 86.4 9.987496 61 9.386438 91. 10.613562 23666 97162 19 42 374s.L 86.3 987465 6.1 386987 91.4 613013 2368497165 18 43 374970 86.2 987434.1 38736 91.3 612464 2371297148 17 44 3754871861 9874036 6.2 388084 91.2 611916 23740197141 16 45 376003 86.0 987372 5.2 388631 91.1 611369123769 971341i 46 376519 869 987341 389178 91. 610822 23797197127 14 47 37703|5 987310 5.2 *389724 9 610276 123825 97120 13 48 377549 85 987279 5.2 390270 90.8 609730 123853 97113 12 49 378063 85'6 987248 5.2 390815 90.8 609185 23882197106 11 50 378677 85'4 9872.17 6 391360 90.7 608640 23910197100110 51 9.379089 853 9.987186 6'2 9.391903 90.6 10.608097 23938197093 9 52 379601 85-2 9871566.2 392447 90.6 607653 23966 97086 8 53 380113 861 987124 52 392989 90.~ 607011 23995 97079 7 54 380624 85_ 9870921.2 393531 90.31 606469 24023 97072 6 56 381134 987061 5'2 394073 90.2 605927 24051 9706S 5 6 381643 849 9870301 52 394614 90.1 605386 24079197058 4 57 382152 84.8 986998 5'2 395154 90.0 604846 24108197051 3 84.7 9.8699 58 382661.7 9869675.1 395694 89.9 604306 24136197044 2 59 383168 84-6 986936 6.2 396233 89.8 603767 24164197037 1 60 383676 45 986904 6.2 396771 89.7 603229 24192 97030 0 Cosine. Sine. I Cotang. TLn.. N.sine.. 76 Degrees. TABLE II. Log. Sines and Tangents. (140) Natural Sines. 35 L Sine. Di. o"' Cosine. D. 1(Y' Tang.'D. 10 Cotang. IN. sine. N.cos.0 9.383675 84 9.986904 9. 39f771 896 10.603229 24192197030 60 1 384182 843 986873 6.3 397309 896 602691 24220 97023 69 2 384687 842 986841 397846 895 602154 2424997015 58 3 385192 81'1 986809.3 398383 601617 124277 97008 57 4 385697 810 986778 398919 89 601081 24305 97001 56 6 386201 83.9 986746 63 399455 89 2 600545 2433396994 655 6 386704 986714 399990 600010 24362 96987 64 7 387207 837 986683 400624 89 599476 2439096980 63 8 387709 3.6 986661.3 401058 89. 598942 24418196973 52 9 388210 83 986619.3 401691 8898 598409 24446 96966 51 10 388711 8354 986587 402124 88:7 597876 24474196959 60 83.4 5.3 88.7 11 9.389211 83. 9.98656 402656 88 6 10.597344 1'24503 96952 49 12 389711 832 986523 403187 885 596813 24531 96945 48 13 39O210 831 986491 403718 884 696282 2455996937 47 14 390708 3. 986459 404249 88. 695751 24587 96930 46 15: 391206 8 986427 5. 404778 88.2 595222 24615196923 465 16 391703 82798639 405308 1 94692 24644 96916 44 1 7 392199 82*6 986363 * 4 06836 88.O o94164 24672 96909 43 18 392695 825 986331 406364 8. 93636 2470 96902 42 19 393191 24 986299 406892 87.8 593108 2472896894 41 20 39368 2 986266 407419 87 592581 2475696887 40 82.3 5.4 87.7 10.592055 24784'96880 89 21 9,394179 82.9896234 4 7946 87.6 2478496880 9 22 394673 1 986202 i 4 408471 87.6 591529 24813 96873 38 23 3945166 82: 986169 564 408997 87.4 691003 2484196866 37 2. 395166 9 86169 408997 24 3956568 8. 986137.4 409521 87.4 590479 24869 96858 36 25 396160 81.8 986104 410045 87. 589955 248979681 35 26 396641 81.7 986072 5.4 410669 87.3 589431 24925696844 34 27 397132 817 986039 4 411092 87.12 88908 2495496837 33 28 397621 816 986007 411616 87.O 58838 24982 96829 32 29 398111 81.S 986974 5:4 412137 87.0 587863 2501 96822 31 30 398600 814 986942 6 412658 86.9 87342 2503896815 30 31 9 399088 9,985909.413179 868 10.5882 25066 6807 29 32 399575 81.3 985876 413699 86:7 586301 25094 96800 28 81.2 4.6 86.6 33 400062 986843 5G8 414219 68781 25122 967938 27 4147386 8'.8 585262 25151986 26 34 400549 986811 54 414738 25151 96786 26 35 40103 8 986778 5. 417 86.4 84743 25179 6778 25 36 401520 80 98574 5. 415776 86.4 584225 25207 96771 24 37 402005 8 985712 5 416293 86' 583707 25235 96764 23 38 402489 80-7 986679:. 6 416810 86 1 583190 25263 96766 22 39 402972 80.6 986646 685 417326 86O 682674 25291 96749 21 40 403465 806 986613. 417842 86.' 58218 125320 96742 20 41 9.403938 803 (9.9856580 5.6 9418358 85.8 10.581642 25348 96734 19 42 404420 802 985547 418873 8042 44 0 8.~7 581127 i2537696727 18 43 404901 801 985514 6 419387 86. 580613 26404 96719 17 44 406382 800 6986480 6.5 419901 86.6 680099 26432 96712 16 45 405862 986447 * 420415 85. 679585 26460 96705 16 46 406341 79*8 986414 5 6 420927.'4 679073 2548896697 14 47 406820 986380 421440. 578560 25516:96690 13 797 546 585. )3,5545o96682 12 48 407299 77 986347 6 421952 86. 678048 254696682 12 49 4077717796 985314 6 422463 8.2 f 77537 25731 96675 11 60 40254 796 985280 6 422974 5. 77026 2560196667 10 61 9,4082731 4794 9986247 56 9.423484 86.0 10.576516 25629 96660 9 62 409207 79:4 986213 5 6 423993 84. 676007 2567 9663 8 2 49 7 9.3 6 4 84.8 63 409682 792 985180 5.6 424650 11 84. 67497 25686 9664 7 64 410157 79:1 985146 5 6 4269011 84 674989 125713496638 6 66 410632 79.0 986113: 66 426219 84.6 574481 2574196630 5 66 411106 789 985079.6 42602734 84. 673733 2576696623 4 67 411579 78.8 986046 5.6 426534 84.4 573466. 25798 9661 3 68 412052 78 7 985011 6 6 427041 84.3 672959 25826 96608 2 59 412524 786 984978 427547 84. 72463 125854 96600 1 60 412996 984944 428052 6719481 26882s6093 0 Cosine. Sine. Cotang. Tang. N.:co.- in. 75 Degrees. 23 9.412996 7.984944 7 9.42807582 108 671948 926882 96593'60 1 414675 98491 67 428557 671443 25910 9685 59 413938 7 984876 7 429062 84 670938 1253896678'68 3 414408 9783 984842 42966 8. 570434 125966 96570 57 4 41484 78 984808 67 430070 89 669930 2599496562 i56 5 415347 78 984774 43067 83 569427 2,6022 656 55 1 7 983 5.7 83.8 64 4 6 41561 78. 984740 5.7 431075 838 668925 2606096547 54 7 41628 779 984706 5 431577 83 568423 26079 9654053 8 4167 77 984672 57 432079 836 567921 2610q796632 52 9 417217 984637 432680 567420 13615 96224 51 10 4176842 77. 984603 67 4080 83 66920 26163 96517 40 11 9.41810 77 9.9845629 57 9.4380 83-2 10.666420 26191 96609 49 12 418615 77 984636.7 434080 832 65920 26219 96602 4 13 41907 779 984500 5. 434679 566421 26247 96494 47 14 419544 984461 5.8 43 507 8. 564922 267 96486 46 16 429007 72 984432 8 4576 83. 564424 26368 6479 145 6 77.2 94060 8 4082.9 16 420470 776.1 984397 5.8 460734 2.8 663927 16331 96471 i44 17 420933 984368 436570 8 563430 26365996463 43 77.0 5088 82.8 18 429396. 984329 6.8 43 706.7262933 2687 96456 34 19 4187 98494 437963 82.6 662437 26417 96448 41 2 422318 767 984269 6 438085 561941 6 264439640 0.40 21 9.422773.984224.438448 8524 10.561446 26471 96433 39 22 43238 76 984193 81' 64390 48 560962 26600 96424 26 23'4 76,6 9841 6,8 439E 823 0457 2652896417 37.24t 424156 765984120 5.8 440036 8 59964 2656 96410, 36 25 424661,5 76. 984086, 44052629 559471'2658496402 36 26 422073 984050 4410232 8 558978 26612 06394 34 27 4 755~ 98394015 8': 914 668486 26640 9638633 28 4287 7 * 983981' 442006 557994 26668 96379, i32 29 464437 988 946 8 442497 81 55 7563 26696 96371 381 30 426899 76 983911 6.8 44,988 5657012 2672496363 30 31 9,427359 75.2 9838756. 44 8 9 10.556621 26697 96355 229 983849 443968 55603', 2678096347.28 33 4 7 982986805 444468 1 6555542 26808 963401 27 34 428717 765. 98377 58 447 0 81.5 56053 26836 96286 26 34 429. 70 317 76.6983735 2.9 81.43 42 4329 74.0, 997 444351 88 565 11 26864096324 18 43 423977 7 984 9 3749 4 44 9923 6540677 268928 6316 214 744 4300 74.8 983664 4 986411 806 3589 271106 96308 23 I 1 48 1678 174| 2 5|69 | 81.2 | 88 435062 745- 983629.9 462898: 5310 26948 96301;22 30 490098 7. 02060. 7 2744 96246 1 39 43078 74.6 983 694 6 447384 8 5526163 26976 962938 21 40 43142 79 983358 6.9 447870; 5521380 27004 96285:2 4.99 323',4 9 33 6 10.551644 927032 9627719 48 436329' | 983487 44541. 45119 57 0 27 628 9 2 618 74.9 980.3 648 49 436462 7 983452 6. 465326 80 550674 27088 961261 17 98841'6 449810 0,6 50190 9711,6 96253 16 45 4336987 983382 6 0 42 54706 27144 96216 15 6.436346 7 983345 45777 56498223 27172 9638'14 47 434569 74. 983309 6. 4532660 5487463 27200 13 48 4362416 7 983273 45743 8. 548257 27228196222 12 56 9.436353 7. 9983166. 9.4531 W W.5468414 27.312 96198 9 62 436798 4 983132 9' 45'3668. 463 32 27340 96190 8 53 439242 983094 6,.0 46148 545852 27368:6182 7 74.0 79.9 54 437686,.983058 4' 4628 545372 27396] 6174, 6 555 97439014 82960 W966 2748096160 60 440338 982842 457496 542504 27564196126 0 Cos.ine. Sie. Cotang. = Tang N. os. Nine......74 Degrees..... TABLE II. Log. Sines and Tangents. (160) Natural Sines. 37 sine. D. 10"1 Cosine. D.101 Tang. D. 101" Cotang. N. sine. N. cos. 0 9.410338 9 9.982842 60 9.457496 4 10.642504 27564 96126 60 1 440778 733 982805 6.0 457973 79.3 642027 27592196118 59 2 441218 732 982769 1 458449 793 41551 2762096110 58 3 441658 982733 61 458925' 2 541075 27648 96102 57 7341 6.1 79.49 4 442096 73 1 982696 61 9400 79 1 840600 27676 96094 56 6 442535 73.0 982660 6'1 459875 79.1 540125 27704196086 55 6 442973 7-29 982624 61 460349 790 39661 27731 96078 54 7 443410 728 982587 61 460823 789 5839177 27759 96070 53 8 443847 982551 461297 53870 27787 96062 52 444284 761 78 278156'054 9551 9 4 44284 727 982514 6'1 461770 78'9 538230 927815 96054 5 1 io 444720 72:6 982477 6.1 462242 5. 37758 27843 96046150 11 9.445155 72.6 9.982441 6.1 9.462714 78 10.537286 2787196037 49 12 445590 72.6 982404 61 7463186 7865 636814 27899 96029 48 13 446025 72-4 9824367:1 463668 785 636342 27927 96021 47 14 446459 723 982331 6. 464129 78' 535871 27955 96013 46 15 446893 982294 6 1 464599 78.4 535401 27983 96005 45 16 447326 72.l 982257 6.1 465069 78.3 34931 28011195997 44 17 447759 72. 982220 6'2 465539 78. 534461 28039 95989 43 18 448191 72'0 982183 62'466008 78.2 533992 28067195981 42 18 448191 982183 2 466476 1_ 533992 19 - 448623 71290 6 78. 533524 28095 95972 41 20 449054 1'8 982109 6,2 466945 78. 533055 28123196964 40 1 9.449485 717 9.98207 6:2 9.467413 77.9 10.532587 28150 95956 39 22 449915 7 98203 467880 77 8 632120 28178 95948 38 23 450345 71 981998 6.2 468347 77.8 531653 28206195940 37 24 450776 71. 981961 6.2 468814 77. 531186 28234 95931 36 25 4514 7104 981924 62 469280 777 530720 28262 95923 35 71.4 62 77.6 2 6 451632 71. 981886 6.2 469746 776 30254 282909591 34 71.3 6.9, 775 7 4520671.3 981849 6. 470211 77. 529789 28318 95907 33 28 45248060 71. 981812.2 470676 77. 9324 283469898 32 71.2 9817741 6.2' 29 452915 71.1 981774 6.2 471141 77. 528859 28374 95890 31 43342 98173 6.2 471606 77.3 i28395 2840295882 30 3 9 453768 71. 9981699 6.9 472068 77.3 10 527932 28429 95874 29 3129.453768 7529.981699 6.2[ 2 454194 71.0 981662 6.3'472532 77.2 5627468 28467 19866 28 33 454619 70.9 981626 6.3 472995 77.1 i27005 28486 95857 27 34 455044 70.8 981587 6.3 47347 77.1 526643 28613 96849 26 35 45469 70.7 98149 63 473919 770 526081 2854196841 26 3 70.5 6.3 4 5283951 37 463167().6 981474 6.3 474842 76. 525158 f'2840'95824823 38 456739 981436 475303 7 24697 28625 96816 22 70.4 6.3 76.1 524237 28657295807 39 457162 70.4 - 981399 6.3 475763 767 24237 286595807 21 40 457584 981361 6.3 476223 76.7 523777 28680 9799 20 41 9.458006 7 9 981323 63 9 476683 76.6 10523317 870895791 19 458427 70.2 981286 6.3 477142 76. 22868 28736 95782 18 43 458848 70.1 98124 6.3 477601 76 22399 28764 95774 17 44 459268 70.1 981209 6;3 478069 76.4 521941 28792 95766 16 46 459688 7.0 981171 6. 478617 76.3 521483 28820 9577 15 46 460108 6.9 981133 6.3 478975i 76.3 621025 28841 95749 14 48 460946 69. 9107 479889 76.1 211 2890395732 12 49 461364 69.7 981019 6.4 48034 76. 519655 2893195724 11 50 461782 69.6 980981 76.0 519199 28959195715 10 51 9.462199 6.9.980942 6.4 481267 10 518743 2898795707 9 52 4626i6 69.5 980904 6.4 481712;.9 518288 29015 95698 8 63 463032 69. 980866 6.4 482167 75.8 617833 2904295690 1 54 463448 69.3 980827 6.4 482621.7 17379 29070 95681 6 4563864'69.3 9801789 6;4 4483076 757 16925 2909895673 55 463864 1'2' 2 56 464279 69U2. 980750 6.4 483529 75.6 516471 29126 95664 4 57 464694 69. 980712 6.4 483982 75.5 51,60i8 29154 95656 3 58 465108 6. 980673 6.4 484436 7 156 2918295647 2 9 465522 69.0 980635 6.4 484887 754 51 113 292095639 1 68.9 6.4 u 75~ 760 6d 465935 98059 6 485339 14661 29247 95630 0 _ Cosine. Sine. Cot1ang. - lran6. 1 N c. s399ine. E 73 Degrees._ 38 Log. Sines and Tangents. (170) Natural Sines. TABLE II. Sine. D. 10" Cosineg. D. 10" Cotang. I;N.sineN. cos. 0 465935 9.980596 6 4 9.486339 10,614661 29237 9630 60 1 466348 68.8 980558 6.4 485791 75. 514209 29265 96622 59 2 466761 68.8 980519 6'S 486242 751 513758 29293 95613 58 3 467173 6.7 980480 486693 5.1 13307 2932195605 57 4 467685 68.6 980442 6.6 487143 7550 612857 29348 95596 56 6 467996 68.5 980403 6.6 487593 612407 29376 965588 6,5 6 468407 68.6 980364 66 488043 7 611967 29404 6579 64 7 468817 68.4 980325 6 6 488492 74'8 511508 29432 95571 53 8 469227 68.3 980286 66 488941 74 611069 29460 95562 52 9 469637 68. 980247 6 489390 7 6 510610 29487965554 61 10 470046 2 980208 489838 510162 29515 96545 50 11 9.470455.681 9.980169 6'6 9.490286 74.6 10.609714 29543 95536 49 12 470863 68.,'980130 6.' 490733 74 609267 29671 95628 48 13 471271 68. 980091 806 491180 74 i 08820 29599 95519 47 14 471679 67.9 980052 6 491627 7 08373 29626 95511 46 14 4716795 66 491 16 472086 67.8 980012 6.6 492073 74 507927 29654 95502 45 67.8 19 6074 81 29682 95493 44 16 472492 67.8 979973 6'5 49S2E19 74' 507481 29682 95493 44 17 472898 67.7 979934 6'6 492965 74'2 607035 2971095486 43 18 473304 67.6 979895 6'6 493410 7451 06590 29737195476 42 19 473710 67.6 979855 6.6 493854'74 506146 29765 95467 41 20 4741.15 67. 979816 6 6 494299 74'0 605701 29793 956459 40 21 9,474519 67,4 9.979776 6'6.494743:4 0 10.605257 29821 95450 39`2 7 674 6.6 74.... 22 474923 67.4 979737 6'6 495186 504814 29849 95441 38 23 476327 67. 979697 6*6 496630 7. 504370 29876 5433 37 24 4175730 67.2 979658 6'6 496073 7378 503927 29904 95424 36 26 476133 67.2 979618 6'6 496616 603485 29932195416 35 26 476536 67.1 979579 6'6 496957 736 503043 2996095407 34 27 476938 6. 979539 6 49739973.6 602601 29987 95398 33 28 477340 66.9 979499 6 497841 73.6 602159 30016 95389 32 29 477741 66.8 979469 6'6 468282 501718 30043 96380 31 30 478142 66.8 979420 6'6 498722 601278 30071 95372 30 31 9.478542 66.7 9,979380 *66 9.499163 733 10.500837 30098195363 29 32 478942 66.7 979340 6'6 499603 73.3 600397 30126 95354 28 33 479342 66.6 979300 6.7 600042 73'2 499958 30154195345 27 66.5 6.7 73.2 34 479741 66. 979260 6 7 600481 499519 30182 95337 26 36 480140 66. 979220 6 700920 73.1 499080 30209 95328 25 36 480539 66.4 979180 6'7 601359 73. 498641 30237 95319 24: 37 67 773.0 498 37 480937 66.3 979140' 7 601797 498203 30265 95310 23 38 481334 66.3 979100 6.7 502236 753. 497766 3029292 301 22 39 481731 66.2 979059 6.7 502672 497328 30320 9293 21 40 482128 66.1 979019 67 603109 72.8 496891 30348 95284 20 41 9.4824 72.81 41 9.482525 660 9.978979 6'. 9.503546 7,4964 54 30376 96275 19 42 482921 66.0 978939 6'7 603982 7.7 496018 30403 5266 18 43 488316 66.9 978898.7 604418 72. 495582 3043195257 17 44 483712 6 978868 7 604854 72.6 495146 30469 95248 16 65.8 6'7 508972 45 484107 65.8 978817 6'7 505289 494711 30486 95240 15 46 484501 6.7 978777 6 7 605724 72.6 494276 30514 5231 14 47 484895 65.7 978736'7 606159 493841 30542 96222 13 47 48 5895 6'7 48 485289 65.6 978696 687 606693 724 493407 30570 95213 11 49 485682 6:.5 978655 6.8 607027 72.3 492973 30597 95204 11 40 486807 66.6 978615 6.8 607460 72.2 492540 306251 95196 1 10 05 486075 51 9.486467 66.4 9.978674 6 9.607893 72.210.492107 3066395186 9 52 486860 66.3 978633 6.8 608326 72.1 491674 30680 95177 8 53 487261 65.3 978493 6.8 608759 72.1 491241 3070895168 7 64 487643 66.2 978462 6.8 609191 72.0 490809 30736 95159 6 55 488034 65.1 978411 6 09622 71.9 490378 30763 5150 5 66 488424 65.1 978370 6.8 610054 71.9 489946 30791 95142 4 57 488814 65.0 978329 6.8 610486 71.8 489515 30819 95133 3 58 489204 65.0 978288 6.8 610916 71.8 489084 30846 95124!2 59 489593 6491 978247 6.8 611346 71.7 488654 30874 95116 1 60 489982 64.8 978206 6.8 611776 1.6 488224 30902 95106 0 Cosine. ISine. Cotang. Tang. N. cos. N.sine. 72 D)egrees.. t,. — TABLE II. Log. Sines and Tangents. (18~) Natural Sines. 39. Sine. D. 10" Cosine. D. 10" Tanl. D. 10" Cotang. iN.sine.N. eos. D. _ - - -N.sine.N. eos. 0 9.489982 4 9.978206 6.8 9.511776 71 6 10.48822430902 95106 60 1 490371 68 978165 512206 4877941 30929 95097 a59 2 490759 64.7 978124 68 612635 71. 487365 30957 95088 58 3 491147 46 978083 9 513064 486936!3098595079 57 4 491535 646 978042 6.9 513493 714 486507 31012995070 56 5 491922 5 978001 6. 13921 71 486079 31040 95061 55 6 492308 977959 6.9 14349.3 485651 31068 95052 54 7 49269564.4 977918.9 614777 71. 485223 31095 95043 53 8 493081 644 977877 69 516204 71.2 484796 31123 95033 52 9 493466 3 977835 69 616631 71.2 484369 31151 95024 61 10 493851 62 977794 6. 61607 71.1 483943 31178 95015 50 10 4938164:2 51 10.483516 3120695006 49 11 9.494236 64 9.977752 6 9.616484.0 48316 31206 95006 49 12 494621 641 977711 516910 483090 31233 94997 48 13 495005 640 977669 6.9 517335 709 482665 31261 94988 47 14 495388 39 977628 6:9 517761 70 482239 31289 94979 46 15 495772 6 977586 69 618185 70.8 481815 31316 94970 45 16 496154 63 977544 18610.8 481390 31344 94961 44 17 496537 977503 7.0 519034 70.7 480966 31372 4952 43 18 496919 637 977461 519458 480542 3139994943 42 19 497301 636 977419 0 51988 7 480118 31427 94933 41 20 497682 636 977377 70 0305 70.5 479695 3145494924 40 21 9.498064 9.977335 7 9.20728 7 10.479272 3148294915 39 22 498444 63.5 977293 7 0 521151 70.4 478849 3151094906 38 23 498825 63.4 977251 7. 2173 70.3 478427 3153794897 37 24 499204 63.4 977209 0 1995 7.3 478005 31565 94888 36 25 499584 633 977167 70 22417 70.3 477583 31593 94878 35 26 499963 9771257 2838.2 477162 3162094869 34 27 00342 63.2 977083 0 3259.2 476741 3164894860 33 0 52359 170 28 50072631 977041 3680 1 476320 31675 94851 32 9 1099 63.1 976999 7.0 24100 70.1 475900 31703 94842 31 29 63001099 976999 00 30 501476 62. 976957 7 0 524520 7060 475480 31730]94832 30 31 9.501854 6 9.976914 7 0 9.624939 69.9 10.475061 3175894823 29 32 502231 62. 976872. 52539 699 474641 3178694814 28 33 02607 62.8 976830 7.1 62778 69.8 474222 31813 94805 27 34 02984 62.8 976787 7.1 26197 69.78 473803 31841 94795 26 34 502984 976787 5266197 35 03360 627 97674 7 1 526615 69.7 473385 31868 94786 25 36 03735 2.6 976702 71 27033 69 472967 31896 94777 24 37 504110 976660 1 27451.6 472549 31923 94768 23 38 04485 62.5 6 976617 1 627868 69.6 4721321 31951 94758 22 39 504860 62.5 976574 71 528285 69.5 471715 31979 94749 21 40 505234 62. 976532 7. 8702 6 471298 32006 94740 20 41 9.505608 62.976489 1 9.529119 69.10.470881 32034 94730 19 42 05981 62.3 9764467.1 9535 69.3 470465 32061 94721 18 43 506354 62.2 976404 7:1 629950 69.3 470050 32089 94712 17 44 506727 6.2 976361 7 1 630366 69.3 469634 32116 94702 16 45 507099 62. 976318 7:1 530781.2 469219 32144 94693 15 46 07471 62.0 97675 7 1196 69.1 468804 32171 94684 14 06 1 976275 7.1 531196 47 507843 62.9 976232 7.2 531611 69.1 468389 32199 94674 13 48 508214 61.9 976189 72 3202 690 467975 3222794665 12 49 08585 61.8 976146 72 32439 69.0 467561 32250194656 11 50 508956 6 976146 72 32839 51 9;089536 61.8 976103 7. 532853 68.9 467147 32282 94646 10 61 9.609326 6 9.976060.72 533266 68.8 10.466734 32309 94637 9 52 509696 976017 2 33679 688 466321 32337 94627 8 3 10065 61. 97974 72 634092 68 465908 32364 94618 7 64 510434 6 97930 7 53404 465496 32392 94609 6 B55 510803 61.5 976887 7.2 534916 68.7 465084 32419 94599 5 56 511172 61.5 975844 7.2 63328 68.6 464672 32447 94590 4 67 511540 61. 97800 2 35739 68 464261 32474 94580 3 58 5119'07 975800 4 5 7 39 8 11907 61 9757Ei7 636150 463850 32502 94571 2 9 1227 975714 61. 7.2 636561 68. 463439 32529 94561 1 60 512642 975670 2 536972 463028 32557 94552 0 Cosine. Sine. sCotang. Tang. N. cos. N.sine. 71 Degrees. 40 Log. Sines and Tangents. (19O) Natural Sines. TABLE II. _ Sine. Di. IO" Cosine. 1. l1o Tang D. 1t Cotang. -N. sine. N. cos. 0 9.512642 61 2 9.975670 73 9.36972 68 4 10.463028 1132557 94552 ]60 1 513009 6. 975627 3 537382 8 3 462618 32584 94542 659 2 513375 61. 975583 537792 68 462208 32612 94533 58 3 513741 6. 975539 538202 682 461798 32639 94523 57 4 514107 6 9 975496 538611 682 461389 32667 94514 56 5 514472 60.9 976452 539020 681 400980 32694 94504 55 6 514837 60.8 975408 7 539429 681 460571 32722 94495 54 7 515202 608 975365 673 539837 68. 460163 32749 94485 53 8 515566 60-7 975321 540245 68 459755 32777 94476 52 9 515930 60.7 976277 7'3 540653 67. 4.59347 32804 94466 51 10 516294 60:6 975233 541061 67 9 458939 32832 94457 50 11 9.516657 60. 9.975189 9.541468 67.8 10,458532 3285994447 49 12 517020 975145 541875 67 458125 3288794438 48 13 517382 60-4 975101 542281 677 457719 32914 94428 47 14 517745 60Q4 975057 542688 677 457312 32942 94418 46 15 618107 975013 543094 456906 32969 94409 45 16 518468 0.3 974969 43499 67,6 456501 32997 94399 44 17 518829 60 974925 543905 67 456095 33024 94390 43 60.2 455690 37051 94380642 18 619190 601 974880 544310 676 455690 33051 9480 42 19 519551 601 974836 544715 674 455285 33079 94370 41 20 619911 600 974792 7'4 645119 6 4 464881 3310694361 40 21 9.520271 60:0 9.974748 7*4 9. 4524 673 10.454476 33134 94351 39 22 520631 60. 974703 74 45928 673 454072 33161 94342 38 23 520990 974659 646331 67 453669 33189 94332 37 24 21349 598 974614 7 46735 67 43265 3321694322 36 59.8 7,4 67,2 25 521707 59 8 974570 547138 67,1 452862 3324494313 36 26 522066 974525 547540 67.1 452460 33271 943Q3 34 27 522424 69.7 974481 74 47943 67.1 452057 33298 94293 33 28 522781 59 6 974436 4 548345 67,0 451655 33326 94284 32 29 523138 974391 548747 67.0 451253 33353 94274 31 30 523495 9. 974347 5. 649149 66.9 450851 33381 94264 30 31 9.52385.62 9 9.974302 9,649550 10.45040 33408 94254 29 32 524208 9 *4 974257 549951 66.8 450049 33436 94245 28 33 624564 5 3 974212 7 550352 66.8 449648 33463 94235 27 34 524920 9 74167 550752 667 449248 33490 94225 26 35 5i2527i5 9 974122 551152 448848 33518 94215 25 36 525630 59.2 974077 551552 *6 448448 33545 94206 24 37 525984 69.1 974032 6 1952 66. 448048 33573 94196 23 38 26339 9.1 973987 552351 66.5 447649 33600 94186 2 59.0 7,5 66.594 39 52e693 59'0 973942 552750 6 447250 133627194176 21 40 527046 589. 973897 7.5 553149 66 446851 33656 94167 20 41 9.527400 58.,973852 9.53548 664 10.4464213368294157 19 42 527753:., 973807 7 5539466 42 527753 58.9'973807 7 Eib396 66 4 446054 3371094147 18 43 528105 88 973761 5444 6. 445656 33737 94137 17 44 5284558.8 973716 7 5 54741 66.3 445259 33764194127 16 45 928810 58.7 973671 6 6555139 66 2 444861 33792[94118 1 46 529161 58 973625 76 55536 444464 ]33819194108 14 47 529513 6 973580 6 559336. 444067 133846 94098 13 48 529864 58.6 973536 7,6 556329 66.1 443671 33874194088 12 49 63021 58.5 973489 7,6 556725 66,0 443275 33901194078 11 50 530565 68.5 973444 557121 66. 442879 33929194068 10 51 9.530916 58.4 973398 7 9.557517 65.9 10,442483 33956 94058 9 52 531265 58.4 973352 7 56 557913 65.9 442087 33983 94049 8 53 531614 68.3 973307 7.6 558308 6. 441692 34011 94039 7 64 531963 58.2 973261 7,6 658702 6,8 441298 34038194029 6 M5 5323128 973215 76 59097 44(0903 3406594019 6 56 632661 1'.:973169 7, 59491 65 7 440509 34093194009 4 57 533009 581 973124. 5988 6, 44011 93999 3 58 633357 i80 973078 560279 656 4397211 3414793989 2 59 33704' 973032.6 60673 439327 1134175 93979 1 60 340527 972986 61066 65.5 4389341l34202.3969 0 Cosinine. Sine. Cotang. Tang. N. eos.lN.sine. 70 Degrees. TABLE II. Log. Sihxes and T'angentsa. (200) Natural Siese. 41 Sine. p. 10" Cosine. D. 10" Tang. D. 10' Cotaug. N.sine. N. eos. o 9.534052 E 89.972986 79561066 655 10.438934 34202 93969 60 1 534399'7 972940 7 661459 65 4 438541 34229 93959 59 2 534745 972894 7 61851 654 438149 I342Ei793949 68 67.7' 797 65A 3 635092 972848 7 62244 65 437756,3428493939 3957 4 535438 57 6 9728092 I77 i562636 653 4437364 34311 93929 56 65 35783 57.6 972755 7J7 i63028 65 3 436972'34339 93919 55 6 536129 972709 r 68419 436581 34866193909 64 7 5364746 972663 563811 436189 34393. 93899 53 8 536818 574 972617 664202 651 435798 34421193889 52 9 537163 57.34972570 7.7 564592 65.1 4354081 34448 93879 51 10 537507 7.3 972524 7t7 564983 65:0 4350171 34475 93869 50 1 19.537851 57 2 9.972478 7 9.56566373 650 10.434627 34503 93859 49 12 538194 57 2 972431 78 665763 649 434237 34530 93849 48 13 538538 71 972385 78 666153 69 433847!34557 93839 47 14 538880 57 1 972338 7.8 66542 649 433458 3458493829 46 15 539223 570 972291 78 566932 648 433068 84612 93819 45 16 539565 70 972245 78 67320 432680 34639 93809 44 17 539907 6'9 972198 7 48 67709 647 482291!34666 93799 43 18 640249 69 972151 8 A68098 64 431902 18469493789 42 19 540591) 5668 972105 78 668486 6467 431514 34721 93779 41 20 540931 568 972058 78 668873 646 431127 3474893769 40 21 9. 641272 67 9.972011 7,8 9 569261 64.6 10.430739 34775i 93759 39 22 541613 667 971964 78 569648 45 430352 34803 93748 38 23 541953 566 971917 78 570036 645 429965 3483093738 37 24 642293 666 971870,78 670422 644 429578 84857 93728 86 25 642632 6.5 971823 7.8 70809 644 429191 34884 93718 35 26 542971 565 971776 8.571195 428805 34912 93708 34 27 543310 564 971729 9 71581 643 428419 3493993698 33 28i 543649'56 971682 7 71967 642 428033 3496693688 32 29 543987'6A 971635 9 672352 642 427648 34993 (9377 31 30 544325 971588 572738 427262 35021 93667 30 663 3 7.9 64.2! 31 9544663 562 9.971540 79 9.6731234 1 10.426877 35048 93657 29 32 545000 562 971493 7.9 673607 426493 3076193647'8 33 545338 56 971446 73892 426108'36102 93637 92'i 34 5 46674 5 1 971398. 674276 640 425724 35130 93626 26 34 646674 1, 9 71398 35 46011 56 97151 74660 639 425340 35156793616 26 56.0 7.9 63 9 36 646347 971303 575044 424956 35184 93606 924 37 546683 56. 971256 79 67427 639 424573 352111 93596:3 55,9 7.9 63,9 38 547019 971208 9 75810 638 424190 35239 93585 22 39 547354 558 971161 7 76193 638 423807 36266 93575 21 40 547689 558 971113 5,9 576676 6 8 423424 35293 935656 20 4 4.84 799 6387 41 9. 48024 9.971066 9576958 10.423041 36320 93555 19 42, 548359 971018 677341 422659 35347 93544 18 43V 548693 6 970970 8 5677723 636 422277 35375 9334 17 44: 549027 55 6 970922 80 578104 636 421896 3654)2 9352,1 16 45 8i4936 7 0 578486 635 421514 3542993514 15 46 549693 970827 78867 6 421133 3545693503 14 47 550026 5'4 970779 579248 63. 420752 35484 93493 13 48 550359 9. 970731 8.0 79629 63.4 420371 35511 93483 12 49 65;3069. 1.4 970683 8.0 580009 63.4 419991 365538 93472 11 50 651024 55'3 970635.0 580389 633 419611 35665 93462 10 51 9.551356 5,.97086 80 9.580769 633 10.419231 3559293452 9 52 551687 552 970538 0 81149 418851 35619 93441 8 53 552018 55 2 970490 80 581628 632 418472 35647 93431 7 54 552349 5'1 970442 80 581907 632 418093 36674 93420 i 551 662680, 970394. 682286 1 417714 35701 93410 5 56 553010 s.0 970345 81 582665 631 417335 3572893400 4 57 553341'5.0 970297 81 83043 630 416957 3 5756 93.389 3 58 bi53670 9 970249 1 583422 630 416578 35782 93379 2 59 6554000 5 970200 583800 416200 35810 93368 1 60 654329 970162 584177 416823 35837 93368 0 Cosine. Sine. Cotang. Tang. IN. cos. N.sine. 69 Degrees, -. 42 Log. Sines and Tangents. (210) Natural Sines. TABLE I1. Sine.. 10" Cosine. D. 10" Tang. D.10" Cotang. N.sine. N. cos. 0 9.554329 548 9.970152 81 9.684177 62 9 10.415823 35837 93358 i60 1 554658 64'8 970103 8.1 84555 62.9 415445 35864 93348 59 2 554987. 970055 81 584932 628 415068 3B89193337 15 3 5553156 4.7 970006. 585309 62. 414691 [35918 93327 57 4 555643 54.6 969957. 585686 62. 414314 3594593316 566 5 555971 64'6 969909 8.1 586062 627 413938 39739330655 6 556299 B4.5 969860 8. 586439 62.7 413561 36000 9396 54 7 556626 B4. 969811 1 86815 626 413185 36027 9385 53 8 556963 64.4 969762 8. 687190 62.6 412810 36054 9327452 9 557280 64.4 969714 8.1 587566 62 412434 36081 93264 o1 10 567606 64.3 969665 8.1 687941 62 41259 3610893253 50 11 9.557932. 9.969616 8.2 9.588316 62 10.411684 36135 93243 49 12 558258 64.3 969567 8.2 688691 62"4 411309 36162 93232[ 48 13 558583 64.2 969518 8.2 589066 62.4 410934 36190193222 4 I 14 658909 54'2 969469 8.2 589440 62'3 410560 36217 93211 46 15 559234 54.1 969420 8.2 589814 62-3 410186 36244 93201 45 16 559558 4.'1 969370 8.2 590188 623 409812 36271 93190 44 17 559883 540 969321 8.2 9062 409438 36298 93180 43 18 60207 54.0 969278.2 90935.2 409065 36325 93169 42 19 560531 63.9 969223 8.22 691308 62. 408692 36 93169 41 20 660855 639 969173 8.2 591681 62.1 408319 36379 93148 40 21 961178 9.969124 8.2 9.692064 10.407946 36406 93137 39 22 561501 63'8 96907 8.2 9246 621 407674 3643493127 38 23 561824 63.8 969025 8.2 592798 62.0 407202 36461 93116 37 24 562146 63.7 968976 593170 62.0 406829 36488 93106 36 25 562468 53.7 968926 8.2 593542 61.9 406458 36516 93095 35 26 562790 63.6 968877 8.3 593914 61. 406086 36642930434 53 6 8.3 61 8 27 63112 53 968827.3 9428618 406715 3669 3074 33 28 563433 968777 8.3 594656 61.8 405344 36696 3063 32 29 563755 968728.3 596027 61.8 404973 36623 052 31 30 564075 3. 4 968678 8.3 596398 61.7 404602 36650 9342 30 31 9.564396 63.4 9.968628 8.3 9 595768 61.7 10. 404232 36677 93031 29 32 664716 53.3 968578 8.3 596138 61.6 403862 36704 93020 28 33 565036 968528 6965608 61 403492 36731 93010 27 34 665356 968479 8.3 696878 61.6 403122 36758192999 26 36 565676 53.2 968429 8.3 697247 61.6 402753 3678692988 25 36 565995 968379 8.3 697616 61 402384 3681297824 37 566314 53.1 9683298.3 5979856 61.5 40215 3683912967 23 88 566632 65.1 968278 681 983.54 61 401646 36867 92956 22 39 566951 63.1 968228 8.3 698722 61.4 401278 36894 92945 21 40 567269 53.0 968178 8.4 599091 614 400909 36921 9293 20 41 9.567587 52,9 9.968128 8.4 9 61.3 10 400541 36948 2926 19 42 567904. 968078 8.4 699827 61.3 400173 369756 2913 18 29 8.4 60019, 43 568222 62.8 96802 8.4 600194 61. 399806 37002192902 17 44 568539 652.8 967977 8.4 600562 61-.2 899438 370299 2892 16 45 5688566 528 967927 8.4 600929 61.1 399071 370i6 92881 14 46 569172 52.7 967876 8.4 601296 61.1 398704 37083 92870 14 47 569488 62.7 967826 8.4 601662 61 398338 37110 92859 13!4 527 8.4 |6020 48;69804 62.6 967775 8.4 602029 61.1 397971 37137 2849 12 49 570120 52.6 967725 8.4 602395 61.0 397606 37164 9838 11 50 570435 52.5 967674 8.4 602761 61*0 397239 37191 92827 10 il 9.570751 52.5 9.967624 8.4 9.603127 60 10.396873 37218 92816 9 62 571066 2.4 96773 84 603493 60.9 396507 37245 28061 8 3571380| 967522| 8. 603858| li3 571380 52.4 95 7522 8. 603828 609 896142 37272 92794 7 64 571695 62.3 967471.6 604223 895777 37299 92784 6 65 672009 52.3 967421 8.5 604588 60.8 395412 37326 927731 56 572323 62.3 967370 8.6 604953 60.71 39047 37353 92762 4 57 572636 52.3 967319 8.5 605317 60.7 394683 37380 9271 3 58 572950 62.2 967268 8. 605682 60.7 394318 37407 92740 2 59 573263 52.2 967217. 606o046 6061 93954 37434 92729 1 60 573575 62.1 967166 606410 393590 37461 2718 0 Cosine. Sine. Cotang. ang. N. cos. N. 68 Degrees. r TABLE II. Log. Sines and Tangents. (220) Natural Sines. 43' Sine. D. In" Cosine. D. 10" Tang. D. 10" Cotang. ic8~ 17,.2, A3-. 0 13 800892.8 889,168 172 9114724 43 0 088276 6322577476 47 14 801047 88906 911982 088018 6824877458, 46 25.8 17'43 0 9 451 15 80120 2 888961 9122404 087760 63271.774699 4 25 8 S M-2,43.0 16 801;3562 888858 91.2498 4 087502 63293 77421, 44 25,..~7 1.7.2;43..0 1 17 801511 2l I 888755 1,2 912756 43 087244 6331.6177402 43 18 801665 2 888651 913014 086986, 63338 /7384- 42 25!,~7 1 42.9 19 804.819' 888548 913271 086729' 63361 77366 41 25,.-7 1, 17..2 1 42.9 20 801973 888444 173 91-3529 086471 6338377347 40' 21 9.8021'.888341 9.913787, 4'. 086213 63406J77329 39 25.17. 42.99i 22 802282 888237 17 914044 085956 63428 77310! 38 23 802436 i.6 888134 17 914302 429 085698 63451 77292 37 25,..~6''17 3 42,9 24 802589.6 888080'1 914560 0854401 63473 77273, 36 25-..6 1~77.S- i 42,.9 25i 82743 25 887926 17. 914817.9 085183 6349677255 35 25 o~6 3~7~,3 42.9 26 800897 56 887822 915075 084925 63518 77236 3427 803050 26. 887718 17 916332 42.9 084668 6354077218 33 28 8032042 887614 915590 084410- 68563 77199 32 29 803357 26 887610 173 915847 2.9 084153 6358577181 31'9 80835' 87~ 11:4 i i 30 8085114 1 887406 916104 42.9 083896 63608 77162 30' 25!.15 1,7;4 49,.9 1 31 ~,9.803664.887302.916362 0. 083638' 63630 177144 29 25t5i r17. 36242.9 1. 32 806817 887108 916619' 083381 63653 77125 28 2Ei.~~~~17 42.9 33 803970 88709317. 91687-7 083123 63675,77107 27 9,6.. ~t5 17. 4 42,9 34 804423 886989 917134 082866 68698 77088 26 2&.6 ~E'1;7.4 4,35i 804276' 886885 7 917391- 42 082609 63720 77070 25 i25. 4~~~174, 42,9 02616727514 36 804428954; 886780 917648' 0823 637427051 24 t %A, 1'fi..4 42.9371 37 804581. 2 886676 917905 - 082095 63765 77033 23 38 804734 2. 886571 918163 4 i081837 63787 77014 22'25i.4 -17.44, 42.8l 39' 804886 886466 918420 42t 081580 63810176996 211 40 8050839.4 886362 918677 4 08133 638326977 20' 471il.6 — glg 842. 10~-8106 63854 7969,' 1 41 9.805191.886257. 918934 0081066,638546959 42 805343; 886162 7. 919191 42 080809116387,776940 18 43 8054952. 886047 9.6 19448 " 080552 16389976921 17 44 25,36.~ 17.6 42.8:4 80564,7 885942 910705' 080295, 63922 76903 16 25-.3 1799125 4 21,& 45i 805799 2. 885837 919962 080088{ 63944 86884 15 125.3 17.5.. Aw. 8 46' 8059.51 885732 920219 07978b 63966 76866 14 17.5 1 42.8, 389 71 47 806103 2 885627 920476 42.8 079524 639897i847 1.,25.3 17.5 42.848 806254 885522 17.5 920733 4 079267 64011 76828 12;2.5,,3 Q~.8 49 806406 885416 920990 42. 0791010. 6403 76810 1150 80655,7 885311- 921,247 4.8" 078763-,1 64056 76791 1,0% i25.2ri ~~17.6 91168 4.8 51 9.806709.885205 17. 991503 4 0.07849:7. 64078 76772 925..2 II 17.6t 491.8 2 806860. 8851007.6 921760,:42. 078240' 6410076754 8' 53 807011. 884994 922017- 077983 6412376735,1'7 61 25,.2,1.642.8 54: 8071,63 884889 922274-i4 077726 64145 7671,7 6 55 125,2!~~~~ ~~~17.6.42.8 80 /3314 884783 922530:. 077470, 64167 76698 5 25.2 A17.6'491. 56 807465 884677 1 922787 077213 64419076679- 4 9.5.1;16 42..8 7 80761 25.1 88457217.6 923044 42.8 076956 6421:276661 3 58 807766 884466 923300. 076700, 6423476649,, 25.1 17. 42.8 a60~ 424761-1f 9 807917 25.1 884360 923557 076443 6425676623:' 1i 25 1 417.61 42-7 0 60 808067 884254 923813 076187 64279 7660447'-'Co-in.RO. S in.C~ -C-ootang.._:-'Tnn. T (: jN.g.co.INS.sine.; 50 Degrees. TABLE II. Log. Sines and Tangents. (400) Natural Sines. Sine. ID. 10" Cosine. D. 10' Tang.'D. 10" Cotang. N.sine. N. cos. 0 9.808067 2 1 9.884254 17 7 9,923813 42 7 10.076187 64279 76604 60 1 808218 251 884148 17 7 924070 427 075930 64301 76586 59 2 808368' 251 884042 17 7 924327 427 075673 64323 76567 58 3 808519 250 883936 17.7 924583 2.7 075417 64346 76548 57 4 808669 260 883829 17.7 924840 427 075160 64368 76530 56 6 808819 2:6 0 883723 17.7 925096 42.7 074904 64390 76511 55 6 808969 25.0 883617 17.7 925352 427 074648 64412 76492 64 7 809119 26.0 883610 17.7 925609 427 074391 64435 76473 63 8 809269 25.0 883404 17 925865 42.7 074135 64457 76455 62 9 809419 49 883297 17, 926122 427 073878 64479 76436 51 10 809569 24'9 883191 17.8 926378 42:7 073622 6450176417 50 11 9.809718 24 9.883084 17.8 9.926634 42'7 10.073366 64524 76398 49 12 809868 49 882977 17 8 926890 427 073110 64546 76380 48 13 810017 24 9 882871 17 8 927147 42'7 072853 64568 76361 47 14 810167 882764 1 927403 7 072597 64590 76342 46 15 810316 24.9 882657 17 8 927669 42 072341 64612 76323 4 16 810465 8825 1 9276 4207 16 810465 248 882660 17 8 9279156 427 072085 6463 76304 44 17 810614 24 882443 17.8 928171 427 071829 6465776286 43 18 810763 24.8 882336 17,8 928427 42'7 071573 64679 76267 42 19 810912 24,8 882229 179 928683 427 071317 6470176248 41 20 811061 248 882121 17 9 928940 427 071060 64723 76229 40 21 9.811210 24,8 882014 17.9 9929196 42.7 10,070804 64746 76210 39 22 811358 24,7 881907 17 9 929452 2,7 070548 64768 76192 38 23 811507 24,7 881799 17.9 929708 427 070292 647907617337 24 811655 47 881692 17:9 929964 42 070036 648127615436 25 811804 247 881584 17,9 90220 6 069780 64834 76135 35 26 811952 47 881477 1. 930475 42 069525 64856676116 34 27 812100 24,7 881369. 930731 069269 64878 76097 33 28 812248 24,7 881261 18 930987 426 069013 64901 76078 32 29 812396 246 1 8:0 9324 29 8196 246 88116 1. 931243 426 068757 64923 76069 31 30 812644 246 881046 18.0 931499 426 068501 64945676041 30 31 981269 4,6 9880938 18.0 9.931755 46 10.068245 64967 76022 29 32 812840 24 6 880830 18.0 932010 42 06 067990 64989 76003 28 33 81S988 24.6 880722 18 0 932266 42 76 067734 65011 75984 27 34 813136 24,6 880613 18,0 932522 6 067478 66088 75966 26 35 813283 24,6 880505 18,0 932778 42 6 067222 650556 75946 25 36 83430 24,6 880397 18.0 933033 066967 65077756927 24 37 813578 24,5 880289 18.0 933289 426 066711 610075908 23 38 813725 24, 880180 18.1 933545 066455 66122 76889 22 39 813872 24,6 880072 18.1 933800 6 066200 651447 5870 21 40 814019 24 879963 18. 934056 065944 65166 75851 20 41 9 814166 24, 9 879855 18.1 934311 426 10.065689 65188 75832 19 42 814313 24.6 879746 18.1 934567 26 066433 65210 76813 18 43 814460 24,6 879637 18.1 934823 42,6 065177 6523275794 17 44 8i4607 244 879529 1 93078426 064922 6525475775 16 45 814753 24.4 879420 181 935333 426 064667 65276 75766 15 46 814900 24 4 879311 18 1 935489 426 0644111 6529876738 14 47. 816046 244 879202 18.1 935844 6 064156 65320 75719 13 48 815193 24,4 879093 18,2 936100 426 063900 65342 75700 12 49 815339 244 878984 18'2 936355 426 063645 65364 76680 11 50 815485 24.4 878876 18.2 936610 6 063390 65386 7661 10 51 9 815631 878766 182 936866 426 10063134 65408 756642 9 52 815778 24,3 878656 18.2 937121 42 062879 65430 76623 8 53 8159244.3 878547 18 937376 42.5 062624 65452 76604 7 54 816069 43 878438 18.2 937632 062368 6547476585 6 55 816215 878328 182 937887 425 062113 65496 7566 6 66 816361 24.3 878219 18.2 938142 42;5 06188 65518 7665547 4 57 816507 24.3 878109 18.3 938398 425 061602 66540 76528 68 816652 24.2 877999 18.3 938653 425 061347 165562 75o09 2 59 816798 24.2 877890 18.3 938908 42S 061092 6558475490 1 60 816943 24.2 877780 18,3 939163 060837 65606 75471 0 I CoCile. Sine, Cotang; C Tang..!. cos. N.sine. 49 Degrees. 62 Log. Sines and Tangents. (410) Natural Sines. TABLE II. Sine. D. 10"' Cosine. D. 10" Tang. ID. 10" Cotang. N. sine. N. cos. - 0 9. 816943 9. 877780 18 9.939163 4 10060837 65606 75471 60 1 817088 24.2 877670 939418 42.5 060582 65628 75452 59 2 817233 24.2 877560 18.3 939673 42.5 060327 65650 75433 58 3 817379 24.2 877450 18.3 939928 42.5 060072 65672 75414 57 4 817524 24.2 877340 18.3 940183 42.5 0598171 6569475395 56 5 817668 24.1 877230 18.3 940438 42.5 059562 65716 75375 55 18;.44 42.5 0 6 817813 24.1 877120 940694 059306 65738 7356 154 7 817958 24.1 877010 18.4 940949 425 059051 65759 75337 53 24~.1 877042.5 8 818103 24.1 876899 18.4 941204 42.5 058796 65781 75318 52 9 818247 24.1 876789 18.4 941458 08542 65803 75299 51 10 818392 24.1 876678 18.4 941714 425 058286 65825 76280 50 11 9.818536 24.1 9876568 18.4 9.941968 42.5 10.058032 65847 75261 49 12 818681 24.0 876457 18.4 942223 057777 65869 75241 48 24.0 876341.4 425 13 818825 24 87634718.4 942478 42.5 057522 65891 75222 47 14 818969 24.0 876236 18. 942733 42.5 057267 65913 75203 46 15 819113 24.0 876125 18.5 942988 4225 057012 65935 75184 45 16 819257 24. 0 876014 1.5 943243 42. 056757 65956 75165 44 17 819401 875904 18. 943498 5 056502 65978 75146 43 24.0 18.5 42.5 18 819545 24.0 875793 18.5 943762 42:5 056248 66000 75126 42 19 819689 3.9 875682 18.5 944007 05993 66022 75107 41 23.9 875571 42.5 O44 20 819832 87557118.5 944262 425 055738 66044 75088 40 21 19.819976 23.9.87549 5 9.944517 425 10.055483 66066 75069 39 22 820120 23.9 87348 18. 944771 424 055229 166088 75050 38 23 820263 23.9 875237 18.5 945026 42.4 054974 166109 75030 37 24 820406 23.9 875126 18. 945281 42.4 054719 I 66131 75011 36 25 820550 23.9 875014 18.694535 424 054465 66153 74992 35 26 820693 23.8 874903 945790 054210 66175 74973 34 23,8 188 6 49-4 27 820836 23.8 874791 18.6 94604 424 05395 166197 7493 33 28 820979 874680 186 946299 053701 166218 74934 32 29 8211223.8 874568 18.6 946554 42.4 053446 166240 74915 31 23.8 18.6 42.4 46 30 82126523 874456 6 946808 053192 166262 74896 30 31 9821407 23.8 9.87434 6 424 10.052937 66284 74876 29 32 821550 2 874232 18.7 947318 052682 i 66306 74857 28 23.8 18.7 - ) 7 48 33 821693 238 8774121 18.7 947672 42.4 0524281 66327 74838 27 34 821835 23.7 874009 187 947826 42.4 052174! 66349 74818 26 35 821977 23.7 873896 18.7 948081 42.4 051919 it66371 74799 25 23.7 873 7 42.4 36 822120 23.7 873784 18.7 948336 42.4 051664 66393 74780 24 37 822262 3.7 873672 948590 42.4 0514101 66414 74760!23 38 822404 23.7 873560 18.7 948844 42.4 051156 66436 74741 22 39 822546 23.7 873448 8.7 949099 42.4 050901 66458 74722 21 23.6 2318.7 9496 40 82.2688 2233 76 873335 1 8 * 77 949353 42.,4 050647 1 66480 74703 20 41 9.822830 23.6 9.873223 18.7 9949607 42 4 10.050393 166501 74683 19 42 822972 83.6 73110 18 949862 424 0501381 66523 74663 18 43 823114 23.6 872998 18.8 950116 424 049884 66545 74644 17 44 823255 23.6 872885 18.8 950370 424 049630( 66566 74625 16 |45 823397 23.6 872772 18.8 950625 42.4 049375 66588 74606 15 46 823539 23.6 872659 8.8 950879 4 049121 l 66610 714586 14 47 823680 23.6 872547 18,8 951133 424 048867 66632 74567 13 48 823821 23.6 872434 18.8 951388 42.4 048612 66653 74548 12 49 823963 23 872321 18.8 951642 42.4 048358 166675 74522 11 50 824104 23.6 872208 18.8 951896424 048104 Ii 66697 74509 10 51 9.824245 23.5 89 72095 18.8 95210 424 10.0478501 66718174489 9 52 824386 23.5 871981 18.9 952405 42 4 0475951 66740,74470 8 53 824627 23.5 871868 18.9 952659 42.4 047341 166762 74451 7 54 824668 23.5 871755 19 952913424 047087 1166783174431 6 55 824808 23.4 871641 18.9 953167 046833 1 66805174412 5 56 824949 23.4 871528 18.9 953421 42.3 046579 166827174392 4 23.4 18,9 42.3 6 57 825090 23.4 871414 18.9 953675 42.3 0463256 66848174373 3 58 825230 23.4 871301 18.9 953929 423 046071 166870174353 2 59 825371 23.4 871187 18. 954183 423 045817 66891 74334 1 60~ 82655 1 23-1 871073 18,9 954437 045563 }66913174314 0 I Cosine. I I Sine. Cotang. Tang. jj N. cos.N.sine. 48 Degrees............:.,... _ TABLE II. Log. Sines and Tangents. (42)' Natural Sines. 63 Sine. D. 10o" Cosine. D. 10" Tang. D. 10' Cotang. N. sine. N. cos. 0 9.825511 23 4 9.871073 190 9.964437 4 3 10.045563 66913 74314 60 1 825651 23.3 870960 954691 42. 045309 66936 74295 69 2 825791 23 870846 9649 423 045056 66956 74276 68 3 825931 233 870732 19.0 956200 42.3 044800 66978 74256 57 4 826071 233 870618 19. 454 044546 66999 74237 56 5 826211 23 3 870504 9 955707 42.3 044293 67021 74217 55 6 826351 870390 190 955961 423. 044039;67043 74198 54 7 826491 23*3 870276 190 956216 423 043785 67064 74178 53 8 826631 23 3 870161 190 956469 3 043531 67086 74159 52 9 826770 232 870047 9. 956723 42.3 043277 67107 74139 51 10 826910 23'2 869933 956977 42.3 043023 67129 74120 50 11 9.827049232 9.869818 1 9.957231 42.3 10.042769 67151 74100 49 12 827189 2. 869704 9567485 42.3 042515 67172 74080 48 13 827328 23.2 869589 19.1 967789 42.3 042261 67194 74061 47 23.2 869 195 7 403 14 827467 232 869474 191 7993 42.3 042007 67215 74041 46 16 827606 23.2 869360 1911 958246 42.3 041754 67237 74022 45 16 827745 232 869246 958500 42.3 041500 67258 74002 44 23.217 827819.1 42.3 17 827884 23-1 869130 9'1 968764 041246 67280 73983 43 18 828023 23 869015 19.2 959008 42.3 040992 67301173963 42 19 828162 231 868900 192 959262 42.3 040738 67323 73944 41 20 828301 231 868785 192* 959516 42.3 040484 67344 73924 40 21 9.828439 923.1 868670 1 9.959769 42.3 10.040231 67366 73904 39 22 828578 231 868555 19.2 960023.3 039977 67387 73885 38 23.1 868555192 960023 23 828716 23:1 868440 960277 42.3 039723 6740913865 37 24 828855 19.2 42.3 03946 24 828855 23 0 868324 19.2 960531 42. 039469 67430 73846 36 256 828993 0 868209 19.2 960784 42. 039216 6742 73826 3 26 829131 23*0 868093 19.2 961038 42.3 038962 67473 73806 84 27 829269 23 0 867978 19.3 961291 42.3 038709 67495 73787 33 28 829407 23-0 867862 19.3 961545 423 3845 67616 73767 32 29 829545 230 867747 13 961799 038201 67538 73747 31 30 829683 23~0 867631 19.3 962052 42.3 037948 67559 73728 30 81 9.829821 229 9.867515 19.3 9.962306 42.3 10.037694 ]67680 3708 29 32 829959 867399 193 962660 42.3 037440 67602 73688 28 33 830097 22.9 867283 19 962813 2.3 037187 67623 73669 27 34 830234 83216283 19 963067 42.3 036933 67645 73649 26 830372 9 867051 193 963320. 036680 67666 73629 25 36 830509 22.9 866935 19.4 963574 42.3 036426 67688 73610 24 37 830646 229 86670 19.4 963827 42-3 036173 67709173590 23 38 830784 866703 964081 4 035919 67730!73570 22 39 830921 228 586 194 964335 42.3 035665 67752 73551 21 40 831058 866470 9 64588 42.2 035412 67773731311 20 41 9.831195 8 9.866353 194 9.964842 422 10.03518 67796 73511 19 42 831332 228 866237 965095 42.2 034905 67816173491 18 43 831469 228 86610 19.20 965349 422 034661 67837173472 17 4. 034398 6874 44 831606 866004 1 96042 034398 67869!73452 16 45 831742 22:8 865887 195 9658565 4 034145 67880 73432 15 46 831879 865770 19.6 966109 42.2 033891 67901 73413 14 47 832015 22. 86653 195 966362 42.2 033638 67923 73393 13 227..4. 0336384 48 832152 22 866536 966616 033384 67944 73373 12 49 832288 2 865 419 5 966869 22 033131 67965 73353 11 50 832425 86302 967123 4'~ 032877!67987 73333 10 51 9832561 9227.865185 195 9.967376 10.032624 68008 73314 9 52 832697 865068 967629 42.2 032371 680973294 8 232.7 19538 42.2 032117 6 832833 227 864950 195 967883 032117 68051 73274 7 54 832969 86483319 968136 42.2 031864 68072 73254 6 55 833105 226 86471 196 968389 42.2 031611 6809373234 56 833241 226 864598 196 968643 4.o2 031367 68115'73215 4 57 833377 226 864481 196 968896 42.2 031104 68136 73195 3 58 833512 226 864363 196 969149 42.2 030851 68157 73175 2 59 833648 2 68649245 16969403 42.2 030597 68179 73155 1 60 833783 864127. 969656 030344 6820073135 0 — Cosine.! Sine. I.ota.ng. - Tang.. N. co.N.sine. — I 47 Degrees. 64 Log. Sines and Tangents. (430) Natural Sines. TABLE II. _Sine. D. 10" Cosine. D). 10" Tan,,. D. 10"j Cotang. INsine. N. cos. 0 9.833783 6 9.864127 9.969656 10.030344 168200 73135 60 1 833919 22.5 864010 196 969909 422 030091 68221 73116 59 2 834054 22.5 863892 197 970162 4292 029838 1 68242 73096 58 3 834189 22.5 863774 19.7 970416 42 029584 68264 73076 57 4 834325 225 863656 197 970669 422 029331 68285 73056 56 5 834460 22.5 863538 197 970922 422 029078 68306 73036 55 6 834595 225 863419 19 7 971175 42.2 028825 68327 73016 54 7 834730 226 863301 197 971429 422 028571 68349 72996 53 8 834865 225 863183 197 971682 422 028318 68370 72976 52 9 834999 22. 863064 19.7 971935 42.2 028065 68391 72957 61 10 835134 22.4 862946 19.8 972188 42.2 027812 68412 72937 50 11 9.835269 22.4 9.862827 198 9.972441 42.2 10,027559 68434172917 49 12 835403 22.4 862709 198 972694 422 027306 68455 72897 48 13 835538 22.4 862590 972948 42 027052 68476 72877 47 14 836672 22.4 862471 19.8 973201 422 026799 68497 72857 46 15 835807 22.4 862353 198 973454 422 026546 68518 72837 45 161 835941 22.4 862234 198 973707 422 026293 68539 72817 44 17 836075 22.4 862115 198 973960 42.2 026040 6856172797 43 18 836209 22.3 861996 198 974213 422 025787 6858272777 42 19 836343 3 861877 18 974466 4 2 025534 68603 72757 41 20 836477 22.3 861758 19.8 974719 42,2 025281 68624 72737 40 21 9.836611 22.3 9.861638 19.9 9.974973 42.2 10.025027 68645,72717 39 2 2836745 22.3 861519 19.9 975226 42'2 024774 68666 72697 38 23 836878 22.3 861400 19 976479 4' 2 024521 6868872677 37 24 837012 861280 199 975732 4. 024268 68709 72657 36 22.2 19.9 42,5 7 25 837146 861161 975985 024015 6873072637 35 26 887279 22.2 861041 976238 42 2 023762 68751 72617 34 27 837412 22.2 860922 19.9 976491 V.2 023509 68772 72597 33 28 837546 22.2 860802 19.9 976744 42.2 023256 68793 72677 32 29 837679 22.2 860682 199 976997 4'.2 023003 68814 72557 31 30 837812 22.2 860562 20.0 977250 4' 2 022750 68835 72537 30 31 9.837945 22.2 9860442 9.977503 42 2 10.022497 68857 72517 29 32 838078 22.2 860322 200 977756 42 022244 68878 72497 28 33 838211 22.1 860202 20.0 978009 42.' 021991 68899 72477 27 34 838344 222. 860082 7862 021738 689207247 26 36 838477 22.1 859962 2 978515 42.2 021485 68941 72437 25 36 838610 2.1 859842 200 978768 42. 021232 68962 72417 24 37 838742 22.1 859721 20. 979021' 2. 020979 68983 72397 23 38 838875 22.1 859601 201 979274 42.2 020726 6900472377 22 39 839007 22.1 859480 20.1 979627 42.2 020473 69025 72357 21 40 839140 22.1 859360 20.1 979780 42.2 020220 69046 72337 20 41 9.839272 9 859239.1 9980033 42,2 10.019967 69067 72317 19 42 839404 859119 201 980286 42.2 019714 6908872297118 43 839536 22.0 858998 2 980538 4.2 019462 6910972277 17 44 839668 922.0 858877 20.1 980791 42,2 019209 69130172257 16 45 839800 220 8587562 981044 2 018956 69151172236 15 46 839932 22.0 858635 20.2 981297 41 018703 69172 72216 14 47 840064 22.0 858514 2 98150 42,1 018450 69193 72196 13 48 840196 2 9 858393 20 981803 42.1 018197 69214 72176 12 49 840328 219 868272 20, 982056 42.1 017944 692356 72156 11 60 840459 21.9 858151 20.2 982309 42.1 017691 6925672136 10 51 9.840591 21.9 858029 20.2 982562 42.1 10 017438 69277 72116 9 52 84079 21.9 857908 20.2 982814 42.1 017186 69298 72095 8 653 840854 21.9 857786 20.2 983067 42.1 016933 69319 72075 7 64 840985 21.9 857665 20.2 983320 42.1 016680 69340 72055 6 55 841116 21. 857543 20.3 983573 42.1 016427 6936172035 5 56 841247 21.8 857422 20.3 983826 42.1 016174 69382 72015 4 57 841378 2 857300 984079. 015921 69403 71995 3 58 841509 218 857178 20.3 984331 42.1 015669 6942471974 2 591 8 68 20.3 42.1 59 81640 857056 984584 015416 69445 71954 1 60 841771 21.8 856934 20.3 984837 1 015163 6946671934 0 Gcsine. Sine. O Cotang. Ta ing. ii N. cos. N.sine. 46 Degrees. TABLE II. Log. Sines and Tahngents. (440) ]NAual Snea. n5 Sine. D. 10" Cosine. D. 10'1 Tang. D. 10' Cotang. N. sinei N.,cos. 9.841771 9.856934 2.984837 42 10.015163 6 946 71934 60 21.8 20.3 o 42.6 1 8419022 856812 985090 1 014910 6948771914 69 21 84232103 203 42.198 2k 842033 8 8566902 985343' 0.14657 6952917187394 5 4i 842294721..8 5 20..4 9 482.1 3 8421 21 6668' 985 596 42 1 014404 69579 71873 7 42294 85644620 9858482 014152 69549171853 56 84 2174 204] 42. 842424 2 856323.9486101' 013899 69570171833 55 21825. 8o61 9863542. 7 842685 856078 9865 42 1 013646 659171813 54 7 842865 21 7 8056 248 986607 1' 013393.69612717.92 53 8; 842815 8'5956.98686 11 421 013140 69633 71772 52, 9 84294621'85833 9 87412 1 l!1 012888 69654 71752 51 10( 843076 21.7 855711 20 98736 02635 667 71732 0 21.7 11 20..5; 9873 [ 0;02639 6467 11 9..8432 6 21. 7 855488 20 987618 4 2. 1100232.969 71711 49 12 843336 216 855465 987871 1 12129 697717169 48 1 8 f21.6 ~'~ 0, 5 421 0121291 69717,76 91 48 13 843466 865342 0 988123 42.1 011877; 6973 7A671 47 14 84359 21 6 855219 988376 011624 6975871650 46 15 843725 21 6 855096 988629 42,1 011371 6977071630 45 16 843855 84973 20. 988882 01118 6980 7160 44 17 843984 685 4850 989134 010866 698200 71590 43 18 84411421 854727 989387 42 1 01613 69842 71.69 i42 19 8442432 85246.03 989640 010368) 69862 71649 41 20 844372:21 854480 989893 42 1 010107 16983: 715294 21 9.844502 2. 854356 990145 1110009855 69904 7508 39 22 844631 21 84233 9.990398' 4 1 009602.6992D 71488 38 23 844760 21 5 854109 20.6 990651 42 1 009349 69946 71e168 37 24, 844889 21 5 853986 990903 4 009U97 6996671447 36 2Si 84591 21,5 ~ 20,6' 42.. 14 25' 845018 2L 853862 206 991166 ~ 1 008844eL[.9987 71427 J3, 26i 845147 21 853738 99140.9 4.108591 7000871407 34 27i 845276 853614 991662 42 008338 17,0024171386] 33 28, 845405 214 5o34900 991914' 4 1 0080861 7004971366 32 29' 845533 853366 992167 42 1 00783.3 70070 71345 1 30a 845662 21.. 8532420 992420 42 1 07580 7 l0091i 71325 30 31 9.845790 21 9 853118.7 992672 4 110.007328 01 7012,7130 29 32, 845919 214 2994 9.92925 4 I 007075 701321,184 28 33 846047 1.4 852869.7 993178 42 00682 7053712642 34i 846175 21 852745 993430 1 06570 7017712483.26 35: 846304 21. 2620 20 993.683' 06317 701965171223 25 36 846432 852496 993936 4 UQ 006064 7021 7 102 3 24 37K 846o60 21..3 852371 20 994189 42. 005811 7'023671182 23 21.3 2O.. 8 42.1' 006569 j 7 125~7 42 1 38 846688 2 852247 994441 ~ 005559 1 70257171162 22 3.9. 846816 852122 994694 42 00o3.06 7,0277i71141 21 40; 846944 21 831997 994947. 1 005053 70298 71121 20 419.847071 21.39.851872 9,99.5199 41.-0o04301' 319711,0019 -')I,'i 310 2..8 0 42.1:'39 1'! 40. 8471 9.9, 20 278 9.95452 42; 846719-9;13 851747 9 2452 1 004548 70339'171080 18 43. 847 2 1.37 851622 208 995705 00425 703601171a59 17 44; 84744 21.. 3 81497 995957 42 1 0040431 70311'71039! i6 45' 84782 1.2 51372 99621 42 9 70401 71019 15 46 847709 21.2 851246 996463 0035371 70422170998114 37 2,' 20,39 I42rO 47 847836 851121 99671. 42 1 003285 7.0443!76978 13 48 847964 21. 0996 9 996968 42 00321 7046370957 12 49 848091 21.2, 850870 9 97221 4 2 002779170484:70937 1,50 848218 8 0'745 9.97473 4. 1.0025271 70505,70916 10 6519.84834 21.2 9 850619 9 997726 42 10.002274 1705257089t 9 52 84847 21.2 850493 997979 421 002021 7054670875 8 63 84899 21.1 0368 21.0 998231 421 00179 Ii 70567:70855 7 64 848726 21. 850242 998484 4. 001516 70587 70834 6 55o 848852 21. 850116 998737 2 001263 70608i7081.3 561 848979 21.1 849990 21.1 998989 42.1 001011 70628 70798 4 57 849106 21 849864 999242 4 000768 7064970772 3 58; 849232 849738 999495 000505 70670170752 2 59 849359 1.1 849611 999748 42 0023 1170690t70731 1 4948 o21.0 7 421; 0002 60 849485 21 84948 0.00000 000000oooooo 70711011 Co-i', _I Sine.. Cotang. Tang. N. cols...sil' 45 Degree s, 66 LOGARITHMS TABLE III. LOGARITHMS OF NUMBERS, FROM I TO 110, INCLUDING TWELVE DECIMAL PLACES. N. Log. N. Log. 1 0. 000 000 000 000 36 1. 556 302 500 767 2 0. 301 029 995 644 37 1. 568 201 724 067 3 0. 477 121 254 720 38 1. 579 783 596 617 4 0. 602 059.991 328 39 1. 591 064 607 264 5 0. 698 970 004 336 40 1. 602 059 991 328 6 0. 778 151 250 384 41 1. 612 783 846 720 7 0. 845 098 040 014 42 1. 623 249 290 398 8 0. 903 089 986 992 43 1. 633 468 455 579 9 0. 954 242 509 440 44 1. 643 452 676 486 10 1. 000 000 000 000 45 1. 653 212 513 775 11 1. 041 392 685 158 46 1. 662 757 831 682 12 1. 079 181 246 048 47 1. 672 097 857 936 13 1. 113 943 352 309 48 1. 681 241 237 376 14 1. 146 128 035 678 49 1. 690 196 080 028 15 1. 176 091 259 059 50 1. 698 970 004 336 16 1. 204 119 982 656 51 1. 707 570 176 098 17 1. 230 448 921 378 52 1. 716 003 243 635 18 1. 255 272 505 103 53 1. 724 275 869 601 19 1. 278 753 600 953 54 1. 732 393 759 823 20 1. 301 029 995 664 55 1. 740 362 689 494 21 1. 322 219 294 734 56 1 748 188 027 006 22 1. 342 422 680 822 57 1. 755 874 855 672 23 1. 361 727 836 076 58 1. 763 427 993 563 24 1. 380 211 241 712 59 1. 770 852 011 642 25 1. 397 940 008 672 60 1. 778 151 250 384 26 1. 414 973 347 971 61 1. 785 329 835 011 27 1. 431 363 764 159 62 1. 792 391 689 492 28 1. 447 158 031 342 63 1. 799 340 549 454 29 1. 462 397 997 899 64 1. 806 179 973 984 30 1. 477 121 254 720 65 1. 812 913 356 643 31 1. 491 361 693 834 66 1. 819 543 935 542 32 1. 505 149 978 320 67 1. 826 074 302 701 33 1. 518 513 939 878 68 1. 832 508 912 706 34 1. 531 478 917 042 69 1. 838 849 090 737 35 1. 544 068 044 350 70 1. 845 098 040 014 OF NUMBERS. 67 N. Log. N. Log. 71 1. 851 258 348 719 91 1. 959 041 392 321 72 1. 857 332 496 431 92 1. 963 787 827 346 73 1. 863 322 860 120 93 1. 968 482 948 554 74 1. 869 231 719 731 94 1. 973 127 853 600 75 1. 875 061 263 392 95 1. 977 723 605 289 76 1. 880 813 592 281 96 1. 982 271 233 040 77 1. 886 490 725 172 97 1. 986 771 734 266 78 1. 892 094 602 690 98 1. 991 226 075 692 79 1. 897 627 091 290 99 1. 995 635 194 598 80 1. 903 089 986 992 100 2. 000 000 000 000 81 1. 908 485 018 879 101 2. 004 321 373 783 82 1. 913 813 852 384 102 2. 008 600 171 762 83 1. 919 078 092 376 103 2. 012 837 224 705 84 1. 924 279 286 062 104 2. 017 033 339 299 85 1. 929 418 925 714 105 2. 021 189 299 070 86 1. 934 498 451 244 106 2. 025 305 865 265 87 1. 939 519 252 619 107 2. 029 383 777 685 88 1. 944 482 672 150 108 2. 033 423 755 487 89 1. 949 390 006 645 109 2. 037 426 497 941 90 1. 954 242 509 439 110 2. 041 392 685 158 LOGARITHMS OF THE PRIME NUMBERS FROM 110 TO 1129. INCLUDING TWELVE DECIMAL PLACES. N.. Log. N. Log. 113 2. 063 078 443 483 197 2. 294 466 266 162 127 2. 103 803 720 956 199 2. 298 853 076 410 131 2. 117 271 295 656 211 2. 324 282 455 298 137 2. 136 720 567 156 223 2. 348 304 863 222 139 2. 143 014 800 254 227 2. 356 025 857 189 149 3. 173 186 268 412 229 2. 359 835 482 343 151 2. 178 976 947 293 233 2. 367 355 922 471 157 2. 195 899 653 409 239 2. 378 397 902 352 163 2. 212 187 604 404 241 2. 382 017 042 576 167 2. 222 716 471 148 251 2. 399 673 721 509 173 2. 238 046 103 129 257 2. 409 933 123 332 179 2. 252 853 030 980 263 2. 419 955 748 490 181 2. 257 678 574 869 269 2. 429 752 261 993 191 2. 281 033 367 248 271 2. 432 969 290 877 193 2. 285 557 309 008 277 2. 442 479 768 999 68 LOGARITHMS N. Log. N. Log. 281 2. 448 706 319 906 601 2. 778 874 471 998 283 2. 451 786 435 523 607 2. 783 188 691 074 2.93 2. 466 867 523 5662 613 2. 787 460 556 130 307 2. 487 138 375 477 617 2. 790 285 164 033 311 2. 492 760 389 026 619 2. 791 690 648 987 313 2. 495 544 337 550 631 2. 800 029 359 232 317 2. 501 059 267 324 641 2. 806 858 879 634 331 2. 519 827 993 783 64 2. 808 210 973 921 337 2. 627 629 883 034 647 2. 810 904 280 666 347 2. 540 329 475 079 653 2. 814 912 981 274 349 2. 642 826 426 673 659 2. 818 885 490 409 353 2. 547 774 138 015 661 2. 820 201 459 485 359 2. 555 094 447 578 673 2. 828 015 064 226 367 2. 564 666 064 254 677 2. 830 588 667 946 373 2. 571 708 831 809 683 2. 834 420 703 630 379 2. 678 639 209 957 691 2. 839 477 902 551 383 2. 683 198 773 980 701 2. 845 718 017 237 389 2. 589 949 601 323 709 2. 850 646 235 112 397 2. 598 790 506 763 719 2. 856 728 890 383 401 2. 603 144 372 687 727 2. 861 534 410 855 409 2. 611 723 298 019 733 2. 865 103 970 639 419 2. 622 214 022 971 739 2. 868 643 643 162 421 2. 624 282 095 835 743 2. 870 988 813 759 431 2. 634 477 268 999 751 2. 875 639 937 004 433 2. 636 488 015 871 757 2. 879 095 879 497 439 2. 642 464 620 242 761 2. 881 384 656 769 443 2. 646 403 726 235 769 2. 885 926 339 800 449 2. 652 246 388 777 773 2. 888 179 493 917 457 2. 659 916 200 054 787 2. 895 974 732 358 461 2. 663 700 925 389 797 2. 901 458 321 400 463 2. 665 580 991 012 809 2. 977 948 459 773 467 2. 669 317 881 008 811 2. 909 020 854 210 479 2. 680 335 513 415 821 2. 914 343 157 120 487 2. 687 528 961 120 823 2. 915 399 835 203 491 2. 691 081 487 026 827 2. 917 505 509 487 499 2. 698 100 545 623 829 2. 918 -554 530 558 503 2. 701 567 985 083 839 2, 923 761 960 830 509 2. 706 717 782 345 853 2. -930 949 031 163 521 2. 716 837 623 304 857 2. 932 980 821 917 523 2. 718 502 688 873 859 2. 933 993 163 838 541 2. 733 197 265 134 863 2. 936 010 794 546 547 2. 737 987 326 358 877 2. 942 999 693 360 557 2. 745 855 195 192 881 2. 944 975 908 412 663 2. 750 508 395 940 883 2. 945 960 703 512 569 2. 755 112 178 598 887 2. 947 923 619 839 571 2. 756 636 108 333 907 2. 957 607 287 059 577 2. 761 175 813 171 911 2. 959 518 376 972 687 2. 768 638 004 455 919 2. 963 815 513 609 593 2. 773 054 693 364 929 2. 968 015 713 997 599 2. 777 427 803 257 937 2. 971 739 690 780 OF NUMBERS. 69 N. Log. N. Log. 941 2. 973 589 620 234 1039 3. 016 615 647 558 947 2. 976 349 979 055 1049 3. 020 775 488 195 953 2. 979 092 900 639 1051 3. 021 602 716 026 967 2. 985 426 474 084 1061 3. 025 715 383 898 971 2. 987 219 229 907 1063 3. 026 533 264 523 977 2. 989 894 659 717 1069 3. 028 977 705 205 983 2. 992 553 512 733 1087 3. 036 229 513 712 991 2. 996 073 604 003 1091 3. 037 824 749 671 997 2. 998 695 158 313 1093 3. 038 620 157 372 1009 3. 003 891 170 203 1097 3. 040 206 627 571 1013 3. 005 609 445 427 1103 3. 042 575 512 437 1019 3. 008 174 244 007 1109 3. 044 931 546 149 1021 3. 009 025 742 086 1117 3. 048 053 173 103 1031 3. 013 258 660 430 1123 3. 050 379 756 239 1033 3. 014 100 321 518 1129 3. 052 693 942 370 It is not necessary to extend this table, as the logarithm of any one of the higher numbers can be readily computed by the following formula, which may be found in any of the standard works on algebra, namely: Log. (z+l)=log. z+0.8685889638 ( 22. ) The result will be true to ten decimal places for all numbers over 1000, and true to twelve decimals for all numbers over 2000. The logarithms of composite numbers can be determined by the combination of logarithms already in the table, and the prime numbers from the formula. Thus, the number 3083 is a prime number, find its logarithm, true to ten places of decimals. We first find the logarithm of 3082. By factoring this number, we find that it may be composed by the multiplication of 46 into 67. Log. 46..............1. 662 757 8316 Log. 67..............1. 826 074 3027 Log. 3082............3. 488 832 1343 Now, Log. 3083=3.4888321343+ 856658896 38 We give a few additional prime numbers: 1151 1223 1291 1373 1451 1511 1153 1229 1297 1381 1453 1523 1163 1231 1301 1399 1459 1531 1171 1237 1303 1409 1471 1543 1181 1249 1307 1423 1481 1549 1187 1259 1319 1427 1483 1553 1193 1277 1321 1429 1487 1559 1201 1279 1327 1433 1489 1567 1213 1283 1361 1439 1493 1571 1217 1289 1367 1447 1499 1579 70 AUXILIARY LOGARITHMS. AUXILIARY LOGARITHMS. N. Log. N. Log. 1. 009 0. 003 891 170 203 1. 0009 0. 000 390 576 304 1. 008 0. 003 461 527 188 1. 0008 0. 000 347 233 698 1. 007 0. 003 030 465 635 1. 0007 0. 000 303 836 798 1. 006 0. 002 597 985 739 1. 0006 0. 000 260 435 561 1. 005 0. 002 166 071 750 A 1. 0005 0. 000 217 099 966 B 1. 004 0. 001 733 722 804 1. 0004 0. 000 173 690 053 1. 003 0. 001 300 943 017 1. 0003 0. 000 130 268 803 1. 002 0. 000 867 721 529 1. 0002 0. 000 086 850 213 1. 001 0. 000 434 077 479 1. 0001 0. 000 043 427 277J N. Log. 1. 00009 0. 000 039 084 741 1. 00008 0. 000 034 742 166 1. 00007 0. 000 030 399 546 1. 00006 0. 000 026 056 884 1. 00005 0. 000 021 714 178 1. 00004 0. 000 017 371 430 1. 00003 0. 000 013 028 638 1. 00002 0. 000 008 685 802 1. 00001 0. 000 004 342 923 (a) 1. 000001 0. 000 000 434 294 (b) 1. 0000001 0. 000 000 043 429 (c) 1. 00000001 0. 000 000 004 343 (d) 1. 000000001 0. 000 000 000 434 (e) 1. 0000000001 0. 000 000 000 043 (f) Number. Log. 0. 4342944819............ — 1. 637 784 298 This decimal number is the modulus of our system of logarithms. Its logarithm is very useful in correcting other logarithms, as may be seen in the Chapter on Logarithms. T RAVERSE TABLE. 71 Y Deg. 1 Deg. | 1/ Deg. 2 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 1.00 0. 01 1. 00 0. 02 1.00 0.03 1.00 0. 03 2 2. 00 0. 02 2. 00 0. 03 2. 00 0. 05 2. 00 0. 07 8 3. 00 0. 03 3. 00 0. 05 3.00 0.08 3.00 0. 10 4 4. 00 0. 03 4. 00 0. 07 4. 00 0. 10 4. 00 0. 14 6 5. 00 0. 04 5. 00 0. 09 5. 00 0. 13 5. 00 0. 17 6 6. 00 0. 05 6. 90 0. 10 6. 00 0. 16 6. 00 0. 21 7 7. 00 0. 06 7.00 0. 12 7.00 0. 18'7.00 0. 24 8 8. 00 0. 07 8. 00 0. 14 8. 00 0. 21 7. 99 0. 28 9 9. 00 0. 08 9. 00 0. 16 9. 00 0. 24 8. 99 0. 31 10 10. 00 0.09 10. 00 0. 17 10.00 0.26 9. 99 0. 35 11 11. 00 0. 10 11. 00 0. 19 11.00 0.28 10. 99 0. 38 12 12. 00 0. 10 12. 00 0. 21 12. 00 0. 31 11. 99 0. 42 13 13. 00 0. 11 13. 00 0.23 13. 00 0. 34 12. 99 0. 45 14 14. 00 0. 12 14. 00 0. 24 14. 00 0. 37 13. 99 0. 49 15 15. 00 0. 13 15. 00 0. 26 14. 99 0. 39 14. 99 0. 52 16 16. 00 0. 14 16. 00 0.28 15.99 0. 42 15. 99 0. 56 17 17. 00 0. 15 17. 00 0. 30 16. 99 0. 45 16. 99 0. 59 18 18.00 0. 16 18.00 0. 31 17. 99 0.47 17. 99 0. 63 19 19. 00 0. 17 19. 00 0. 33 18. 99 0. 50 18. 99 0. 66 20 20.00 0. 17 20.00 0. 35 19. 99 0. 52 19. 99 0. 70 21 21.00 0. 18 21.00 0. 37 20. 99 0. 55 20. 99 0.73 22 22.00 0. 19 22. 00 0. 38 21. 99 0. 58 21. 99 0. 77 23 23. 00 0.20 23. 00 0. 40 22. 99 0.60 22. 99 0.80 24 24. 00 0. 21 24. 00 0. 42 23. 99 0. 63 23. 99 0. 84 25 25 00 0. 22 25. 00 0.44 24. 99 0. 65 24. 98 0.87 26 26. 00 0.23 26.00 0.45 25. 99 0. 68 25. 98 0. 91 27 27. 00 0. 24 27. 00 0. 47 26. 99 0. 71 26. 98 0. 94 28 28. 00 0.24 28, 00 0. 49 27. 99 0. 73 27. 98 0. 98 29 29. 00 0. 25 29. 00 0. 51 28. 99 0. 76 28. 98 1. 01 30 30.00 0.26 30.00 0.52 29. 99 0. 19 29. 98 1.05 35 35. 00 0. 31 34. 99 0. 61 34. 99 0. 92 34. 98 1. 22 40 40. 00 0. 35 39. 99 0. 70 39. 99 1. 05 39. 98 1. 40 45 45. 00 0. 39 44. 90 0. 79 44. 99 1. 18 44. 97 1. 57 50 50. 00 0.44 49. 99 0. 87 49. 98 1. 31 49. 97 1. 74 55 55. 00 0. 48 54. 99 0. 96 54. 98 1. 44 54. 97 1. 92 60 60. 00 0. 52 59. 90 0. 05 59. 98 1. 57 59. 96 2. 09 65 65. 00 0.57 64. 99 1. 13 64. 98 1. 70 64. 96 2. 27 70 70. 00 0. 61 69. 99 1. 22 69. 98 1. 83 69. 96 2. 44 75 75. 00 0. 65 74. 99 1. 31 74. 97 1. 96 74. 95 2. 62 80 80. 00 0. 70 79. 99 1. 40 79. 97 2. 09 79. 95 2. 79 85 85. 00 0. 74 84. 99 1. 48 84. 97 2. 23j 84. 95 2. 97 90 90.00 0. 79 89. 99 1.57 89. 97 2. 36 89. 95 3. 14 95 90. 00 0. 83 94. 99 1. 66 94. 97 2. 49 94. 94 3. 32 100 100. 00 0. 87 99. 98 1. 75 99. 97 2. 62 99. 94 3. 49 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 891, Deg. 89 Deg. l 88/ Deg. 88 Deg. 7 tAYA t RAV tSE TABLE. 2/4 Deg. 3 Deg. 32 Deg. 4 Deg. i" - _....... 4,,, D Lat.' Dep. Lat. Dep. Lat. lDep. Lat. Dep. i 1,00 0. 04 1. 00 0. 05 1. 00 0:06 i. 00 0. 07 2 I. 00 0.09 2. 00. 10 2 00 0. 12 f.00 0. 14 3 3.00.1 3. 00 0. 16. 99 0. 18 2.99 0.21 4 4.00 0. 17 9. 99 0;21. 99 0.24 9.99 a.28 5 5.00 0. A2 4. 99 0.26 4. 99 0;31 4.99 0.35 6 6,99 0.26 5. 99 0.131 5. 99 0. 37 5i 99 0.42 17. 99 0. 31 6. 99 O. 37 6. 99 0. 43 6; 98 O. 49 8 1.99 0.35'1.99 0.42- la99 0;49' It98 0.66 9 8. 99 0. 39 8. 99 O. 47 8. 98 0. 55 8:98 0. 63 10 9. 99 0.44 9. 99 0,ti 9.98 0.61 9. 98 0.70 11 10. 99 0.48 10. 98 O. 58 10.98 0. 67 10. 91 0. 77 12 11.99 0. 5 11. 98 0. 63 i. 98 0. 73 il. 917. 84 i3 12.99 0.57 12. 98 0.68 1i299 0.79 12l 91 0.91 14 13.99 0, 61 13. 98 0.'73 19. 97 0. 85 1,' 91 0. 98 15 14, 99 0. 65 14. 98 0. 19 i4. 97 O. 92 14. 96 i. 05 16 15, 99 0. 0 15. 98 0 84 15. 91 O. 98 I15: 96 i. 17 16. 98 0. 74 16.98 0. 9 i. 97 1.04 16. 96 i. 19 18 11, 98 0.79 11. 98 0. 94 17. 9 i. 10 17. 9 i.2 6 19 18. 98 0, 83 18. 98 o. 99 18. 96 i. 16 18. 95 i. 83 20 19.98 0. 87 19. 97 1.05 19.96 1.22 19. 95 i. 40 21 20: 98 0, 92 20. 97 1. 1 20. 96 1. 28 20 95 1. 46 22 21. 98 I0,96 21.91 1. 15 211.96 i.34.i 95 i.53 23 22. 98 1.00 o 12.97 1. 20 22. 96 1. 40 22. 94 1. 60 24 23.98 1.05 s3.97 1.26,23.96 1.47 23.94 i. 67 25 24. 98 t. 09 24; 97 i. 1 24. 95 1i 53 244 94 1. 74 26 25.96 1. 13 25. 96 i. g o 5.95 1.59 25.94 1.8i 27 26. 9 1. 18 26. 96 1. 41 26. 95 1. 65 26. 93 1, 88 28 27. 97 1. 22 27. 96 i. 41 27. 95 1. l1 217 9 1. 95 29 28. 91 1.26 2 8.96 1.62 2s., 95 1; 17 28.93 2.02 30 29. 97 1.31 2996 1.57 29. 94 1i 83 29.931 2. 09 35 34. 97 1. 5' 34. 95 1. 83 34. 93 2. t4 34.91 2. 44 40 39. 96 1. 7 39. 95 2. 09 39. 93 2. 44 39. 9d 2. 79 45 44.96 1. 96 44. 94 2. 3 44. 92 2.75 489 3. 14 50 49. 95 2. 18 49. 93. 62s 49. 91 9. 05 49. 88 3. 49 55 54.95 2. 40 54. 92 2.88 54.90 3. 86 1 54. 8 3.84 60 59. 94 2. 62 59. 92 3. 14 59. 89 39 66 59. 83 4. 19 65 64. 94 2. 84 04. 91 3. 40 64. 88 },97 64. 84 4. 53 70 09. 93 3.05 69. 90. 66 d9. 87 4. 27 6g9 8 4. 88 75 74. 93 3. 27 74. 90 3. 93 14. 86 4. 58 74. 82 5; 29 80 79.92 3. 49'79.89 4. 19 79. 85 4. 88. 79,81 t'58 85 84.92 3.71 84. 88 4.45 84, 84 5. 19 84. 79 5. 99 90 89. 91 3. 93 89. 98 4. 71 89. 83 5, 49 89. 78 0. 28 95 94. 91 4. 14 94. 8 4. 97 94. 82f 5. 80 94,'7 6. 63 100 99. 91 4. 36 99. 86 5. 23 99. 81 6. 10 99. 76 6. 98 Dep' Lat. Dep. Lat. Dep. t Lat. Dep. Lat 871/2 beg, 8 Deg. 86:bteg. 86 Deg..~fe., Dp..s..,, TRAtRISE TABLt; 7L il 42 Deg. 6 beg, 6$ Deg, 6 beg. Lat. Dep. Lat. Dep. L. Dep. Lat. Dep. Lat Dep; 1 1.900 0.8 1.00 0.09 1d00 0 l O. 99 0.10 2 1. 99 0. 16 1, 99 0. 11 1.99 0. 19 1 99 0; 21 3 2. 99 O. 24; 99 0. 26 2, 99 0.29 2. 98 0. 31 4 3. 99 O. 31 3, 98 0. 35 3. 98 0. 38,90 0. 41 5 4. 98 O. 39 4 98 O. 44 4. 98 0. 48 4. 97 0. 52t 6 5. 98 0. 47 5, 98 0.;5 5. 97 0; 58 B. 71 0. 63 7 6. 98 0. 55 6,91 0.61 6. 97 0,67 0.. 96 0. 7 8 7. 98 0. 63 7,97 0.70 7.96 0,76 7.96 0.84 9 8. 97 0. 71 8, 97 0.; 78 8, 96 0, 86. -,95 O. 94 10 9.97 0.78 9. 96 0,87 9.95 0,96 9..9 1. 0P 11 10.97 0.86 10.90 0.96 10, 95; 05 10.94 1.1 12 11. 94 0. 94 115 1. 05. 94 1.15 11. 25 139 12. 90 1.02 12, 95. 1.1 12. 94 i. 25 2.; 93 1., 36 14 13. 90 1. 10 13, 95 1 22 13. 94 1i 34 13. 92 1.: 40 15 14. 95.- 18 14, 94 1, 31 14. 93 t. 44 14, 92 1. 57 16 15.95 1.26 15. 94 1, 39 15. 93 1: 15, 91. t; 61 17 16. 95 1. 33 16, 94 1, 48 16. 92 1. 63 O16 91 1. 7& 18 17. 94 1. 41 17. 93 1. 51 17.92 1.g 7 17. 90 1. 88 19 18.94 1.49 18,93 1.60 18,91i 182 I8. 90 1,99t 20 19. 94 1.51 19. 92 1, 74 1 19, 91 1. 92 19. 89 2. 09 21 20. 94 1. 65 20, 92 1. $3 20, 90 2. 01 20. 88 2. 20 22 21. 9 1.73 21. 92. 92 2. 21o 2.11 2; 88. 2 30 23 22, 93 1. 80 22, 91 2. 00 22. 89 2. 0- 22. 81 2, 40 24 23.i93 1.88 23 91 2. 09g 2. 89 2. 30 23. 81 2.51 25 24. 92 1.90 24. 90 2. 18 24, 88 2. 40 24. 86 2.; 61 26 25. 92 2. 04 25, 90 2. 27 25, 88 I. 49 25. 286. 72 7 26. 92 2. 12 26. 90 2. g 26.88 2. 59 26.85 2. 82 28 27. 91 2. 20 27. 89 2. 44 27. 817 2.68 21. 8$' 2. 93; 29 28 91 2. 28 28. 89 2. 53 28. 8r7 2. 7.8 28. 84 3. 0e3 30 29. 21 2. 5 29. 89 2.61 29. 8. 2.8s 29. 84 3. 14 35 34. 8 2. 75 34. 87 3. O5 34. 84 3. 35 34. 81 39..66 40 39. 88 3. 14 39. 85 3. 49 39. 82 3 83 39. 78 4. 18 45 44, 86 3. 53 44. 83 3, 92 44. 79 4. 31 44. 75 4. 70 50 49. 85 3. 92 49. 81 4. 306 49. 77 4. 79 49. 73; 5;2 55 54. 85 4, 0 54. 79 4. 79 54,75 5. 21 54.70 5. 15 60 59. 84 4. 45 9. 77 5, 29 59. 72 5.75 59. (f 0.27 65, 64.. 82 4. 82 04. 75 5. 67 64. 70 6. (2 3 4. 64 6.'9 70 69. 81 5. 19 69. 73 6, 10 69. 06 6.71 6-9. 62 7.. 32 75 74. 79 5, 65 74. 71 6. 54 74. 65 7. 19 14.- 59 1. 84 80 1 79. 78 S. 93 79. 70 6, 97 79. 6 7. 67 76. 50 8. 36 85 84, 77 6. 30 84, 68 7. 41 84. 01 8. 15 84. 53 8. 88 90 89. 75 6. 67 89. 66 7. 84 89,590 8. 6 89. 51 9.41 95 94. 74 7. 04 94. 64 8. 28 94. 56 9. 11 94. 48 9.. 93 100- 99. 7 7, 41 99, 62. 98 72 99. 54 9. 58 99. 45 10. 43 - 6/ lYeL. Dep. La. Dep. at. el at. Dep. Lt. De85 g.! 85 Deg. 8s Deg. 84 Deg. _..,.. 9....,,J,..~~~~~~~~~i 74 TRAVERSE TABLE. 6Y Deg. 7 Deg. 7 Deg. 8 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.99 0. 11 0.99 0.12 0.99 0.13 0.99 0. 14 2 1. 99 0. 23 1.99 0. 24 1. 98 0.26 1. 98 0. 28 3 2. 98 0. 34 2. 98 0. 37 2 97 0. 39 2. 97 0. 42 4 3. 97 0. 45 3. 97 0. 49 3. 97 0. 52 3. 96 0. 56 5 4. 97 0. 57 4. 96 0. 61 4. 96 0.65 4. 95 0. 70 6 5. 96 0. 68 5. 96 0. 73 5. 95 0.78 5. 94 0. 84 7 6. 96 0. 79 6. 95 0. 85 6. 94 0. 91 6. 93 0. 97 8 7. 95 0. 91 7. 94 0. 97 7. 93 1.04 7. 92 1. 11 9 8. 94 1. 02 8. 93 1. 10 8. 92 1. 17 8. 91 1. 25 10 9. 94 1. 13 9. 93 1. 22 9. 91 1.31 9. 90 1. 39 11 10. 93 1. 25 10. 92 1. 34 10. 91 1.44 10. 89 1. 53 12 11. 92 1. 36 11. 91 1. 46 11. 90 1, 57 11. 88 1. 67 13 12. 92 1. 47 12. 90 1.58 12. 89 1.70 12.87 1. 81 14 13. 91 1. 59 13. 90 1. 71 13. 88 1. 83 13. 86.1. 95 15 1i4. 90 1. 70 14. 89 1.83 14. 87 1. 96 14. 85 2. 09 16 15. 90 1. 81 15. 88 1. 95 15. 86 2. 09 15. 84 2. 23 17 16. 89 1. 92 16. 87 2. 07 16. 85 2. 22 16. 83 2. 37 18 17. 88 2. 04 17. 87 2. 19 17. 85 2.35 17. 82 2. 51 19 18. 88 2. 15 18. 86 2. 32 18. 84 2.48 18. 82 2. 64 20 19. 87 2. 26 19. 85 2. 44 19. 83 2.61 19. 81 2. 78 21 20. 87 2. 38 20. 84 2. 56 20. 82 2. 74 20. 80 2. 92 22 21. 86 2. 49 21. 84 2. 68 21. 81 2.87 21. 79 3. 06 23 22. 85 2. 60 22. 83 2. 80 22. 80 3.00 22. 78 3. 20 24 23.85 2. 72 23. 82 2. 92 23. 79 3.13 23. 77 3. 34 25 24. 84 2. 83 24. 81 3. 05 24. 79 3.26 24. 76 3. 48 26 25. 83 2. 94 25.81 3. 17 25. 78 3. 39 125. 75 3. 62 27 26. 83 3. 06 26. 80 3. 29 26. 77 3.52 26. 74 3. 76 28 27. 82 3. 17 27. 79 3. 41 27. 76 3.65 27. 73 3. 90 29 28. 81 3. 28 28. 78 3. 53 28. 75 3.79 28. 72 4. 04 30 29. 81 3. 40 29. 78 3. 66 29. 74 3.92 29. 71 4. 18 35 34. 78 3. 96 34. 74 4. 27 34. 70 4. b7 34. 66 4. 87 40 39. 74 4. 53 39. 70 4. 87 39. 66 5. 22 39. 61 5. 57 45 44. 71 5. 09 44. 67 5. 48 44. 62 5.'87 44. 56 6. 26 50 49, 68 5. 66 49. 63 6. 09 49. 57 6. 53 149. 51 6. 96 55 54. 65 6. 23 54. 59 6. 70 55. 58 6. 70 54. 46 7. 65 60 59. 61 6. 79 59. 55 7. 31 59. 55 7.31 159. 42 8. 35 65 64. 58 7. 36 64. 52 7. 92 64. 52 7. 92 64. 37 9 05 70 69. 55 7. 92 69. 48 8. 53 69. 48 8. 53 69. 32 9. 74 75 74. 52 8. 49 74. 44 9. 14 74.44 9. 14 74. 27 10. 44 80 79. 49 9. 06 79. 40 9. 75 79. 40 9. 75 79. 22 11. 13 85 84. 45 9. 62 84. 37 10. 36 84. 37 10. 36 84. 17 11. 83 90 89. 42 10. 19 89. 33 10. 97 89. 33 10. 97 89. 12 12. 53 95 94. 39 10. 75 94. 29 11. 58 94. 29 11.58 94. 08 13. 22 100 99. 36 11. 32 99. 25 12. 19 99. 25 12. 19 99. 03 13. 92 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 83~2 Deg. 83 Deg. 82/2 Deg. 82 Deg. TRAVERSE TABLE. 75 8 Deg. 9 Deg. 9 e. Deg. 10 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0. 99 0. 15 0.99 0.16 0. 99 0. 17 0. 98 0.17 2 1.98 0.30 1.98 0.31 1.97 0.33 1.97 0.35 3 2. 97 0. 44 2. 96 0. 47 2. 96 0. 50 2. 95 0. 52 4 3. 96 0. 59 3. 95 0. 63 3. 95 0. 66 3. 94 0. 69 5 4.95 0. 74 4. 94 0. 78 4.93 0. 83 4. 92 0. 87 6 5. 93 0. 89 5. 93 0. 94 5. 92 0. 99 5. 91 1.04 7 6. 92. 03 6.91. 10 6. 90 1. 16 6. 89 1. 22 8 7. 91 1. 18 7. 90 1. 25 7. 89 1. 32 7. 88 1. 39 9 8. 90 1. 33 8. 89 1. 41j1 8. 88 1. 49 8. 86 1. 56 10 9. 89 1. 48 9. 88 1. 56 9. 86 1. 65 9. 85 1. 74 11 10. 88 1. 63 10. 86 1. 72 10. 85 1. 82 10. 83 1.91 12 11. 87 1. 77 11. 85 1. 88 11. 84 1. 98 11. 82 2. 08 13 12. 86 1. 92 12. 84 2. 03 12. 82 2. 15 12. 80 2. 26 14 13. 85 2. 07 13. 83 2. 19 13. 81 2. 31 13. 79 2. 43 15 14. 84 2. 22 14. 82 2. 35 14. 79 2. 48 14. 77 2. 60 16 15. 82 2. 36 15. 80 2. 50 15. 78 2. 64 15. 76 2. 78 17 16. 81 2.1 16.79 2.66 16.77 2.81 16. 74 2.95 18 17. 80 2. 66 17. 78 2. 817.75 2.97 17. 73 3. 13 19 18. 79 2. 81 18. 77 2. 97 18. 74 3. 14 18.71 3. 30 20 19.78 2.96 19.75 3. 13 19.73 3.30 19.70 3.47 21 20. 77 3. 10 20. 74 3. 29 20.71 3. 47 20. 68 3. 65 22 21. 76 3. 25 21. 73 3. 44 21. 70 3.63 21.67 3. 82 23 22. 75 3. 40 22. 72 3. 60 22. 68 3. 80 22. 65 3. 99 24 23. 74 3. 55 23.70 3. 75 23. 67 3.96 23.64 4. 17 25 24. 73 3. 70 24. 69 3. 91 24. 66 4. 13 24. 62 4. 34 26 25. 71 3. 84 25. 68 4. 07 25. 64 4. 29 25. 61 4. 51 27 26. 70 3. 99 26. 67 4. 22 26. 63 4. 46 26. 59 4. 69 28 27. 69 4. 14 27. 66 4. 38 27. 62 4. 62 27. 57 4. 86 29 28. 68 4. 29 28. 64 4.4 28.60 4. 79 28.56 5. 04 30 29. 67 4. 43 29. 63 4. 69 29. 59 4. 95 29. 54 5. 21 35 34. 62 5. 17 34. 57 5. 48 34. 52 5. 78 34. 47 6. 08 40 39. 56 5. 91 39. 51 6. 26 39. 45 6. 60 39. 39 6. 95 45 44. 51 6. 65 44. 45 7. 04 44. 38 7. 43 44. 32 7. 81 50 49. 45 7. 39 49. 38 7. 82 49. 32 ) 8. 25 49. 24 8. 68 55 54. 40 8. 13 54. 32 8. 60 54. 25 9. 08 54. 16 9. 95 60 59. 34 8. 87 59. 26 9. 39 59. 18 9..90 59. 09 10. 42 65 64.29 9. 61 64. 20 10. 17 64. 11 10. 73 64. 01 11. 29 70 69. 23 10. 35 69. 14 10. 951 69. 04 11. 551 68. 94 12. 16 75 74. 18 11. 09 74. 08 11. 731 73. 97 12. 38 73. 86 13. 02 80 79. 12 11. 82. 79. 02 12. 51 78. 90 13. 20 78. 78 13. 89 85 84. 07 12. 56 83.95 13. 30 83. 83 14. 03 83. 71 14. 76 90 89. 01 13. 30 88. 89 14. 08 88. 77 14. 85 88. 63 15. 63 95 93. 96 14, 04 93. 83 14. 86 93. 70 15. 68 93. 56 16. 50 100 98. 90 14. 78 98. 77 15. 64 98. 63 16. 50 98. 48 17. 36 Dep. Lat.-I Dep. Lat. Dep. Lat. Dop. Lat. 8112 Deg.! 81 Deg. 80/4 Deg. 80 Deg. 26 76 TRAVERSE TABLE. _ _ _ _ _ _ _ I _ _ _ _ _I 1. 0. 98 Q. 18 0. 98 0. 19 0. 98 0.20 0o. 98 0. 21 2 1. 97 0. 36 1. 96 0. 38 1. 96 0.40 1. 96 0. 42 3 2. 95 0.55 2.94 0.57 2.94 0.60 2. 93 0.62 4 3.93 0. 73 3.93 0.76 3. 92 0. 80 3.91 0. 83 5 4. 92 0. 91 4. 91 0. 95 4. 90 1.00 4. 89 1. 04 6 5. 90 1. 09. 89 1. 14 5. 88 1.20 5. 87 1. 25 7 6. 88 1. 28 6. 87 1. 34 6. 86 1.40 6. 85 1. 46 8 7. 87 1. 46 7. 85 1. 53 7. 84 1.59 7. 83 1.66 9 8. 85 1. 64 8. 83 1. 72 8. 82 1.79 8. 80 1. 87 10 9. 83 1.82 9.82 1. 91 9. 80 1.99 9. 78 2. 08 11 10.82 2. 00 10. 80 2.10 10.78 2. 19 10. 76 2. 29 12 11. 80 2. 19 11.78 2 29 11. 76 2.39 11.74 2. 49 13 12.78 2. 37 12. 76 2.48 12.74 2.59 12.72 2.70 14 13. 77 2. 55 13. 74 2. 67 13. 72 2.79 13.69 2. 91 15 14. 75 2.73 14. 72 2.87 14.70 2.99 14.67 3. 12 16 15.73 2. 92 15.71 2.86 15. 68 3.19 15.65 3. 33 17 16. 72 3. 10 6.69 3. 05 16. 66 3. 39 16. 63 3. 53 18 17. 70 3. 28 17.67 3.24 17. 64 3.59 17. 61 3. 74 19 18. 68 3. 46 18.65 3. 43 18.62 3.79 18.58 3. 95 20 19. 67 3. 64 19. 63 3.63 19. 60 3.99 19.56 4. 16 21 20. 65 3. 83 20. 61 3. 82 20. 58 4. 13 20. 54 4. 37 22 21.66 4.01 21 4.0121.56 4. 39 21.52 4.57 23 22. 61 4. 19 22.58 4. 20 22.54 4. 59 22.50 4. 78 24 23.604 4.37 23.56 4.39 23.52 4.78 23.48 4.99 25 24.58 4. 56 24.54 4.58 24.50 4.98 24.45 5.20 26 25. 56 4. 74 25. 52 4.77 25.48 5.18 25. 43 5. 41 | 27 26. 55 4. 92| 26. G0 4. 96 26. 46 5. 38 26. 41 5.61 28 27. 53 5. 10 27. 49 5.15 27. 44 5.58 2 7. 39 5. 82 29 28.5 1 5. 28 28. 47 5. 34 28.42 5.78 28. 37 6.03 30 29.50 5. 4 29.5.5. 72 29.40 5.98 29.34 6.24 35 34.411 6. 38 34. 36 6. 681 34. 30 6.98 34.24 7. 28 40 39. 33 7. 29 39.27 7.631139.20 7.97 39.13 8.32 45 44.25 8. 20 44.17 8. 59 144. 10 8.97 44.02 9.36 50 49. 16 9. 11 49.08 9.54 49. 00 9.97 48.91 10. 40 55 54. 08 10. 02 53. 99 10. 49 53. 90 10.97 53. 80 11.44 60 59. oo 10.93 58.90 11.451 58. 80 11.96 58. 69 12.47 65 63.91 11.85 63. 81 12. 40 63.70 12.96 63.58 13.51 70 68. 83 12. 76 68. 71 13. 36 I68. 59 13.96 68. 47 14. 55 75 73. 74 13. 67 73. 62 14. 31 73. 49 14.95 73. 36 15. 59 80 78. 66 14. 58 78. 53 15. 26 78. 39 15. 95 78. 25 16. 63 85 83. 58 15. 49 83. 44 16. 22 83. 29 16.95 83. 14 17. 67 90 88. 49 16. 40 88. 35 17. 17 88. 19 17.94 88. 03 18. 71 95 93. 41 17. 311193. 25 18. 13 93. 09 18. 94 192. 92 19. 75 100 98. 33 18. 22 98. 16 19. 08 97. 99 19. 94 97. 81 20. 79 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 79|| 2 Deg. 79 Deg. 78~/ Deg. 78 Deg.,....~~~~~~~~~~~~~~I TRAVERSE TABLE. 77 1232 Deg. 13 Deg. 132 Deg. 14 Deg. Lat. | Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.98 0. 22 0.97 0.23 0. 97 0.23 0.97 0.24 2 1.95 0. 43 1.95 0. 46 1. 95 0. 47 1.94 0.48 3 2. 93 0. 65 2. 92 0. 67 2. 92 0. 70 2. 91 0. 73 4 3. 91 0. 87 3. 90 0. 90 3. 89 0. 93 3. 88 0. 97 5 4. 88 1. 08 4. 87 1. 12 4. 86 1. 17 4. 85 1. 21 6 5. 86 1. 30 5. 85 1. 35 5. 83 1. 40 5. 82 1. 45 7 6.83 1. 52 6. 82 1.57 6.81 1. 63 6.79 1. 69 8 7. 81 1. 73 7. 80 1. 80 7. 78 1. 87 7. 76 1. 94 9 8. 79 1. 95 8. 77 2. 02 8. 75 2. 10 8. 73 2. 18 10 9. 76 2. 16 9. 74 2. 25 9. 72 2. 33 9 70 2. 42 11 10.74 2. 38 10.72 2. 47 10.70 2.57 10. 67 2. 66 12 11.72 2. 60 11.69 2.70 11.67 2. 80 11. 64 2.90 13 12.69 2. 81 12. 67 2. 92 12.64 3. 03 12. 61 3. 15 14 13. 67 3. 03 13. 64 3. 15 13. 61 3. 27 13. 58 3. 39 15 14.64 3. 25 14. 62 3. 37 14. 59 3. 50 14. 55 3. 63 16 15.62 3. 46 15. 59 3. 60 15. 56 3. 74 15.52 3. 87 17 16. 60 3. 68 16. 57 3. 82 16. 53 3. 97 16. 50 4. 11 18 17. 57 3. 90 17. 54 4. 05 17. 50 4. 20 17. 47 4. 35 19 18.55 4. 11 18. 51 4. 27 18. 48 4.44 18.44 4.60 20 19.53 4. 33 19. 49 4. 50 19. 45 4. 67 19.41 4. 84 21 20. 50 4. 55 20. 46 4. 72 20. 42 4. 90 20. 38 5. 08 22 21.48 4. 76 21. 44 4. 95 21. 39 5. 14 21. 35 5. 32 23 22. 45 4. 98 22. 41 5. 17 22. 36 5. 37 22. 32 5. 56 24 23. 43 5. 19 23. 38 5. 40 23. 34 5. 60 23. 29 5. 81 25 24. 41 5. 41 24. 36 5. 62 24. 31 5. 84 24.26 6. 05 26 25. 38 5. 63 25. 33 5. 85 25. 28 6. 07 25. 23 6 29 27 26. 36 5. 84 26. 31 6. 07 26. 25 6. 30 26. 20 6. 53 28 27. 34 6. 06 27. 28 6. 30 27. 23 6. 54 27. 17 6. 77 29 28. 31 6. 28 28. 26 6. 52 28.20 6. 77 28. 14 7. 02 30 29. 29 6. 49 29. 23 6. 75 29. 17 7. 00 29. 11 7. 26 35 34. 17 7. 58 34. 10 7. 87 34. 03 8. 17 33. 96 8. 47 40 39. 05 8. 66 38. 97 9.00 38. 89 9. 34 38. 81 9. 68 45 43. 93 9. 74 43. 85 10. 12 43. 76 10. 51 43. 66 10. 89 50 48. 81 10. 82 48. 72 11. 25 48. 6f 11. 67 48. 51 12. 10 55 53. 70 11. 90 53. 59 12. 37 53. 48 12. 84 53. 37 13. 31 60 58. 58 12. 99 58. 46 13. 50 58. 34 14. 01 58. 22 14. 52 65 63. 46 14. 07 63. 33 14. 62 63. 20 15. 17 63. 07 15. 72 70 68. 34 15. 15 68. 21 15. 75 68. 07 16. 34 67. 92 16. 93 75 73. 22 16. 23 73. 08 16. 87 72. 93 17. 50 1172. 77 18. 14 80 78. 10 17. 32 77. 95 18. 00 77. 79 18. 68 77. 62 19. 35 85 82. 99 18. 40 82. 82 19. 12 82. 65 19. 841 82. 48 20. 56 90 87. 87 19. 48 87. 69 20. 25 87. 51 21. 01 87. 33 21. 77 95 92. 75 20. 56 92. 57 21. 37 92. 38 22. 18 92. 18 22. 98 100 97. 63 21. 64 97. 44 22. 50 97. 24 23. 34[ 97. 03 24. 1 9 Dep. Lat. Dep. Lat. Dep. Lat. 1)p. Lat. 77Y2 Deg. 77 Deg. 763/ Deg. 1I 76 Deg. 78 TRAVERSE TA-BLE. t U14y2 Deg. 15 Deg. 1512 Deg. 16 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. I Dep. 1 0.97 0.25 0. 97 0.26 0. 96 0.27 0.96 0.28 2 1. 94 0. 50 1. 93 0. 52 1. 93 0.53 1. 92 0. 55 3 2. 90 0. 75 2. 90 0. 78 2. 89 0.80 2. 88 0. 83 4 3. 87 1. 00 3. 86 1. 04 3. 85 1.07 3. 85 1. 10 5 4. 84 1. 25 4. 83 1. 29 4.-82 1.34 4. 81 1. 38 6 5.81 1.50 5.80 1.55 5. 78 1.60 5. 77 1.65 7 6. 78 1. 75 6. 76 1. 81 6.75 1.87 6. 73 1. 93 8 7. 75 2. 00 7. 73 2. 07 7. 71 2. 14 7. 69 2. 21 9 8.71 2. 25 8. 69 2. 33 8. 67 2. 41 8. 65 2. 48 10 9. 68 2. 50 9. 66 2. 59 9. 64 2.67 9. 61 2. 76 11 10. 65 2. 75 10. 63 2. 85 10. 60 2.94 10. 57 3. 03 12 11. 62 3. 00 11. 59 3. 11 11. 56 3.21 11. 54 3. 31 13 12.59 3.25 12.56 3. 36 12. 53 3.47 12.50 3.58 14 13. 55 3. 51 13. 52 3. 62 13. 49 3.74 13. 46 3. 86 15 14..52 3. 76 14. 49 3.88 14. 45 4. 01 14. 42 4. 13 16 15. 49 4. 01 15. 45 4. 14 15. 42 4.28 15. 38 4. 41 17 16. 46 4. 26 16. 42 4. 40 16t.38 4.54 16. 34 4. 69 18 17. 43 4. 51 17. 39 4.66 17. 35 4. 81 17. 30 4. 96 19 18. 39 4. 76 18. 35 4. 92 18. 31 5.08 18. 26 5. 24 20 19. 36 5. 01 19. 32 5. 18 19. 27 5.34 19. 23 5.51 21 20. 33 5. 26 20. 28 5. 44 20. 24 5.61 20. 19 5. 79 22 21. 30 5. 51 21. 25 5. 69 21. 20 5.88 21. 15 6. 06 23 22. 27 5. 76 22. 22 5. 95 22. 16 6. 15 22. 11 6. 34 24 23. 24 6. 01 23. 18 6. 21 23. 13 6. 41 23. 07 6. 62 25 24. 20 6.26 24. 15 6. 47 24. 09 6.68 24. 03 6. 89 26 25. 17 6. 51 25. 11 6. 73 25. 05 6.95 24. 99 7. 17 27 26. 14 6. 76 26. 08 6. 99 26. 02 7. 22 25. 95 7. 44 28 27. 11 7..01 27. 05 7. 25 26. 98 7.48 26. 92 7. 72 29 28. 08 7. 26 28. 01 7. 51 27. 95 7.75 27 88 7. 99 30 29. 04 7. 51 28.98 7. 76 28. 91 8.02 28. 84 8. 27 35 33. 89 8.76 33. 81 9. 06 33. 73 9. 35 33. 64 9. 65 40 38. 73 10.02 38. 64 10. 35 38. 55 10. 69 38. 45 11. 03 45 43.57 11.27 43. 47 11.65 43. 36 12.03 43.26 12.40 50 48. 41 12. 52 48. 30 12. 94 48. 18 13. 36 48. 06 13. 78 55 53. 25 13. 77 53. 13 14. 24 53. 00 14.70 52. 87 15. 16 60 58. 09 15. 02 57. 96 15. 53 57. 82 16.03 57. 68 16. 54 65 62. 93 16. 27 62. 79 16. 82 62. 64 17.37 62. 48 17. 92 70 67. 77 17. 53 67. 61 18. 12 67. 45 18.71 67. 29 19. 29 75 72. 61 18. 78 72. 44 19.41 72. 27 20.04 72. 09 20. 67 80 77. 45 20. 03 77. 27 20. 71 77. 09 21. 38 76. 90 22. 05 85 82. 29 21. 28 82. 10 22. 00 81. 91 22. 72 81. 71 23. 43 90 87. 13 22. 53 86. 93 23. 29 86. 73 24. 05 86. 51 24. 81 95 91. 97 23. 79 91. 76 24. 59 91. 54 25.39 91. 32 26. 19 100 96. 81 t25. 04 96. 59 25. 88 96. 36 26.72 96. 13 27. 56 Dep. L-at. Dep. Lat. Dep. Lat. Dep. Lat. 75/2 Deg. 75 Deg. 74 14 Deg. 74 Deg. TRAVERSE TABLE. 79 16Y4 Deg. 17 Deg. 12 Deg.18 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 96 0. 28 0. 96 0. 29 0. 95 0. 30 0. 95 0. 31 2 1.92 0. 57 1. 91 0. 58 1. 91 0. 60 1.90 0. 62 3 2. 88 0. 85 2. 87 0. 88 2. 86 0. 90 2 85 0. 93 4 3. 84 1. 14 3. 83 1. 17 3. 81 1. 20 3. 80 1. 24 5 4.79 1.-42 4. 78 1. 46 4. 77 1. 50 4. 76 1 55 6 5. 75 1. 70 5.74 1.75 5.72 1.80 5.71 1.85 7 6.71 1.99 6.69 2..05 6.68 2.10! 6.66 2.16 8 7. 67 2. 27 I7. 65 2. 34 7. 63. 2. 41 7. 61 2. 47 9 8. 63 2. 56 8. 61 2. 63 8. 58 2. 71 8. 56 2. 78 10 9. 59 2. 84 9. 56 2. 92 9. 54 3. 01 9 61 3. 09 11 10. 55 3. 12 10. 52 -3. 22 10. 49 3. 31 10. 46 3. 40 12 11.51 3. 41 11.48 3.51 11.44 3.61 11.41. 3. 71 13 12. 46 3. 69 12. 43 3. 80 12. 40 3..91 12. 36 4. 02 14 13. 42 3. 98 13. 39 4. 09 13. 35 4. 21 13. 31 4..33 15 14. 38 4.26 14. 34 4. 39 14. 31 4. 51 14. 27 4.64 16 15. 34 4. 54 15. 30 4. 68 15. 26 4. 81 15, 22 4. 94 17 16. 30 4. 83 16. 26 4. 97 16. 21 5. 11 16. 17 5. 25 18 17. 26 5. 11 17. 21 5.26 17. 17 5. 41 17. 12 5. 56 19 18. 22 5. 40 18. 17 5 56 18. 12 5i'71 18. 07 5. 87 20 19. 18 5. 68 19. 13 5.85 19.07 6.01 19, 02 6.18 21 20. 14 5. 96 20. 08 6. 14 20. 03 6. 31 19. 97 6. 49 22 21. 09 6. 25 21. 04 6. 43 20.98 6. 62 20,92 6 80 23 22. 05 6. 53 21. 99 6. 72 21. 94 6. 92 21. 87 7. 11 24 23. 01 6. 82 22. 95 7. 02 22. 89 7. 22 22. 83 7. 42 25 23. 97 7. 10 23. 91 7. 30 23. 84 7. 52 23. 78 7. 73 26 24. 93 7. 38 24. 86 7. 60 24. 80 7. 82 24. 73 8. 03 27 25. 89 7. 67 25. 82 7. 89 25. 75 8..12 25. 68 8. 34 28 26. 85 7. 95 26. 78 8. 19 26. 70 8. 42 26. 63 8. 65 29 27. 81 8. 24 27. 73 8. 48 27. 66 8. 72 27. 58 8. 96 30 28.76 8.52 28.69 8.77 28. 61 9. 02 28.53 9. 27 35 33. 56 9. 94 33. 47 10. 23 33. 38 10. 62 33. 29 10. 82 40 38. 35 11. 36 38. 25 11. 69 38. 15 12. 03 38. 04 12. 36 45 43. 15 12. 78 43. 03 13. 16 42. 92 13. 53 42. 80 13. 91 50 47. 94 14. 20 47. 82 14. 62 47. 69 15. 04 47. 55 15. 45 55 52. 74 15. 62 52. 60 16. 08 52. 45 16. 54 52, 31 17. 00 60 57. 53 17. 04 57. 38 17. 54 57, 22 18. 04 57. 06 18. 54 65 62. 32 18. 46 62. 16 19. 00 61, 99 19. 55 61. 82 20. 09 70 67. 12 19. 88 66. 94 20. 47 66. 76 21. 05 66. 57 21. 63 75 71. 91 21. 30 71. 72 21. 93 71. 53 22- 55 71. 33123. 18 80 76. 71 22. 72 76. 50 23. 39 76. 30 24. 06 76. 08 124. 72 85 81. 50 24. 14 81. 29 24. 85 81, 07 25. 56 80. 84 126. 27 90 86. 29 25.56 86. 07 26. 31 85. 83 27.06 85. 60 27. 81 95 91. 09 26. 98 90. 85 27. 78 90. 60 28. 57 90. 35 29. 36 100 95. 88 28. 40 95. 63 29. 24 95. 37 30. 07 95. 11 30. 90 Dep. Lat. Dep. Lat. Dep. Lat. D p. Lat. 7312 Deg. 73 Deg. 7212 Deg. 72 Deg. 80 TRAVERSE TABLE. w 18Y Deg. 19 Deg. 19Y Deg. 20 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.95 0. 32 0.95 0. 33 0. 94 0.33 0. 94 0. 34 2 1. 90 0.63 1. 89 0. 65 1. 89 0. 67 1. 88 0. 68 3 2. 84 0. 95 2. 84 0. 98 2. 83..00 2. 82 1.03 4 3.79 1.27 3.78 1.30 3.77 1.34 3.76 1.37 5 4.74 1.59 4.73 1.63 4. 71 1.67 4.270 1.71 6 5. 69 1. 90 5. 67 1. 95 5. 66 2. 00 5. 64 2. 05 7 6. 64 2. 22 6. 62 2.-28 6. 60 2.34 6. 58 2. 39 8 7. 59 2. 54 7. 56 2. 60 7. 54 2. 67 7. 52 2. 74 9 8. 53 2. 86 8.51 2.93 8.48 3.01 8.46 3.08 10 9. 48 3. 17 9. 46 3. 26 9. 43 3.-34 9. 40 3. 42 11 10.43 3. 49 10.40 3. 58 10. 37 3.67 10. 34 3. 76 12 11. 38 3. 81- 11. 35 3. 91 11. 31 4.01 11. 28 4. 10 13 12.33 4. 12 12. 29 4. 23 12. 25 4. 34 12. 22 4. 45 14 13.28 4. 44 13. 24 4. 56 13.20 4. 67 13. 16 4. 79 15 14. 22 4. 76 14. 18 4. 88 14. 14 5.01 14. 10 5. 13 16 15. 17 5..08 15. 13 5. 21 15. 08 5. 34 15. 04 5. 47 17 16. 12 5. 39 16. 07 5. 53 16. 02 5.67 15.97 5. 81 18 17. 07 5. 71 17. 02 5. 86 16.- 97 6.01 16. 91 6. 16 19 18. 02 6. 03 17. 96 6. 19 17. 91 6.34 17. 85 6. 50 20 18. 97 6. 35 18. 91 6.51 18.85 6.68 18.79 6.84 21 19.91 6. 66 19. 86 6. 84 19. 80 7. 01 19. 73 7. 18 22 20. 86 6. 98 20.80'7.16 20. 74 7. 34 20.67 7. 52 23 21. 81 7. 30 21.75 7. 49 21. 68 7. 68 21. 61 7. 87 24 22. 76 7. 62 22.69 7. 81- 22. 62 8.01 22. 55 8. 21 25 23. 71 7. 93 23.64 8. 14 23.57 8. 34 23.49 8. 55 26 24. 66 8. 25 24. 58 8. 46 24. 51 8.68 24. 43 8. 89 27 25. 60 8.57 25. 63 8. 79 25.45 9. 01 25.37 9. 23 28 26. 55 8. 88 26. 47 9. 12 26. 39 9. 34 26.31 9,. 58 29 27. 50 9. 20 27. 42 9. 44 27. 34 9.68 27.25 9. 92 30 28. 45 9. 52 28. 37- 9. 77 28. 28 10. 01 28. 19 10. 26 35 33. 19 11. 11 33. 09 11. 39 32.99 11.68 32.89 11. 97 40 37. 93 12. 69 37. 82 13. 02. 37. 71 13. 35 37. 59 13. 68 45 42. 67] 14. 28 42. 55 14. 65 42. 42 15. 02 42. 29 15. 39 50 47. 42 15. 87 47. 28 16. 28 47. 13 16. 69 46. 98 17. 10 55 52. 16 17- 45 52. 00 17. 91 51. 85 18. 36 51. 68 18. 81 60 56. 90 19. 04 56. 73 19. 53 56. 56 20.03 56. 38 20. 52 65 61. 64 20. 62 61. 46 21. 16 61. 27 21.70 61. 08 22. 23 70 66. 38 22. 21 66. 19 22. 79 67. 98 23. 37 65. 78 23. 94 75 71. 12 23. 80 70. 91 24. 42 70. 70 25. 04 70. 48 25. 65 80 75. 87 25. 38 75. 64 26. 05 75. 41 26. 70 75. 18 27.-36 85 80. 61 26. 97 80. 37 27. 67 80. 12 28. 37 79. 87 29. 07 90 85. 35 28. 56 85. 10 29. 30 84. 84 30. 04 84. 57 30. 78 95 90.09 30. 14 89. 82 30. 93 89. 55 31.71 89. 27 32. 49 100 94. 83 31. 73 94. 55 32. 56 94. 26 33.38 93. 97 34. 20 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 7113 Deg. 71 Deg. 701 Deg. 70 Deg. TRAVERSE TABLE. 81 2 20 Deg. 21 Deg. 21. Deg. 22 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. | ep. 1 0. 94 0. 35 0.93 0. 36 0. 93 0. 37 0. 93 0. 37 2 1. 87 0. 70 1. 87 0. 72 1. 86 0. 73 1. 85 0. 75 3 2. 81 1. 05 2.$80 1.08 2.79 1. 10 2. 78 1. 12 4 3. 75i. 40 3. 73 1. 43 3. 72 1. 47 3. 71 1. 50 5 4. 68 1.75 4. 67 1.79 4. 65 1. 83 4. 64 1. 87 6 5. 62 2. 10 5. 60 2. 15 5. 58 2. 20 5.56 2. 25 7 6.56 2. 45 6.54 2.51 6.51 2.57 6.49 2.62 8 7. 49 2. 80 7. 47 2. 87 7. 44- 2. 93 7. 42 3. 00 9 8. 43 3. 15 8. 40 3. 23 8. 37 3. 30 8. 34 3. 37 10 9. 37 3. 50 9. 34 3. 58 9. 30 3. 67 9 27 3. 75 11 10. 30 3. 85 10. 27 3. 94 10. 23 4. 03 10. 20 4. 12 12 11. 24 4. 20 11. 20 4. 30 11. 17 4. 40 11. 13 4. 50 13 12. 18 4. 55 12. 14 4. 66 12. -10 4. 76 12. 05 4.87 14 13. 11 4. 90 13. 07 5. 02 13. 03 5. 13 12. 98 5. 24 15 14. 05 5. 25 14. 00 5. 38 13. 96 5.50 13. 91 5. 62 16 14. 99 5. 60 14. 94 5. 73 14. 89 5. 86 14. 83 5. 99 17 15. 92 5. 95 15. 87 6. 09 15. 82 6. 23 15. 76 6. 37 18 16. 86 6. 30 16.80 86.45 16. 75 6. 60 16. 69 6. 74 19 17. 80 6. 65 17. 74 6. 81 17. 68 6. 96 17. 62 7. 12 20 18.73 7. 00 18. 67 7. 17 18.61 7. 33 18. 54 7. 49 21 19. 67 7. 35 19. 61 7. 53 19. 64 7. 70 19. 47 7. 87 22 20. 61 7. 70 20. 54 7. 88 20. 47 8. 06 20. 40 8. 24 23 21. 54 8. 05 21. 47 8. 24 21.40 8. 43 21. 33 8.62 24 22. 48 8. 40 22. 41/ 8. 60 22. 33 8. 80 22. 25 8. 99 25 23. 42 8. 76 23. 34 8. 96 23. 26 9. 16 23. 18 9. 37 26 24. 35 9. 11 24. 27 9. 32 24. 19 9. 53 24. 11 9. 74 27 25. 29 9. 46 25. 21 9. 68 25. 12 9. 90 25. 03 10. 11 28 26. 23 9. 81 26. 14 10. 08 26. 05 10. 26 25. 96 10. 49 29 27. 16 10. 16 27. 07 10. 39 26. 98 10. 63 i26. 89 10.86 30 28. 10 10. 51 28. 01 10. 75 27. 91 11. 001 27. 82 11. 24 35 32. 78 12. 26 32. 68 12. 54 32. 56 -12. 83 32. 45 13. 11 40 37. 47 14. 01 37. 34 14. 331 37. 22 14. 66 37. 09 14. 98 45 42. 15 15. 76 42. 01 16. 13 41. 87 16. 49 41. 72 16. 86 50 46. 83 17. 51 46. 68 17. 92 146. 52 18. 33 146. 36 18. 73 55 51. 52 19. 26 51. 35 19. 71 51. 17 20. 166 51. 00 20. 60 60 56. 20 21. 01 56. 01 21. 50 55. 83 21. 99 55. 63 22. 48 65 60. 88 22. 76 60. 68 23. 29 60. 48 23. 8'2 60. 27 24. 35 70 65. 57 [24. 51 65. 35 25. 09 65. 13 25. 66 64. 90 26. 22 75 70. 25 26. 27 70. 02 26. 888 69. 78 27. 49 69. 54 28. 10 80 74. 93 28. 02 74. 69 28. 67 74. 43 29. 32 74. 17 29. 97 85 79. 62 29. 77 79. 35 30. 461 79. 09 31. 15 78. 81 31. 84 90 84. 30 31. 52 84. 02 32. 25 83. 74 32. 99 83. 45 33. 71 95 98. 98 33. 27 88. 69 34. 04 88. 39 34. 82 88. 08 35. 59 100 93. 67 35. 02 93. 36 35.84 93. 04 36.65 92. 72 37. 46 Dep. Lat. Dep. Lat. Dep. Lat. |ep. tat. 691, Deg. 69 Deg. 1 681 Deg. 68 Deg. 82 TRAVERSE TABL-E....22/2 Deg. 23 Deg. 231, Deg. 24 Deg. c Lat. -Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 90 0. 38 0. 92 0. 39 0. 92 0. 40 0. 91 0. 41 2 1.85 0.77 1.84 0.78 1. 83 0.80 1. 83 0. 81 3 2. 77 1. 15 2. 76 1. 17 2. 75 1.20 2. 74 1.22 4 3. 70 1. 53 3. 68 1. 56 3. 67 1.59 3. 65 1. 63 5 4. 62 1.91 4. 60 1. 95 4. 59 1. 99 4. 57 2. 03 6 5. 64 2. 30 5. 52 2. 34 5. 50 2.39 5. 48 2. 44 7 6. 47 2. 68 6. 44 2. 74 6. 42 2. 79 6. 39 2. 85 8 7. 39 3. 06 7. 36 3. 13 7. 34 3. 19 7. 31 3. 25 9 8. 31 3. 44 8. 28 3. 52 8. 25 3.59 8.22 3. 66 10 9. 24 3. 83 9. 20 3. 91 9. 17 3. 99 9 14 4. 07 11 10. 16 4.21 10. 13. 4. 30 10. 09 4. 39 10. 05 4. 47 12 11. 09 4. 59 11. 05. 4. 69 11. 00 4. 78 10. 96 4. 88 13 12. 01 4. 97 11. 97 5. 08 11. 92 5. 18 11. 88 5. 29 14 12. 93 5. 36 12. 89 5. 47 12. 84 5.58 12.'79 5. 69 15 13. 86 5. 74 13. 81 5. 86 13. 76 5.98 13. 70 6. 10 16 14. 78 6. 12 14. 73 6. 25 14. 67 6. 38 14. 62 6. 51 17 15. 71 6. 51 15. 65 6. 64 15. 59 6.78 15. 53 6. 92 18 16.63 6. 89 16. 57 7. 03 16. 51 7. 18 16. 44 7. 32 19 17. 55 7. 27 17. 49 7. 42 17. 42 7. 58 17. 36 7. 73 20 18. 48 7. 65 18. 41 7. 81 18. 34 7.97 18.27 8. 13 21 19. 40 8. 04 19. 33 8.21 19. 26 8.37 19. 18 8. 54 22 20. 33 8. 42 20. 25 8. 60 20. 18 8. 77 20. 10 8. 95 23 21. 25 8. 80 21. 17 8. 99 21. 09 9.17 21.01 9. 35 24 22. 17 9. 18 22. 09 9. 38 22. 01 9.57 21.93 9. 76 25 23. 10 9. 57 23. 01 9. 77 22. 93 9.97 22. 84 10. 17 26 24. 02 9. 95 23. 93 10. 16 23. 84 10. 37 23. 75 10. 58 27 24. 94 110. 33 24. 85 10. 55 24. 76 10.77 24. 67 10. 98 28 25. 87 i 10. 72 25. 77 10. 94 25. 68 11. 16 25. 58 11. 39 29 26. 79 11. 10 26. 69 11. 33 26. 59 11.56 26. 49 11. 80 30 27. 72 11.48 27. 62 11.52 27. 51 11.96 27. 41 12. 20 35 32. 34 13. 39 32. 22 13. 68 32. 10 13. 96 31. 97 14. 24 40 36. 96 15. 31 36. 82 15. 63 36. 68 15. 95 36. 54 ]6. 27 45 41. 57 17. 22 41. 42 17. 58 41. 27 17.94 41. 11 18. 30 50 46. 19 19. 13 46. 03 19. 54 45. 85 19. 94 1 45. 68 20. 34 55 50. 81 21. 05 50. 63 21. 49 50. 44 21. 93 1 50. 24 22. 37 60 55. 43 22. 96 55. 23 23. 44 55. 02 23. 92 1 54. 81 24. 40 65 60. 05 24. 87 59. 83 25. 40 59. 61 25. 92 159. 38 26. 44 70 64. 67 26. 79 64. 44 27. 35 64. 19 27. 91 63. 95 28. 47 75 69. 29 28. 70 69. 04 29. 30 68. 78 29. 91 68. 52 30. 51 80 73. 91 30. 61 73. 64 31. 26 73. 36 31.90 73. 08 32. 54 85 78. 53 32. 53 78. 24 33. 21 77. 95 33. 89 77. 65 34. 57 90 83. 15 34. 44 82. 85 35. 17 82. 54 35.89 82. 22 36. 61 95 87. 77 36. 35 87. 45 37. 12 87. 12 37.88 86. 79 38. 64 100 92. 39 38. 27 92. 05 39. 07 91. 71 39.87 91. 35 40. 67 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 67Y/ Deg. 67 Deg. 66Y/ Deg. 66 Deg. TR AVERSE TABLE. 83 244 Deg. 2 Deg. 25 Deg. 26 Deg. Lat. Dep Lat Dep. Lat. Depep. Lat. t. Dep. 1 0.91 0.41 0. 91 0. 42 0. 90 0.43 0. 90 0. 44 2 1.82: 0. 83 1.81. 0.85 1.81.,86 1.80 0.88 3 2. 3 1. 24 2. 72 1. 27 2.71 1.29 2. 70 1.32 4 3.64 1. 66 3.63 1.69 3.i61 1.72j 3.60 1.75 5 4.55 2.07 4.53 2.11 4. 51 2.,15 4.49 2.19 6 5.46. 49 5.44 2.54 5.42 2.58 5.39 2.63 7 6. 37 2. 90 6. 34 2. 96 6. 32 3. 01 6.29 3. 07 8 7.28 3. 32 7. 25 3.38 7.22 344 7.19 3.51 9 8. 19 3. 73 8. 16 3. 80 8. 12 3 87 8.09 3. 95 10 9. 10 4. 15 9. 06 4. 23 9. 03 4. 31 8. 99 4. 38 11 10.01 4.56 9. 97 4. 65 9.93 4.74 9. 89 4,.82 12 10. 92 4. 98 10. 88 5. 07 10. 83 5- i7 10. 79 5. 26 13 11, 83 5.39 11.78 5. 49 11.73 5 60 11.68 5.70 14 12.74; 5.81 12.69, 5.92 1: 2.64 6-.03 12.58 6.14 15 13.65:6.22 13.59 6.34 13.54 6.46 13.48 6.58 16 14. 56 6, 64 14. 50 6. 76 14, 44 6. 89 14. 38 7. 01 17 15.47 7.05 15.41 7. 18 15. 34 7. 92 15. 28 7. 45 18 16. 38 7. 46 16. 31 7., 61 16. 25 7. 75 16. 18 1T. 89 19 17. 19 7. 88. 17. 22 8. 03 17. 15 8- 18 17. 08 8. 33 20 18.20 8.29 18. 13 8.45 18.05 8.61 17.98 8:.77 21 19. 11 82 71 19. 03 8, 87 18.95 9. 04 18. 87 9. 21 22 20. 02 9. 12 19. 94 9. 30 19. 86 9.47 19. 77 9. 64 23 20. 93 9. 54 20, 85 9. 72 20. 76 9. 90 20. 67 10. 08 24 21. 84 9. 95 21. 75 10. 14 21. 66 10.-33 21. 57 10.52 25 22. 75 10. 37 22. 66 10. 57 22. 56 10. 76 22. 47 10. 96 26 23. 66 10. 78 23. 56 10.99 23- 47 11. 19 23. 37 11. 40 27 24. 57 1I. 20 24. 47 11. 41. 24. 37 11. 62 24. 27 11. 84 28 25. 48 11. 61 25. 38 11. 83 25. 27 12.: 05 25. 17 12. 27 29.26. 39 12.- 03 26. 28 12. 26 26. 17 12. 48 26. 06 12. 71 30 27. 30 12. 44 27. 19 12. 68 27. 08 12. 92 26. 96 13. 15 35 31. 85 14. 51 31. 72 14. 79 31. 59 15. 07 31. 46 15. 34 40 36. 40 16. 59 36. 25 16. 90 36. 10 17. 22 35. 95 17. 53 45 40. 95 18. 66 40. 78 19. 02 40. 62 19. 37 40. 45 19. 73 50 45. 50 20. 73 45. 32 21. 13 45. 13 21i. 3 44. 94 21. 92 55 50. 05 22. 81:49. 85 23. 24 49. 64 23. 68. 49. 43 24. 11 60 54. 60 24. 88 54. 38 25, 36 54. 16 25- 83 53. 93 26. 30 65 59. 15 26.96 58. 91 27. 47 58. 67. 27. 98 58. 42 28. 49 70 63. 70 29. 03:63. 44 29. 58 63. 18 30. 14 62. 92 30. 69 75 68. 25 31. 10 ~67. 97 31. 70 67. 69 32. 29 67. 41 32. 88 80 72. 80.33. 18 72. 50 33. 81 72. 21 34. 44 71. 90 35. 07 85 77, 35 35. 25 77. 04 35. 92 76. 7/2 36. 59 76. 40 37. 26 90 81. 90 37. 32 81, 57 38, 04 81. 23 38. 75 80. 89 39. 45 95 86. 45 39. 40 86. 10 40. 15 85. 75. 40. 90 85. 39 41. 65 100 91. 00 41. 47 90. 63 42. 26 90. 26 43. 05 89. 88 43. 84 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 65V Deg. 66 Deg. 6412 Deg. 64 Deg. 27 Lat. | Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 89. 0. 45 0.o89 0.45 1 0.o89 0.o46 0o. 88 o. 47 2 1. 79 0. 89 1. 78 0. 91 1. 77 0. 92 1. 77 0. 94 3 2. 68 1. 34 2. 67 1. 36 2. 66 1.39 2. 65 1. 41 4 3..58. 78 3. 56 1.82 3. 55/. 1.85 3. 53 1. 88 5 4. 57 2.23 4.45 2. 27 4.44 2.31 4.41 2.35 6 5. 37 2. 68 5. 35 2. 72 5. 32 2. 77 5. 30 2. 82 7 6. 26 3, 12 6.324 3. 18 6.21 3.23 6. 18 3.29 8 7. 16 3. 47 7. 13 3. 63 7. 10 3.69 7. 06 3. 76 9 8.(05 4.02 8.02 4.09 7.98 4.16 7.95 4.23 10 8.{95 4.46 8.91 4.54 8.87 4.62 8.83 4.69 11 9. 84 4.91 9. 80 4. 99 9. 76 5.08 9. 71 5. 16 12 10.74 5. 35 10. 69 5. 45 10. 64 5.54 10. 60 5. 63 13 11.63 5. 80 11. 58 5.90 11. 53 6. 00 11. 48 6. 10 14 12. 53 6.25 12. 47 6. 36 12. 42 6.49 12. 36 6. 57 15 13. 42 6. 69 13. 37 6. 81 13. 31 6.93 13. 24 7. 04 16 14. 32 7. 14l 14. 26 7. 26 14- 19 7. 39 14. 13 7. 51 17 15. 21 7. 59 15. 15. - 72 15i 08 7.85 15. 01 7. 98 18 16. 11 8. 03 16. 04 8. 17 15 97 8.31 15. 89 8. 45 19 17. 00 8.48 16.93 8.63 16.85 8.!77 16. 78 8. 92 20 17. 90 8. 92 17. 82 9. 08 17. 74 9.23 17. 66 9. 39 21 18. 79 9. 37 18. 71 9. 53 18. 63 9. 70 18. 54 9. 86 22 19. 69 9. 82 19. 60 9. 99 19. 51 10. 16 19. 42 10. 33 23 20. 58 10. 26 20. 49 10. 44 20- 40 10.62 20. 31 10. 80 24 21. 48 10. 71 21. 38 10. 90 21. 29 11. 08 21. 19 11. 27 25 22. 37 11.15 22. 28 11.35 22.18 11. 54 22.07 11.74 26 23. 27 11. 60 23. 17 11/ 80 23. 06 12. 01 22. 96 12. 21 27 24. 16 12. 05 24. 06 12. 26 23. 95 12. 47 23. 84 12. 68 28 25. 06 12. 49 24. 95 12 71 24. 84 12. 93 24. 72 13. 15 29 25. 95 12. 94 25. 84 13. 17 25. 72 13. 39 25. 61 13. 61 30 26. 85 13. 39 26.- 73 13. 62 26 61 13. 85 26. 49 14. 08 35/ 31. 32 15. 62 31. 19 15. 89 31. 05 16. 16 30. 90 16. 43 40 35. 80 17. 85 35. 64 18. 16 35. 48 18. 47 35. 32 18. 78 45 40. 27 20. 08 40. 10 20.43 39- 92 20.78 39. 73 21. 13 50 44. 75 22. 31 44. 55 22. 70 44. 35 23. 09[1 44. 15 23. 47 55 49.: 22 24. 54 49. 01 24. 97 48. 79 25. 40 1'48. 56 25. 82 60 53. 70 26. 77 53. 46 27. 24 53. 22 27.701 52. 98 28. 17 65 58. 17 29. 00 57. 92 29. 51 57. 66 30. 01 57. 39 30. 62 70 62. 65 31.:23 62. 37 31. 78 62. 09 32.32 61. 81 32. 86 75 67. 12 33. 46 66. 83 34. 05 66. 53 34.63 66. 22 35. 21 80 71. 59 35. 70 71. 28 36. 32 70. 96 36. 94 70. 64 37. 56 85i'76. 07 37. 93 75. 74 38. 509 75. 40 39. 25 75. 05 39. 91 90 80. 54 40. 16 80.'19 40. 86 79. 83 41.56 79. 47 42. 25 95 85. 03 42. 39 84. 65 43. 13 84. 27 43.87 83. 88 44. 60 100 89. 49 44. 62 89. 10 45. 40 88. 90 46. 17 88. 29 46. 95 Dep. Lat..Dep. Lat. Dep. Lat. Dep. Lat. 63f Deg. 63'Deg. 623 Deg. 62 Deg. TRAVERSE TABLE. 85 ~ ~28Y Deg. 29 Deg. 293 Deg. 30 Deg. I -. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.88 0. 48 0.87 0.48 0.87 0. 49 0. 87 0. 50 2 1.76 0. 95 1. 75 0. 97 1. 74 0. 98 1. 73 1. 00 3 2. 64 1. 43 2. 62 1. 45 2. 61 1. 48 2. 60 1.50 4 3. 52 1. 91 3. 50 1. 94 3. 48 1. 97 3. 46 2. 00 5 4. 39 2. 39 4. 37 2. 42 4. 35 2. 46 4. 33 2. 50 6 5. 27 2. 86 5. 25 2. 91 5. 22 2. 95 5. 20 3. 00 7 6. 15 3. 34 6. 12 3. 39 6. 09 3. 45 6. 06 3. 50 8 7. 03 3. 82 7. 00 3. 88 6. 96 3. 94 6. 93 4. 00 9 7. 91 4. 29 7. 87 4. 36 7. 83 4. 43 7. 79. 4. 50 10 8. 79 4. 77 8. 75 4. 85 8. 70 4. 92 8. 66 5. 00 11 9. 67 5. 25 9. 62 5. 33 9. 57 5. 42 9. 53 5. 50 12 10. 55 5. 73 10. 50 5. 821 10. 44 5. 91 10. 39 6. 00 13 11.42 6. 20 11. 37 6. 30 11. 31 6. 40 11. 26 6. 50 14 12. 30 6. 68 12. 24 6. 79 12. 18 6. 89 12. 12! 7. 00 15 13. 18 7. 16 13. 12 7. 27 13. 06 7. 39 12. 99 7. 50 16 14. 06 7. 63 13. 99 7. 76 13. 93 7. 88 13.86. 8.00 17 14. 94 8. 11 14. 87 8. 24 14. 80 8. 37 14. 72 8. 50 18 15. 82 8. 59 15. 74 8. 73 15. 67 8. 86 15. 59 9. 00 19 16. 70 9. 07 16. 62 9. 21 16. 54 9. 36 16. 45 9. 50 20 17. 58 9. 54 17. 49 9. 70 17. 41 9. 85 17. 32 10. 00 21 18. 46 10. 02 18. 37 10. 18 18. 28 10. 34 18. 19 10. 50 22 19. 33 10. 50 19. 24 10. 67 19. 15 10. 83 19. 05 11. 00 23 20. 21 10. 97 20. 12 11. 15 20. 02 11. 33 19. 92 11. 50 24 21. 09 11. 45 20. 99 11. 64 20. 89 11. 82 20. 78 12. 00 25 21. 97 11. 93 21. 87 12. 12 21. 76 12. 31 21. 65 12. 50 26 22. 85 12. 41 22. 74 12. 60 22. 63 12. 80 22. 52 13. 00 27 23. 73 12. 88 23. 61 13. 09 23. 50 13. 30 23. 38 13. 50 28 24. 61 13. 36 24. 49 13. 57 24. 37 13. 79 24. 25 14. 00 29 25. 49 13. 84. 25. 36 14. 06 25. 24 14. 28 25. 11. 14. 50 30 26. 36 14. 31 26. 24 14. 54 26. 11 14. 77 25. 98 15. 00 35 30. 76 16. 70 30. 61 16. 97 30. 46 17. 23 30. 31. 17. 50 40 35. 15 19. 09 34. 98 19. 39 34. 81 19. 70 34. 64 20. 00 45 39. 55 21. 47 39. 36 21. 82 39. 17 22. 16 38..97. 22. 50 50 43. 94 23. 86 43. 73 24. 24 43. 52 24. 62 43. 30. 25. 00 55 48. 33 26. 24 48. 10 26. -66 47. 87 27. 08 47. 63. 27. 50 60 52. 73 28. 63 52. 48 29. 09 52. 22 29. 55 51. 96 30. 00 65 57. 12. 31. 02 56. 85 31. 51 56. 57 32. 01 56. 29 32. 50 70 61. 52 33. 40 61. 22 33. 94 60. 92 34. 47 60. 62 35. 00 75 65. 91 35. 79 65. 60 36. 36 65. 28 36. 93 64. 95 37. 50 80 70. 31 38. 17 69. 97 38. 78 69. 63 39. 39 69. 28 40. 00 85 74. 70 40. 56 74 34 41. 21 73. 98 41. 86 73. 61 42. 50 90 79. 09 42. 94 78. 72. 43. 63 78. 33 44. 32 77. 94 45. 00 95 83. 49 45. 33 83. 09 46. 06 82. 68 46. 78 82. 27 47. 50 100 87. 88 47. 72 87. 46 48. 48 87. 04 49. 24 86. 60 50. 00 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 61Y2 Deg. i 61 Deg. 60y2 Deg. 60 Deg. 86 TRAVERSE TABLE. p 30 Deg. 81 Deg. 81Y Deg. 32 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 86 0. 51 0.86 0. 51 0.85 0. 52 0. 85 0. 53 2 1. 72 1. 02 1.71 1. 03 1.71 1. 04 1.70 1. 06 3 2.58 1.52 2.57. 55 2.56 1. 57 2. 54 1. 59 4 3. 45 2. 03 3. 43 2. 06 3. 41 2. 09 3. 39 2. 12 5 4. 31 2. 54 4. 29 2. 58 4. 26 2.61 4. 24 2. 65 6 5. 17 3.05 5. 14 3.09 5. 12 3. 13 5. 09 3. 18 7 6. 03 3. 55. 6.00 3. 61 5.97 3.66 5.94. 3. 71 8 6. 89 4. 06 6. 86 4. 12 6. 82 4. 18 6. 78 4. 24 9 7. 75 4. 57 7. 11 4. 64 7. 67 4. 70 7. 63 4. 77 10 8. 62 5.:08 8.57 5. 15 8.53 5.22 8.48 5. 30 11 9. 48 5. 58 9. 43 5. 67 9. 38 5.75 9. 33 5. 83 12 10. 34 6. 09 10. 29 6. 18 10. 23 6. 27 10. 18 6. 36 13 11.20 6.60 11. 14 6. 70 11.08 6. 79 11.02 6. 89 14 12.06 7. 11 12.00 7.21 11.94 7. 31 11.87 7. 42 15 12. 92 7. 61 12. 86 7. 73 12. 79 7. 84 12. 72 7. 95 16 13.79 8. 12 13.71 8. 24 13. 64 8. 36 13. 57 8.48 17 14. 65 8. 63 14. 57 8. 77 14. 49 8. 88 14. 42 9. 01 18 15. 51 9. 14 15.43 9.27 15. 35 9. 40 15.26 9. 54 19 16. 37 9. 64 16. 29 9. 79 16. 20 9. 93 16. 11 10. 07 20 17. 23 10, 15 17. 14 10. 30 17. 05 10. 45 16. 96 10. 60 21 18. 09 10. 66 18. 00 10. 82 17. 91 10.97 17. 81 11. 13 22 18. 96 11. 17 18. 86 11. 33 18. 76 11.49 18. 66 1i. 66 23 19. 82 11. 67 19. 71 11. 85 19. 61 12. 02 19. 1il 12. 19 24 20. 68 12. 18 20. 57 12 36 20. 46 12. 64 20. 35 12. 72 25 21. 54 12. 69 21. 43 12. 88 21. 32 13. 06 21. 20 13. 25 26 22. 40 13. 20 22. 29 13. 39 22. 17 13. 58 22. 05 13. 78 27 23. 26 13. 70 23. 14 13. 91 23. 02 14. 11 22. 90 14. 31 28 24. 13 14. 21 24. 00 14. 42 23. 87 14. 63 23. 75 14. 84 29 24. 99 14. 72 42. 86 14. 94 24. 73 15. 15 24. 59 15. 37 30 25. 85 15.23 25. 71 15. 45 25. 58 15. 67 25. 44 ]5.90 35 30. 16 17. 76 30. 00 18. 03 29. 84 18. 29 29. 68 18. 55 40 34. 47 20. 30 34. 29 20. 60 34. 11 20. 90 33. 92 21. 20 45 38. 77 22. 84 38. 57 23. 18 38. 37 23.51 38. 16 23. 85 50 43. 08 25. 38 42. 86 25. 75 42. 63 26. 12 42. 40 26. 50 65 47. 39 27. 91 47. 14 28. 33 46. 90 28. 74 46. 64 29. 15 60 51. 70 30. 45 51. 53 30. 90 51. 16 31. 35 50. 88 31. 80 65 56. 01 32. 99 55. 72 33. 48 55.42 33.96 55. 12 34. 44 70 60. 31 35. 53 60. 00 36. 05 59. 68 36. 57 59. 36 37. 09 75 64. 62 38. 07 64. 29 38. 63 63. 95 39. 19 63. 60 39. 74 80 68. 93 40. 60 68. 57 41. 20 68. 21 41. 80 67. 84 42. 39 85 73. 24 43. 14 72. 86 43. 78 72. 47 44. 41 72. 08 45. 04 90 77. 55 45. 68 77. 15 46. 35 76. 74 47. 02 76. 32 47. 69 95 81. 85 48. 22 81. 43 48. 93 81. 00 49. 64 80. 56 50. 34 100 86. 16 50. 75 85. 72 61. 50 85. 26 52. 25 84. 80 52. 59 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 592 Deg. 59 Deg. 68Y2 Deg. 58 Deg. TRAVERSE TABLE. 87. 832Y Deg. 33 Deg. 33Y Deg. 34 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 84 0. 54 0. 84 0.54 0. 83 0.55 0. 83 0.56 2 1.69 1. 07 1. 68 1. 09 1. 67 1. 10 1. 66 1. 12 3 2. 53 1. 61 2. 52 1. 63 2.50 1. 66 2. 49 1. 68 4 3. 37 2. 15 3. 35 2. 18 3. 34 2. 21 3. 32 2. 24 6 4. 22 2. 69 4. 19 2. 72 4. 17 2. 76 4. 15 2. 80 6 5. 06 3. 22 5. 03 3. 27 5. 00 3. 31 4. 97 3. 36 7 5. 90 3. 76 5. 87 3. 81 5. 84 3. 86 5. 80 3.91 8 6. 75 4. 30 6. 71 4. 36 6. 67 4. 42 6. 63 4. 47 9 7. 59 4. 84 7. 55 4. 90 7. 50 4. 97 7. 46 5. 03 10 8. 43 5. 37 8. 39 5. 45 8. 34 5. 652 8. 29 5. 59 11 9. 28 5. 91 9. 23 5. 99 9. 17 6. 07 9. 12 6. 15 12 10. 12 6. 45 10.06 6. 54 10. 01 6. 62 9. 95 6. 71 13 10.96 6. 98 10.90 7. 08 10. 84 7. 18 10. 78 7. 27 14 11.81 7.52 11.74 7. 62 11.67 7.73 11. 61 7. 83 15 12. 65 8. 06 12.58 8. 17 12.51 8.28 12. 44 8.39 16 13. 49 8. 60 13. 42 8. 71 13. 34 8. 83 13. 26 8.95 17 14. 34 9. 13 14. 26 9. 26 14. 18 9. 38 14. 09 9.51 18 15. 18 9. 67 15. 10 9. 80 15. 01 9. 93 14. 92 10. 07 19 16. 02. 10. 21 15. 93, 10. 35 15. 84 10. 49 15. 75 10. 62 20 16. 87 10. 75 16. 77{ 10. 89 16. 68 11. 04 16. 58 11. 18 21 17.71 11.28 17.61 11.44 17. 51 11. 59 17. 41 11.74 22 18. 55 11. 82 18. 45 11. 98 18. 35 12. 14 18. 24 12. 30 23 19. 40 12. 36 19. 29 12. 53 19. 18 12. 69 19. 07 12. 86 24 20. 28 12. 90 20. 13 13. 07 20. 01 13. 25 19. 90 13. 42 25 21. 08 13. 43 20. 97 13. 62 20. 85 13. 80 20. 73 13. 98 26 21. 93 13. 97 21. 81 14. 16 21.68 14. 35 21. 55 14. 54 27 22. 77 14. 51 22. 64 14. 71 22. 51 14. 90 22. 38 15. 10 28 23. 61 15. 04 23. 48 15. 25 23. 35 15. 45 23. 21 15. 66 29 24. 46 15. 58 24. 32 15. 97 24. 18 16. 01 24. 04 16. 22 30 25. 30 16. 12 25. 16 16. 34 25. 02 16. 56 24. 87 16. 78 35 29. 52 18. 81 29. 35 19. 06 29. 19 19. 32 29. 02 19. 57 40 33. 74 21. 49 33. 55 21. 79 33. 36 22. 08 33. 16 22. 37 45 37. -95 24. 18 37. 74 24. 51 37. 52 24. 84 37. 31 25. 16 50 42. 17 26. 86 41. 93 27. 23 41. 69 27. 60 41. 45 27. 96 55 46. 39 29. 55 46. 13 29. 96 45. 86 30. 36 45. 60 30. 76 60 50. 60 32. 24 50. 32 32. 68 50. 08 33. 12 49. 74 33. 55 65 54. 82 34. 92 54. 51 35. 40 54. 20 35. 88 53. 89 36. 35 70 59. 04 37. 61 58. 72 38. 12 58. 37 38. 641 58. 03 39. 14 75 63. 25 40. 30 62. 90 40. 85 62. 54 41. 40 62. 18 41. 94 80 67. 47 42. 98 67. 09 43. 57 66. 71 44. 15 66. 32 44. 74 85 71. 69 45. 67 71 29 46. 29 70. 88 46. 91 70. 47 47. 53 90 75. 91 48. 36 75. 48 49. 02 75. 05 49. 67 74. 61 50.33 95 80. 12 51. 04 79. 67 51. 74 79. 22 54. 43 78. 76 53. 12 100 84. 34 53. 73 83. 87 54. 46 83. 39 55. 19 82. 90 55. 92 Dep. Lat. Dep. Lat. Dep. Lt. Dop. Lat. 5712 Deg. 57 Deg. 66Y2 Deg. 56 Deg. 88 TRAV:E R SE TABLE. 34% Deg. 35 Deg. 3532 Deg. 88 Deg...... 8 eg. I e'....... at,,eg............... a L at. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.82 0.57 0.82 0.57 0.81 0.58 0.81 0.59 2 1.65 1.13 1.64 1.15 1.63 1.16 1.62 1.18 3 2. 47 1. 70 2. 46 1. 72 2. 44 1.74 2. 43 1. 76 4 3. 30 2. 27 3. 28 2.29 3.6 2. 32 3. 24 2. 35 5 4. 12 2. 83 4. 10 2. 87 4. 07 2. 90 4. 05 2. 94 6 4. 94 3. 40 4. 91 3. 44 4.88 3.48 4. 85 3. 53 7 5. 77 3.96 5.73 4.01 5.70 4.06 5.66 4.11 8 6. 59 4. 53 6. 55 4. 69 6. 51 4.5 6.47 4. 70 9 7.42 5.10 7.37 5. 16 7.33 5.23 7.28 5.29 10 8.24 5.66 8. 19 5. 74 8. 14 5.81 8.09 5.88 11 9.07 6.23 9.01 6.3 1 8.Q96 6.39 8.90 6.47 12 9.89 6,80 9.83 6,88 9.77 6.97 9.71 7.05 13 10.71 7. 36 10.65 7.46 10.58 7.55 10.52 7.64 14 11. 4 7. 93 11.47 8. 03 11.40 8. 13 11. 33 8. 23 15 12. 36 8. 50 12.29 8. 60 12. 21 8. 71 12. 14 8. 82 16 13. 19 9. 06 13. 11 9. 18 13.03 9. 29 12. 94 9. 40 17 14. 01 9. 63 13. 93 9. 75 13. 84 9. 87 13. 75 9. 99 18 14. 83 10. 20 14. 74 10. 32 14. 65 10. 45 14. 56 10. 58 19 15. 66 10. 76 15. 56 10. 90 15. 47 11. 03 15. 37 11. 17 20 16. 48 11. 33 16. 38 11. 47 16. 28 11. 61 16. 18 11. 76 21 17. 31 11. 89 17. 20 12. 05 17. 10 12. 19 16. 99 12. 34 22 18. 13 12. 46 18. 02 12. 62 17. 91 12. 78 17. 80 12. 93 23 18. 95 13. 03 18. 84 13. 19 18. 72 13. 36 18. 61 13. 52 24 19. 78 13. 59 19. 66 13. 77 19. 54 13. 94 19. 42 14. 11 25 20. 60 14. 16 20. 48 14. 34 20. 35 14. 52 20. 23 14. 69 26 21. 43 14. 73 21. 30 14, 91 21. 17- 15. 10 21. 03 15. 28 27 22. 25 15. 29 22. 12 15. 49 21. 98 15. 68 21. 84 15. 87 289 23. 08 15. 86 22. 94 16. 06 22. 80 16. 26 22. 65 16. 46 29 23. 90 16. 43 23. 76 16. 63 23. 61 16. 84 23. 46 17. 05 30 24. 72 16. 99 24. 57 17. 21 4. 42 17. 42 24.27 17. 63.356'28.84- 19. 82 28.672008 28. 49 20. 32 28. 32 20. 57 ~40 32. 97 22. 66 32. 77 22. 94 32. 56 23. 23 32. 36 23. 51 45- 37. 09 25. 49 36. 86- 5. 81 36. 64 26. 13 36. 41 26. 45 50 41. 21- 28. 32 40. 96 28. 68 40. 71 29. 04 40. 45 29. 39 55 45. 33 31. 15 45. 05 31. 55 44. 78 81. 94 44. 50 32. 23 60 49. 45 33. 98 49. 15 34. 41 48. 85 34. 84 48. 54 35. 27 65 53. 57 36. 82 53. 24 37. 28 52. 92 37. 75 52. 59 38. 21 70 57. 69 39. 65 57. 34 40. 15 56. 99 40. 65 56. 63 41. 14 75 61. 81 42. 48 61. 44 43. 02 61. 06 43. 55 60. 68 44. 08 80 65. 93 45. 31 65. 53 45. 89 65. 13 46. 46 64. 72 47. 02 85 70. 05 48. 14 69. 63 48. 75 69. 20 49. 36 68. 77 49. 96 90 74. 17 50. 98 73. 72 51. 62 73. 27 52. 26 72. 81 52. 90 95 78. 29 53.-81 77. 82. 54. 49 77. 34 55. 17 76. 86 55. 84 100 82. 41 56. 64 81. 92 57. 36 81. 41 58. 07 80. 90 58. 78 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 6612 Deg. 55 Deg. 654~ Deg. 64 Deg. TRAVERSE TABLE. 89 St 362 Deg. 37 Deg. 3 e2 Dog. 38 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat, Dep. 1 0.80 0.59 0.80 0.60 0.79 0 61 0. 79 0. 62 2 1. 61 1. 19 1. 60 1. 20 1. 59 1. 22 1. 58 1. 23 3 2. 41 1. 78 2. 40 1. 81 2. 38 1 83 2. 36 1, 85 4 3. 22 2. 38 3. 19 2. 41 3. 17 2- 43 3. 1. 2. 46 5 4.02 2.97 3.99 3.01 3.97 3 04 3.94 3.08 6 4. 82 3.57 4. 79 3.61 4. 76 3.65 4. 73 3.69 7 5. 63 4. 16 5.59 4. 21 5. 55 4 20 5. 52 4. 31 8 6. 43 4. 76 6. 39 4. 81 6. 35 4. 87 6.30 4.93 9 7.23 5. 35 7. 19 5. 42 7. 14 5. 48 7. 09 5. 54 10 8.04 5. 95 7. 99 6. 02 7. 93 6. 09 7. 88 6. 16 11 8. 84 6. 54 8. 78 6. 62 8. 73 6- 70 8. 67 6. 77 12 9. 65 7. 14 9.58 7. 22 9, 52 7. 31 9. 46 7. 39 13 10.45 7. 73 10. 38 7. 82 10.31 7. 91 10.24 8.00 14 11.25 8. 33 11. 18 8. 43 11. 11 8- 52 11, 03 8. 62 15 12. 06 8. 92 11. 98 9. 03 11.90 9. 13 11. 82 9.23 16 12. 86 9. 62 12. 78 9. 63 12. 69 9.74 12. 61 9. 85 17 13. 67 10. 11 13. 58 10. 23 13. 49 10. 35 13. 40 10. 47 18 14. 47 10. 71 14. 38 10. 83. 14. 28 10' 96 14. 18 11. 08 19 15. 27 11. 30 15. 17 11. 43 15. 07 11. 57 14. 97'11. 70 20 16. 08 11. 90 15. 97 12. 04 15.,87 12. 18 15. 76 12. 31 21 16. 88 12. 49 16. 77 12. 64 16. 66 12. 78 16. 65 12. 93 22 17. 68 13. 09 17. 57 13. 24 17. 45 13. 39 17. 34 13. 54 23 18. 49 13. 68 18. 37 13. 84 18.25 14. 00 18. 12 14. 16 24 19. 29 14. 28 19. 17 14. 44 19. 04 14. 61 18. 91 14. 78 25 20. 10 14. 87 19. 97 15. 0 19. 83 15. 22 19. 70 15. 39 26 20. 90 15. 47 20. 76 15. 65 $0. 63 15. 83 20. 49 16. 01 27 21. 70 16. 06 21. 56 16. 25 21. 42 16. 44 21. 28 16. 62 28 22. 51 16. 65 22. 36 16. 85 22. 21 17. 05 22. 06 17. 24 29 23. 31 17. 25 23. 616 17. 45 23. 01 17- 65 22. 85 17, 85 30 24. 12 17. 84 23. 96 18. 05 23. 80 18. 26 23. 64 18. 47 35 28. 13 20. 82 27. 95 21. 06 27.77 21. 31 27. 568 21. 55 40 32. 15 23. 79 31. 95 24. 07 31. 73 4. 35 31. 52 24. 63 45 36. 17 26. 77 35. 94 27. 08 35. 70 27. 39 35. 46 27. 70 50 40. 19 29. 74 39. 93 30. 09 39. 67 30 44 39. 40 30. 78 55 44. 21 32. 72 43. 92 33. 10 43. 63 33. 48 43. 34 33. 86 60 48. 23 35. 69 47. 92 36. 1 1 47. 60 36. 53 47. 28 36, 94 65 52. 25 38. 66 51 91 39. 121 51.57 39.57 51.22 40, 02 70 56. 27 41. 64 55. 90 42. 131 55. 53 42. 61 55. 16 43. 10 75 60..29 44. 61 59. 90 45. 14 59. 50 45. 66 59. 10 46. 17 80 64. 31 47. 59 63. 89 48. 15 63. 47 48. 70 63. 04 49. 25 85 68. 33 50. 56 67. 48 51. 15 67. 43 51. 74 66. 98 52. 33 90 72. 35 53. 63 71. $8 54. 16 71. 40 54. 79 70. 92 55. 41 95 76. 37 66. 51 75. t7 57. 17 75. 37 57. 83 74. 86 58. 49 100 80. 39 59. 48 79. 86 60. 18 79. 34 60. 88 78. 80 61. 57 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. - 31 Deg. 63 Deg. 6234 Deg. 62 Deg. 90 TRAVERSE TABLE. ~. 381/ Deg. 39 Deg. 391 Deg. 40 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 78 0. 62 0.78 0. 63 0.77 0.64 0.77 0.64 2 1. 57 1.24 1.55 1. 26 1.54 1.27 1.53 1.29 3 2. 35 1. 87 2. 33 1. 89 2. 31 1.91 2. 30 1. 93 4 3. 13 2. 49 3. 11 2. 52 3. 09 2. 54 3. 06 2. 57 5 3. 91 3. 11 3.89 3. 15 3. 86 3. 18 3.83 3. 21 6 4. 70 3. 74 4. 66 3. 98 4. 63 3. 82 4. 60 3. 86 7 5. 48 4. 36 5. 44 4. 41 5. 40 4. 45 5. 36 4. 50 8 6. 26 4. 98 6. 22 5.03 6. 17 5.09 6. 13 5. 14 9 7. 04 5. 60 6. 99 5. 66 6. 94 5. 72 6. 89 5. 79 10 7. 83 6. 23 7. 77 6. 29 7. 72 6. 36 7. 66 6. 43 11 8.61 6. 85 8.55 6.92 8.49 7.001 8. 43 7.07:12 9. 39 7. 47 9. 33 7. 55 9. 26 7. 63 9. 19 7. 71 -13:10. 17 8. 09 10. 10 8. 18 10. 03 8. 27 9. 96 8. 36 14 10. 96 8. 72 10. 88 8. 81 10. 80 8. 91 10. 72 9. 00 15 11. 74 9. 34 11. 66 9. 44 11. 57 9. 54 11. 49 9. 64 16 12. 52 9. 96 12. 43 10. 07 12. 35 10. 18 12. 26 10. 28 17 13. 30 10. 58 13. 21 10. 70 13. 12- 10. 81 13. 02 10. 93 18 14. 09 11. 21 13. 99 11. 33 13. 89 11. 45 13. 79 11. 57 19 14. 87 11. 83 14. 77 11. 96 14. 66 12. 09 14. 55 12. 21 20 15. 65 12. 45 15. 54 12. 59 15. 43 12.72 15. 32 12. 86 21 16. 43 13. 07 16. 32 13. 22 16. 20 13. 36 16. 09 13. 50 22 17. 22 13. 70 17. 10 13. 84 16. 98 13. 99 16. 85 14. 14 23 18. 00 14. 32 17. 87 14. 47 17. 75 14. 63 17. 62 14. 78 24 18. 78 14. 94 18. 65 15. 10 18. 52 15. 27 18. 39 15. 43 25 19. 57 15. 56 19. 43 15.73 19. 29 15. 90 19. 1 5 16. 07 26 20. 35 16. 19 20. 21 16. 36 20. 06 16. 54 19. 92 16. 71 27 21. 13 16. 81 20. 98 16. 99 20. 83 17. 17 20. 68 17. 36 28 21. 91 17. 43 21. 76 17. 62 21. 61 17. 81 21. 45 18. 00 29 22. 70 18. 05 22. 54 18. 25 22. 38 18. 45 22. 22 18. 64 30 23. 48 18. 68 23. 31 18. 88 23. 15 19. 08 22. 98 19. 28 35 27. 39 21. 79 27. 20 22. 03 27. 01 22. 26 26. 81 22. 50 40 31. 30 24. 90 31. 09 25. 17 30. 86 25. 44 30. 64 25. 71 45 35. 22 28. 01 34. 97 28. 32 34. 72 28. 62 34. 47 28. 93 50 39. 13 31. 13 38. 86 31. 47 38. 58 31. 80 38. 30 32. 14 55 43. 04 34. 24 42. 74 34. 61 42. 44 34. 98 42. 13 35. 35 60 46. 96 37. 35 46. 63 37. 76 46. 30 38. 16 45. 96 38. 57 65 50. 87 40. 46 50. 51 40. 91 50. 16 41. 35 49. 79 41. 78 70 54. 78 43. 58 54. 40 44. 05 54. 01 44. 53 53. 62 45. 00 75 58. 70 46. 69 58. 29 47. 20 57. 87 47. 71 57. 45 48. 21 80 62. 61 49. 80 62. 17 50. 35 61. 73 50. 89 61. 28 51. 42 85 66. 52 52. 91 66. 06 53. 49 65. 59 54. 07 65. 11 54. 64 90 70. 43 56. 03 69. 94 56. 64 69. 45 57. 25 68. 94 57. 85 95 74. 35 59. 14 73. 83 59. 79 73. 30 60. 43 72. 77 61. 06 100 78. 26 62. 25 77. 71 62. 93 77. 16 63. 61 76. 60 64. 28 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 51/2 Deg. 51 Deg. 503 Deg. 50 Deg. TRAVERSE TABLE. 91 t1 40Y Deg. 41 Deg. 41i Deg. 42 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. I 0.776 0. 65 0. 75 0 66 0. 75 0. 66 0. 74.. 67 2 1. 52 1. 30 1. 51 1. 31 1. 50 1. 33 1. 49 1. 34 3 2. 28 1.95 2.26 1. 97 2. 25 1. 99 2. 23 2. 01 4 3. 04 2. 60 3.02 2. 62 3. 00 2. 65 2. 97 2. 68 5 3. 80 3. 25 3. 77 3. 28 3. 74 3. 31 3. 72 3. 35 6 4. f56 3. 90. 4.53 3. 94 4. 49 3. 98 4. 46 4.01 7 5. 32 4. 55 5.28 4. 59 5. 24 4. 64 5.20 4. 68 8 6. 08 5. 20 6. 04 5. 25 5. 99 5 30 5. 95 5. 35 9 6. 84 5. 84. 6. 19 5. 90 6. 74 5. 96 6. 69 6. 02 10 7.60 6. 49 7.55 6.56 7. 49 6. 63 7. 43 6. 69 11 8.36 7. 14 8.30 7. 22 8. 24 7- 29 8. 17 7. 36 12 9. 12 7. 79 9. Q6 7. 87 8. 99 7. 95 8. 92 8. 03 13 9. 89 8. 44 9. 81 8. 53 9. 74 8. 61 9. 66 8. 70 14 10. 65 9. 09 10.5 7 9. 18 10. 49 9. 28 10. 40 9.37 15 11. 41 9. 74, 11. 32 9. 84 11. 23 9. 94 11. 15 10.04 16 12. 17 10. 39 12. 08 10. 50 11. 98 10. 60 11. 89 10. 71 17 12. 93 11. 04 12. 83 11. 15 12. 73 11.26 12. 63 11. 38 18 13. 69 11. 69 13. 58 11. 81 13. 48 11- 93 13. 38 12. 04 19 14. 45 12. 34 14. 34 12. 47 14. 23 12. 59 14. 12 12. 71 20 15. 21 12. 99 15. 09 13. 12 14. 98 13- 25 14. 86 13. 38 21 15. 97 13. 64 15. 85 13. 78 15. 73 13. 91 15. 61 14. 05 22 16. 73 14. 29 16. 60 14. 43 16. 48 14. 58 16. 35 14. 72 23 17. 49 14. 94 17. 36 15. 09 17. 23 15- 24 17. 09 15.39 24 18. 25 15. 59 18. 11 15. 75 17. 97 15- 90 17. 84 16. 06 25 19. 01 16. 24 18. 87 16. 40 18. 72 16. 57 18. 58 16. 73 26 19. 77 16. 89 19. 62 17. 06 19. 47 17. 23 19. 32 17. 40 27 20. 53 17. 54 20. 38 17. 71 20. 22 17 89 20. 06 18. 07 28 21. 29 18. 18 21. 13 18. 37 20. 97 18- 55 20. 81 18. 74 29 22. 05 18. 83 21. 89 19.03 21. 72 19- 22 21. 55 19. 40 30 22. 81 19. 48 22. 64 19. 68 22. 47 19. 88 22. 29 20. 07 35 26. 61 22. 73 26. 41 22.-96 26. 21 23. 19 26. 01 23. 42 40 30. 42 25. 98 30. 19 26. 24 29. 96 26. 50 29. 73 26. 77 45 34. 22 29. 23 33. 96 29. 52 33. 70 29. 82 33. 44 30. 11 50 38. 02 32. 47 37. 74 32. 80 37. 45 33. 13 37. 16 33. 46 55 41. 82 35. 72 41. 51 36. 08 41. 19 36. 44 40. 87 36. 80 60 45. 62 38. 97 45. 28 39. 36 44. 94 39. 76 44. 59 40. 15 65 49. 43 42. 21 49. 06 42. 64 48. 68 43. 07 48. 30 43. 49 70 53. 23 45. 46 52. 83 45. 92 52. 43 46. 38 52. 02 46. 84 75 57. 03 48. 71 56. 60 49. o20 56. 17 49. 70 55. 74 50. 18 80 60. 83 51. 96 60. 38 52. 48 59. 92 53. 01 59. 45 53. 53 85 64. 63 55. 20 64. 15 55. 76 63. 66 56. 32 63. 17 56. 88 90 68. 44 68. 45 67. 92 59. 05 67. 41 59. 64 66. 88 60. 22 95 72. 24 61. 70 71. 70 62. 33 71. 15 62. 95 70. 60 63. 57 100 76. 04 64. 98 75. 47 65. 61 74. 90 66. 26 74. 31 66. 91 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 491/ Deg. 49 Deg. 4812 Deg. 48 Deg. 92 TRAVERSE TABLE. 422 Deg. 43 Deg. 434 Deg. 44 Deg. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.74 0.68 0.73 0.68 0.73 0.69 0.72 0.69 2 1.47 1.35 1.46 1.36 1.45 1.38 1.44 1.39 3 2. 21 2. 03 2. 19 2. 05 2. 18 2. 07 2. 16 2. 08 4 2.95 2. 70 2.93 2. 73 2.90 2. 75 2. 88 2. 78 5 3.69 3. 38 3. 66 3. 41 3. 63 3. 44 3.60 3. 47 6 4.42 4. 05 4. 39 4. 09 4. 35 4. 13 4. 32 4. 17 7 5. 16 4. 73 5. 12 4. 77 5. 08 4. 82 5. 04 4. 86 8 5. 90 5. 40 5. 85 5. 46 5. 80 5.51 5.75 5. 56 9 6. 64 6. 08 6. 58 6. 14 6. 53 6. 20 6.47 6. 25 10 7. 37 6. 76 7. 31 6. 82 7. 25 6. 88 7. 19 6. 95 11 8. 11 7. 43 8. 04 7. 50 7. 98 7. 57 7. 91 7. 64 12 8.85 8. 11 8. 78 8. 18 8.70 8. 26 8. 63 8.34 13 9. 58 8. 78 9. 51 8. 87 9. 43 8. 95 9. 35 9.03 14 10.32 9.46 10.24 9.55 10. 16 9.64 10.07 9.73 15 11.06 10. 13 10. 97 10. 23 10. 88 10. 33 10. 79 10. 42 16 11. 80 10. 81 11. 70 10. 91| 11. 61 11. 01 11. 51 11. 11 17 12. 53 11. 48 12. 43 11. 59 12. 33 11. 70 12. 23 11. 81 18 13. 27 12. 16 13. 16 12. 28 13. 06 12. 39 12. 95 12. 50 19 14. 01 12. 84 13. 90 12. 96 13. 78 13. 08 13. 67 13. 20 20 14. 75 13. 51 14. 63 13. 64 14. 51 13. 77 14. 39 13. 89 21 15. 48 14. 19 15. 36 14. 32 15. 23 14. 46 15. 11 14. 59 22 16. 22 14. 86 16. 09 15. 00 15. 96 15. 14 15. 83 15. 28 23 16. 96 15. 54 16. 82 15. 69 16. 88 15. 83 16. 54 15. 98 24 17. 69 16. 21 17. 55 16. 37 17. 41 16. 52 17. 26 16. 67 25 18. 43 16. 89 18. 28 17. 05 18. 13 17. 21 17. 98 17.37 26 19. 17 17. 57 19. 02 17. 73 18. 86 17. 90 18. 70 18. 06 27 19. 91 18. 24 19. 75 18. 41 19. 59 18.59 19. 42 18.76 28 20. 64 18. 92 20. 48 19. 10 20. 31 19. 27 20. 14 19. 45 29 21. 38 19. 59 21. 21 19. 78 21. 04 19. 96 20. 86 20. 15 30 22. 12 20. 27 21.-94 20. 46 21. 76 20. 65 21. 58 20. 84 35 25. 80 23. 65 25. 60 23. 87 25. 39 24. 09 25. 18 24. 31 40 29. 49 27. 02' 29. 25 27. 28 29. 01 27. 63 28. 77 27. 79 45 33. 18 30. 40 32. 91 30. 69 32. 64 30. 98 32. 37 31. 26 50 36. 86 33. 78 36. 57 34. 10 36. 27 34. 42 35. 57 34. 73 55 40. 55 37. 16 40. 22 37. 51 39. 90 37. 86 39. 96 38. 21 60 44. 24 40. 54 43. 88 40. 92 43. 52 41. 30 43. 16 41. 68 65 47. 92 43. 91 47. 54 44. 33 47. 15 44. 74 146. 76 45. 15 70 51. 61 47. 29 51. 19 47. 74 50. 78 48. 18 50. 35 48. 63 75 55. 30 50.67 54. 85 51. 15 54. 40 51. 63 53. 95 52. 10 80 58. 98 54. 05 58. 51 54. 56 58. 03 55. 07 57. 55 55. 57 85 62. 67 57. 43 62. 17 57. 97 61. 66 58. 51 61. 14 59. 05 90 66. 35 60. 80 65. 82 61. 38 65. 28 61. 95 64. 74 62. 52 95 70. 04 64. 18 69. 48 64. 79 68. 91 65. 39 68. 34 65. 99 100 73. 73 67. 56 73. 14 68. 20 72. 54 68. 84 71. 93 69. 47 Dep. Lat. Dep. Lat. Dep. Lat. Dep. tat. 472 Deg. 47 Deg. |46 Deg. 46 Deg. TRAVERSE TABLE. 93 i: 441 Deg. 45 Deg. 2 Lat. Dep. Lat. Dep. 1 0.71 0.0. 70 0.71 0 71 2 1. 43 1. 40 1. 41 1. 41 3 2. 14 2. 10 2. 12 2. 12 4 2. 85 2. 80 2. 83 2. 83 5 3. 57 3. 50 3. 54 3. 54 6 4. 28 4. 21 4. 24 4. 24 7 4. 99 4. 91 4. 95 4. 95 8 5. 71 5. 61 5. 66 5. 66 9 6. 42 6. 31 6. 36 6. 36 10 7. 13 7. 01 7.07 7.07 11 7. 85 7. 71 7. 78 7. 78 12 8. 56 8. 41 8. 49 8. 49 13 9. 27 9. 11 9. 19 9. 19 14 9. 99 9. 81 9. 90 -9. 90 15 10. 70 10. 51 10. 61 10. 61 16 11.241 11. 21 11.31.31 17 12. 13 11. 92 12. 02 12. 02 18 12. 84 12. 62 12. 73 12. 73 19 13. 55 13. 32 13. 43 13. 43 20 14. 26 14. 02 14. 14 14. 14 21 14. 98 14. 72 14.-85 14. 85 22 15. 69 15. 42 15. 56 15. 56 23 16. 40 16. 12 16. 26 16. 26 24 17. 12 16. 82 16. 97 16. 97 25 17. 83 17. 52 17. 68 17. 68 26 18. 54 18. 22 18. 38 18. 38 27 19. 26 18. 92 19. 09 19. 09 28 19. 97 19. 63 19. 80 19. 80 29 20. 68 20. 33 20. 51 20. 51 30 21. 40 21. 03 21. 21 21.21 35 24. 96 24. 53 24. 75 24. 75 40 28. 53 28. 04 28. 28 28. 28 45 32. 10 31. 54 31. 82 31.82 50 35. 66 35. 05 35. 36 35. 36 65 39. 23 38. 55 38. 89 38. 89 60 42. 79 42. 05 42. 43 42. 43 65 46. 36 45. 56 45. 96 45. 96 70 49. 93 49. 06 49. 50 49. 50 75 53. 49 52. 57 53. 03 53. 03 80 57. 06 56. 07 56. 57 56. 57 85 60. 63 59. 58 60. 10 60. 10 90 64. 19 63. 08 63. 64 63. 64 95 67. 76 66. 59 67. 18 67. 18 100 71. 33 70. 09 70. 71 70. 71 Dep. Lat. Dep. Lat. 45Y Deg. 45 Deg. 94 Meridional Parts TABLE IV. 0~ 1 20 33 40 5~ 60 70 80 9 100 110 12 13~ 140 150 0 0 60 120 1801 240 300 361 421 48' 5 42 603 664 726 787 848 910 1 1 61 121 181 241 301 362 422 483 543 604 665 726 788 850 911 2 2 62 122 1832 242 302 363 423 484i 544 605 666- 727 789 851 913 3 3 63 123 183 243 303 364 424 485 545[ 606 667 728 790 852 914 4 4 64 124 184 244 304 365] 425 486 546 607 668 729 791 853 915 5 5 66 1265 185 245 305 366 426 487 647 608 669 730 792 864 916 6 6 66 126 186 246 306 367 427 488 548 609 670 731 793 855 917 7 7 671 127. 187 247 307 368 428 489 549 610 671 732 794 856 918 8 8 68 128 188 248 308 369 429 490 550 611 672 734 795 857 919 9 9 69 129 189 249 309 370 430 491 551 612 673 735 796 858 920 10 10 70 130 190 250/ 310 371 431 492 552 613 674 736 797 859 921 11 1 1 71 131 191 2561 311 372 432 4931 553 614 676 737 798 860 922 12 12 72 132 192 262 312 373 433 494 554f 615 676 738 799 861 923 13 13 73 133 193 253 313 374 434 495 5556 616 677 739 800 862 924 14 14 74 134 194 254 314 375 435 496 556 017 678 740 801 863 925 15 15 75 135 195 255 315 376 436 497 E557 618 679 741 802 864 926 16 16 76,136 196 266 316 377 4371 468 558 619 680 742 803 865 927 17 17 77 137 197 257 317 378 438 499 559 620 681 743 804 366 928 18 18 78 138 198 258 318 379 439 500 560 621 682 7441 805 867 929 19 19 79 139 199 259 319 380 440 601 661 622 683 745 806 868 930 20 20 80 140 200 260 320 381 441 502 562 623 684 746 807 869 931 21 21 81 141 201 26111 21 382 442 603 564 624 685 747 808 870 932 22 22 82 142 202 262 322 383 44 604565 625[ 687 748 809 871 933 23 23 83 1431 20 63 33 3384 444 50 i5 626 688 749 810 872 934 24 24 84 1441204 264 824 385446 5065671627 689 750 811 873 935 25 25 85 146 205 265 325 386 446 507 568 628 690 751 812 874 936 26 26 86 146 206 266 326[ 387 447 508 569 629 691 752 313 875 937 27 27 87 147 207 267 327 388 448 509 570 631 692 753 815 876 938 28 28 88 148 208 268 328 389 449 510 571 632 693 754[ 816 877 939 29 29 89 149 209 269 330 390 450 511 57'2 633 694 7655 817 878 941 30 30 90 150 21 270 331 391 451 12 573 634 695[ 7561 818 8791 942 31 31 91 161 211 271 332 392 452 613 574 665 696 7571 8191 8801 943 32 32 92 152 212 272 333 393 453 514 575 636 697 758[ 820 882 944 33 33 93 163 213 273 334 394 454 515 576 637 698 759 821 883 945 34 34 94 154 214 274 335 39.5 455 516 577 638 699 7601 822 884 946 36 35 95 16655 215 275 336 396 456 517 578 639 700 761 8231 885 947 36 36 96 156 216 276 337 397 457 518 579 640 701 7621 824 8861 948 37 37 97 157 217 277 338 398 458 519 580 941. 702 7631 8251 887 949 38 38 98 158 218 278 339 399 459 520 581 642 703 764 826] 8881 950 39 39 99 159 219 27 340 4 460 521. 582 645 704 765 827 889 951 40 401 100 160 21 280 341 401 4611522 583 644' 705] 7661 8281 8901 952 41 411 101 161 221 281 342 40 462 523 584 640 706 767 829 891 953 42 42 102 162 225 282 343 4631 524 58,5 [646[ 707 7681 830 8921 954 43 43 103 163 223 283 344 404 464] 625 586 6 708 769 831 893 965 44 441041 164 294 28 345 40546 587 648 709 770 832 894 956 45 41 105] 165 225 285 346 4( 466 527 588 649[ 710 7711 8331 895 957 46 46 1061 166 226 286 347 407 467 5328 589 650, 7111 721 834 896 958 47 47 1071 167 227 287 348 408 4681 62 590 651 712 773 835 897 959 48 48 1081 168 228 288 349 409 469 530 591 652 713 774[ 836[ 8981 960 49 49 109 169 229 289 350 410 470 531 5i92 663 714 775 837 899 961 50 50 110 170 230 290 351 411 471 5632 5931654 716 777 838 900 962 51 61 111 171 231 291 352 41-2 472 6331 94 665'[ 716 778[ 839 901 963 52 62 112 172 232 292 363 413 473 4 63 5956 656f 717' 7791 8401 902[ 964 53 53 113| 173 233| 2931 354 414 474 5356 596 67[: 718 7801 841 903 965 54 54 1141 174 234 294 355 415 4761 36 597 658 719 781[ 842 904 966 55 55 115 175 235 2995 36 416 477 537 5981 659 720 7821 843 905 968 66 56 116| 176 236 296 357| 417 478| 538 599 660 721 783 844 606 969 57 67 1171 177 237 297 358 418 479 539 6001 661 722 784 845 907 970 58 58 1181 178 238 298 359 419 480 540 6011 6621 723 7851 846 908 971 59 59 1191 179 2391 299 3601 4201 481| 541 6021 6631 724 7861 847 909 972 TABLE IV. Meridional Parts. 95 w 160 170 180 190 200 210 22o 23~ 240 250 260 270_. 28~ 0 973 1035 1098 1161 1225 1289 1354 1419 1484 1550 1616 1684 1751 1 974 1036 1099 1163 1226 1290 1355 1420 1485 1551 1618 1685 1752 2 975 1037 1100 1164 1227 1291 1356 1421 1486 1552 1619 1686 1753 3 976 1038 1101 1166 1228 1292 1357 1422 1487 1553 1620 1687 1755 4 977 1039 1102 1166 1229 1293 1358 1423 1488 1554 1621 1688 1756 5 978 1041 1103 1167 1230 1295 1359 1424 1490 1556 1622 1689 1757 6 979 1042 1105 1168 1232 1296 1360 1425 1491 1557 1623 1690 1758 7 980 1043 1106 1169 1233 1297 1361 1426 1492 1558 1624 1692 1759 8 981 1044 1107 1170 1234 1298 1362 1427 1493 1559 1625 1693 1760 9 982 1045 1108 1171 1235 1299 1363 1428 1494 1560 1626 1694 1761 10 983 1046 1109 1172 1236 1300 1364 1430 1495 1561 1628 1695 1762 11 984 1047 1110 1173 1237 1301 1366 1431 1496 1562 1629 1696 1764 12 985 1048 1111 1174 1238 1302 1367 1432 1497 1563 1630 1697 1765 13 986 1049 1112 1175 1239 1303 1368 1433 1498 1564 1631 1698 1766 14 987 1050 1113 1176 1240 1304 1369 14341 1499 1565 1632 1699 1767 15 988 1051 11114 1177 1241 1306 1370 14351 1500 1567 1633 1700 1768 16 989 1052 1115 1178 1242 1306 1371 1436 1502 1568 1634 1701 1769 17 990 1053 1116 1179 1243 1307 1372 1437 1503 1569 1635 1703 1770 18 991'1054 1117 1181 1244 1308 1373 1438 1504 1570 1637 1704 1772 19 993 1055 1118 1182 1245 1310 1374 1439 1505 1571 1638 1705 1773 20 994 1056 1119 1183 1246 1311 1375 1440 1506 1572 1639 1706 1774 21 995 1057 1120 1184 1248 1312 1376 1441 1507 1573 1640 1707 1775 22 996 1058 1121 1185 1249 1313 1377 1443 1508 1674 1641 1708 1776 23 997 1059 1122 1186 1250 1314 1379 1444 1509 1575 1642 1709 1777 24 998 1060 1123 1187 1251 1315 1380 1445 1510 1577 1643 1711 1778 25 999 1161 1125 1188 1252 1316 1381 1446 1611 1578 1644 1712 1780 26 1000 1063 1126 1189 1253 1317 1382 1447 1513 1579 1645 1713 1781 27 1001 1064 1127 1190 1254 1318 1383 1448 1514 1580 1647 1714 1782 28 1002 1065 1128 1191 1255 1319 1384 1449 1515 1681 1648 1715 1783 29 1003 1066 1129 1192 1256 1320 1385 1450 1516 1582 1649 1716 1784 80 1004 1067 1130 1193 1257 1321 1386 1451 1517 1583 1650 1717 1785 31 1005 1068 1131 1194 1258 1322 1387 1452 1518 1584 1661 1718 1786 82 1006 1069 1132 1195 1259 1324 1388 1453 1519 1585 1652 1720 1787 83 1007 1070 1133 1196 1260 1325 1389 1455 1520 1586 1653 1721 1789 34 1008 1071 1134 1198 1261 1326 1390 1456 1521 1588 1654 1722 1790 35 1009 1072 1135 1199 1262 1327 1392 1457 1522 1589 1656 1723 1791 36 1010 1073 1136 1200 1264 1328 1393 1458 1524 1590 1657 1724 1792 31 1011 1074 1137 1201 1265 1329 1394 1459 1525 1591 1658 1725 1793 38 1012 1075 1138 1202 1266 1330 1395 1460 1526 1592 1659 1726 1794 39 1013 1076 1139 1203 1267 1331 1396 1461 1527 1593 1660 1727 1795. 40 1014 1077 1140 1204 1268 1332 1397 1462 1528 1594 1661 1729 1797 41 1015 1078 1141'1205 1269 1333 1398 1463 1629 1695 1662 1730 1798 42 1016 1079 1142 1206 1270 1334 1399 1464 1530 1696 1663 1731 1799 43 1018 1080 1144 1207 1271 1335 1400 1465 1531 1598 1664 1732 1800 44 1019 1081 1145 1208 1272 1336 1401 1467 1532 1699 1666 1733 1801 45 1020.1082 1146 1209 1273 1338 1402 1468 1533 1600 1667 1734 1802 46 1021 1084 1147 1210 1274 1339 1403 1469 1535 1601 1668 1736 1803 47 1022 1085 1148 1211 1275 1340 1405 1470 1536 1602 1669 1736 1805 48 1023 1086 1149 1212 1276 1341 1406 1471 1587 1603 1670 1737 1806 49 1024 1087 1150 0213 1277 1342 1407 1472 1538 1604 1671 1739 1807 50 1025 1088 1151 1215 1278 1343 1408 1473 1539 1605 1672 1740 1808 51 1026 1089 1152 1216 1280 1344 1409 1474 1540 1606 1673 1741 1809 52 1027 1090 1153 1217 1281 1345 1410 1475 1541 1608 1675 1742 1810 53 1028 1091 1154 1218 1282 1346 1411 1476 1542 1609 1676 1743 1811 54 1029 1092 1155 1219 1283 1347 1412 1477 1543 1610 1677 1744 1813 65 1030 1093 11566 1220 1284 1348 1413 1479 1544 1611 1678 1746 1814 66 1031 1094 1157 1221 1285 1349 1414 1480 1546 1612 1679 1747 1815 57 1032 1096 1158 1222 1286 1350 1415 1481 1547 1613 1680 1748 1816 58 1033 1096 1159 1223 1287 1352 1416 1482 1548 1614 1681 1749 1817'59 1034 1097 1160 1224 1288 1353 1418 1483 1549 1615 1682 1750 1818......... 96 Aleridional Parts,'ABLd IV. _ 29o0 300~ 3I 301 3~ 8 0 3~ 350 36~ o7~ 30 380 390 400 410 10 819 1888 1958 2028 2100 2171 2 244 2318 2393 24681 2545 2623 2702 1 1821 1890 19591 20301 2101 21731 2246 2319 2394 2470 2546 2624 2703 2 1822 1891 1960 2031 2102 2174 2247 2320 2395 2471 2648 2625 2704 3 1823 1892 1962 2032 2103 2176 2248 2322 2396 2472 2549 2627 2706 4 1924 1893 1963 2033 2104 2176 2249 2323 2398 2473 2550 2628 2707 6 1825 1894 1964 2034 2105 2178 2250 2324 2399 2475 2551 2629 2708 6 1826 1895 1965 2035 2107 2179 2252 2325 2400 2479 2553 2631 2710 1827 1896 1966 2037 2108 2180 2253 2327 2401 2477 2554 26321 2711 8 1829 1898 1967 2038 2109 2181 2254 2328 2403 2478 2555 2633 2712 9 1830 1899 1969 2039 2110 2182 2255 2329 2404 2480 2557 2634 2714 10 1831 1900 1970 2040 2111 2184 2257 2330 2405 2481 2558 2636 2715 11 1832 1901 1971 2041 2113 2185 2258 2332 2406 2482 2559 2637 2716 12 1833 1902 1972 2043 2114 2186 2259 2333 2408 2484 2560 2638 2718 13 1834 1903: 1973 2044 2115 2187 2260 2334 2409 2485 2562 2640 2719 14 1835 1905 1974 2045 2116 2188 2261 2335 2410 2486 2563 2641 2720 15 1837 1906 1976 2046 2117 2190 2263 2337 2411 2487 2564 2642 2722 16 1838 1907 1977 2047 21191 2191 2264 2338 2413 2489 2566 2644 2723 17 1839 1908 1978 2048 21'20 2192 2265 2339 2414 2490 2567 2645] 2724 18 1840 1909 1679 2050 2121 2193 2266 2340 2415 2491 2568 2646 2726 19 1841 1910 1980 2051 2122 2194 2268 2342 2416 2492 2569 2648 2727 20 1842/ 1912 1981 2052 2123 2196 2269 2343 2418 2494 2571 2649 2728 21 1843 1913 1983 2053 12125 2197 2270 2344 2419 2495i 2572 2650 2729 22 1845 1914 1984 2054 2126 2198 2271 2345 2420 2496 2573 2551 2731 23 1846 1915 1985 2056 2127 2199 2272 2346 2422 2498 2575 2653 -2732 24 1847 1916 1986 2057 2128 2200 2274 2348 2423 2499 2576 2654 2733 25 1848 1917 1987 2058 2129 2202 2275[ 2349 2424 2500[ 2577. 26556 2735 29 1849 1918 1988 2059 2131 2203 2266 2350 2425 2501 2578 2657 2736 27 1850 1920 1990 2060 2132 2204 2267 23:51 2427 2503 2580] 2658 2737 28 1852 1921 1991 2061 2133 2205 2279 2353 2428 2504 2581 2659: 2739 29 1853 1922 1992 2063 2134 2207 2280 2354 2429 2505 2582 2661 2740 30 1854 1923 1993 2064 2135] 2208 2281 2355 2430 2506 2584 2662 2742 31 1855 1924 1994 2065/ 2137 2209 2232 2356 2432 2508 2585'-2663 2743 32 1856 1925 1995 2066 2138 2210 2283 2358 2433 2509 2686 2665 2744 33 1857 1927 1997 2067 2139 2211 2285 2359 2434 2510 2588 2666 2746 34 1858 1928 1998 2069 21'40 22913 2286 2360 2435 2512 2589 2667 2!747. 35 1360 1929 1999 2070 2141 2214 2287 2361 2437 2513 25901 2669 2748 36 1861 1930 2000 2071 2143 2215 2288 2363 24389 2514 2591 2670 2760 37 18929 1931 2001 2072 2144 2216 2290 2364 2439 25i15 2593 2671 2751 38 1863 1932 2002 2073 2145{ 2217 2291 2366 2440 2517 2594 2673 2752 39 1864 1934 2004 2075 2146 2219 2292 2366 2442 2518 25965 2674 2754 40 1865 1935 2005 2076 2147 2220 2293 2368 2443 2519 2597 2675/ 2755 41 1866 1936 2006 2077 2149 2221 2295] 2369 2444 2521 2598 2676 2756 42 1868 1937 2077 2078 2150 2222 2296 2370 2445] 2522 2599 2678 2758 43 1869 1938 2008 2079 2151 2224 2297 2371 2447 29523' 2601 2679 2759 44 1870 1939 2010 2080 21521 2225[ 2298 2373 2448 2524 2602 2680 2760 45 1871 1941 2011 2082 2153 2226 2299 2374 2449 2526 2603 2682 2762 46 1872 1942 2012 2083 2155 2227 2301 2375 2451 2527 2604 2683 2763 47 1873 1943 2013 2084 2156 2228 2302 2376 2452 2528 2606 2684 2764 48 1875 1944 2014 2085 2157 2230 2303 2378 2453 2530 2907 2636 2766 49 1876 1945 2015 2086 2158 2231 2304 2379 2454 2531 2608 2687 2767 50 1877 19461 2017 2088 2159 2232 2306 2380 2456 2532 2610 2688 2768 51 1878 1948 2018 2089 2161 2233 2307 2381 2457 2533 2611 2690 2770 52 1879 1949 2019 2090 2162 2235 2308 2383 2458 2535 2612 2691 2771 53 1880 1950 2020 2091 2163 2236 2309 2384 2459 2636 2614 26929 2772 54 1881 1951 2021 2092 2164 12237 2311 2385 2461 2537 2615 2694 2774 55 1883 1952 2022 2094 2165 2238 2312 2386 246212538 2616 2695 2775 56 1884 1963 92024 2095 2167 2239 2313 2388 2463 2540 2617 2696 2776 57 1885 1955 2025 2096 2168 2241 2314 2389 2464 2541 2619 2698 2778 58 1886 1956 2026 2097 2169 2242 23161 2390 2466 2542 2620 26991 2779 59 1887 1957 2027 2098 2170 2243 2317 2391 2467 2544 2621 2700 2780 TABLE IV. Meridional Parts. 97 42~ 430 440~ 450 4~0 47~0 480~ 49~ 50~ 510 52~ 530~ 540 0 2782 2863 2946 3030 3116 3203 3292 3382 3474 3569 3665 3764 386-5 1 2783 2864 2947 3031 3117 3204 3293 3384 3476 3570 3667 3765 3866 2 2734 2866 2949 3033 3118 3206 3295 3385 3478 3572 3668 3767 3868 3 2786 2867 2950 3034 2120 3207 3296 3387 3479 3573 3670 3769 3870 4 2787 2869 2951 3036 3121 3209 3298 3388 3481 3575 3672 3770 3871 5 2788 2870 2953 3037 3123 3210 3299 3390 3482 3577 3673 3772 3873 6 2790 2871 2954 3038 3123 3212 3301 3391 3484 3578 3675 3774 3375 7 2791 2873 2956 3040 3126 3213 3302 3393 3485 3580 3677 3775 3877 8 2792 2874 2957 3041 3127 3214 3303 3394 3487 3582 3678 3777 3878 9 2794 2875 2958 3043 3129 3216 3305 3396 3488 3583 3680 3779 3880 10 2795 2877 2960 3044 3130 3217 3306 3397 3490 3585 3681 3780 3882 11 2797 2878 2961 3046 3131 3219 3308 3399 3492 3586 3683 3782 3883 12 2798 2880 2963 3047 3133 3220 3309 3400 3493 3588 3685 3784 3885 13 2799 2881 2964 3048 3134 3222 3311 3402 3495 3590 3686 3785 3887 14 2801 2882 2965 3050 3136 3224 3312 3403 3496 3591 3688 3787 3889 15 2802 2884 2967 3051 3137 3225 33141 3405 3498 3593 3690 3799 3890 16 2803 2885 2968 3053 3139 3226 3316 3407 3499 3594 3691 3790 3892 17 2805 2886 2970 3054 3140 3228 3317 3408 3501 3596 3693 3792 3894 18 2806 2888 2971 3055 3142 3229 3319 3410 3503 3598 3695 3794 3895 19 2807 2889 2972 3057 3143 3231 3320 3411 3504 3599 3696 3795 3897 20 2809 2891 2974 3058 3144 3232 3322 3413 3506 3601 3698 3797 3899 21 2810 2892 2975 3060 3146 3234 3323 3414 3507 3602 3699 3799 3901 22 2811 2893 2976 3061 3147 3235 3325 3416 3509 3604 3701 3800 3902 23 2813 2895 2978 3063 3149 3237 3326 3417 3510 3606 3703 3802 3904 24 2814 2896 2979 3064 3150 3238 3328 3419 3512 3607 3704 3804 3906 25 2815 2897 2981 3065 3152 3240 3329 3420 3514 3609 3706 3806 3907 26 2817 2899 2982 3067 3153 3241 3331 3422 3515 3610 3708 3807 3909 27 2818 2900 2983 3068 3155 3242 3332 3423 3517 3612 3709 3809 3911 28 2820 2902 2985 3070 3156 3244 3334 3425 3518 3614 3711 3811 3913 29 2821 2903 2986. 3071 3157 3245 3335 3427 3520 3615 3713 3812 3914 30 2822 2904 2988 3073 3159 3247 3337 3428 3521 3617 3714 3814 3916 31 2824 2906 2989 3074 3160 3248 3338 3430 3523 3618 3716 3816 3918 32 2825 2907 2991 3075 3162 3250 3340 3431 3525 3620 3717 3817 3919 33 2826 2908 2992 3077 3163 3251 3341 3433, 3526 3622 3719 3819 3921 34 2828 2910 2993 3078 3165 3253 3343 3434 3528 3623 3721 3821 3923 35 2829 2911 2995 3080 3166 3254 3344 3436 3529 3625 3722 3822 3925 36 2830 2913 2996 3081 3168 3256 3346 3437 3531 3626 3724 3824 3926 37 2832 2914 2998 3083 3169 3257 3347 3439 3532 3628 3726 3826 3928 38 2833 2915 2999 3084 3171 3259 3349 3440 3534 3630 3727 3827 3930 39 2834 2917 3000 3085 3172 3260 3350 3442 3536 3631 3729 3829 3932 40 2836 2918 3002 3087 3173 3262 3352 3543 3537 3633 3731 3831 3933 41 2837 2919 3003 3088 3175 3263 3353 3445 3539 3634 3732 383~ 3935 42 2839 2921 3005 3090 3176 3265 3355 3447 3540 3636 3734 3834 3937 43 2840 2922 2006 3091 3178 3266 3356 3448 3542 3638 3736 3836 3938 44 2841 2924 3007 3093 3179 3268 3358 3450 3543 3639 3737 3838 3940 45'2843 2925 3009 3094 3181 3269 3359 3551 3545 3641 3739 3839 3942 46 2844 2926 3010 3095 3182 3271 3361 3453 3547 3643 3741 3841 3944 47 2845 2928 3012 3097 3184 3272 3362 3454 3548 3644 3742 3843 3945 48 2847 2929 3013 3098 3185 3274 3364 3456 3550 3646 3744 3844 3947 49 2848 2931 3014 3100 3187 3275 3365 3457 3551 3647 3746 3846 3949 50 2849 2932 3016 3101 3188 3277 3367 3459 3553 3649 3747 3848 3951 51 2851 2933 3017 3103 3190 3278 3368 3460 3555 3651 3749 3849 3952 52 2852 2935 3019 3104 3191 3280 3370 3462 3556 3652 3750 3851 3954 63 2854 2936 3020 3105 3192 3281 3371 3464 3558 3654 3752 3853 3956 54 2855 2937 3021 3107 3194 3283 3373 3465 3559 3655 3754 3854 3958 55 2856 2939 3023 3108 3195 3284 3374 3467 3561 3657 3755 3856 3959 56 2858 2940 3024 31101 3197 3286 3376 3468 3562 3659 3757 3858 3961 57 2859 2942 3026 31111 3198 3287 3378 3470 3564 3660 3759 3860 3963 681 2860 2943 3027 3113 3200 3289 3379 3471 3566 3662 3760 3861 3964 59 2862 2944 3029 31141 3201 3290 3381 34731 3567 3664 37621 38631 3966.... _.... 98 Meridional Parts. TABLE IV.' 550 660 570 680 590 600 610 620 630 640 650 660 670 O 3968 4074 4183 4294 4409 4527 4649 4775 4905 5039 5179'5324 5474 1 3970 4076 4184 4296 4411 4529 4651 4777 4907 5042 5181 5326 5477 2 3971 4077 4186 4298 4413 4531 4653 4779 4909 5044 5184 5328 5479 3 3973 4079 4188 4300 4415 4533 4655 4781 4912 5046 5186 5331 5482 4 3975 4081 4190 4302 4417 4535 4657 4784 4914 5049 5188 5333 5484 5 3977 4083 4192 4304 4419 4537 4660 4786 4916 5051 5191 5336 5487 6 3978 4085 4194 4306 4421 4539 4662 4788 4918 5053'5193 6338 5489 7 3980 4086 4195 4308 4423 4541 4664 4790 4920 5055 5195. 5341 5492 8 3982 4058 4197 4309 4425 4543 4666 4792 4923 5058 5198 5343 5495 9 3984 4080 4199 4311 4427 4545 4668 4794 4925 5060 5200 5346 5497 10 3985 4092 4202 4313 4429 4547 4670 4796 4927 5062 5203 5348 5500 11 3987 4094 4203 4315 4431 4549 4672 4798 4929 5065 5205 53B51 5502 12 3989 4095 4205 4317 4433 4551 4674 4801 4931 5067 5207 5353 5505 13 3991 4097 4207 4319 4434 4553 4676 4805 4934 5069 5210 5356 5507 14 3992 4099 4208 4321 4436 4555 4678 4808 4936 65071 5212 5358 5510 15 3994 4101 4210 4323 4438 4557 4680 4807 4938 5074 5214 5361 5513 16 3996 4103 4212 4325 4440 4559 4682 4809 4940 5076 5217 5363 5515 17 3998 4104 4214 4327 4442 4562 4684 4811 4943 5078 5219 5366 5518 18 3999 4106 4216 4328 4444 4564 4687 4814 4945 5081 5222 5368 5520 19 4001 4108 4218 4330 4446 4566 4689 4816 4947 5083 5224 5371 5523 20 4003 4110 4220 4332 4448 4568 4691 4818 4949 5085 5226 5373 5526 21 4005 4112 4221 4334 4460 4670 4693 4820 4951 508 56229 5376 5528 22 4006 4113 4223 4336 4452 4572 4695 4822 4954 6090 5231 5378 5531 23 4008 4116 4225 4338 4454 4574 4697 4824 4956 5092 5234 5380 5533 24 4010 4117 4227 4340 4456 4576 4699 4826 4958 5095 5236 5383 5536 25 4012 4119 4229 4342 4458 4578 -4701 4829 4960 5097 5238 5385 5539 26 4014 4121 4231 4344 4460 4580 4703 4831 4963 5099 5241 5388 5541 27 4015 4122 4232 4346 4462 4582 4705 4833 4965 5102 5243 5390 56544 28 4017 4124 4234 4347 4464 4584 4707 4835 4967 5104 5246 5393 5546 29 4019 4126 4236 4349 4466 4586 4710 4837 4969 5106 6248 5395 5549 30 4021 4128 4238 4351 4468 4588 4712 4839 4972 5108 5250 5398 55562 31 4022 4130 4240 4353 4470 4590 4714 4842 4974 5111 5253 5401 5554 32 4024 4132 4242 4355 4472 4592 4716 4844 4976 5113 5255 5403 5557 33 4026 4t33 4244 4357 44Y4 4594 4718 4846 4978 6115 5258 5406 5559 34 4028 4135 4246 4359 4476 4696 4720 4848 4981 5118 5260 5408 5562 35 4029 4137 4247 4361 4478 4598 4722 4850 4983 6120 5263 5411- 5565 36 4031 4139 4249 4363 4480 4600 4724 4852 4985 6122 5265 5413 5567 37 4033 4141 4251 4365 4482 4602 4726 48655 4987 5125 5267 5416 5570 38 4035 4142 4253 4367 4484 4604 4728 4857 4990 5127 5270'5418 5573 39 4037 4144 4255 4369 4486 4606 4731 4859 4992 5129 5272 5421 5575 40 4038 4146 4257 4370 4488' 4608 4733 4861 4994 5132 5276 5423 5578 41 4040 4148 4259 4372 4490 4610 4735 4863 4996 5134 5277 5426 5580 42 4042 4160 4250 4374 4492 4612 4736 4865 4999 5136 5280 5428 5583 43 4044 4152 4262 4376 4494 4614 4739 4868 5001 5139 5282 54311 65586 44 4045 4153 4264 4378 4495 4616 4741 4870 5003 5141. 5284 5433 5588 456 4047 4155 4266 4380 4497 4618 4743 4872 5005 5143 5287 5436 5591 46 4049 4157 4268 4382 4499 4620 4745 4874 5008 5146 6289 5438 5594 47 4051 4159 4270 4384 4601 4623 4747 4876 5010 5148 5292 5441 5596 48 4052 4161 4272 4386 4503 4625 4750 4879 56012 5151 5294 5443 5599 49 4054 4162 4274 4388 4605 4627 4752 4881 56014 5153 5297 5446 5602 50 4056 4164 4275 4390 4507 4629 4764 4883 56017 5155 5299 5448 5604 51 4058 4166 4277 4392 45609 4631 4756 4886 5019 5158 5301 5451 5607 52 4060 4168 4279 4394 4511 4633 4758 4887 5021 6160 5304 5454 5610 53 4061 4170 4281 4396 4513 4635 4760 4890 5023 5162 5306 5456 5612 54 4063 4172 4283 4398 4515 4637 4762 4892 6026 5165 6309 5458 5615 55 4065 4173 42865 4399 4517 4639 4764 4894 56028 5167 5311 5461 5617 56 4067 4175 4287 4401 4519 4641 4766 4896 56030 56169 5314 5464 5620 57 4069 4177 4289 4403. 4521 4643 4769 4898 5033 5172 5316 5466 5623 58 4070 4179 4291 4405 4523 4645 4771 4901 5035 5174 5319 5469 5625 59 4072 4181 4292 4407 4626 4647 47738 4903 5037 5176 5321 5471 6628............. TABLE IV. Meridional Parts. 99 _' 680 690 700 710 72 730 740 7 0 7o 77_ 7so8 7o0 800 10 5631579 5-966 6146- 6335 6534 6746 6970 7210 7467 77451 8046 8376 1 5633 5797 5969 6149 6338 6538 6749 6974 7214 7472 7749 8051 8381 2 5636 5800 5972 6152 6341 6541 6753 6978 7218 7476 7754 8056 8387 3 5639 6803 5975 61556 6346 6545 6767 69829 7222 7481 7759 8061] 8393 4 5642'806 5978 6158 6348 6548 6760 6986 7227 7485 7764 8067 8398 6 5644 5809 5981 6161 6351 6562 6764 6990 7231 7490 7769 8072 8404 6 6646 5811 5984 6164 6354 6555 6768 6994 7235 7494 7774 8077 8410 7 5650 5814 5986 6167 6368 6558 6771 6997 7239 7498 7778 8083 8416 8 56562 5817 5989 6170 6361 6562 6775 7001 7243 7503 7783 8088 8422 9 5655 6820 5992 6173 6364 6565 6779 7005 7247 7507 7788 8093 8427 10 5658 5823 5995 6177 6367 6569 6782 7009 7252 7512 7793 8099 8433 11 5660 5825 5998 6180 6371 6572 6786 7013 7256 7516 7798 8104 8439 12 5663 5828 6001 6183 6374 6576 6790 7017 7260 7521 7803 8109 8445 13 5666[ 5831 6004 6186 6377 6579 6793 7021 7264 7525 7808 8115] 8451 14 5668 5834 6007 6189 6380 9583 6797 7025 7268 7530 7813 8120 8457 15 5671 5837 6010 6192 6384 6586 6801 7029 7273 7535 7817 8125 8463 16 5674 5839 6013 6195 6387 6590 6804 7033 7277 7439 7821 8131 8469 17 5676 5842 6016 6198 63901 6593 6808 7027 7281 7544 7827 8136 8474 18 5679 5845 6019 6201 6394 6597 6812 7041 7285 7548 7832 8141 8480 19 5682 58481 6022 6205 6397 6600 6815 7045 7289 7553 7837 8147 8486 20 5685 5851 6025 6208 6400 6603 6819 7048 7294 7557 7842 8152 8492 21 5687 5854 6028 6211 6403 5607 6823 7052 7298 7562 7847 8158 8498 22 5690 5856 6031 6214 6407 6610 6826 7056 7302 7566 7852 8163 8504 23 5693 5859 6034 6217 6410 6614 6830 7060 7306 7571 7857 8168 8510 24 5696 5862 6037 6220 6413 6617 6834 7064 7311 7576 7862 8174 8516 25 5698 6866 6040 6223 6417 6621 6838 7068 7315 7580 7867 9179 8522 26 5701 5868 6043 6226 6420 6624 6841 7072 7319 7585 7872 8185 8528 27 5704 5871 6046 6230 6423 6628 68451 7076 7323 7589 7877 8190 8634 28 5706 5874 6049 6233 6427 6631 6849 7080 7328 7694 7882 8196 8540 29 5709 5876 6052 6236 6430 6635 6853 7084 7332 7599 7887 8201 8546 30 5712 5879 6055 6239 6433 6639 6856 7088 7336 7603 7892 8207 8552 31 5715 5882 6058 6242 6437 6642 6860 7092 7341 7608 7897 8212 8658 32 5717 5885 6061 6245 6440 6646 6864 7096 7345 7612 7902 8218 8565 33 5720 5888 6064 6249 6443 6649 6868 7100 7349 7617 7907 82238 8571 34 5723 5891 6067 6252 6447 6653 6871 7104 7353 7622 7912 8229 8577 35 5725 5894 6070 6255 6450 6666 6875 7108 7358 7626 7917 8234 8583 36 5728 5896 6073 6258 6453 6660 6879 7112 7362 7631 7922 8240 8689 37 5731 6899 6076 6261 6457 6663 6883 7116 7366 7636 7927 8245 8595 38 5734 5902 6079 6264 6460 6667 6886 7128 7371 7640 7932 8251 8601 39 6736 69056 6082 6268 6463 6670 6890 7124 73756 7646 7937 8266 8607 40 5739 5908 6085 6271 6467 6674 6894 7128 7379 7650 7942 8262 8614 41 5742 5911 6088 6274 6470 6677 6898 7132 7384 7654 7948 8267 8620 42 5745 6914 6091 6277 6473 6681 6901 7136 7388 7659 7953 8273 8626 43 5747 5917 6094 6280 6477 6685 6905 7140 7392 7664 7958 8279 8632 44 5750 5919 6097 6283 6480 6688 6909 7145 7397 7668 7963 8284 8638 45 5753 5922 6100 6287 6483 6692 69i3 7149 7401 7673 7968 8290 8644 46 5756 5925 6103 6290 6487 6695 6917 7153 7406 7678 7973 8295 8651 47 5758 5928 6106 9293 6490 6699 6920 7157 741(0 7683 7978 8301 8657 48 5761 5931 6109 6296 6494 6702 6924 7161 7414 7687 7983 8307 8663 49 5764 5934 6112 6299 6497 6706 6928 7165 7419 7692 7989 8312 8669 50 5767 5937 6115 6303 6500 6710 6932 7169 7423 7697 7994 8318 8676 51l 770 59401 6118 6306 6504 6713 6936 7173 7427 7702 7999 8324 8682 52 5772 59431 6121 6309 6507 6717 6940 7177 7432 7706 8004 8329 8688 53 5775 5946 6124 6312 6511 6720 6943 7181 7436 7711 8009 8335 8695 54 5778 5948 6127 6315 6514 6724 6947 7185 7441 7716 8914 8341 8701 55 5781 5951 6130 6319 6517 6728 6951 7189 7445 7721 8020 8347 8707 56 5783 6954B 6133 6322 6521 6731 6955 7194 7449 772518025 8352 8714 67 5786 5957 6136 6325 6524 6735 6959 7198 7454 7730 8030 8358 8720 58 5789 5960 6140 6328 6528 6738 6963 7202 7458 7735 8035 8364 8726 59 5792 5963 6143 6332 6531 6742 6966 7206 7463 7740 8040 8369 8733 28 100 - 810 Meridional Parts. TABLE IV. 810 82: 830 840 850 0 8739 9145 9606 11317 10765E 1 8745 9163 9614 10146 10776 2 8752 9160 9622 10156 10788 3 8758 9167 9631 10166 10799 4 8766 9174 9639 10175 10811 5 8771 9182 9647 10185 10822 6 8778 9189 9655 10195 10834 7 8784 9197 9664 10205 10846 8 8791 9203 9672 10214 10858 9 8797 9211 9683110224 10869 10 8804 9218 9689 10234 10881 11 8810 9225 9697 10244 10893 12 8817 9233 97 10254 10906 131 8823 9240 9714 10364 10917 14 8830 9248 9723 10273 10929 15] 8836 9255 9731 10283 10941 161 8843 9262 974 10293 10953 171 8849 9270 9748 10303 10965 18 8856 9277 9757 10314 10978 19 8863 9285 9765 10324 10990 20 8869 9292 9774 10334 11002 21 8876 93001 9783 10344 11014 22 8831 9307 97911 10354 11027 23 8889 9315 9800 10364 11039 24 8896 9322 9809 10374 11052 25 8903 9330 9817 10386 11064 26 8909 9337 9826 10396 11077 27 8916 9346 9835 10406 11089 28 8923 9353 9844 10416 11102 29 8930 9360 9852 10426 11115 30 8936 9368 9861 10437 11127 31 8943 9376 9870 10447 11140 32 8950 9383 9879 10457 11153 33 8957/ 9a391 9888 10468 11166 34 8963 9399 9897 10479 11179 36 8970 9407 9906 10489 11192 36 8977 9414 9915i 10500 11206 37 8984 9422 9924 10510 11218 38 8991 9430 9933 106521 11231 39 8998 9438 9942 10532 11244 40 9005 9445 99651 10542 11257 41 901 9453 9960 10553 11270 42 9018 9461 996 10564 11284 43 9025 9469 9978 10575 11297 44 9032 9477 9987 10686 11310 45 9039 9485 9996 10697 11324 46 9046 9493 10005 10608 11337[ 47t 9053 9501 10016 10619 11351 4 960 9509 1,0024 10630 11365 49 9067 95106 17 10033 10641 11378 50 9074 9525 10043 10662 11392 511 9081 9533 10052 10663 11406 52 9088 9541 10061 10674 11420 53 9096 9649 10071 10686 11434 54 9103 9557 10080 10696 11448 55 9110 9566 10089 10708 11462 56 9117 9673 10099 10719 11476 67 9124 9581 10108 10730 11490 58 9131 9689 10118 10742 11504 59 9138 9598 10127 10753 11518 TABLE V. TABLE VII. 101 Dip of the Sea Horizon. Mean Refraction of Celestial Objects. t 1 - t I~ l.it. Alt. Refr Alt. Re Alt. efr. I Alt. Refr. Alt. Itefr @.oU gO 6-,,, o t s t o I 5oo''o o0 E 33 010 0 5 15 20 0 2 35 32 0 1 30 67 24, P'.+ 10 31 32 10 3 10 10 2 24 40 129 68 23 - 7 —11- —,, T 20129 50 20 6 05 2012 2233 0 1 28 69 22 1 09 38 6 4 30 28 23 30 5 00 302 21 20 1 26 70 21 2 124 41 6 18 402700 40 4 56 402 29 401 25 71 19 3 142 44 6 32 50 25 42 50 4 1 502 28 34 0 1 24 72 18 4 1 58 47 645 1 0242911 0 4 4721 02 27 201 23 73 17 5 212 50 65 8 10 23 20 10 4 43 102 26 40 1 22 74 16 6 225 53 7 10 20 22 15 20 4 39 20 2 25 35 01 21 75 16 7 236 66 7 12 3021 15 30 4 34 30 2 24 20 120 76 14 8 2 47 59 7 24 4020 18 40 4 31 40223 401 19 77 13 9 2 657 62 7 45 50 19 25 50 4 27 502 21 3601 18 78 12 10 307 65 7 566 2 018 3512 0 4 23 22 02 20 301 17 79 11 11 3 16 68 8 07 101748 14 20 102 19 37 16 80 10 12 325 71 8 18 2017 04 20 4 16 202 18 301 14 81 9 13 3 33 74 8 28 3016 4 24 30413 2 17 38 01 13 82 8 14 3 41 77 838 40 15 45 40 4 09 4012 16 3011 1 83 7 15 3 49 80 8 48 5015 0911 504 06 502 156 39 01 10 34 6 16 3 56 83 8 58 3 0 14 3413 04 03 23 0214 30 1 09 85 5 17 4 04 86 9 08 10114 04 1014 00 1012 13140 011 08 86 4 18 411 89 9 17 201334 20 3 57 20 2 12 30 1 07 87 3 19 4 17 921 9926 3013 06 30 3 54 302 1 1 1il 015 88 2 20 4 24 95 936 4012 493 51 40210 30104 89 1 21 4 31 98 9 45 50/12 15 50 3 48 50 2 09 420l 103 90 0 22 437 101 9 54 40 511114 03 45 240 208 301 02 23 4 43 104 10 02 1011 29 1013 43 1012 07 43 011 01 24 4 49 10710 11 2011 08 2013 40 202 06 3011 00 25 4 55 110 10 19 3010 48 3013 38 30 2 056 4 00 9 26 6 01 113 10 28 401029 40 35 40204 300 58 27 5 07 116 10 36 5010 11/ 5013 331 502 03 145 010 57 28 513} 119 1044 28 513 119 191044 5 0 9 54 15 013 30 25 02 02 300 56 29 518 122 1028 10016 00 3081 52 128 11 0 20 9 231 -203 26 20 2 00 3010 54 31 52 34 128 11 16 30 9 08 303 24 301 59 47 0053 32 534 131 11 8 403 21 401 68 300 52 33 539 134 1124 0 34 544 137 11 31 50 8 41 5013 19 0 1 578 00 5 35 649 140 11 39 8 2816 0317 2601 6 30 60 1 8 15 1013 15 10 1 5 400 49 20 8 03 2013 12 201 55 300 49 TABLE VI. 30 7 15 3013 1 30 1 54 0 048 Dip of the Sea Horizon at 401 40 40{3 08 401 53 30047 differen. 50 7 30 503 06 501 52 51 00 46 dierent Distances from it. 7 0 7 20 17 03 0427 01 51 30045 10. 7 11 103 03 15 150 52 00 44 Dist. Hight of Eye in Ft. 20 7 02 2013 01 301 491 3010 44 in Miles. 5 10 16520125 30 30 6 63 3012 59 45 1 483 0 43 7-T TTT-T~~ 406 451 402 57128 01 47 300 42 1112213414 68 50 6 371 5012 55I 16 1 461 4 o0 41 611 17222834 8 06 2918 02 54 301 4 6 0040 1 6 22 10 2 62 46 1 44 O 6 0 38 4 8 1215 19 231 4 6 9112115 7 20 615 202 51 29 0 142 7 0 37 3 6 7 9112114 306 08 3012 491 20 1 41158 010 35 1I 3 4 6 8 9112 401 6 01 40 2471 4011 4059 034 2 2 3 6 6 8 0 50551 02 4630 01 38 5 60 010 33 21 2 3 5 6 7 8 9 5 9119 01244 2011 37 61 010 32 3 2 3 4 5 6 7 10 5 42 1012 43 4011 36162 010 30 34 2 3 4 5 6 6 20 65 46 2012 411 310 35 63 00 29 4 2 3 4 4 5 6 30 5 41 3012 40 20 1 33164 010 28 5 2 3 4 4 5 401 5 25 4012 38 401 32165 ol0 26 6 24 3 4{ 4 5 5 501 5 502 37132 01 31 66 010 25