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Cornell University Library 
 TA 545.S66 
 
 Smithsonian geographical tables 
 
 3 1924 004 447 110 
 
Cornell University 
 Library 
 
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 tine Cornell University Library. 
 
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 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004447110 
 
^tmti^sioman ^mtisceuanemisi CoIlKtiansi 
 
 854 
 
 SMITHSONIAN 
 
 GEOGRAPHICAL TABLES 
 
 PREPARED BY 
 
 R. S. WOODWARD 
 
 CITY OF WASHINGTON 
 PUBLISHED BY THE SMITHSONIAN INSTITUTION 
 
 1894 
 
 D 
 
The Riverside Press, Cambridge, Mass., U.S.A. 
 Electrotyped and Printed by H. O. Houghton & Co. 
 
ADVERTISEMENT. 
 
 In connection with the system of meteorological observations established by 
 the Smithsonian Institution about 1850, a series of meteorological tables was 
 compiled by Dr. Arnold Guyot, at the request of Secretary Henry, and was pub- 
 lished in 1852 as a volume of the Miscellaneous Collections. 
 
 A second edition was published in 1857, and a third edition, with further 
 amendments, in 1859. 
 
 Though primarily designed for meteorological observers reporting to the 
 Smithsonian Institution, the tables were so widely used by meteorologists and 
 physicists that, after twenty-five years of valuable service, the work was again re- 
 vised, and a fourth edition was published in 1884. 
 
 In a few years the demand for the tables exhausted the edition, and it appeared 
 to me desirable to recast the work entirely, rather than to undertake its revision 
 again. After careful consideration I decided to publish the new work in three 
 parts : Meteorological Tables, Geographical Tables, and Physical Tables, each 
 representative of the latest knowledge in its field, and independent of the others ; 
 but the three forming a homogeneous series. 
 
 Although thus historically related to Doctor Guyot's Tables, the present work 
 is so entirely changed with respect to material, arrangement, and presentation, 
 that it is not a fifth edition of the older tables, but essentially a new publication. 
 
 The first volume of the new series of Smithsonian Tables (the Meteorological 
 Tables) appeared in 1893. The present volume, forming the second of the 
 series, the Geographical Tables, has been prepared by Professor R. S. Woodward, 
 formerly of the United States Coast and Geodetic Survey, but now of Columbia 
 College, New York, who has brought to the work a very wide experience both in 
 field work and in the reduction of extensive geodetic observations. 
 
 S. P. Langley, Secretary. 
 
PREFACE. 
 
 In the preparation of the following work two difficulties of quite different 
 kinds presented themselves. The first of these was to make a judicious selec- 
 tion of matter suited to the needs of the average geographer, and at the same 
 time to keep the volume within prescribed limits. Of the vast amount of 
 material available, much must be omitted from any work of limited dimen- 
 sions, and it was essential to adopt some rule of discrimination. The rule 
 adopted and adhered to, so far as practicable, was to incorporate little material 
 already accessible in good form elsewhere. Accordingly, while numerous ref- 
 erences are made in the volume to such accessible material, an attempt has 
 been made wherever feasible to introduce new matter, or matter not hitherto 
 generally available. 
 
 The second difficulty arose from the present uncertainty in the relation of the 
 British and metric units of length, or rather from the absence of any generally 
 adopted ratio of the British yard to the metre. The dimensions of the earth 
 adopted for the tables are those of General Clarke, published in 1866, and now 
 most commonly used in geodesy. These dimensions are expressed in English 
 feet, and in order to convert them into metres it is necessary to adopt a ratio of 
 the foot to the metre. The ratio used by General Clarke, and hitherto gener- 
 ally used, is now known to be erroneous by about one one hundred thousandth 
 part. The ratio used in this volume is that adopted provisionally by the Office 
 of Standard Weights and Measures of the United States and legalized by Act 
 of Congress in 1866. But inasmuch as a precise determination of this ratio is 
 now in progress under the auspices of the International Bureau of Weights and 
 Measures, and inasmuch as the value for the ratio found by this Bureau will 
 doubtless be generally adopted, it has been thought" best in the present edition 
 to restrict quantities expressed in metric measures to limits which will require 
 no change from the uncertainty in question. In conformity with this decision 
 the dimensions of the earth are given in feet only, and, with a few unimportant 
 exceptions, to which attention is called in the proper places, tables giving quan- 
 tities in metres are limited to such a number of figures as are definitely known. 
 
VI PREFACE. 
 
 It is a matter of regret that, owing to the cause just stated, less prominence 
 has been given in the tables to metric than to British units of length. On the 
 other hand, it seems probable that the more general use of British units will 
 meet the approval of the majority of those for whose use the volume is designed. 
 
 The introductory part of the volume is divided into seven sections-under the 
 heads, Useful Formulas, Mensuration, Units, Geodesy, Astronomy, Theory of 
 Errors, and Explanation of Source and Use of Tables, respectively. In pre- 
 senting the subjects embraced under the first six of these headings an attempt 
 was made to give only those features leading directly to practical applications 
 of the principles involved. It is hoped, however, that enough has been given of 
 each subject to render the work of value in a broader sense to those who may 
 desire to go beyond mere applications. 
 
 The most of the calculations required in the preparation of the tables were 
 made by Mr. Charles H. Kummell and Mr. B. C. Washington, Jr. Their work 
 was done with skill and fidelity, and it is believed that the systematic checks 
 applied by them have rendered the tables they computed entirely trustworthy. 
 Mention of the particular tables computed by each of them is made in the 
 Explanation of Source and Use of Tables, where full credit is given also for 
 data not specially prepared for the volume. 
 
 The Appendix to the present volume is that prepared by Mr. George E. Cur- 
 tis for the Meteorological Tables. Its usefulness to the geographer is no less 
 obvious and general than to the meteorologist 
 
 The proofs have been read independently by Mr. Charles H. Kummell and 
 the editor. The plate proofs, also, have been read by the editor ; and while it 
 is difficult to avoid errors in a first edition of a work containing many formulas 
 and figures, it is believed that few, if any, important errata remain in this volume. 
 
 R. S. Woodward. 
 Columbia College, New York, N. Y., June 15, 1894. 
 
CONTENTS. 
 
 USEFUL FORMULAS. 
 
 PAGE 
 
 1. Algebraic Formulas xiii 
 
 a. Arithmetic and geometric means xiii 
 
 b. Arithmetic progression xiii 
 
 0. Geometric progression xiii 
 
 d. Sums of special series xiii 
 
 e. The binomial series and applications xiv 
 
 f. Exponential and logarithmic series xiv 
 
 g. Relations of natural logarithms to other logarithms .... xv 
 
 2. Trigonometric Formulas xv 
 
 a. Signs of trigonometric functions xv 
 
 b. Values of functions for special angles xv 
 
 c. Fundamental formulas xv 
 
 d. Formulas involving two angles xvi 
 
 e. Formulas involving multiple angles xvi 
 
 f. Exponential values. Moivre's formula xvi 
 
 g. Values of functions in series xvii 
 
 h. Conversion of arcs into angles and angles into arcs .... xvii 
 
 3. Formulas for Solution of Plane Triangles xviii 
 
 4. Formulas for Solution of Spherical Triangles xx 
 
 a. Right angled spherical triangles xx 
 
 b. Oblique angled triangles xx 
 
 5. Elementary Differential Formulas xxi 
 
 a. Algebraic xxi 
 
 b. Trigonometric and inverse trigonometric xxi 
 
 6. Taylor's and Maclaurin's Series xxii 
 
 a. Taylor's series xxii 
 
 b. Maclaurin's series xxii 
 
 c. Example of Taylor's series xxii 
 
 d. Example of Maclaurin's series xxiii 
 
 7. Elementary Formulas for Integration xxiii 
 
 a. Indefinite integrals xxiii 
 
 b. Definite integration xxvi 
 
 MENSURATION. 
 
 I. Lines xxviii 
 
 a. In a circle xxviii 
 
 b. In regular polygon xxviii 
 
 c. In ellipse xxix 
 
vm CONTENTS. 
 
 2. Areas xxix 
 
 a. Area of plane triangle xxix 
 
 b. Area of trapezoid xxix 
 
 c. Area of regular polygon xxx 
 
 d. Area of circle, circular annulus, etc xxx 
 
 e. Area of ellipse xxx 
 
 f. Surface of sphere, etc xxxi 
 
 g. Surface of right cylinder xxxi 
 
 h. Surface of right cone xxxi 
 
 i. Surface of spheroid xxxi 
 
 3. Volumes xxxii 
 
 a. Volume of prism xxxii 
 
 b. Volume of pyramid xxxii 
 
 c. Volume of right circular cylinder xxxii 
 
 d. Volume of right cone with circular base xxxii 
 
 e. Volume of sphere and spherical segments xxxii 
 
 f. Volume of ellipsoid xxxiii 
 
 UNITS. 
 
 1. Standards of Length and Mass xxxiv 
 
 2. British Measures and Weights xxxvii 
 
 a. Linear measures xxxvii 
 
 b. Surface or square measures xxxviii 
 
 c. Measures of capacity xxxviii 
 
 d. Measures of weight xxxix 
 
 3. Metric Measures and Weights xl 
 
 4. The C. G. S. System of Units xlii 
 
 GEODESY. 
 
 1. Form of the Earth. The Earth's Spheroid. The Geoid . . xliii 
 
 2. Adopted Dimensions of Earth's Spheroid xliii 
 
 3. Auxiliary Quantities xliii 
 
 4. Equations to Generating Ellipse of Spheroid xliv 
 
 5. Latitudes used in Geodesy xliv 
 
 6. Radii of Curvature xlv 
 
 7. Lengths of Arcs of Meridians and Parallels of Latitude xlvi 
 
 a. Arcs of meridian xlvi 
 
 b. Arcs of parallel xlix 
 
 8. Radius-Vector of Earth's Spheroid 1 
 
 9. Areas of Zones and Quadrilaterals of the Earth's Surface 1 
 
 10. Spheres of Equal Volume and Equal Surface with Earth's 
 
 Spheroid Hi 
 
 11. Co-ordinates for the Polyconic Projection of Maps . . . Hii 
 
 12. Lines on a Spheroid Ivi 
 
 a. Characteristic property of curves of vertical section .... Ivi 
 
 b. Characteristic property of geodesic line Ivii 
 
CONTENTS. IX 
 
 13. Solution of Spheroidal Triangles Ivii 
 
 a. Spherical or spheroidal excess Iviii 
 
 14. Geodetic Differences of Latitude, Longitude, and Azimuth Iviii 
 
 a. Primary triangulation Iviii 
 
 b. Secondary triangulation Ix 
 
 15. Trigonometric Leveling Ixi 
 
 a. Computation of heights from observed zenith distances ... Ixi 
 
 b. Coefficients of refraction Ixiii 
 
 c. Dip and distance of sea horizon Ixiii 
 
 16. Miscellaneous Formulas Ixiii 
 
 a. Correction to observed angle for eccentric position of instrument Ixiii 
 
 b. Reduction of measured base to sea level Ixiv 
 
 c. The three-point problem Ixiv 
 
 17. Salient Facts of Physical Geodesy Ixv 
 
 a. Area of earth's surface, areas of continents, area of oceans . . Ixv 
 
 b. Average heights of continents and depths of oceans .... Ixv 
 
 c. Volume, surface density, mean density, and mass of earth . . Ixv 
 
 d. Principal moments of inertia and energy of rotation of earth . Ixvi 
 
 ASTRONOMY. 
 
 1. The Celestial Sphere. Planes and Circles of Reference . Ixvli 
 
 2. Spherical Co-ordinates Ixvii 
 
 a. Notation Ixvii 
 
 b. Altitude and azimuth in terms of declination and hour angle . Ixviii 
 
 c. Declination and hour angle in terms of altitude and azimuth . Ixix 
 
 d. Hour angle and azimuth in terms of zenith distance .... Ixix 
 
 e. Formulas for parallactic angle Ixix 
 
 f. Hour angle, azimuth, and zenith distance of a star at elongation Ixx 
 
 g. Hour angle, zenith distance, and parallactic angle for transit of 
 
 a star across prime vertical Ixx 
 
 h. Hour angle and azimuth of a star when in the horizon, or at the 
 
 time of rising or setting Ixxi 
 
 i. Differential formulas Ixxii 
 
 3. Relations of Different Kinds of Time used in Astronomy . Ixxii 
 
 a. The sidereal and solar days Ixxii 
 
 b. Relation of apparent and mean time Ixxiii 
 
 c. Relation of sidereal and mean solar intervals of time .... Ixxiii 
 
 d. Interconversion of sidereal and mean solar time Ixxiii 
 
 e. Relation of sidereal time to the right ascension and hour angle 
 
 of a star Ixxiv 
 
 4. Determination of Time Ixxiv 
 
 a. By meridian transits Ixxiv 
 
 b. By a single observed altitude of a star Ixxvi 
 
 c. By equal altitudes of a star Ixxvii 
 
 5. JDetermination of Latitude Ixxvii 
 
 a. By meridian altitudes Ixxvii 
 
 b. By the measured altitude of a star at a known time .... Ixxviii 
 
 c. By the zenith telescope Ixxix 
 
X CONTENTS. 
 
 6. Determination of Azimuth Ixxix 
 
 a. By observation of a star at a known time Ixxix 
 
 b. By an observed altitude of a star Ixxxi 
 
 c. By equal altitudes of a star Ixxxi 
 
 THEORY OF ERRORS. 
 
 1. Laws of Error Ixxxiii 
 
 a. Probable, mean, and average errors Ixxxiv 
 
 b. Probable, mean, average, and maximum actual errors of inter- 
 
 polated logarithms, trigonometric functions, etc Ixxxv 
 
 2. The Method of Least Squares Ixxxvi 
 
 a. General statement of method Ixxxvi 
 
 b. Relation of probable, mean, and average errors Ixxxviii 
 
 c. Case of a single unknown quantity Ixxxix 
 
 d. Case of observed function of several unknown quantities . . xc 
 
 e. Case of functions of several observed quantities xciii 
 
 f. Computation of mean and probable errors of functions of ob- 
 
 served quantities xcv 
 
 EXPLANATION OF SOURCE AND USE OF TABLES. 
 
 Tables i and 2 xcix 
 
 Table 3 . xcix 
 
 Table 4 xcix 
 
 Tables 5 and 6 xcix 
 
 Tables 7 and 8 c 
 
 Table 9 c 
 
 Tables 10 and 11 c 
 
 Table 12 c 
 
 Tables 13 and 14 c 
 
 Tables 15 and 16 ci 
 
 Table 17 ci 
 
 Table 18 cii 
 
 Tables 19-24 cii 
 
 Tables 25-29 ciii 
 
 Table 30 ciii 
 
 Table 31 civ 
 
 Tables 32 and 33 civ 
 
 Tables 34 and 35 civ 
 
 Tables 36 and 37 civ 
 
 Table 38 civ 
 
 Table 39 civ 
 
 Table 40 civ 
 
 Table 41 cv 
 
 Table 42 cv 
 
CONTENTS. XI 
 
 TABLES. 
 
 TABLB PAGE 
 
 1. For converting U. S. Weights and Measures — Customary to 
 
 Metric 2 
 
 2. For converting U. S. Weights and Measures — Metric to Cus- 
 
 tomary 3 
 
 3. Values of reciprocals, squares, cubes, square roots, cube roots, 
 
 and common logarithms of natural numbers 4-2,2 
 
 4. Circumference and area of circle in terms of diameter d . . . 23 
 
 5. Logarithms of numbers, 4-place 24-25 
 
 6. Antilogarithms, 4-place 26-27 
 
 7. Natural sines and cosines .... 28-29 
 
 8. Natural tangents and cotangents 30-31 
 
 9. Traverse table (differences of latitude and departure) .... 32-47 
 
 10. Logarithms of meridian radius of curvature in English feet . . 48-56 
 
 1 1 . Logarithms of radius of curvature of normal section in English feet 5 7-65 
 
 12. Logarithms of radius of curvature (in metres) of sections oblique 
 
 to meridian 66-67 
 
 13. Logarithms of factors for computing spheroidal excess of triangles 
 
 (unit ■= English foot) 68 
 
 14. Logarithms of factors for computing spheroidal excess of triangles 
 
 (unit = the metre) 69 
 
 15. Logarithms of factors for computing differences of latitude, longi- 
 
 tude, and azimuth in secondary triangulation (unit = English 
 
 foot) 70-73 
 
 16. Logarithms of factors for computing differences of latitude, longi- 
 
 tude, and azimuth in secondary triangulation (unit = the metre) 74-77 
 
 17. Lengths of terrestrial arcs of meridian (in English feet) .... 78-80 
 
 18. Lengths of terrestrial arcs of parallel (in English feet) .... 81-83 
 
 19. Co-ordinates for projection of maps, scale = 1/250 000 . . . . 84-91 
 
 20. Co-ordinates for projection of maps, scale =^ 1/125 000 .... 92-101 
 
 21. Co-ordinates for projection of maps, scale := 1/126 720 . . . . 102-109 
 
 22. Co-ordinates for projection of maps, scale = 1/63 360 .... 110-121 
 
 23. Co-ordinates for projection of maps, scale = 1/200000 .... 122-131 
 
 24. Co-ordinates for projection of maps, scale = 1/80 000 .... 132-141 
 
 25. Areas of quadrilaterals of the earth's surface of 10° extent in lati- 
 
 tude and longitude 142 
 
 26. Areas of quadrilaterals of the earth's surface of 1° extent in lati- 
 
 tude and longitude 144-145 
 
 27. Areas of quadrilaterals of the earth's surface of 30' extent in lati- 
 
 tude and longitude • . . 146-148 
 
 28. Areas of quadrilaterals of the earth's surface of 15' extent in lati- 
 
 tude and longitude 150-154 
 
 29. Areas of quadrilaterals of the earth's surface of 10' extent in lati- 
 
 tude and longitude 156-159 
 
 30. Determination of heights by the barometer (formula of Babinet) . 160 
 
 31. Mean astronomical refraction 161 
 
Xll CONTENTS. 
 
 32. Conversion of arc into time : 162 
 
 33. Conversion of time into arc 163 
 
 34. Conversion of mean time into sidereal time 164 
 
 35. Conversion of sidereal time into mean time 165 
 
 36. Length of 1° of the meridian at different latitudes (in metres, 
 
 statute miles, and geographic miles) 166 
 
 37. Length of 1° of the parallel at different latitudes (in metres, stat- 
 
 , ute miles, and geographic miles) 167 
 
 38. Interconversion of nautical and statute miles 168 
 
 39. Continental measures of length, with their metric and English 
 
 equivalents 168 
 
 40. Acceleration (g) of gravity on surface of earth and derived func- 
 
 tions 169 
 
 41. Linear expansions of principal metals 170 
 
 42. Fractional change in a number corresponding to a change in its 
 
 logarithm 170 
 
 APPENDIX. 
 
 Numerical Constants 171 
 
 Goedetical Constants 171 
 
 Astronomical Constants . 172 
 
 Physical Constants 172 
 
 Synoptic conversion of English and Metric Units — 
 
 English to Metric 173 
 
 Metric to English 174 
 
 Dimensions of physical quantities 175 
 
 INDEX 177 
 
USEFUL FORMULAS. 
 
 I. Algebraic. 
 
 a. Arithmetic and geometric means. The arithmetic mean of n quanti- 
 ties a, 6, c, . . . is 
 
 -^(« + d + . + ...); 
 
 their geometric mean is 
 
 {a b c. . .)". 
 
 A case of special interest is 
 
 b. Arithmetic progression. If a is the first term, and a-\-d, a-\- 2 d, 
 <7 -|- 3 ^, . . . are the successive terms, the «th or last term z is 
 
 z-:=. a -\- (n — i) d. 
 The sum s of the n terms of this series is 
 
 sz=\{a-\-z)n^{a-\-\(^~-i)d}n 
 =^ {z — i (n — i) d} ft 
 
 c. Geometric progression. If a is the first term, and a r, a r^, . . . aie the 
 successive terms, the «th or last term z is 
 
 z = a r"-\ 
 The sum of the n terms is 
 
 a (y"— i ) r z — a __ z (f— i) 
 
 If 
 
 r— I r— I (/•— i) ^-1- 
 
 r < I and n ^co, 
 a 
 
 d. Sums of special series. 
 
 I + 2+3 + 4 + --- + " =i«(«+i) 
 
 2+4 + 6 + 8 + ... + 2 « = «(«+i) 
 
 i+3 + 5-i-7-l---- + (2«— i)= «' 
 
 i"+ 2''+ 3^+ 4^+ • ■ . + «=" = J « (« + I) (2 « + i) 
 
 i'+ 2'+ 3'+ 4'+ . . ■ + «' = i «" (« + i)^ 
 
XIV USEFUL FORMULAS. 
 
 e. The binomial series and applications. 
 
 For a > b. 
 
 (JL^) ..-. 
 
 n {n— i) {n — 2) „ , ,. , 
 1.2.3 ' 
 
 For X < I, 
 
 , n{n~ i) , , n (n — T.) (n — 2) . . 
 (i.±xy=^±nx + \.a ' ^' ± ^ ^77:^ -' ^' + . . . 
 
 \ -\- X 
 
 I 
 
 ^\— x-\-x'^ — o(?-\-x^— ... 
 = 1 ■+-x^x^+x^+x^-\-. . 
 
 (^ J ^y = i + 2a: + 3^=+4^'+S^*+--- 
 
 {1 -^ xf = I -^ i X- i x' + ^x' - jh ^'+ ■ ■ ■ 
 
 (i _ ^)i = I _ 1 a; - i aj^" - -jJb- jc' - T^H •^^^^ - • • • 
 
 (^ ^ ^)i = I - i ^ + I *^ - A *' + A\ ^* - ■ • ■ 
 
 ^^^^=1 + ^^ + 1 .^= + ,^^''+1^^* + .... 
 
 f. Exponential and logarithmic series. 
 For — 00 < a: < 00, 
 
 ^^i ^^1.2 ' 1.2. 3 ' 1.2.3.4 ' 
 
 The number e is the base of the natural or " Napierian " system of logarithms. 
 For X = + I, the above series gives 
 
 e = 2.718281828459 .... 
 
 In the natural system the following series hold with the limitations indicated : 
 
 ^=. + '^x 
 
 '^ 1.2 
 
 ^ 1.2.3 
 
 — 
 
 00 < X < 00; 
 
 
 log (i-{-x) = x 
 
 x' ,x' _ 
 2 "•" 3 
 
 x' x' 
 4 "^ S 
 
 log (1 -x)=—x 
 
 x' x' 
 
 :r^ *^ 
 
 2 3 
 
 4 S 
 
 ^ < I ; 
 
 iog«=,{j^+i(i^)'+i(:-^y+*(:-f^)'+-} 
 
 o < a; < 00 ; 
 
 X -\- y \ y ( y \^ ( y \^ ~\ 
 
 ^°g "T~ = ^ I 2x+y + * U^+j/j +* iTl^+7J +• ■ • [ 
 f<(2x+y)\ 
 
USEFUL FORMULAS. 
 
 XV 
 
 g. Relations of natural logarithms to other logarithms. 
 
 £ =^ base of any system, 
 JV= any number, 
 Z = log iVto base B = log^JV, 
 I = log iV" to base e = logeiV; 
 Then 
 
 Z = / log^ = Ijlog^B, 
 
 logoff = i/log«5 = /t, say, which is called the modulus of the system whose base 
 is £. In the common, or Briggean system, 
 
 /* = logioi? = 0.43429448 .... 
 log /A = 9-6377843 — 10- 
 
 2. Trigonometric Formulas. 
 a. Signs of trigonometric functions. 
 
 Function. 
 
 1st Quadrant. 
 
 2d Quadrant. 
 
 3d Quadrant. 
 
 4th Quadrant. 
 
 sine 
 
 cosine .... 
 tangent . . . 
 cotangent . . . 
 
 + 
 
 + 
 + 
 + 
 
 + 
 
 + 
 + 
 
 + 
 
 b. Values of functions for special angles. 
 
 
 0° 
 
 90° 
 
 180° 
 
 270° 
 
 360° 
 
 30° 
 
 45° 
 
 60° 
 
 sine .... 
 
 
 
 + 1 
 
 
 
 — I 
 
 
 
 h 
 
 ^V2 
 
 iVs 
 
 cosine . . . 
 
 + 1 
 
 
 
 — I 
 
 
 
 + 1 
 
 W3 
 
 l^/2 
 
 i 
 
 tangent . . . 
 
 
 
 00 
 
 
 
 00 
 
 
 
 i^s 
 
 I 
 
 V3 
 
 cotangent . . 
 
 00 
 
 
 
 00 
 
 
 
 00 
 
 V3 
 
 I 
 
 4V3 
 
 c. Fundamental formulas. 
 
 sin^ a -(- cos' a = I, 
 cos a sec 0=1, 
 
 sin a 
 
 tan tt =^ ■ 
 
 tan a cot a ^ I, 
 sin a cosec a = i, 
 
 cos a 
 
 cos a 
 
 cot a = 
 
 sm a 
 
 I + tan* a = ——2— =^ sec* a, 
 ' cos" o ' 
 
 versed sin a : 
 
 I -\- cot* a : 
 
 I — COS a. 
 
 -2— ^^= coseC a, 
 
XVI USEFUL FORMULAS. 
 
 d. Formulas involving two angles. 
 
 sin (a ± 13)^ sin a cos y8 ± cos u. sin 13, 
 cos (a ± /3) ^ cos a cos yS T sin u sin /3. 
 
 tan (a ± /3) = (tan a ± tan 13) /{i T tan a tan P), 
 
 cot (a ± y3) =^ (cot a cot ^ T l)/(c0t a ± COt /3). 
 
 sin a -[- sin /8 ^ 2 sin J(a -|- )8) cos ^(a — ;8), 
 sin I* — sin )8 = 2 cos i(a -|- ^) sin J(a — jS). 
 
 cos a -|- cos /8 = 2 cos i(a -}- )8) COS i(a — yS), 
 
 cos a — COS p^ — 2 sin ^(a -\- /3) sin J^(a — ;8). 
 
 sin (a ± fi) 
 tan u. ± tan « = ^^ 51 
 
 '^ cos a COS /3 
 
 sin (y8 ± a) 
 
 cot a ± cot S = ■ ^ • — i- 
 
 "^ sm a sin /3 
 
 2 sin a sin |8 = cos (a — p) — cos (a -|- /3), 
 
 2 cos a cos P = COS (a — /8) + COS (a -|- P), 
 
 2 sin u, COS p = sin (<» — )S) -j- sin (a -j- ;8). 
 
 sin a + sin /3 , , ,^~. , , „\ 
 
 -^ ^ ■ Q = tan i(o 4- iS) cot Ka — /S), 
 
 sin a — sin yS 2^ 1 f/ 2^ r/i 
 
 cos a -|- cos ^ . . . . 
 ' a = — cot Ua -\- B) cot i(a — 8). 
 
 cos a — COS )8 2v I f'/ 2\ f/ 
 
 e. Formulas involving multiple angles. 
 
 sin 2 a = 2 sin u. cos a, 
 
 sin 3 a = 3 sin u, cos^ a — sin' a. 
 
 cos 20.=^ cos" a — sin"" 0=1 — 2 sin" a = 2 cos" a— i, 
 
 cos 3 a = C0S° a — 3 sin" a COS a. 
 
 sin a I — cos a /l — COS a\i 
 
 tan i a = I _|_ COS a sin a V -(- cos a) ' 
 
 2 tan a ^ cot" 0—1 
 
 tan 2 a =: 3 — ' cot 2 a :^ > 
 
 "" ^ I — tan" a 2 cot a 
 
 2 tan i a I — tan" i o 
 sin a = i |-i — ' COS a = j — 7 — r-; 
 
 I + tan" i a I + tan" ^ a 
 
 2 sin" a = I — cos 2 a, 2 COs" a = I -)- COS 2 a, 
 
 4 sin' a = 3 sin u, — sin 3 a, 4 cos° = 3 cos a -)- cos 3 o. 
 
 f. Exponential values. Moivre's formula. 
 
 e = base of natural logarithms, 
 z^V — I, 2"= — I, ?^ — i, 1*^=1, etc. 
 cos ;t; = i (tf*" + ^ -*"), sin a; = ^V ("f^ — ^~*"), 
 
 cos zjc ^ ^ (^-'^ 4" ^)> sill ^'•^ = 2V («"""' — O' 
 
 (cos .»: ± z sin xy^ = cos »zx ± i sin /«x. 
 
USEFUL FORMULAS. 
 
 g. Values of functions in series. 
 
 For X in arc the following series hold within the limits indicated. 
 
 *"0 mA lyl 
 
 sm X = X — T — i 
 
 6 I20 5040 
 
 COS ;c=i _1_ . . 
 
 2 ' 24 720 ' ' 
 
 — 00 <:<:<-)- 00. 
 
 tan X = * + J ^» + T^V ^' + 3Vf *' + . . . , 
 sec ^ = I + i ^= + A ^* + ^ ^s + . . . , 
 
 — J7r<a:< + ^T. 
 
 I 
 
 cotx = -^(i -ix^-^x'-^^^x"- ...), 
 cosec X = i (i + i ^= + g^^ x^ + T-^^ ^' + . . . ), 
 
 arc sin X = :c + i ^' + ^5 *5 _|_ ^^^ ^7 _j_ . _ ^ 
 
 ^S *,B „7 *„9 
 
 arc tan :«r = a;— • — .... 
 
 — I < j; < -j- I. 
 J? = sin X 4- ^ sin' x -\- ^ sin« a: + -j-f ^ sin^ x -\- . . . , 
 
 — ^■T<X<-\-i-!r. 
 
 * = tan X — J tan' x -\- ^ tan' x — ^ tan' a; -|- . . . , 
 -4,r<a:< + i,r. 
 
 log sin a: = log ^ - /. (J ^2 4- tJ^ x^ + 5^3^ :c= + . . . ), 
 .a; positive and < ir, 
 /t := modulus of common logarithms. See p. xv. 
 
 log tan x=zlogx-\-fx. (1 x^ -\- ^^ x* -\- ^f f ^ x^ -\- . . . ), 
 X positive and < ^ tt. 
 
 h. Conversion of arcs into angles and angles into arcs. 
 
 Denote by x°, x', and x" respectively the angle (in degrees, minutes, or sec- 
 onds) corresponding to the arc x. Then by equality of ratios 
 
 360° 360 X 60' 360 X 60 X 60" 2j7r 
 
 o~ f 77 J 
 
 whence 
 
 180° 
 
 X = X > 
 
 , 180 X 60' 
 
 TT 
 
 18 X 6 X 60" 
 
 x!' = X • 
 
Xviii USEFUL FORMULAS. 
 
 Put i8o° 
 
 IT 
 
 Then 
 
 z^ p° z=. number of degrees in the radius, 
 ^ ° ^ — - ^ p' = number of minutes in the radius, 
 ^ ° ^ — 5-^^ — — = p" = number of seconds in the radius. 
 
 IT 
 
 x° ^x p°, x' ^x p', x" = X p". 
 
 p° = 57.''295779S, log p° — 1.75812263, 
 
 p' — 3437-'74677, log p' = 3-53627388, 
 
 p" = 2o6264."8o6, log p" = 5.31442513. 
 
 3. Formulas for Solution of Plane Triangles. 
 
 a, d, c = sides of triangle, 
 a, ^, y := angles opposite to a, b, c, respectively, 
 A =■ area of triangle, 
 r ■=■ radius of inscribed circle, 
 R = radius of circumscribed circle, 
 s = i (a -i- 6 -\- c). 
 
 sin a sin /3 sin y ' 
 
 a-^b cos y + ^ cos ^, b =^ c cos a-\- a cos y, c = a cos P -\- b cos a. 
 
 „ . a b c 
 
 r = 4 i? sm J a sm ^ ^ sm i y = ^^ ^ - 
 
 (a + b) cos ^ (a -j- )8) = ^ cos J- (a — j8), 
 
 (d! — b) sin ^ (a -j- ;S) :^ <r sin ^ (a — y8). 
 
 a -\- b tan i (a -|- j8) ^ tan \ y 
 
 a — b tan J (a — ^) tan i (a — jS)* 
 
 «" = /J'^ + <:' — 2 (5 f cos a = (^ + c)2 — 4 /5 ^ cos'' i a. 
 
 sint.^y'gZSEEZ), c„,j.=^/ZE3. 
 
 J (j — d) 
 
 a'' sin 18 sin y „„ . . „ . 
 
 A ^-igabBva y = ■ ^2 7?'' sin a sin a sin y 
 
 '^ ' 2 sin a '^ ' 
 
 ^ r" cot ^ a cot ^ /? cot ^ y = \J s (s —■ a) (s — b) (s — c) 
 = rs:^\abc JR. 
 
In right angled triangles let 
 
 Then 
 
 USEFUI. FORMULAS. 
 
 a = altitude, 
 b = base, 
 <r = hypothenuse, 
 y = 9°°- 
 
 a^c sin a = <: cos p = b tan a := ^ cot /8, 
 b ^= c sin |8 = f cos o ^ a tan ^ ^ a cot a. 
 
 ^ = ^«i5^^a^ cot a = J ^'^ tan «, =: ;J r' sin 2 u. 
 Table for solution of oblique triangles. 
 
 XIX 
 
 Given. 
 
 Sought. 
 
 Formula. 
 
 a, b,C 
 
 a 
 
 
 sin|a = ^(^-^K"-^), ^=^(«-t-^ + .), 
 
 cosia-y/^(^-'^), 
 
 tania-t/(^-/)(^-^). 
 V J (j- — a:) 
 
 ^ = y/j-(j _ a) (j- _ ^) (j _ ^). 
 
 a, b, a 
 
 /8 
 
 y 
 
 c 
 A 
 
 sin /3 =: i5 sin aja. 
 
 When ay- b, fi <. 90° and but one value results. When b > a, 
 p has two values. 
 
 V=l8o°-(a + y3). 
 <r = fl! sin 7/sin a. 
 ^ = ^ a 3 sin y. 
 
 a, a, 13 
 
 b 
 
 y 
 
 c 
 A 
 
 /5 = « sin ;8/sin ci. 
 
 y = 180° - (a + ;8). 
 
 <: = fl! sin y/sin u = <? sin (a -f- ;8)/sin a. 
 
 ^ = J « ^ sin y = J a^ sin j8 sin y/sin a. 
 
 a,b, y 
 
 u. 
 
 A 
 
 a sin y 
 
 d — a cos y 
 ^(« + /3) = 9°°-i7- 
 tani(a-^) — ^_^^cotiy. 
 
 r — (a^ _|- ^2 _ 2 a ,5 cos y)*, 
 
 = {(a-\-by-4ab cos^ ^ y}*, 
 
 =z{(a-by-\-4<z&s^ri'iyV, 
 
 = (a — ^)/cos <;!>, where tan ^ = 2 v^a ^ sin ^ y/(a — b), 
 
 = a sin y/sin a. 
 ^ := ^ « i5 sin y. 
 
USEFUL FORMULAS. 
 
 4. Formulas for Solution of Spherical Triangles. 
 
 a. Right angled spherical triangles. 
 
 a, b,c = sides of triangle, c being the hypotenuse, 
 a,l3,y = angles opposite to a, b, c, respectively, 
 y = 9°°- 
 
 sin a = sva. c sin a, sin ^ = sin r sin j8, 
 
 tan a = tan c cos /3, tan ^ = tan ^ cos a, 
 
 = sin b tan a, = sin a tan ^ ; 
 
 cos a = cos a sin ;8, cos ^ = cos b sin a ; 
 
 cos c = cos a cos 3 ^ cot a cot ;8. 
 
 b. Oblique angled triangles. 
 
 a, b, c-=^ sides of triangle, 
 a, p,y ::= angles Opposite to a, b, c, respectively, 
 s=^\{a-\-b-\-c), 
 
 a = i (a + ^ + y), 
 
 £=a-|-j8-j-y — 180° = spherical excess, 
 ^ = surface of triangle on sphere of radius r. 
 
 sin a sin b sin c 
 
 sin a sin /? sin y' 
 
 cos a = cos b cos c -\- sin b sin c cos a, 
 
 — cos a- cos (cr — a) „ COS ((T — S) COS (<r — y) 
 ^i-^ i '^ = sin /3 sin y ' ^°^^ * "^ = sin ^in y ^' 
 
 „ — cos <r cos (cr — a) 
 
 tan^ i a = 7 o\ H -\- 
 
 ^ COS (o- — p) COS (o- — y) 
 
 _ sin (j - b) sin (j - <r) _ sin .f sin (^ - a) 
 
 sm i a — ■■ — 7 — ■■ 1 cos i- a — : — -, — : > 
 
 ^ sin b s\n c ^ sm b sin c 
 
 sin (j — V) sin Cj — c) 
 
 tan'' J a = ^ ■ I \ ■ 
 
 ■^ sin J sm [s — a) 
 
 cot \ a cot i ^ + cos y 
 
 cot i £ = ; : » 
 
 -* sin y 
 
 tan" J € = tan \ s tan J (j — a) tan i (s — b) tan ^ (j — f). 
 
 Napier's analogies. 
 , . , ,, cos \(a — S) , , , ,. sin 4- (a — S) 
 
 tanH^ + ^) = cosH'^ + /) ^^" ^'' *^" ^ ^'^ ~ ^) = sinHa + ^r ^"^'' 
 
 1 / I o\ COS i (fl — ^) sin i (a — ^) 
 
 tanH-+^) = 35ry^^cotiy, tan H- - ^) = ^HTHM^) ^°H 7- 
 
USEFUL FORMULAS. XXI 
 
 Gauss's formulas. 
 
 cos i (a 4" /^) *^°s i ^ = cos \{a -\- b^ sin \ y, 
 sin J (a -j- j8) cos i f = cos ^ {a — b) cos J y, 
 
 cos J- (a — ^) sin ^ f =: sin is (a -\- b) sin J- y, 
 sin J (a — ;8) sin J^ f =: sin J (« — (5) cos J y. 
 
 5. Elementary Differential Formulas. 
 
 a. Algebraic. 
 
 «, V, IV, . . . =■ variables subject to differentiation, 
 a, b, c, . . . ^= constants. 
 
 </(« -\- u) = du, d{a u) = a du, 
 d(u -\-v-{-w-\-...)=^du-{-dv-\-dw-\-..., 
 d{u v)^u dv -\- V du, 
 
 (du.dv.dw, \ 
 
 d(u V w . . .)^=\ — H r • • • I 
 
 ,(u\ V du — u dv du u I. 
 
 \v) 1^ V v' 
 
 U V w . 
 
 dv 
 
 (a -{- b u\ b h — a g 
 
 '^\^+g^)~{A-\-guy ^^- 
 
 dv 
 
 dv"^ n if-'^ dv, dslv = ,— 
 
 da" = a" log a dv, de" = e" dv 
 
 (e = base of natural logarithms), 
 d log V = dvjv. 
 
 dF{u,v, w . . .)-^ du-\-^du^^dw + . . . . 
 
 b. Trigonometric and inverse trigonometric. 
 
 ^sin X = cos X dx, dcos x = — sin x dx, 
 
 dtan X ^= sec'^ x dx, dcot x ^ — cosec^ x dx, 
 
 dsec X = sec^ x sin x dx, ^cosec x = — cosec'' x cos x dx. 
 
 dlog sin x = cot x dx, dlog cos a; = — tan x dx. 
 
 , . , dx dx 
 
 aarc sin x = ± , „ » izarc cos x 
 
 VI — x^ VI — ^ 
 
 dx dx 
 — 1 — 51 «arc cot X ■:^ , — 2' 
 
 dx , dx 
 
 dare tan x = — j — s' "arc cot x := 
 
XXU USEFUL FORMULAS. 
 
 6. Taylor's and Maclaurin's Series. 
 
 a. Taylor's series. 
 
 If u-=.f{x-^H), any finite and continuous function oi x -\- h, h being an 
 arbitrary increment to x; and if dujdx, d^u/dx^ . . . are finite and deter- 
 minate, 
 
 /i' , A' , 
 
 u 
 
 ^f(x + h) ^f(x) +/' ix) h +/" (*) - +/'" ix) 
 
 2 ■ •' ^ ' I. 2. 3 ■ 
 
 vihexe. f(x),f' {x), f" (x), . . . are the va\a&s oi /{x -\- h), du/dx, d^ujdx^ . . . 
 when h-=^o. This is Taylor's series or theorem. The remainder after the first 
 n terms in h is expressed by the definite integral 
 
 ^ ^r*"- ix -\- h - z) ^' dz. 
 
 I. 2. 3 ... « J-' *" ' ' 
 
 o 
 
 b. Maclaurin's series. 
 
 If in Taylor's series we make a: = o, and ^ = jc, the result is 
 
 u =f{x) =f{p) +/' (o) X +/" (o) 4 +/'" (o) -^°- + . . . , 
 
 I .2 •' ^ ' I. 2. 3 ' 
 
 vihtre f (p), f (p), f" {o), ... are the values oif(x), dujdx, d^u/dx^, . . . when 
 « := o. This is Maclaurin's series or theorem. The remainder after the first n 
 terms in x is expressed by the definite integral 
 
 1-2.3.. 
 
 -j'r + ^(x-z)^dz. 
 
 c. Example of Taylor's series. 
 
 u =f(x -\-k) = log {x + h). 
 
 f{x) = \QgX, 
 
 du I 
 
 -Tx — T^h' f C'^) = + ^~S 
 
 d^u I 
 
 -d^—-{x^hy /"{x)=-x-\ 
 
 d^u , 2 
 
 -d^—^(x + hf f"{x) = +2X-\ 
 
 Hence for common logarithms, /u being the modulus, 
 
 log {x -\- h) = log jc 4- III (.^-1 h-\x-'' h'-^^x-^ h!"-.. .), 
 and the sum of the remaining terms is 
 
 h 
 
 ~ 77773 J {x-{-h-zf ^° '^^- 
 
USEFUL FORMULAS. xxiii 
 
 Since x is the least value oi{x-\- h — z) within the limits of this integral, the 
 sum of the remaining terms is negative, and numerically 
 
 <J/^ 
 
 (I)' 
 
 If, for example, (hlx) = i/ioo, the remainder in question is less than 
 J X 0.434 X io~', or about one unit in the ninth place of decimals. 
 
 d. Example of Maclaurin's series. 
 
 u =z/(x) = sin X. 
 
 /(°)=o, 
 
 •^ — cos X, / (o) =: + I, 
 
 ^^2 = sm X, 
 
 
 
 
 /'(o)=°, 
 
 
 
 
 
 f"(o)=-J.. 
 
 Hence 
 
 
 _|_ 
 
 
 x" 
 
 /(x) — sm a: — ■^ ~ I . 2 . 
 
 3 
 
 T^ 
 
 I 
 
 .2.3.4.5 •• 
 
 and the sum of the remaining terms is 
 
 
 
 
 
 X 
 
 
 
 
 
 -fjfsia(x 
 
 
 — 
 
 ^) 
 
 z' 
 
 dz. 
 
 If g- is the greatest value of sin (x — z) within the limits of this integral the 
 remainder in question is negative and numerically 
 
 If, for example, x = tt/S (the arc of 30°), ^ := J, and the remainder is numeri- 
 cally less than 0.0000143. 
 
 7. Elementary Formulas for Integration. 
 a. Indefinite integrals. 
 I adx ^ai dx = ax -\- C. 
 
 ^fix) dx -f ^4, (x) dx = f{/(x) + <i> (x)} dx. 
 
 If a; =: <^ (y), and dx = <^' (y) dy, 
 
 f/(x) dx = J/{<^ (7)} f (y) dy. 
 
 d_ 
 dy 
 
 f/(x,y)dx = f^dx. 
 
Xxiv USEFUL FORMULAS. 
 
 Since d{uv) = udv -\- vdu, 
 
 I udv =.uv — \ vdu ; and 
 
 if « = f^x) and z/ = ^ {x), 
 
 JX-) '-^ dx = A-) ^ (-) -> (-)#^^ '^-* 
 
 jdxj/(x, y) dy =jdyjy{x, y) dx. 
 I dx \f{x) dx:=^ X i/{x) dx — i xfix) dx, 
 
 n-\-i ^ 
 
 ^ia + bxTdx^^^+^'^-C. 
 /-J = log.t+C, /^ = 3-Mog(. + ^.). 
 
 Jl?^~x-^^' J {a-\- bxf= ~ b{a-\-bx) + ^- 
 
 /^/a: , ^ r — '^•^ , ^ 
 
 J I ^2 = arc tan x -\- C, I ^i ^a = arc cot * + C. 
 
 /dx 
 — -j—r-^^ (ab)~^ arc tan {b/a)* x -\- C, for « and b both positive, 
 
 = (ab)~* arc cot (^/i?)* ^ -|~ ^> ^°^ '^ ^'^'^ '^ both negative, 
 
 ( — ab^ — bx 
 = i(- «^)~* log (_ aby -\- bx + ^' ^°^ ^^ negative. 
 
 /dx b -\- ex 
 
 a^^bx^cx- = ('^^ - ^"^'^ ^''^ *^" (SF::^)* + ^' *°'' «^ - -^^ > °' 
 
 C(a -\- x'f dx = i X (a -\- x^i -{- Is a log {^c + (a + ar^*} + C. 
 fid' -xydx = lx (a^ -xy-\-\ a" arc sin ^ + C. 
 
 !"(« + bxy dx = ^(a + bx)i/b + a 
 
 * This is the formula for Integration by parts. 
 
 t Natural logarithms are used in this and the following integrals. For relation of natural to 
 common logarithms see section i, g. 
 
USEFUL FORMULAS. 
 
 J*(« -{-2 dx-\- cxy dx=i(d-\- cx) {a-\-2bx-^ cx^'lc 
 
 ■^\(ac- V^jc^ia + 2bx + cx^i dx + C. 
 C(a -\- bx)-i (ix=2(a-{- bx^jb + C. 
 Jia + ^x) (a + bx)-i dx = %{7,ab-2aji-{-p bx) {a + bxy/b'' + C. 
 
 j (a" — x^)-i dx=± arc sin - + C, 
 
 X , ^ 
 = =F arc cos - -f- C, 
 
 (a~\-x\i , „ 
 —^ \ Ar C. 
 
 C(a + xyi dx = log {x + (« 4- xj} + C, 
 
 = ilog 
 
 a; -|- (3 -|- :r^)* 
 
 X — (a ■ 
 
 ypy- 
 
 + c. 
 
 C(a -{-2 bx-\- cxY^ dx = -j=p log {b-\-cx-\- {ac-\- bcx + c'xy)-\- C, for <r > o, 
 
 b -\- cx 
 
 — ~ / — arc sin /ts u 
 
 y/— <r {b^ — acf 
 
 C, for f <o. 
 
 faVa; = a'^/log a + C, jV/Zx = e' -\- C. 
 
 I log a: (/* ^ a; log x — x-\- C. 
 
 Jilog xfx-^dx = ^j^ (log x)» + i + C. 
 
 I sin :c (/a; = — cos x -\- C, I cos x d?x = sin j; -|- C. 
 
 I sin" X ^x = ^ ;«; — :| sin 2 X "1- C, J cos^ x dx= ^x -{-^ sin 2 x-\- C. 
 
 I tan X dx-=^ — log cos x -)- C, | cot x dx=z log sin j; -)- C". 
 
 /dx , , „ C dx , 
 
 ^j^ = log tan i x+ C, j ^^^ = log tan H^+ i -) + C. 
 
 I -:—i — = — cot X -\- C. I — 2 — = tan x -\- C. 
 
 J sin-* X J cos^ a: ^^ 
 
 /' . , , a s,m bx — b cos bx , „ 
 «°^ sin bx dx = ^g _,_ ^^3 tf°^ + C. 
 
 e^ cos ^jc (/x = 
 
 a' + b^ 
 
 a cos ^a: -(- ^ sin bx 
 a^-\-b^ 
 
 e^-\- C. 
 
 S' 
 
 I arc sin x dx^ x arc sin x ±{-i — x'')^ -\- C. 
 
 I arc cos x dx=- x arc cos x^ (1 — x°)* + C. 
 I arc tan xdx^::^ x arc tan x — \ log (i -j- xF) -\- C. 
 j arc cot X dx = X arc cot ^ + i log (i -|- x"^ -j- C. 
 
USEFUL FORMULAS. 
 
 b. Definite Integration. 
 
 n b c n 
 
 \ <j> (x) dx -^^ \ 4> (x) dx -\- \ ^ (x) dx -\- . . . {<j> (x) dx. 
 
 a b 
 
 b a 
 
 \<^{x)dx-^ — \<^ (x) dx. 
 a b 
 
 a a 
 
 I </> (a;) dx^\^ {a — 00) dx. 
 o o 
 
 If 4> (x) = <li (— x), an " even function " of x, 
 
 a o a 
 
 I ^ {x) dx=l cjy (x) dx = ^ j ^ (x) dx. 
 o — a — a 
 
 If <^ (a:) = — <^ (— x), an " odd function " of x, 
 
 a -\-a 
 
 1 <t> (x) dx= t ^ (x) dx, and | ^ (x) dx = o. 
 — a o — a 
 
 If A be tlie greatest and B the least value of ^ (x) within the limits a and b, 
 
 b 
 A(b — a)>C^ (xydx > B {b — a), 
 a 
 
 a formula useful in determining approximate values of integrals. See, e. g., 
 section 6, d. 
 b 
 
 If U=: \ 'k (x) dx, 
 
 a 
 
 '^^_ JL/\ ^"—^^l\ 
 
 00 
 
 r dx _ r dx 
 
 J i+x^~J iJrx-' — ^'^- 
 
 o 
 
 I 00 
 
 o 
 
 00 
 
 /jfE. = WVW, /^ = i.. 
 
 o 
 
USEFUL FORMULAS. xxvii 
 
 O 
 
 00 
 
 OO 00 
 
 o o 
 
 Ce-"'^' x-"dx= 1.3.5... (2 «-i)a-»(2 a)-"' + '^V7r. 
 o 
 
 00 
 
 p-' "= x-i dx = ^{kIo). 
 o 
 
 I sill mx sin nx dx ■=. | cos w:c cos «;i; </.« = o. 
 o o 
 
 when m and n are unequal integers. 
 
 ■ff 
 
 /» 2 wz 
 
 sin »m; cos nx ax ^ 2 _ — 2> for w and « integers and m — n odd, 
 
 o 
 
 = o, for m and « integers and m — n even. 
 
 TV IT 
 
 I sin^ mx dx = I cos° wz;c ^a: = J n-, for m an integer. 
 
 ^- 
 
 I sin" X dx = \ cos" ^ (/a: = j (i — «'') *^"~^' ^ar. 
 000 
 
 CO 00 
 
 /sin X J /'cos X , /, , ^ 
 
 o o 
 
 00 00 
 
 1 sin x' dx :=l cos x^ dx = ^ V(w/2). 
 o o 
 
 00 
 
 Ce - "' ==" cos2l>xdx=zie-^''' "'>' V(ir/a). 
 o 
 
 00 
 
 Ig-a2x2 gjjj 2^a; (/.»;:= o. 
 
 o 
 
MENSURATION. 
 
 I. Lines. 
 a. In a circle. 
 
 r = radius of circle, 
 c = length of any chord, 
 s = arc subtended by c, 
 a = angle corresponding to s, 
 
 h = height of arc s above c, or perpendicular distance from middle point of 
 arc to chord. 
 
 Circumference ■=■2 irr, 
 
 77 = 3.14159265, log 77 = 0.49714987, 
 277 = 6.28318531, log 2 77 = 0.79817987. 
 
 ^ = 2 r sin J a, J = r a. 
 
 Length of perpendicular from center on chord 
 = r cos \ a 
 
 =.{.-i(5^)-.(^)*-A(^y- 
 
 A^r (1 — cos i a) 
 = 2 r sin^ J a 
 
 = u{(9'+*©*+ri.(9'+. 
 
 = »?{.+«©■+...}. 
 "=«{'+•(*)■+•••}■ 
 
 b. In regular polygon. 
 
 r = radius of inscribed circle, 
 ^= radius of circumscribed circle, 
 n =^ number of sides, 
 s :=: length of any side, 
 /3 = angle subtended by s, 
 p = perimeter of polygon. 
 
MENSURATION. 
 
 ^ = 36o7«, 
 
 J = 2 r tan ^ yS = 2 i? sin J j8, 
 
 p=^ns-:=2nr tan ^ ^8 = 2 « ^ sin J /8. 
 
 See table under c, below. 
 
 c. In ellipse. 
 
 a = semi-axis major, 
 
 b = semi-axis minor, 
 
 e = eccentricity = (i — b^laF)^, 
 
 P= perimeter of ellipse, 
 
 « = (3 - b)l(a. + b) 
 
 XXIX 
 
 i+y/r^^~4 "^8 "^ 64^' 
 
 Distance from centre to focus = a «, 
 
 Distance from focus to extremity of major axis z= a {1 — e), 
 
 Distance from focus to extremity of minor axis ^ a. 
 
 P=,r(«+3)(l + i«^ + TV«' + ^k/ + .--) 
 
 = IT (a + 3) ^, say, where ^ stands for the series in n. The values of ^ cor- 
 responding to a few values of n are : — 
 
 n 
 
 1 
 
 n 
 
 ? 
 
 
 
 1. 0000 
 
 o-S 
 
 1-0635 
 
 O.I 
 
 1.0025. 
 
 0.6 
 
 1.0922 
 
 0.2 
 
 1. 0100 
 
 0.7 
 
 1. 1267 
 
 0-3 
 
 1.0226 
 
 0.8 
 
 1. 1677 
 
 0.4 
 
 1.0404 
 
 0.9 
 
 i-aiSS 
 
 
 
 I.O 
 
 1.2732 
 
 • 2. Areas. 
 
 a. Area of plane triangle. 
 
 (See table on p. xix.) 
 
 b. Area of Trapezoid. 
 
 b^ = upper base of trapezoid, 
 bi = lower base of trapezoid, 
 a = altitude of trapezoid, or perpendicular distance between bases. 
 
 Area = ^ (bi-{- b^ a. 
 
MEJfSURATION. 
 
 c. Area of regular polygon. 
 
 A = area, 
 r, J? ^ radii of inscribed and circumscribed circles, 
 s = length of any side, 
 n = number of sides, 
 (3 =: angle subtended hy s = 36o°/«. 
 
 A = nr'tanil3 = in Ji^ sin P — ins^ cot i^. 
 
 Table of Values. 
 
 n 
 
 & 
 
 A 
 
 A 
 
 R 
 
 » ^ 
 
 3 
 
 I20° 
 
 0.4330 j'' 
 
 1.2990 R'^ 
 
 0.5774 J- 
 
 1. 7321 i? 
 
 4 
 
 9° 
 
 1. 0000 
 
 2.0000 
 
 0.7071 
 
 I.4I42 
 
 5 
 
 72 
 
 1-7205 
 
 2.3776 
 
 0.8507 
 
 1-1756 
 
 6 
 
 60 
 
 2.5981 
 
 2.5981 
 
 I.OOOO 
 
 I.OOOO 
 
 7 
 
 5if 
 
 3-6339 
 
 2.7364 
 
 1-1524 
 
 0.8678 
 
 8 
 
 45 
 
 5.8284 
 
 2.8284 
 
 1.3066 
 
 0.7654 
 
 9 
 
 40 
 
 6.1818 
 
 2.8925 
 
 1. 4619 
 
 0.6840 
 
 lO 
 
 36 
 
 7.6942 
 
 2.9389 
 
 I.6I80 
 
 0.6180 
 
 II 
 
 32A 
 
 9-3656 
 
 2-9735 
 
 1.7747 
 
 0-5635 
 
 12 
 
 30 
 
 II. 1962 
 
 3.0000 
 
 1-9319 
 
 0.5176 
 
 13 
 
 28A 
 
 13.1858 
 
 3.0207 
 
 2.0893 
 
 0.4786 
 
 14 
 
 2Sf 
 
 15-3345 
 
 3-0372 
 
 2.2470 
 
 0.4450 
 
 IS 
 
 24 
 
 17.6424 
 
 3-0505 
 
 2.4049 
 
 0.4158 
 
 i6 
 
 22^ 
 
 20.1094 
 
 3-0615 
 
 2.5629 
 
 0.3902 
 
 d. Area of circle, circular annulus, etc. 
 
 r = radius of circle, 
 
 d = diameter, 
 
 a = angle of any sector, 
 rj, ^2 = smaller and greater radii of an annulus. 
 
 Area of circle ^ tt r^ = ^ tt d^, 
 'r = 3-14 159 265, log n- = 0.49 714987. 
 
 Area of sector :^ ar', for a in arc, 
 
 =:ir r^ (a/360), for a in degrees. 
 
 Area of annulus ::= tt (r/ — ri'^. 
 
 e. Area of ellipse. 
 a, i = semi axes respectively 
 <? = eccentricity = (a^ — d^^/a 
 = {(a -\-6)(a- b)yia. 
 
MENSURATION. XXXI 
 
 Area of ellipse = tt a ^, 
 
 ■=i-K c^ COS <^, if « ^ sin <^. 
 
 f. Surface of sphere, etc. 
 
 r = radius of sphere, 
 <^j, ^2 = latitudes of parallels bounding a zone, 
 £ = spherical excess of a spherical triangle 
 = sum of spherical angles less i8o°, 
 
 Total surface ^ 4. tt r'. 
 
 Surface of zone :^ 2 tt r^ (sin <^2 — sin ^1), 
 
 = 4 TT /-'^ cos J (<^2 + i>i) sin i ((^2 — ^i)- 
 
 Surface of spherical triangle =: r' e, for e in arc, 
 
 = r^ c/p", for £ in seconds, 
 p" = 206 264.8", log p" = 5.31 442 513. 
 
 g. Surface of right cylinder. 
 
 r = radius of bases of cylinder, 
 h = altitude of cylinder. 
 
 Area cylindrical surface ^= 2 tr r h. 
 
 Total surface =1 2 tt r (r -\- k). 
 
 h. Surface of right cone. 
 
 r = radius of base, 
 h = altitude, 
 f = slant height. 
 
 Conical surface = irrs^Trr(h'^-\- r^, 
 Total surface = tt r (j- -}- ^)- 
 
 i. Surface of spheroid. 
 
 a, b = semi axes, 
 
 e = eccentricity = {(a -{- 6) (a — b)Yla. 
 
 Surface of oblate spheroid ^ 2 ir a' \^ '^ 2 e '°^ \i — e) \ 
 
 = ^^ a" i-i - \ e" - ^ ^ - ^ e^ - . . .). 
 
 (■ . ,, , . arc sin e ) 
 Surface of prolate spheroid = 2 ira^ i. (j. — ey-\- ^ )■ 
 
 z= j^iT a b {\ — \ ^ — -^ ^ — t\^ e^ — . ■ ■)■ 
 
 * The logarithm in this formula refers to the natural or "Napierian" system. For areas of 
 zones and quadrilaterals of an oblate spheroid, see pp. 1-lii. 
 
XXXll MENSURATION. 
 
 3. Volumes. 
 
 a. Volume of prism. 
 
 A = area of base, h = altitude, F= volume. 
 
 V=Ah. 
 
 For an oblique triangular prism whose edges a, b, c are inclined at an angle a 
 to the base, 
 
 F= \ {a-\-b-\-c) A€m. a. 
 
 b. Volume of pyramid. 
 
 A = area of base, h = altitude, V= volume. 
 
 F= ^AA. 
 
 For a truncated pyramid whose parallel upper and lower bases have areas Ai 
 and Ai respectively and whose distance apart is A, 
 
 V=lh {A, + sjA, A, + A,). 
 
 The volume of a wedge and obelisk may be expressed by means of the volumes 
 of pyramids and prisms. 
 
 c. Volume of right circular cylinder. 
 
 r = radius of base, A = altitude, F= volume. 
 
 F= 77 r' h. 
 
 77 = 3.14159265, log 77 = 0.49 714987. 
 
 For an obliquely truncated cylinder (having a circular base) whose shortest and 
 longest elements are h^ and h^ respectively, 
 
 F= i 77 r^ {h^ + h^. 
 
 For a hollow cylinder the radii of whose inner and outer surfaces are r^ and r^ 
 respectively, and whose altitude is h, 
 
 d. Volume of right cone with circular base. 
 
 r = radius of base, h = altitude, V^ volume. 
 V— lirr'^h. 
 
 For a right truncated cone the radii of whose upper and lower parallel bases 
 are r-i and r^ respectively, and whose altitude is h, 
 
 F= J ,7 /^ (^1+ r^ n + rf). 
 
 e. Volume of sphere and spherical segments. 
 
 r = radius of sphere, h = altitude of segment, F= volume. 
 
MENSURATION. XXXIU 
 
 For the entire sphere 
 
 F:^ I IT r* ^ 4.1888 r" approximately. 
 (For ir and log tt see c above.) 
 
 For a spherical segment of height k 
 
 V=irh^{r — \K). 
 
 For a zone, or difference in volume of two segments whose altitudes are h-^ and 
 hi respectively 
 
 V=irr (hi — h^~\-K (y^i — /^?) 
 
 = i ^ A /5 (3 ^ + 3 ^ + A y^^), 
 where r^ and r^ are the radii of the bases of the zone and ^ h = h^ — h^. 
 
 f. Volume of ellipsoid. 
 
 a, b, c^ semi axes, F= volume. 
 F= %iT a b c. 
 
 For an ellipsoid of revolution about 
 
 the a-axis, V=%-jr a b\ 
 the ^axis, V=. % ir a^ b. 
 
UNITS. 
 
 I. Standards of Length and Mass. 
 
 The only systems of units used extensively at the present day are the British 
 and metric. The fundamental units in these systems are those of time, length, 
 and mass. From these all other units are derived. The unit of time, the mean 
 solar second, is common to both systems. 
 
 The standard unit of length in the British system is the Imperial Yard, which 
 is defined to be the distance between two marks on a metallic bar, kept in the 
 Tower of London, when the temperature of the bar is 60° F. 
 
 The standard unit of mass in the British system is the Imperial Pound Avoirdu- 
 pois. It is a cylindrical mass of platinum marked " P. S. 1844, i lb.," preserved 
 in the office of the Exchequer at Westminster. 
 
 In the metric system the standard unit of length is the Metre, now represented 
 by numerous platinum iridium Prototypes prepared by the International Bureau 
 of Weights and Measures. 
 
 The standard of mass in the metric system is the Kilogramme, now represented 
 by numerous platinum iridium Prototypes prepared by the International Bureau 
 of Weights and Measures. 
 
 Both systems of units have been legalized by the United States. Virtually, how- 
 ever, the material standards of length and mass of the United States are cer- 
 tain Prototype Metres and certain Prototype Kilogrammes. The present status 
 of the two systems of units so far as it relates to the United States is set forth 
 in the following statement from the Superintendent of Standard Weights and 
 Measures, bearing the date April 5, 1893. 
 
 Fundamental Standards of Length and Mass.* 
 
 " While the Constitution of the United States authorizes Congress to ' fix the 
 standard of weights and measures,' this power has never been definitely exer- 
 cised, and but little legislation has been enacted upon the subject. Washington 
 regarded the matter of sufficient importance to justify a special reference to it in 
 his first annual message to Congress (January, 1790), and Jefferson, while Secre- 
 tary of State, prepared a report at the request of the House of Representatives, in 
 which he proposed Quly, 1790) 'to reduce every branch to the decimal ratio 
 already established for coins, and thus bring the calculation of the principal 
 affairs of life within the arithmetic of every man who can multiply and divide.' 
 The consideration of the subject being again urged by Washington, a committee 
 
 * Bulletin 26, U. S. Coast and Geodetic Survey. Washington : Government Printing Office, 
 1893. Published here by permission of Dr. T. C. Mendenhall, Superintendent Coast and Geo- 
 detic Survey. 
 
UNITS. XXXV 
 
 of Congress reported in favor of Jefferson's plan, but no legislation followed. 
 In the mean time the executive branch of the Government found it necessary to 
 procure standards for use in the collection of revenue and other operations in 
 which weights and measures were required, and the Troughton 82-inch brass 
 scale was obtained for the Coast and Geodetic Survey in 1814, a platinum kilo- 
 gramme and metre, by Gallatin, in 182 1, and a Troy pound from London in 1827, 
 also by Gallatin. In 1828 the latter was, by act of Congress, made the standard 
 of mass for the Mint of the United States, and although totally unfit for such pur- 
 pose it has since remained the standard for coinage purposes. 
 
 " In 1830 the Secretary of the Treasury was directed to cause a comparison to 
 be made of the standards of weight and measure used at the principal custom- 
 houses, as a result of which large discrepancies were disclosed in the weights and 
 measures in use. The Treasury Department, being obliged to execute the consti- 
 tutional provision that all duties, imposts, and excises shall be uniform throughout 
 the United States, adopted the Troughton scale as the standard of length ; the 
 avoirdupois pound to be derived from the Troy pound of the Mint as the unit of 
 mass. At the same time the Department adopted the wine gallon of 231 cubic 
 inches for liquid measure and the Winchester bushel of 2 150-42 cubic inches for 
 dry measure. In 1836 the Secretary of the Treasury was authorized to cause a 
 complete set of all weights and measures, adopted as standards by the Depart- 
 ment for the use of custom-houses and for other purposes, to be delivered to the 
 Governor of each State in the Union for the use of the States respectively, the 
 object being to encourage uniformity of weights and measures throughout the 
 Union. At this time several States had adopted standards differing from those 
 used in the Treasury Department, but after a time these were rejected, and finally 
 nearly all the States formally adopted by act of legislature the standards which 
 had been put in their hands by the National Government. Thus a good degree 
 of uniformity was secured, although Congress had not adopted a standard of 
 mass or of length other than for coinage purposes as already described. 
 
 " The next and in many respects the most important legislation upon the subject 
 was the Act of July 28, 1866, making the use of the metric system lawful through- 
 out the United States, and defining the weights and measures in common use in 
 terms of the units of this system. This was: the first general legislation upon the 
 subject, and the metric system was thus the first, and thus far the only system 
 made generally legal throughout the country. 
 
 " In 1875 an International Metric Convention was agreed upon by seventeen 
 governments, including the United States, at which it was undertaken to establish 
 and maintain at common expense a permanent International Bureau of Weights 
 and Measures, the first object of which should be the preparation of a new inter- 
 national standard metre and a new international standard kilogramme, copies of 
 which should be made for distribution among the contributing governments. 
 Since the organization of the Bureau, the United States has regularly contributed 
 to its support, and in 1889 the copies of the new international prototypes were 
 ready for distribution. This was effected by lot, and the United States received 
 metres Nos. 21 and 27, and kilogrammes Nos. 4 and 20. The metres and kilo- 
 grammes are made from the same material, which is an alloy of platinum with ten 
 per cent of iridium. 
 
XXXVl UNITS. 
 
 "On January 2, 1890, the seals which had been placed on metre No. 27 and 
 kilogramme No. 20, at the International Bureau of Weights and Measures near 
 Paris, were broken in the Cabinet room of the Executive Mansion by the Presi- 
 dent of the United States, in the presence of the Secretary of State and the 
 Secretary of the Treasury, together with a number of invited guests. They were 
 thus adopted as the National Prototype Metre and Kilogramme. 
 
 " The Troughton scale, which in the early part of the century had been tenta- 
 tively adopted as a standard of length, has long been recognized as quite un- 
 suitable for such use, owing to its faulty construction and the inferiority of its 
 graduation. For many years, in standardizing length measures, recourse to copies 
 of the imperial yard of Great Britain had been necessary, and to the copies of 
 the metre of the archives in the Ofifice of Weights and Measures. The standard 
 of mass originally selected was likewise unfit for use for similar reasons, and 
 had been practically ignored. 
 
 "The recent receipt of the very accurate copies of the International Metric 
 Standards, which are constructed in accord with the most advanced conceptions 
 of modern metrology, enables comparisons to be made directly with those stand- 
 ards, as the equations of the National Prototypes are accurately known. It has 
 seemed, therefore, that greater stability in weights and measures, as well as much 
 higher accuracy in their comparison, can be secured by accepting the international 
 prototypes as the fundamental standards of length and mass. It was doubtless 
 the intention of Congress that this should be done when the International Metric 
 Convention was entered into in 1875 ; otherwise there would be nothing gained 
 from the annual contributions to its support which the Government has con- 
 stantly made. Such action will also have the great advantage of putting us in 
 direct relation in our weights and measures with all civilized nations, most of 
 which have adopted the metric system for exclusive use. The practical effect 
 upon our customary weights and measures is, of course, nothing. The most care- 
 ful study of the relation of the yard and the metre has failed thus far to show 
 that the relation as defined by Congress in the Act of 1866 is in error. The 
 pound as there defined, in its relation to the kilogramme, differs from the impe- 
 rial pound of Great Britain by not more than one part in one hundred thousand, 
 an error, if it be so called, which utterly vanishes in comparison with the allow- 
 ances in all ordinary transactions. Only the most refined scientific research will 
 demand a closer approximation, and in scientific work the kilogramme itself is 
 now universally used, both in this country and in England.* 
 
 * Note. — Reference to the Act of 1866 results in the establishment of the following : — 
 
 Equations, 
 
 •;6oo 
 r yard = r— - metre. 
 ■' 3937 
 
 I pound avoirdupois == , kilo. 
 
 A more precise value of the English pound avoirdupois is ^. g^ kilo., differing from the above 
 
 by about one part in one hundred thousand, but the equation established by law is sufficiently 
 accurate for all ordinary conversions. 
 
 As already stated, in work of high precision the kilogramme is now all but universally used, 
 and no conversion is required. 
 
UNITS. xxxvii 
 
 " In view of these facts, and the absence of any material normal standards of 
 customary weights and measures, the Office of Weights and Measures, with the 
 approval of the Secretary of the Treasury, will in the future regard the Interna- 
 tional Prototype Metre and Kilogramme as fundamental standards, and the cus- 
 tomary units, the yard and the pound, will be derived therefrom in accordance 
 with the Act of July 28, 1866. Indeed, this course has been practically forced 
 upon this office for several years, but it is considered desirable to make this for- 
 mal announcement for the information of all interested in the science of metrology 
 or in measurements of precision. 
 
 T. C. Mendenhall, 
 
 Superintendent of Standard Weights and Measures. 
 " Approved : 
 
 J. G. Carlisle, 
 
 Secretary of the Treasury. 
 April s, 1893." 
 
 No ratios of the yard to the metre and of the pound to the kilogramme have as 
 yet been adopted by international agreement ; but precise values of these ratios 
 will doubtless be determined and adopted within a few years by the International 
 Bureau of Weights and Measures. In the mean time, it will suffice for most pur- 
 poses to use the values of the ratios adopted provisionally by the Office of Stand- 
 ard Weights and Measures of the United States. These values are — 
 
 I yard ^ f f§f metres, or i metre = %%%i yards, 
 I pound :=: i^8£§ kilogrammes, or i kilogramme ;= f f ^f § pounds. 
 
 These ratios were legalized by Act of Congress in 1866. Expressed decimally 
 these values are * — 
 
 I yard = 0.914402 metres, 1 metre ^ i.093 611 yards, 
 I pound = 0.45 359 kilogrammes, i kilogramme =. 2.20462 pounds. 
 
 The above values of the relations of the standards of the British and Metric 
 systems of units are adopted in this work. Tables i and 2 give the equivalents 
 of multiples of the standard units and also equivalents of multiples of the derived 
 units of surface and volume. These tables are published by the Office of Stand- 
 ard Weights and Measures of the United States, and are here republished by per- 
 mission of the Superintendent of that Office. 
 
 2. British Measures and Weights. 
 
 a. Linear measures. 
 
 The unit of linear measure is the yard. Its principal sub-multiples and multi- 
 ples are the inch ; the foot ; the rod, perch, or pole ; the furlong ; and the mile. 
 The following table exhibits the relations among these measures : — 
 
 * The actual error of the relation of the yard to the metre may be as great as 1/200 000th part, 
 and the actual error of the relation of the pound to the kilogramme as great as i/ioo oooth part. 
 
UNITS. 
 
 Inches. 
 
 Feet. 
 
 Yards. 
 
 Rods. 
 
 Furlongs. 
 
 Miles. 
 
 I 
 
 0.083 
 
 0.028 
 
 0.00505 
 
 0.00012626 
 
 0.0000157828 
 
 12 
 
 1. 
 
 0-333 
 
 0.06060 
 
 0.00151515 
 
 0.00018939 
 
 36 
 
 3- 
 
 I. 
 
 0.1818 
 
 0.004545 
 
 0.00056818 
 
 198 
 
 16.S 
 
 S-S 
 
 I. 
 
 0.025 
 
 0.003125 
 
 7920 
 
 660. 
 
 220. 
 
 40. 
 
 I. 
 
 0.125 
 
 63360 
 
 5280. 
 
 1760. 
 
 320. 
 
 8. 
 
 I. 
 
 Other measures are the — 
 
 Surveyor's or Gunter's chain = 4 rods = 66 feet = 100 links of 7.92 inches 
 each. 
 
 Fathom := 6 feet ; Cable length =120 fathoms. 
 
 Hand = 4 inches ; Palm = 3 inches ; Span = 9 inches. 
 
 b. Surface or square measures. 
 
 The unit of square measure is the square yard. Its relations to the principal 
 derived units in use are shown in the following table : — 
 
 Sq. feet. 
 
 Sq. yards. 
 
 Sq. rods. 
 
 Roods. 
 
 Acres. 
 
 Sq. miles. 
 
 I. 
 
 O.IIII 
 
 0.00367309 
 
 0.000091827 
 
 0.000022957 
 
 
 9- 
 
 I. 
 
 0.0330579 
 
 0.000826448 
 
 0.000206612 
 
 
 272.25 
 
 30.25 
 
 I. 
 
 0.025 
 
 0.00625 
 
 
 10890. 
 
 1210. 
 
 40. 
 
 I. 
 
 0.25 
 
 
 43560- 
 
 4840. 
 
 160. 
 
 4- 
 
 I. 
 
 
 27878400 
 
 3097600. 
 
 102400. 
 
 2560. 
 
 640. 
 
 I. 
 
 c. Measures of capacity. 
 
 The unit of capacity for dry measure is the bushel (2150.4 cubic inches about). 
 The units of capacity for liquid measure are the British gallon (of 277.3 cubic 
 inches about) and the wine gallon (of 231 cubic inches, nominally). The latter 
 gallon is most commonly used in the United States. The following table shows 
 the relations of the sub-multiples and multiples of the bushel and gallon : 
 
UNITS. 
 
 XXXIX 
 
 Dry Measures. 
 
 Liquids. 
 
 Pint 
 
 = ^ bushel. 
 
 Gill 
 
 = ^gan. 
 
 Quart 
 
 = 2 pints =-h " 
 
 Pint = 4 gills 
 
 = i " 
 
 Peck 
 
 — 8 quarts = J 
 
 Quart = 2 pints 
 
 = i " 
 
 Bushel 
 
 = 4 pecks =1 " 
 
 Gallon ^ 4 quarts 
 
 = 1 " 
 
 
 
 Barrel = 31^ gallons 
 
 = 3ii " 
 
 
 
 Hhd. = 2 barrels 
 
 = 63 " 
 
 Besides the above measures of capacity the following volumetric units are 
 used : — 
 
 Cubic foot = 1728 cubic inches. 
 
 Cubic yard ^= 27 cubic feet = 46656 cubic inches. 
 
 Board-measure foot = i square foot X i inch thickness = 144 cubic inches. 
 
 Perch (of masonry) = i perch (16.5 feet) length X i foot height X i-S feet 
 thickness = 24.75 cubic feet ; 25 cubic feet are commonly called a perch for con- 
 venience. 
 
 Cord (of wood) = 8 feet length X 4 feet breadth X 4 feet height. 
 :=: 128 cubic feet. 
 
 d. Measures of weight. 
 
 The unit of weight is the avoirdupois pound. One 7000th part of this is called 
 a grain, and 5760 such grains make the troy pound. The sub-multiples and mul- 
 tiples of these two pounds are exhibited in the following table : — 
 
 Avoirdupois. 
 
 Troy. 
 
 Dram 
 
 
 = 5*^ lb. 
 
 Grain 
 
 
 = 5 tVtt lb. 
 
 Ounce 
 
 ^16 drs. 
 
 = ^ " 
 
 Pennyweight = 
 
 : 20 grs. 
 
 = ^iu 
 
 Pound 
 
 = 16 ozs. 
 
 = %" 
 
 Ounce = 
 
 24 dwt. 
 
 = A " 
 
 Quarter 
 
 = 28 lbs. 
 
 = 2I" 
 
 Pound = 
 
 12 ozs. 
 
 = I " 
 
 Hundred-wt 
 
 = 4qrs. 
 
 = 112 " 
 
 
 
 
 Long ton 
 
 = 20 cwt. 
 
 = 2240 " 
 
 
 
 
 Short ton 
 
 ^= 
 
 = 2000 " 
 
 
 
 
xl 
 
 UNITS. 
 
 3. Metric Measures and Weights. 
 
 As explained in section i above, the standards of length and mass in the 
 metric system are the metre and the kilogramme. Two material representatives 
 of each of these standards are possessed by the United States and preserved at 
 the Office of Standard Weights and Measures at Washington, D. C. 
 
 The standards of length are Prototype Metres Nos. 21 and 27. These are 
 platinum iridium bars of X cross section, and their lengths are defined by lines 
 ruled on their neutral surfaces. Their lengths at any temperature / Centigrade 
 are given by the following equations : — 
 
 Prototype No. 21 = i" -j- 2.''5 -|- 8.>'665 t -\- 0.1^00 100 t^, 
 Prototype No. 27^1"*— i.^^e -]- 8.f 657 ^ + o.''oo 100 t% 
 
 where the symbol /*. stands for one micron, or one millionth of a metre. The 
 
 probable^ errors of these Prototjrpes may be taken as not exceeding ± o.''2, or 
 
 1/5 000 000th of a metre for temperatures between 0° and 30° C. 
 
 The standards of mass are Prototype Kilogrammes Nos. 4 and 20. They are 
 
 cylindrical masses of platinum iridium. Their masses and volumes are given by 
 
 the following equations : — 
 
 Mass. Volume. 
 
 Prototype Kilogramme No. 4=1*" — o."^o7S, 46."''4i8, 
 
 Prototype Kilogramme No. 20 = i*" — o.'"''o39, 46.'"'402, 
 
 where the — 
 
 Symbol kg stands for one kilogramme. 
 
 Symbol mg stands for one milligramme = o.^'oooooi. 
 
 Symbol ml stands for one millilitre = one cubic centimetre. 
 
 The definitive probable error assigned to the Prototype Kilogrammes by the 
 International Bureau is ± o.'^oo2, or 1/500 000 oooth of a kilogramme. 
 
 The act of Congress approved July 28, 1866, authorizing the use of the metric 
 system in the United States, provides that the tables in a schedule annexed shall 
 be recognized " as establishing, in terms of the weights and measures now in use 
 in the United States, the equivalents of the weights and measures expressed 
 therein in terms of the metric system ; and said tables may be lawfully used for 
 computing, determining, and expressing, in customary weights and measures, the 
 weights and measures of the metric system." The following copy of that sched- 
 ule gives the denominations of the multiples and sub-multiples of the measures 
 of length, surface, capacity, and weight in the metric system as well as their 
 legalized equivalents in British units. 
 
UNITS. 
 
 Xli 
 
 Schedule annexed to Act of July 28, 1866. 
 Measures of Length. 
 
 Metric Denominations. 
 
 
 Myriametre 
 Kilometre 
 Hectometre 
 Decametre . 
 Metre . 
 Decimetre 
 Centimetre . . 
 Millimetre 
 
 
 
 
 Values in Metres. 
 
 lOOOO. 
 
 1000. 
 
 100. 
 
 10. 
 
 O.I 
 
 O.OI 
 
 O.OOI 
 
 Equivalents in Denominations in Use. 
 
 6.2137 miles. 
 
 0.62137 mile, or 3280 feet and 10 inches. 
 
 328 feet and i inch. 
 
 393.7 inches. 
 
 39.37 inches. 
 
 3.937 inches. 
 
 0.3937 inch. 
 
 0.0394 inch. 
 
 Measures of Surface. 
 
 Metric Denominations. 
 
 Hectare . 
 Are . . 
 Centare ■ 
 
 Values in 
 Square Metres. 
 
 10000 
 100 
 
 Equivalents in Denominations in Use. 
 
 2.471 acres. 
 
 1 19.6 square yards. 
 
 1550 square inches. 
 
 Measures of Capacity. 
 
 Metric Denominations and Values. 
 
 Kilolitre or stere 
 
 Hectolitre . . 
 
 Decalitre . . . 
 
 Litre . . . 
 
 Decilitre . . . 
 Centilitre 
 Millilitre . 
 
 No. of 
 Litres. 
 
 1000. 
 100. 
 
 0.1 
 
 O.OI 
 O.OOI 
 
 Cubic Measure. 
 
 I cubic metre . . 
 o. r cubic metre 
 10 cubic decimetres . 
 I cubic decimetre . 
 o. I cubic decimetre . 
 10 cubic centimetres . 
 I cubic centimetre 
 
 Equivalents in Denominations in Use. 
 
 Dry Measure. 
 
 1.30S cubic yards 
 2 bus. and 3.35 pks. 
 g.oS quarts . . 
 o.goS quart. . . 
 6.1022 cubic inches . 
 o.6ro2 cubic inch 
 0.061 cubic inch . . 
 
 Liquid or Wine 
 Measure. 
 
 264.17 gallons. 
 26.417 gallons. 
 2.6417 gallons. 
 1.0567 quarts. 
 0.84s ffi'l' 
 0.338 fluid-ounce. 
 0.27 fluid-drachm. 
 
 Measures of W^eight. 
 
 Metric Denominations and Values. 
 
 Names. 
 
 Millier or tonneau 
 
 Quintal 
 
 Myriagramme . . . 
 Kilogramme, or kilo . 
 Hectogramme . 
 Decagramme . . 
 
 Gramme . . . 
 
 Decigramme 
 Centigramme 
 Milligramme 
 
 Number of Weight of what Quantity of Water 
 Grammes. at Maximum Density. 
 
 I 000000. 
 
 looooo. 
 
 loooo. 
 
 1000. 
 
 100. 
 
 10. 
 
 0.1 
 
 O.OI 
 O.OOI 
 
 I cubic metre . . . 
 I hectolitre 
 10 litres . . 
 1 litre . . . . 
 
 I decilitre .... 
 10 cubic centimetres 
 I cubic centimetre . 
 0.1 cubic centimetre 
 10 cubic millimetres 
 I cubic millimetre 
 
 Equivalents in Denominations 
 in Use. 
 
 Avoirdupois Weight. 
 
 2204.6 
 220.46 
 22.046 
 2.2046 
 3-5274 
 0-3527 
 15-432 
 1.5432 
 0.1543 
 0.0154 
 
 pounds. 
 
 pounds. 
 
 pounds. 
 
 pounds. 
 
 ounces. 
 
 ounce. 
 
 grains. 
 
 grains. 
 
 grain, 
 
 grain. 
 
xlii UNITS. 
 
 4. The C. G. S. System of Units. 
 
 The C. G. S. system of units is a metric system in which the fundamental 
 units are the centimetre, the gramme, and the mean solar second. It is the sys- 
 tem now generally used for the expression of physical quantities. 
 
 The most important of the derived units in the C. G. S. system, their equiva- 
 lents in terms of ordinary units, and their dimensions in terms of the fundamen- 
 tal units of length, mass, and time, are given in the Appendix to this volume. 
 
 For an elaborate consideration of the subject of units and their interrelations 
 the reader may be referred to " Units and Physical Constants," by J. D. Everett, 
 London, Macmillan & Co., i2mo, 4th ed., 1891. 
 
GEODESY. 
 
 I. Form of the Earth. The Earth's Spheroid. The Geoid. 
 
 The shape of the earth is defined essentially by the sea surface, which embraces 
 about three fourths of the entire surface. The sea surface is an equipotential 
 surface due to the attraction of the earth's mass and to the centrifugal force of its 
 rotation. We may imagine this surface to extend through the continents, and 
 thus to be continuous. Its position at any continental point is the height at 
 which water would stand if a canal connected the point with the ocean. 
 
 Geodetic measurements show that this surface is represented very closely by 
 an oblate spheroid, whose shorter axis coincides with the rotation axis of the 
 earth. This is called the earth's spheroid. The actual sea surface, on the other 
 hand, is called the geoid. With respect to the spheroid the geoid is a wavy sur- 
 face lying partly above and partly below ; but the extent of the divergence of the 
 two surfaces is probably confined to a few hundred feet. 
 
 2. Adopted Dimensions of Earth's Spheroid. 
 
 The dimensions of the earth's spheroid here adopted are those of General A. 
 R. Clarke, published in 1866, to wit : — 
 
 Semi major axis, a = 20 926 062 English feet. 
 Semi minor axis, (5 =: 20 855 121 " " 
 
 3. Auxiliary Quantities. 
 The following quantities are of frequent use in geodetic formulas : — 
 
 e ^<l^!—^ — , the eccentricity of generating ellipse, 
 
 f = 1 the flattening, ellipticity, or compression, 
 
 a — b 
 
 a-\-b 
 
 b = a sji-e^ = a{i -/) = a YZfn 
 
 I r- ^' JL ^* _U ^^ 4- S ^' O- 
 
 2 n 
 
 i+; 
 
 2 {n — n^ -\- n^ — n* -\- . . .). 
 
xliv GEODESY. 
 
 / 
 
 2 
 
 =7 = ihf) + {h/r + a/)' + (i/)* + . 
 
 4 « 
 
 = (j _j_ „y =■ 4 (« - 2 «' + 3 «' - 4 «' H- • • •)• 
 
 /■'^ ^* is° e^ 
 
 = 4- + 4- + 4- + ;^ + ---- 
 
 2 — ^^ 2 4 8 ' i6 
 
 I » ~r fi^ "t~ ToR ~r 
 
 i + y/i - «^ 4 ' 8 ' 64 ' 128 
 
 The numerical values of the most useful of these quantities and their logarithms 
 
 are — 
 
 log 
 a ^ 20 926 062 feet, 7.3206875, 
 ^ = 20 855 121 feet, 7.3192127, 
 
 ^^ 0.00676866, 7.8305030 — 10, 
 
 m — 0.00339583, 7.5309454 — 10, 
 
 « = 0.00169792, 7.2299162 — 10. 
 
 4. Equations to Generating Ellipse of Spheroid. 
 
 With the origin at the centre of the ellipse, and with its axes as coordinate 
 axes, the equation in Cartesian co-ordinates is 
 
 -^ + -^ = I, (i) 
 
 a and b being the major and minor axes respectively, and x and y being parallel 
 to those axes respectively. 
 
 For many purposes it is useful to replace equation (i) by the two following : — 
 
 x = a cos 0, 
 
 y =z b sva. 6, ^ ^ 
 
 which give (i) by the elimination of d. This angle is called the reduced latitude. 
 See section 5. 
 
 5. Latitudes used in Geodesy. 
 
 Three different latitudes are used in geodesy, namely : (i) Astronomical or 
 geographical latitude ; (2) geocentric latitude ; (3) reduced latitude. The astro- 
 nomical latitude of a place is the angle between the normal (or plumb line) at that 
 place and the plane of the earth's equator ; or when the plumb line at the place 
 coincides with the normal to the generating ellipse, it is the angle between that 
 normal and the major axis of the ellipse. The geocentric latitude of a place is 
 the angle between the equator and a line drawn from the place to the earth's cen- 
 tre ; or it is the angle between the radius-vector of the place and the equator. 
 The reduced latitude is defined by equations (2) in section 4 above. The geo- 
 metrical relations of these different latitudes are shown in Fig. i by the notation 
 given below. 
 
GEODESY. xlv 
 
 In order to express the analytical relations between the different latitudes let 
 
 <^ = the astronomical latitude, 
 ij/ = the geocentric latitude, 
 6 = the reduced latitude. 
 
 Then, referring to equations (i) and (2) under 
 
 
 
 1 
 
 1 
 
 
 ■P^ 
 
 r 
 
 
 b 
 
 ^~^\ 
 
 A 
 
 ^ 
 
 "Uv 
 
 ** 1 
 
 
 V 
 
 \Y 
 
 V 
 
 
 \ 
 
 K 
 
 -^ 
 
 Q 
 
 ^ 
 
 Fig.l. 
 
 section 4 above, and to Fig. i, it appears that 
 
 dx a^y 
 
 tan</, = -^ = + ^, 
 
 tan./r=-^, Xax^6 — % 
 X bx 
 
 Hence 
 
 tan i/f ^ —3- tan <^ = (i — e^ tan <^, 
 
 tan = (i — ey tan (^ = (i — <?^-* tan i/r. 
 
 ^ — ij/ :^ m sin 2 <^ — m^ sin 4 </>-)-... , 
 ^ — 6 = « sin 2 ^ — J «^ sin 4 c^ -|- . . . . 
 
 For the adopted spheroid 
 
 and 
 
 log (i — e^ = 9.9970504, 
 
 1^ — ij/ (in seconds) = 7oo."44 sin 2 <j> — i."i9 sin 4 <^, 
 <j> — 6 (in seconds) = 3So."22 sin 2 4> — o."3o sin 4 ^. 
 
 6. Radii of Curvature. 
 
 p„ ^ radius of curvature of meridian section of spheroid at any point whose 
 
 latitude is <^ =PO, Fig. i, 
 p„ = radius of curvature of normal section perpendicular to the meridian at 
 
 the same point = FQ, Fig. i. 
 Pa =■ radius of curvature of normal section making angle a with the meridian 
 at same point. 
 
 p„ = «(i-^^)(i-^=sin^<^)-i, 
 p„ = a(i-e' sin^ <i>)-^, 
 1 cos'^ a , sin^ a 
 
 P» 
 
 = i (i 4- -^ cos^^ ^ cos= a) (i - e^ sin= #. 
 log (i - e^ sin^ ^)-* — + log (i + n) 
 
 — p, « cos 2<l> 
 
 -\- ^ fhfi^ COS 4<^ 
 
 — ^ p. «' COS 6<j} 
 + .... 
 
 /i =: modulus of common logarithms and n is same as in section 3. For the 
 adopted spheroid — 
 
xlvi GEODESY. 
 
 Radius of curvature of meridian section p„ in feet, 
 log Pm — -{- 7-3I9948Z 
 
 — [4.34482] cos 21^ 
 
 -|- [1.274] COS 4^ 
 
 Radius of curvature of normal section p^ in feet, 
 log p„ = + 7-3214243 
 
 — [3.86770] COS 2^ 
 
 + [0.797] COS 4<j} 
 
 The numbers in brackets in these formulas are logarithms to be added to the 
 logarithms of cos 2^ and cos 4^. The numbers corresponding to the sums of 
 these logarithms will be in units of the seventh decimal place of the first constant. 
 Thus, for tf> = o, 
 
 log p„= 7-3214243 
 - 7373-9 
 
 _± 6^ 
 
 = 7.3206875 = log a. 
 
 7. Length of Arcs of Meridians and Parallels of Latitude. 
 
 a. Arcs of Meridian. 
 
 For the computation of short meridional arcs lying between given parallels of 
 latitude the following simple formulas suffice : 
 
 A<^ = <^2 — "^1, 
 AJ/=p„ A^. 
 
 In these, ^1 and 4>i are the latitudes of the ends of the arc, AM is the required 
 length, and p„ is the meridian radius of curvature for the latitude 4> of the middle 
 point of the arc. The formula for AM implies that A<j> is expressed in parts of 
 the radius. If A(j> is expressed in seconds, minutes, or degrees of arc, the for- 
 mula becomes — 
 
 Meridional distance AM in feet. 
 
 . „. pm Acf> (in seconds) 
 
 206204.8 
 
 Pm, A(j} (in minutes) 
 
 "~ 3437-747 ' 
 
 _ Pm A4> (in degrees) . 
 
 57-29578 ' (2) 
 
 log (1/206264.8) = 4.6855749 — 10, 
 log (1/3437-747) = 6.4637261 — 10, 
 log (1/57-29578) = 8.2418774 — 10. 
 
 ^1) 02 = 6nd latitudes of arc, A0 = 0^ — 0j, 
 p„ = meridian radius of curvature for = ^(^2 + ^,) ; for log pn see Table 10. 
 
GEODESY. xlvii 
 
 The relations (2) will answer most practical purposes when A<^ does not exceed 
 5°. A comparison with the precise formula (3) below shows in fact that the error 
 of (2) is very nearly 
 
 \ ^ Acft" cos 2^ . AM, 
 
 which vanishes for ^ =: 45°, and which for A^ = 5° is at most -zz^ws ^-^> or 
 about II feet. 
 
 Numerical example. Suppose — 
 
 '#^ = 37'' 29'48."i7, 
 'h = 35° 48' 29."89. 
 Then 
 
 't> = K<^2 + <^i) = 36° 39' 09-"°3, 
 
 A4>= </)3 — «^i = 1° 41' l8."28, 
 
 = 6o78."28. 
 
 From the first of (2) 
 
 cons't. log 4.6855749 — 10 
 
 Table 10, log p„ 7.3193112 
 
 log A<^ 3.7837807 
 
 AM= 614705 feet, log AM 5.7886668 
 
 The values of AJIfior intervals of 10", 20" . . . 60", and for 10', 20' . . . 60' are 
 given in Table 17 for each degree of latitude from 0° to 90°. 
 
 For precise computation of long meridional arcs the following formula is ade- 
 quate : — 
 
 AM^ Ao A^ — Ai cos 2<^ sin A<^ 
 -\- A2 cos 4<^ sin 2A<^ 
 
 — A3 cos 6<i> sin 3A<^ (3) 
 
 -|- Ai cos 8<^ sin 4A<^ 
 
 In this, AM, <f>, and A^ have the same meanings as above, and A,^ Ai, . . . are 
 functions of a and if or of a and n. 
 Thus, in terms of a and n, 
 
 A„ = a(i-\- «)-! (i + i «= + ^ «- + ... ), 
 
 ^1 = 3« (i + fi)"^ (n-in^ - . ■ ■), 
 A,= ya(i-\- n)-^ (n'-i^n'- ...), 
 ^3=||a(i+«)-i(«»-...), 
 A,= Uia{-^+nT^(n\- ...). 
 
 Introducing the adopted values of a and n, these constants become — 
 
 log. 
 ^„ = 20 890 606 feet, 7.3199510, 
 Ai= 106 411 feet, 5.0269880, 
 ^2= 113 feet, 2.0528, 
 
 Ai = 0.15 feet, 9.174 — 10. 
 
xlviii GEODESY. 
 
 It appears, therefore, that the first three terms of (3) will give A J/ with an 
 accuracy considerably surpassing that of the constant A,,- In the use of (3) it will 
 generally be most convenient to express A(j> in degrees, and in this case Ao must 
 be divided by the number of degrees in the radius, viz. : 57.2957795 [1.7581226]. 
 Applying this value and writing the logarithms of Aq, Ai, etc., in rectangular 
 brackets in place of A^, Ai, etc., (3) becomes 
 
 Meridional distance AM in feet. 
 
 AM= [5.5618284] A(^ (in degrees) 
 
 — [5.0269880] cos 2<l> sin A(f> (4) 
 
 4" [2.0528] cos 4^ sin 2A(f) 
 
 20 = ^j + 0„ A0 = ^j — 0j, 01, 02 = end latitudes of arc. 
 
 Formula (4) will suffice for the calculation of any portion or the whole of a 
 quadrant. The length of a quadrant is the value of the first term of (4) when 
 <t> = 45° and A<^ = 90°, since all of the remaining terms vanish. 
 
 Numerical examples. — 1°. Suppose 
 
 <^i = 0° and <^2 = 45°- 
 
 Then 2</,= 4S", 
 
 A^ = 45". 
 
 log. 
 cons't 5.5618284 
 45 1-6532125 
 
 I St term 
 
 cos 2<^ 
 sin A^ 
 cons't 
 
 7.2150409 
 
 9.8494850 — 
 9.8494850 — 
 5.0269880 
 
 10 
 10 
 
 ist term -|- 16 407 443 feet 
 
 2d term — 53 205.7 f^^t 2d term 4.7259580 
 
 The third term of the series vanishes by reason of the factor cos 4 ^ = cos 90° 
 = o. The sum of the first two terms, or length of a meridional arc from the 
 equator to the parallel of 45°, is 16 354 237 feet. 
 
 2°. Suppose ^1 =^ 45" and ^2 = 90°. 
 
 Then 2^ = 135", 
 
 A</'= 45°- 
 
 The numerical values of the terms will be the same as in the previous example, 
 but the sign of the second term will h&plus. Hence the length of the meridional 
 arc between the parallel of 45° and the adjacent pole is 16460649 feet. The 
 sum of these two computed distances, or the length of a quadrant, is 32 814886 
 feet. 
 
GEODESY. xlix 
 
 This agrees as it should with the length given by (4) when 2<^ = 90° and Ad. 
 = 90°.* 
 
 b. Arcs of parallel. 
 
 The radius of any parallel of latitude is equal to the product of the radius of 
 curvature of the normal section for the same latitude by the cosine of that lati- 
 tude. That is, see Fig. i, r being the radius of the parallel — 
 
 r = pn cos <f>, 
 
 and the entire length of the parallel is — 
 
 2 TT r = 2 TT p„ cos <f>. 
 
 Designate the portion of a parallel lying between meridians whose longitudes 
 are Aj and X^ by AT', and call the difference of longitude Aj — ' K ^^■ 
 Then — 
 
 Arc of parallel A/' in feet. 
 
 2 7rp„ cos <^ 
 
 ^^= 1296000 ^^ ('" seconds), 
 
 2 irp„ COS ^ .. r • ^ \ / s 
 
 =^ 7 AA. (m mmutes), (i ) 
 
 21600 ^ '' ^ ' 
 
 2 TT p„ COS <t> .... J . 
 
 = 7 AA. (in degrees). 
 
 log (2 w/1296000) = 4.6855749 — 10, 
 log (2 7r/2l6oo) = 6.4637261 — 10, 
 log (2 7r/36o) = 8.2418774 — 10. 
 
 Aj, ^2, = end longitudes of arc, A\ := A^ — \, 
 Pn = radius of curvature of normal section for latitude of parallel ; for log pn see Table 1 1. 
 
 Numerical Exampk. — Suppose ^ ^ 35°, and AA = 72". Then from the third 
 of (9) 
 
 log. 
 cons't 8.2418774 — 10 
 
 Table 11, p„ 7.3211716 
 
 cos <^ 9-9133645 — 10 
 AA 1.8573325 
 
 A/'= 21 564827 feet, A^ 7-3337460 
 
 * The best formula for computing the entire length of a meridian curve is this : 
 
 IT (a + ^) (I + i «2 + A «* + . . .), 
 
 in which a, i, and « are the same as defined in section 2. For the values here adopted — 
 
 log. 
 {i -{- i tfl -\- . . .) 0.0000003 
 (a + b) 7.6209807 
 
 ir 0.4971499 
 
 length 8.1 181 309 
 
 The length of the perimeter of the generating ellipse, or the meridian circumference of the 
 earth, is, therefore — 
 
 131 259 550 feet = 24 859.76 miles. 
 
GEODESY. 
 
 The values of APfor intervals of lo", 20" . . . 60", and for 10', 20' . . . 60' 
 are given in Table 18 for each degree of latitude from 0° to 90°. 
 
 8. Radius- Vector of Earth's Spheroid. 
 
 p = radius-vector 
 
 ■=a(jL — 2e^ sin= 4> + e* sin" <^)J (i - e^ sin^ <^)-». 
 
 a (2 — g") 
 log p = log ^j^,_^ - -\-ix(m - n) cos 2<^ 
 
 — i fi {nfi — n^ cos 4^ 
 -j- ^ //, (»z' — «') cos 6<^ 
 
 For the adopted spheroid 
 
 log (p in feet) = 7.3199520 -)- [3.86769] cos 2^ 
 
 — [1.2737] cos 4"^. 
 
 the logarithms for the terms in ^ corresponding to units of the seventh decimal 
 place. . Thus, for ^ =; o, 
 
 log p = 7.3199520 
 
 + 7373-8 
 
 — 18.8 
 
 = 7.3206875 = log a. 
 
 9. Areas of Zones and Quadrilaterals of the Earth's 
 
 Surface. 
 
 An expression for the area of a zone of the earth's surface or of a quadrilateral 
 bounded by meridians and parallels may be found in the following manner : — 
 
 The area of an elementary zone dZ, whose middle latitude is <^ and whose 
 width is p„ d^, is (see Fig. i), 
 
 dZ = 2 TT r p,„ d^ 
 
 = 2 77 p;„ p„ cos <^ d<l). 
 
 By means of the relations in section 6 this becomes 
 
 dZ =. 2 -n- a^ (1 — e^) -, — — 
 
 cos (/> d<j> 
 
 (i — e^ sin= <ff 
 
 2 I — e^ d (e sin <^) 
 
 2 '^ H 7 o : 2 r\5"" 
 
 (I) 
 
 e {t —e" sin'' 4>f 
 
 The integral of this between limits corresponding to <^i and <^2, or the area of a 
 zone bounded by parallels whose latitudes are ^j and <^2 respectively, is 
 
 Z=ir a' 
 
 I - ^ 
 
 1 
 
 e sin <^2 
 I — e^ sin" </)2 
 
 e sin <^i 
 - e^ sin'' ^1 
 
 4- i Nan \os (^ + e sin .^2) (r - e sin .j,,) 
 
 (2) 
 
GEODESY. 
 
 To get the area of the entire surface of the spheroid, make (^i ^ — J tt and <^2 
 = -[- i 'T in (2). The result is 
 
 Surface of spheroid = 2 tt a^ I i -]- Nap. log ( _ j I. (3) 
 
 For numerical applications it is most advantageous to express (3) in a series of 
 powers of e. Thus, by Maclaurin's theorem, 
 
 I r — . . . j. (4) 
 
 For the calculation of areas of zones and quadrilaterals it is also most advan- 
 tageous to expand (2) in a series of powers of e sin <^i and e sin ^2 and express 
 the result in terms of multiples of the half sum and half difference of ^1 and ^2- 
 Thus, (2) readily assumes the form 
 
 Z ^ 2 TT a" (i — c*^ I (sin <^2 — sin <^i) H — e' (sin' ^2 — sin' ^1) -(- . . . I. 
 
 From this, by substitution and reduction, there results 
 
 wherein 
 
 d cos <^ sin J A<^ — C^ cos 24> sin | A(^ ) ^ 
 
 A</) = ^2 — <f>i, 
 
 If Q be the area of a quadrilateral bounded by the parallels whose latitudes are 
 ^1 and ^2 and by meridians whose difference of longitude is AX, 
 
 ^ 27r 
 
 Hence, using the English mile as unit of length, (5) and (6) give for the 
 
 adopted spheroid — 
 
 Area of quadrilateral in square miles. 
 
 " ( ^1 cos <^ sin ^ A<^ — ^2 cos 30 sin | A.^ ) 
 e = AX (m degrees) j _|_ ,3 cos s<^ sin f A,^ - . . . i ' 
 
 log cf — 5.7375398, (7) 
 
 log ^2= 2.79173, 
 log <^3 = 9-976 — lO- 
 
 <S> = ii.^i + 'Pi ). A<(> = <(>2 — ()>„ 
 (/)j, (/)2 == latitudes'of bounding parallels, 
 
 AX = difference of longitude of bounding meridians. 
 
 » ^1, c^, c, are obtained from C„ C„ C^ respectively by dividing the latter by the number of 
 degrees in the pdius, viz : 57.29578. 
 
Hi GEODESY. 
 
 Numerical examples. — i°. Suppose <^i = o, <^2 = 9°° ^"^^ ^^ = 360°- Then 
 (7) should give the area of a hemispheroid. The calculation runs thus : 
 
 log. 
 
 
 log. 
 
 log. 
 
 
 Ci 5-7375398 
 
 C2 2.79173 
 
 ^3 9.976 
 
 — 10 
 
 COS (^ 9.8494850 — 
 
 10 cos 3 ^ 9.84948^ — 10 
 
 COS 5 ^ 9.849„ 
 
 — 10 
 
 sin \ A<^ 9.8494850 — 
 
 10 sin § A^ 9.84949 — 10 
 
 sin f A^ 9-848„ 
 
 — 10 
 
 360 2.5563025 
 
 360 2.55630 
 
 360 2.556 
 
 
 Sum 7.9928123 
 
 5.o47oo„ 
 
 2.229 
 
 
 Hence — 
 
 
 
 
 
 I St term ^ - 
 
 - 98358591 
 
 
 
 
 2d term = - 
 
 I 11429 
 
 
 
 
 3d term = - 
 
 169 
 
 
 
 Q = sum = 98470189 
 
 Twice this is the area of the spheroidal surface of the earth j z. e., 196 940 378 
 square miles. 
 
 2°. The last result may be checked by (4). Thus, 
 
 — + —+... j = 0.00225928 
 
 log ^i - Y - . . . j = 9-9990177 
 
 logs'' ^7.1961072 
 
 log 4. TT = 1.0992099 
 
 log (196940407) = 8.2943348 
 
 This number agrees with the number derived above as closely as 7-place 
 logarithms will permit, the discrepancy between the two values being about 
 -STSniiswu P^''* of the area. Hence, with a precision somewhat greater than the 
 precision of the elements of the adopted spheroid warrants. 
 
 Area earth's surface = 196 940 400 square miles. 
 
 The areas of quadrilaterals of the earth's surface bounded by meridians and 
 parallels of 1°, 30', 15', and 10' extent respectively, in latitude and longitude, are 
 given in Tables 25 to 29. 
 
 10. Spheres of Equal Volume and Equal Surface with 
 Earth's Spheroid. 
 
 ri =^ radius of sphere having same volume as the earth's spheroid, 
 ^2 = radius of sphere having same surface as that spheroid. 
 
 = a (i _ J ^2 _ ^j ^4 _ ^^^^^, _,_y 
 
GEODESY. 
 
 V 3 IS 35 / 
 
 '^ — >'i = i ae^ (i -\- ^ e^ -{-...) = 0.00113 a, about. 
 ''2 — ''1 = ^ ae* -j^ . . . = 0.00000 1 a, about. 
 
 liii 
 
 
 
 II. Co-ordinates for the Polyconic Projection of Maps. 
 
 In the polyconic system of map projection every parallel of latitude appears on 
 
 the map as the developed circumference of the 
 
 base of a right cone tangent to the spheroid along 
 
 that parallel. Thus the parallel EI^ (Fig. 2) 
 
 will appear in projection as the arc of a circle 
 
 £0J^ (Fig. 3) whose radius 0(? = / is equal 
 
 to the slant height of the tangent cone £J^G 
 
 (Fig. 2). Evidently one meridian and only one 
 
 will appear as a straight line. This meridian is 
 
 generally made the central meridian of the area 
 
 to be projected. The distances along this cen- 
 tral meridian between consecutive parallels are 
 
 made equal (on the scale of the map) to the real A^ 
 
 distances along the surface of the spheroid. The 
 
 circles in which the parallels are developed are 
 
 not concentric, but their centres all lie on the 
 
 central meridian. The meridians are concave 
 
 toward the central meridian, and, except near the corners of maps showing large 
 
 areas, they cross the paral- 
 lels at angles differing little 
 from right angles. 
 
 In the practical work of 
 map making, the meridians 
 and parallels are most ad- 
 vantageously defined by the 
 co-ordinates of their points 
 of intersection. These co- 
 ordinates may be expressed 
 in the following manner : 
 For any parallel, as EOF 
 (Fig. 3), take the origin O 
 at the intersection with the 
 central meridian, and let the rectangular axes of Y (OG) and X {OQ) be re- 
 spectively coincident with and perpendicular to this meridian. Call the interval 
 in longitude between the central meridian and the next adjacent one A\, and 
 denote the angle at the centre G subtended by the developed arc OP by a. 
 
liv GEODESY. 
 
 Then from Fig. 3 it appears that 
 
 :*; = / sin a, 
 y=z2 I sin" \a. 
 
 But from Figs. 2 and 3, 
 
 /=p„COt^, 
 
 /a = r AX = p„ AA, COS <j>, 
 
 whence 
 
 a = AA. sin <^. 
 
 Hence, in terms of known quantities there result 
 
 a; = p„ cot 4> sin (A\ sin <^), / n 
 
 jy = 2 p„ cot <^ sin*" J (A\ sin <^). 
 
 Numerical example. — Suppose ^ = 4°° and AA, = 25° = 90000". 
 
 Then 
 
 log 90000" = 4.9542425, 
 
 log sin 40° := 9.808067s — 10, 
 
 log 5785o."88 = 4.7623100 ; 
 
 AX sin</) = 16° 04' io."88, 
 
 i (AX sin ^) = 8° 02' o5."44. 
 
 log. log- 
 
 sin (AX sin <^) 9.4421760 — 10 sin i (AX sin <^) 9.1454305 — 10 
 
 cot <^ 0.0761865 sin i (AX sin <#.) 9.1454305 — 10 
 
 p„. Table II 7.3212956 cot ^ 0.0761865 
 
 p„. Table 11 7.3212956 
 
 2 0.3010300 
 
 X 6.8396581 y 5-9893731 
 
 ^ = 6912865 feet jy = 975 828 feet. 
 
 The equations (i) are exact expressions for the co-ordinates. But when 
 AX is small, one may use the first terms in the expansions of sin (AX sin 4,) and 
 sin'' J(AX sin <^) and reach results of a much simpler form. 
 
 Thus, 
 
 sin (AX sin </>) = AX sin <f> — ^(AX sin <^)» + . . . , 
 
 sin'' 4(A^ sin ^) = i(AX sin c^)" - ^(AX sin <^)^ + . . . ; 
 
 whence, to terms of the second order, 
 
 a; = p„ AX cos <^ [1 — J(AX sin 4>y], ,s 
 
 y = ip„ (AX)" sin 2<^ [i - ^{AX sin<^)"]. ^ ^ 
 
 If the terms of the second order in these equations be neglected, the value of 
 X will be too great by an amount somewhat less than |^(AX sin ^)" . x, and the 
 value of y will be too great by an amount somewhat less than -^(^^ sin <^5'>ili^ 
 An idea of the magnitudes of these fractions of x and y may be gained from the 
 following table, which gives the values of ^(AX sin </>)" for a few values of the 
 arguments AX and 4>- 
 
GEODESY. 
 
 Values of |-(AA, sin (fif. 
 
 Iv 
 
 AX 
 
 4> 
 
 20° 
 
 40° 
 
 60° 
 
 o 
 
 I 
 
 2 
 
 3 
 
 I/I68000 
 
 1/42000 
 
 I/I8700 
 
 1/47700 
 
 I/II900 
 VS3oo 
 
 1/26260 
 
 1/6560 
 
 1/2920 
 
 It appears from this table that the first terms of (2) will suffice in computing 
 the co-ordinates for projection of all maps on ordinary scales, and of less extent 
 in longitude than 2° from the middle meridian. For example, the value of x for 
 AA, = 2°, and (^ = 40°, and for a scale of two miles to one inch (1/126720), is 
 53.063 inches less i/iigoo part, or about 0.004 inch, which may properly be 
 regarded as a vanishing quantity in map construction. For the computation of 
 the co-ordinates given in the tables 19 to 24, where AA. does not exceed 1°, it 
 is amply sufficient, therefore, to use 
 
 X = p„ AA. cos <l>, 
 
 y = iPn (AA)" sin 2(^. 
 
 (3) 
 
 In these formulas and in (2), if AA is expressed in seconds, minutes, or degrees, 
 it must be divided by the number of seconds, minutes, or degrees in the radius. 
 The logarithms of the reciprocals of these numbers are given on p. xlvi. In the 
 construction of tables like 19 to 24, it is most convenient, when English units are 
 used, to express AA in minutes and x and y in inches. For this purpose, sup- 
 posing log p„ to be taken from Table 11, if j be the scale of the map, or scale 
 factor, equations (3) become — 
 
 Co-ordinates x and y in inches for scale s. 
 
 p„ s AA COS 4>, 
 
 3437-747 
 3 
 
 AA in minutes ; 
 
 log (12/3437-747) = 7-54291 - 10, 
 log (3/(3437-747)") = 3-4046 - 10. 
 
 (4) 
 
 Tables 19 to 24 give the values of x undy for various scales and for the zone of 
 the earth's surface lying between 0° and 80°. 
 
 Numerical example. — Suppose ^ = 40° and AX = 15' ; and let the scale of 
 the map be one mile to the inch, or j = 1/63360. Then the calculation by (4) 
 runs thus : 
 
Ivi 
 
 GEODESY. 
 
 log. 
 
 log. 
 
 cons't 7.54291 — 10 
 
 cons't 3.4046 — 10 
 
 P» 7-32130 
 
 Pn 7-3213 
 
 J 5.19818 — 10 
 
 s 5.1982 — 10 
 
 15 1.17609 
 
 (is)' 2.3522 
 
 COS <^ 9.88425 — 10 
 
 sin 2^ 9-9934 — 10 
 
 X 1. 12273 
 
 y 8.2697 — 10 
 
 In. 
 
 In. 
 
 X = 13.266 
 
 y = 0.01861. 
 
 These values of x and y, it will be observed, agree with those corresponding to 
 the same arguments in Table 22. 
 
 When many values for the same scale are to be computed, log s should, of 
 course, be combined with the constant logarithms of (4). Moreover, since in (4) 
 x varies as AX and y as (A\)^, when several pairs of co-ordinates are to be com- 
 puted for the same latitude, it will be most advantageous to compute the pair cor- 
 responding to the greatest common divisor of the several values of A\ and derive 
 the other pairs by direct multiplication. 
 
 12. Lines on a Spheroid. 
 
 The most important lines on a spheroid used in geodesy are (a) the curve of a 
 vertical section ; (i) the geodesic line ; and (1:) the alignment curve. Imagine two 
 points in the surface of a spheroid, and denote them by /\ and J'i respectively. 
 The vertical plane at I'l containing /'j and the vertical plane at /^ containing 
 I'l give vertical section curves or lines. The curves cut out by these two planes 
 coincide only when -fj and /j are in a meridian plane. The geodesic line is 
 the shortest line joining Pj and /ji a-nd lying in the surface of the spheroid. 
 The alignment curve on a spheroid is a curve whose vertical tangent plane at 
 every point of its length contains the terminal points J^i and ^2. The curve 
 {a) lies wholly in one plane, while (6) and (c) are curves of double curvature. 
 In the case of a triangle formed by joining three points on a spheroid by lines 
 lying in its surface, the curves of class (a) give two distinct sets of triangle 
 sides, while the curves of classes (f) and give but one set of sides each. 
 For all intervisible points on the surface of the earth, these different lines differ 
 immaterially in length ; the only appreciable differences they present are in their 
 azimuths (see formula under b below). Of the three classes of curves the first 
 two only are of special importance. 
 
 a. Characteristic property of curves of vertical section. 
 
 Let ai,2 = azimuth of vertical section at Ii through /j, 
 
 02.1 = azimuth of vertical section at I\ through J^i, 
 
 $1, 6^ = reduced latitudes of J^i and /j respectively, 
 
 81, 82 = angles of depression at /'i and ^2 respectively of the chord joining 
 these points. 
 
 Then the characteristic property of the vertical section curve joining I'l and P^ is 
 
 sin ai2 cos 01 cos Si = sin (a^i — 180°) cos ^2 cos 82. 
 
GEODESY. lyii 
 
 The azimuths ai.j and a^^, it will be observed, are the astronomical azimuths, 
 or the azimuths which would be determined astronomically by means of an alti- 
 tude and azimuth instrument. 
 
 b. Characteristic property of geodesic line. 
 
 Let a'i.2 = azimuth of geodesic line at J'l, 
 
 "■'2.1 = azimuth of geodesic line at ^2, 
 ^i) ^2 = reduced latitudes of F^ and T'a respectively. 
 
 Then the characteristic property of the geodesic line is 
 
 sin ai.2 cos 61 — sin (i8o°— 03.1) cos 0^ = cos $0, 
 
 where 60 is the reduced latitude of the point where the geodesic through ^1 and 
 /a is at right angles to a meridian plane. 
 
 The difference between the astronomical azimuth ajj and the geodesic azimuth 
 a'l 2 is expressed by the following formula : 
 
 «i.2 — a'1.2 (in seconds) = ^V p" ^^ (-) cos^ ^ sin 201,2, 
 
 where s = length of geodesic line T'l ^'2, 
 
 a = major semi-axis of spheroid, 
 e =. eccentricity of spheroid, 
 p" = 206264. "8, 
 
 4> = astronomical latitude of F^, 
 ai.2 = azimuth (astronomical or geodesic) of F^ F^, 
 
 log tV p"\-] ^= 7-4244 — 20, for a in feet. 
 
 Thus, for <^ = o and ai.j = 45°, for which cos^ </> sin 201.2 = 1, the above for- 
 mula gives 
 
 "1.2 — a'1.2 = o."o74, for s z= 100 miles, 
 = 0.296, for s = 200 miles. 
 
 so that for most geodetic work this difference is of little if any importance. 
 13. Solution of Spheroidal Triangles. 
 
 The data for solution of a spheroidal triangle ordinarily presented are the 
 measured angles and the length of one side. This latter may be either a geodesic 
 line or a vertical section curve, since their lengths are in general sensibly equal. 
 Such triangles are most conveniently solved in accordance with the rule afforded 
 by Legendre's theorem, which asserts that the sides of a spheroidal triangle (of 
 any measurable size on the earth) are sensibly equal to the sides of a plane 
 triangle having a base of the same length and angles equal respectively to the 
 spheroidal angles diminished each by one third of the excess of the spheroidal 
 triangle. In other words, the computation of spheroidal triangles is thus made to 
 depend on the computation of plane triangles. 
 
Iviii GEODESY. 
 
 a. Spherical or spheroidal excess. 
 
 The excess of a spheroidal triangle of ordinary extent on the earth is given by 
 
 e (in seconds) ^ p" . 
 
 Pm Pn 
 
 where 5 is the area of the spheroidal or corresponding plane triangle ; p^, p„ are 
 the principal radii of curvature for the mean latitude of the vertices of the tri- 
 angle ; and p" :^ 206 264."8. For a sphere, p^ = p„=: radius of the sphere. 
 
 Denote the angles of the spheroidal triangle by A, B, C, respectively ; the cor- 
 responding angles of the plane triangle by a, p, y (as on p. xviii) ; and the sides 
 common to the two triangles by a, b, c. Then 
 
 S^\ ab wa.y^=\bc svc^ a.-=.\ ca sin j8. 
 
 a = A — \i, P = B — \t, y=<: 
 
 e. 
 
 Tables 13 and 14 give the values of log (fi"l2p^^ for intervals of 1° of astro- 
 nomical or geographical latitude.* 
 
 14. Geodetic Differences of Latitude, Longitude, and 
 
 Azimuth. 
 
 a. Primary triangulation. 
 
 Denote two points on the surface of the earth's spheroid by P^ and F^ respec- 
 tively. Let 
 
 s = length of geodesic line joining Bi and /j, 
 ^1, (f>2 = astronomical latitudes of J\ and Tg, 
 Aj, X2 = longitudes of I\ and I^t 
 
 AA. = A.2 — ^1, 
 
 ai.2 := azimuth of I\ P^ (s) at /j, 
 02,1 = azimuth of /j P^ (i) at P^, 
 e = eccentricity of spheroid, 
 Pm, Pn = principal (meridian and normal) radii of curvature at the point P^. 
 
 Then for the longest sides of measurable triangles on the earth the following 
 formulas will give <^2, A2, and 03.1 in terms of ^j, Xj, ai.2, and s. The azimuths are 
 astronomical, and are reckoned from the south by way of the west through 360°. 
 
 a' := 180° — ai.2, and a2.i =: l8o° -\- a", for ajj <l8o° 
 
 a' = ai.2 — 180°, and oa.i = 180° — a", for a^^ > 180° 
 
 (■) 
 
 " = ^„ ,{ ^ + i 7^. (^J"cOS= <^, COS= a' } • (2) 
 
 t—k YZZ^i cos'' ^1 sin 2a' (3) 
 
 * For the solution of very large triangles and for a full treatment of the theory thereof, consult 
 Die Mathematischen und Physikalischen Theorieen der Hoheren Geoddsie, von Dr. F. R. Helmert. 
 Leipzig, 1880, 1884. 
 
GEODESY. 
 
 tan ^(a" - AX + = ± \^P°l - ^ 7 '? cot ^ a' 
 
 (4) 
 
 sin i(9o° - <^i + ^) ^"' ^ ' 
 
 , , J sin i(a" — a' + Z) 
 
 ■^ - '^^ = ^. sinlK + a' + O {' + tV ^^ cos» K-" - a')}. (5) 
 
 To express 17, f, and <^2 — ^1 in seconds of arc we must multiply the right hand 
 sides of (2), (3), and (5) by p" = 206 264."8. For logarithmic compution of V' 
 and t,", or t] and i in seconds, we may write with an accuracy generally sufficient 
 
 log V = log (p"s/p:) + I -t^, (^)' «=°s' "^i ''°^' <^'' (6) 
 
 log C" = log I (73^ + log {(v'7 cos' <Ai sin 2 a'}, (7) 
 
 where /^ in (6) is the modulus of common logarithms. For units of the 7th deci- 
 mal place of log 17" we have for the adopted spheroid 
 
 Also 
 
 '°g^ (,_^,.y = ^-9^697 -10. 
 
 Similarly, for the computation of the logarithm of the last factor in (5) we have 
 log {i + tV v' C0S= ^(a" - a')} = log {I + -±^^ (r,y cos' ^a" - a')}. 
 Putting for brevity 
 
 the logarithm of the desired logarithm is given to terms of the second order 
 inclusive in ^ by 
 
 log log (i + ?) = log /^ ^ - i- ^ ^. 
 
 For the adopted spheroid 
 
 for units of the seventh decimal place. 
 
 For a line 200 miles (about 320 kilometres) long, the maximum value of the 
 second term in (6) is but 12.6 units in the 7th place of log rj". For the same 
 length of line, the maximum value of C' is o."895, and the maximum value of the 
 logarithm of the last factor in (5), or log (i + ^), is less than 922 units in the 
 seventh place of decimals. 
 
 For computing differences of latitude, longitude, and azimuth in primary 
 triangulation whose sides are 1° (about 70 miles, or 100 kilometres) or less 
 in length, the most convenient means are formulas giving ^2 — ^u ^2 — ^u and 
 
Ix GEODESY. 
 
 ojj — (i8o° — ai.2), in series proceeding according to powers of the distance s. 
 Formulas of this kind with convenient tables for facilitating the computations 
 are given in the Reports of the U. S. Coast and Geodetic Survey* 
 
 b. Secondary triangulation. 
 
 For secondary triangulation, wherein the sides are 12 miles (20 kilometres) or 
 less in length, and wherein differences of latitude and longitude are needed to the 
 nearest hundredth of a second only, the following formulas may suffice. Using 
 the same notation as in the preceding section, the formulas are : — 
 
 <^2 ^ ^1 + '^^J 
 
 X^ = Ai + AA, (i) 
 
 02.1 := 180° -|- tti^ -|- Aa, 
 
 A<^ = — ai s cos ai.2 — ^2 •f^ sin^ ai^, 
 
 AA =: -|- ^1 sec <^i s sin ai.2 — ^2 s^ sin a^^ cos ai.2, (2) 
 
 Aa :^ — (Ti tan ^1 s sin ai.2 -|- ^2 -f^ sin aj.j cos aj^. 
 
 The constants entering the latter equations are defined by the following 
 expressions, wherein p^ and p„ are the principal radii of curvature of the spheroid 
 at the point whose latitude is ^1 and p" = 206 264."8 : 
 
 // // 
 
 Pm Pii 
 
 p" tan <^i , p" sec <;(ii tan <^i p" (1 + 2 tan" <^x) 
 
 «2 , O2 2 , Ci 2 • 
 
 2 Pm Pn Pn 2 Pn 
 
 The logarithms of the factors ai, bi, c^, «2, b^, ^2, are given in Table 15 for the 
 English foot as unit, and in Table 16 for the metre as unit, the argument being 
 the initial latitude <^i for all of them. 
 
 When all of the differences given by (2) are computed, they may be checked 
 by the formula 
 
 sin i(<^2 + <^i) = ^- (3) 
 
 For convenience of reference in numerical applications of the above formulas, 
 (2) may be written thus : 
 
 A<j> = Ai-{- A^, 
 AA = ^1 + B„ 
 
 Aa = Ci + C2, 
 
 in which, for example, A^ and A^ are the first and second terms respectively of 
 A^, due regard being paid to the signs of the functions of ajj. 
 
 Numerical example. The following example will serve to illustrate the use of 
 formulas (i) to (3). The value of log J- is for s in English feet, s being in this 
 case about 12.3 miles. 
 
 ^1 38°54'o8."38 \ 88° 03' 24."is ai.2 43° 01' 46."29 
 Acji — 07' 5o."2i AA +09'20."22 Aa — 05' si."32 
 ^^ 38°46'i8."i7 A2 88°i2'44."3| aj.i 222° 55' S4."97 
 i(<^2+^i) 38° so' i3-"2 7 
 
 » See Appendix 7, Report of 1884, for latest edition of these tables. 
 
GEODESY. Ixi 
 
 log log log log 
 
 s 4.81308 J 4.81308 s sin ai.2 4.647 J sin aj.j 4.647 
 
 COS ai.2 9.86392 sin ai,2 9.83402 J sin ai,2 4.647 J cos ajj 4.677 
 
 '^1 7-9949S sec <^i 0.10890 a^ 0.279 h o-688 
 
 ^i 7-99316 ^2 0.733 
 
 Ai 2.6719s ^1 2.74916 ^2 9-573 -^2 0.012 
 
 sin <^i 9-79795 Q o-o57 
 
 Ci 2.547 1 1 
 
 log 
 
 Ai - 469."84 ^1 + S6i."25 d - 3S2."46 Aa 2.54570 
 
 -^2- o."37 ^2- i."o3 C2+ i."i4 AA. 2.74836 
 
 A^-47o."2i AX + 56o."22 Aa-35i."32 sin ^(^2 + sf-i) 9-79734 
 
 15. Trigonometric Leveling. 
 
 a. Computation of heights from observed zenith distances. 
 
 Let J =^ sea level distance between two points jPj and P^, 
 
 III, ^2 = heights above sea level of P^ and P^, 
 Zi = observed zenith distance of P2 from Pi, 
 02 = observed zenith distance of Pi from P^, 
 
 p = radius of curvature of vertical section at Pi through P2, or at P2 
 through Pi, the curvature being sensibly the same for both for this 
 purpose, 
 C = angle at centre of curvature subtended by s, 
 mi, vti = coefficients of refraction at Pi and P^, 
 A^i, Az2 = angles of refraction at Pi and P^. 
 
 Then, the fundamental relations are 
 
 C = - , A«i = MiC, A«2 = m^C, / s 
 
 ^1 + .^2 + ^Zi -\- Az2 = 180° + C, 
 H2 - Hi = s tan \{Z2 + A02 - % - A^,) (i + ^^ + ^^ + ^+ . . .). (2) 
 
 When the zenith distances Zi and z^ are simultaneous, or when A^i and A^g are 
 assumed to be equal, (2) becomes 
 
 ^2-^i = -tan \{Z2 - zi) (i +=^7^' + I^ + - • •)• (3) 
 
 For the case of a single observed zenith distance Zi, say, and a known or 
 assumed value of »« ^ Wi = m2, the following formula may be applied : 
 
 Z?^ — .Si = J' cot ^1 -| : s' 4" s^ cot'^ ^1. (4) 
 
 The coefficient of refraction m varies very greatly under different atmospheric 
 conditions. Its average value for land lines is about 0.07. The following table 
 gives the values of log ^(i — 2 m) and log (i — m) for values of m ranging from 
 0.05 to o.io. It is taken from Appendix 18, Report of U. S. Coast and Geodetic 
 
Ixii 
 
 GEODESY. 
 
 Survey for 1876. Table 12 taken from the same source gives values of log p 
 needed for use in (3) and (4). 
 
 
 Table of values of log \{i 
 
 — 2 m) and log (i — m). 
 
 
 m 
 
 \o%\{\ — 2ni). 
 
 log (i — m). 
 
 m 
 
 log i(l — 2 m). 
 
 log (l — m). 
 
 0.050 
 
 9.65321 
 
 9.978 
 
 0.07s 
 
 9.62839 
 
 9.966 
 
 51 
 
 65225 
 
 77 
 
 76 
 
 62737 
 
 66 
 
 52 
 
 65128 
 
 77 
 
 77 
 
 62634 
 
 65 
 
 S3 
 
 65031 
 
 76 
 
 78 
 
 62531 
 
 65 
 
 54 
 
 64933 
 
 76 
 
 79 
 
 62428 
 
 64 
 
 0.055 
 
 9.64836 
 
 9-975 
 
 0.080 
 
 9.62325 
 
 9-964 
 
 56 
 
 64738 
 
 75 
 
 81 
 
 62221 
 
 63 
 
 57 
 
 64640 
 
 75 
 
 82 
 
 62118 
 
 63 
 
 58 
 
 64542 
 
 74 
 
 83 
 
 62014 
 
 62 
 
 59 
 
 64444 
 
 74 
 
 84 
 
 61910 
 
 62 
 
 0.060 
 
 9-64345 
 
 9-973 
 
 0.085 
 
 9.61805 
 
 9.961 
 
 61 
 
 64246 
 
 73 
 
 86 
 
 61700 
 
 61 
 
 62 
 
 64147 
 
 72 
 
 87 
 
 61595 
 
 60 
 
 63 
 
 64048 
 
 72 
 
 88 
 
 61490 
 
 60 
 
 64 
 
 63949 
 
 71 
 
 89 
 
 61384 
 
 60 
 
 0.065 
 
 9.63849 
 
 9.971 
 
 0.090 
 
 9.61278 
 
 9-959 
 
 66 
 
 63749 
 
 70 
 
 91 
 
 61172 
 
 59 
 
 67 
 
 63649 
 
 70 
 
 92 
 
 61066 
 
 58 
 
 68 
 
 63548 
 
 69 
 
 93 
 
 60959 
 
 58 
 
 69 
 
 63448 
 
 69 
 
 94 
 
 60853 
 
 57 
 
 0.070 
 
 9-63347 
 
 9.968 
 
 0.095 
 
 9.60746 
 
 9.957 
 
 71 
 
 63246 
 
 68 
 
 96 
 
 60638 
 
 56 
 
 72 
 
 63144 
 
 68 
 
 97 
 
 60531 
 
 56 
 
 73 
 
 63043 
 
 67 
 
 98 
 
 60423 
 
 55 
 
 74 
 
 62941 
 
 67 
 
 99 
 
 60315 
 
 55 
 
 
 
 
 0.1 00 
 
 9.60206 
 
 9-954 
 
 For less precise work one may use equation (4) in the form 
 ^ — ^ = J cot Zi-\- c j", 
 
 (5) 
 
 wherein, if we make m = 0.07 and use for p its average value, or y/ p„ p„, for 
 latitude 45", 
 
 log e = 2.313 — 10 for J in feet, 
 =: 2.829 ~ 10 for J in metres. 
 
 Thus, for a distance (s) of 10 miles the value of the term cs^ in (5) is 57.3 feet. 
 
 : observed in the place of zenit 
 lus : — 
 
 ^2 — -Si = ± J tan ai -|- (T ^, (6) 
 
 If altitudes %, say, are observed in the place of zenith distances z-^, it is most 
 convenient to write (5) thus : — 
 
GEODESY. Ixiii 
 
 where the upper sign is used when a^ is an angle of elevation and the lower sign 
 when tti is an angle of depression. 
 
 b. Coefficients of refraction. 
 
 When «i and z^ are both observed for a given line, a coefficient of refraction may 
 be computed from the assumption of equality of coefficients at the two ends of 
 the line. Thus, equations (i) give 
 
 Az, + Az^ = 180° + C - (2i + z^), 
 or 
 
 s s 
 
 (mi -\-m^-^= 180° + - - (^1 + 22). 
 
 whence 
 
 »Zl + Z«2 ^ 1 — - (^1 -\- Zi — 180°). 
 
 Assuming »«i = Wj = m, and supposing z^ + z^ — 180° expressed in seconds 
 of arc, 
 
 ^ = * { ^ - "^ (^1 + ^^ - ^^''°) 
 
 p" = 206264. "8, log p" = 5.3144251. 
 
 c. Dip and distance of sea horizon. 
 
 Let 
 
 Then 
 
 Ji = height of eye above sea level, 
 
 S = dip or angle of depression of horizon, 
 
 s = distance of horizon from observer. 
 
 8 (in seconds) = 58.82 \/A in feet, 
 
 = 106.54 \//i in metres. 
 
 J (in miles) = 1.317 ^/lia feet. 
 
 J (in kilometres) = 3-839 \/A in metres. 
 
 The above formulas take account of curvature and refraction. They depend 
 on the value 0.0784 for the coefficient of refraction, and are quite as accurate as 
 the uncertainties in such data justify. For convenience of memory, and for an 
 accuracy amply sufficient in most cases, the coefficients of the radicals in the last 
 two formulas may be written f and ^ respectively. 
 
 16. Miscellaneous Formulas. 
 
 a. Correction to observed angle for eccentric position of instrument 
 
 Let C" be the eccentric position of the instrument, and Co the observed value of 
 the angle at that point between two other points A and B. Let C denote the 
 central point as well as the angle AC£ desired. Call the distance CC r and 
 denote the angle ACC by &. Denote the lines BC and AC, which are as- 
 sumed to be sensibly the same as BC and AC, hy a and l> respectively. Then 
 
Ixiv GEODESY. 
 
 p'V sin (e — Co) p"r sin 
 C- Ca (in seconds) = ^ ^ °-^ - '^—^ , 
 
 p" = 2 06 2 64."8, log p" = 5 .3 1 442 5 1 . 
 
 Attention must be paid to the signs of sin {6 — Co) and sin 6, and to the fact 
 that angles are counted from A towards £ through 360°. A diagram drawn in 
 accordance with the above specifications will elucidate any special case. 
 
 b. Reduction of measured base to sea level. 
 
 Let / be the length of the bar, tape or other unit used in measuring the base. 
 Let 4 be the corresponding length reduced to sea level for a height h, this latter 
 being the observed height of /. Then if p denote the radius of curvature of the 
 earth's surface in the direction of the base, 
 
 *=,-^.= (.-:+...)' 
 
 •with sufficient accuracy. Hence, for the whole length of the base, 
 
 2/0 = S/- -'^Ih. 
 
 If L denote the total measured length, Zq the corresponding total sea level 
 length, and ZTthe mean value of the heights h, the above equation gives 
 
 P 
 
 c. The three-point problem. 
 In this problem the positions of three points A, B, C, and hence the elements 
 of the triangle they form, are given together with the two angles APC and BPC 
 at a point P whose position is required. Denote the angles and the sides of the 
 known triangle by A, B, C, and a, b, c, respectively. Also put 
 
 APC=P, BPC=a, 
 PAC^x, PBC=^y. 
 
 Then the sum of the angles in the quadrilateral PACB is 
 
 a + y8 + ^-f^+C=36o°, 
 whence 
 
 \{x ^y)= 180° - Ka + /3 + C). (i) 
 
 Compute an auxiliary angle z from the equation 
 
 « sin fi , s 
 
 tan z = -j— — - ; (2) 
 
 I? sin a 
 
 Then 
 
 tan \{x -y) = tan {z - -45°) tan \ix + y). (3) 
 
 These three equations give all the data essential to a complete determination 
 of the position of P. Any special case should be elucidated by a diagram drawn 
 in accordance with the specifications given above. 
 
GEODESY. 
 
 Ixv 
 
 When the positions of the points A, B, C are given on a map, the position of 
 P on the same map may be found graphically by drawing lines making angles 
 with each other equal to the given angles a. and ;8 from a point on a piece of 
 tracing paper, and then placing this tracing on the map so as to meet the required 
 conditions. This ready method of solving the problem is often sufficient. 
 
 17- 
 
 Salient Facts of Physical Geodesy. 
 
 a. Area of earth's surface, areas of continents, area of oceans.* 
 
 Square miles. 
 
 Total area of earth's surface 196 940 000 
 
 Area continent of Europe 3 820 000 
 
 " " Asia 17230000 
 
 " " Africa 11 480 000 
 
 " " Australia 3 406 000 
 
 " " America 15950000 
 
 Total area of continents 51886000 
 
 Total area of oceans 145054000 
 
 b. Average heights of continents and depths of oceans.f 
 
 Average height of continent of Europe . 
 
 " " " Asia . . 
 
 " " " Africa . 
 
 " " " Australia 
 
 " " " America . 
 Average height of all 
 
 Feet. 
 
 980 
 1640 
 1640 
 
 820 
 
 1340 
 1440 
 
 Feet. 
 
 Metres. 
 300 
 500 
 500 
 250 
 410 
 440 
 
 Metres. 
 
 3680 
 
 3890 
 
 3340 
 
 3440 
 
 Average depth of Atlantic Ocean 12 100 
 
 " " Pacific Ocean 12 700 
 
 " " Indian Ocean 11 000 
 
 Average depth of all 11 300 
 
 c. Volume, surface density, mean density, and mass of earth. 
 
 Volume of earth = 259 880 000 000 cubic miles. 
 
 = I 083 200 000 000 cubic kilometres. 
 
 =: 260 X 1°" cubic miles (about). 
 
 = 108 X 10" cubic kilometres (about). 
 
 Surface density of earth = 2.56 ± 0.16 t 
 Mean density of earth =:: 5.576 ± 0.016. 
 
 * Derived from relative areas given in Helmert's Geoddsie, Band II. p. 313. 
 
 t Helmert's Geoddsie, Band II. p. 313. 
 
 t These densities are given by Professor Wm. Harkness in his memoir on The Solar Parallax 
 and Related Constants. The surface density applies to that portion of the earth's crust which lies 
 above and within a shell ten mUes thick, the lower surface of this shell being ten mUes below sea 
 level. 
 
Ixvi GEODESY. 
 
 Assuming the mass of a cubic foot of water to be 62.28 pounds (at 62° F.)y 
 
 Mass of earth* = 13 284 X 10^^ pounds. 
 
 = 6642 X 10'^ tons (of 2000 lbs.). 
 = 60258 X 10'" kilogrammes. 
 
 d. Principal moments of inertia and energy of rotation of earth. 
 
 M=. mass of earth, 
 
 A = moment of inertia of earth about an axis in its equator, 
 
 C = moment of inertia about axis of rotation, 
 
 a = equatorial axis of earth, 
 
 o) =^ angular velocity of earth, 
 
 =^ (2 T/86164) for mean solar second as unit of time. 
 
 Thent 
 
 ^ =: 0.325 Ma% 
 C = 0.326 Ma\ 
 
 Energy of rotation of earth = ^ lo^C. 
 
 = 0.163 '^'^■^a^- 
 = 504 X 10^' foot-poundals. 
 ^217 X 10^^ kilogramme-metres. 
 ^ 212 X 10°' ergs. 
 
 Jie/erences. 
 
 The most exhaustive treatise on the theory of geodesy is found in " Die Mathe- 
 matischen und Physikalischen Theorieen der Hoheren Geodasie," von Dr. F. R. 
 Helmert. Leipzig : B. G. Teubner ; 8vo, 1880 (vol. i.), 1884 (vol. ii.). An excel- 
 lent work on the practical as well as theoretical features of the subject is " Die 
 geodatischen Hauptpunkte und ihre Co-ordinaten," von G. Zachariae ; autorisirte 
 deutsche Ausgabe, von E. Lamp. Berlin : Robert Oppenheim, 8vo, 1878. Of 
 works in English the most comprehensive is " Geodesy," by A. R. Clarke. Ox- 
 ford : The Clarendon Press, 8vo, 1880. 
 
 * The mass of the earth's atmosphere is about one-minionth part of the entire mass, or about 
 66 X 10" tons. 
 
 t The values of A and C axe those given by Harkness, Uc. cit., but they are here abridged to 
 three places of decimals. 
 
ASTRONOMY. 
 
 I. The Celestial Sphere. Planes and Circles of Reference. 
 
 The celestial sphere is a sphere to which it is convenient to refer stars and 
 other celestial objects. Its centre is assumed to be coincident with the eye of 
 the observer, and the objects referred to it are supposed to lie in its surface. 
 The orientation of this sphere is defined by its equator, which is assumed to be 
 parallel to the earth's equator. The equator is thus the principal plane of refer- 
 ence. Other planes of reference are the plane of the horizon, which is perpen- 
 dicular to the plumb line at the place ; the meridian, which is a plane through 
 the place and the earth's axis of rotation ; the prime-vertical, which is a vertical 
 plane at the place at right angles to the meridian ; and the ecliptic, which is a 
 plane parallel to the plane of the earth's orbit. These planes cut the surface of 
 the sphere in great circles called the equator, the horizon, the meridian, etc. The 
 points on the sphere defined by the intersection of the meridians, or the points 
 where the axis of the equator pierces the sphere, are called the poles. Similarly, 
 the prolongation of the plumb line upwards pierces the sphere in the zenith, and 
 its prolongation downwards pierces the sphere in the nadir. Great circles pass- 
 ing through the zenith are called vertical circles. 
 
 2. Spherical Co-ordinates. 
 
 a. Notation. 
 
 The position of a celestial body may be defined by several systems of co-ordi- 
 nates. The most important of these in practical astronomy are the azimuth 
 and altitude system and the hour angle and declination system. In the first of 
 these the azimuth of a star or other body is the angle between the meridian 
 plane of the place and a vertical plane through the star. It is measured, in gen- 
 eral, from the south around by the west through 360°. The altitude of a star is 
 its angular distance above the horizon, and its zenith distance is the complement 
 of the altitude. In the second system the hour angle of a star is the angle 
 between the meridian plane of the place and a meridian plane through the star. 
 It is measured towards the west. through 360°. The declination of a star is its 
 angular distance above or below the equator ; the complement of the declination 
 is called the polar distance. 
 
 The angular distance of the pole above the horizon is equal to the zenith dis- 
 tance of the equator, or to the latitude of the place. Likewise, the altitude of 
 the equator and the zenith distance of the pole are each equal to the comple- 
 ment of the latitude at any place. 
 
Ixviii ASTRONOMY. 
 
 These quantities are usually designated by the following notation : — 
 
 A = the azimuth of a star or object, 
 h = its altitude, 
 
 z = its zenith distance = 90° — k, 
 t = its hour angle, 
 8 =^ its declination, 
 / = its polar distance = 90° — 8, 
 q = the parallactic angle, or angle at the star between the pole and the 
 
 zenith, 
 ^ = the latitude of the place of observation. 
 
 b. Altitude and azimuth in terms of declination and hour angle. 
 
 The fundamental relations for this problem are — 
 
 sin h = sin <^ sin 8 -|- cos <^ cos 8 cos f, 
 cos A cos ^ = — cos ^ sin 8 -j- sin <^ cos 8 cos f, (i) 
 
 cos ^ sin ^ = cos 8 sin t. 
 
 When it is desired to compute both A and k by means of logarithms, the most 
 convenient formulas are, 
 
 m sin M = sin 8, ,, tan 8 
 
 m cos M-=- cos S cos t, cos r 
 
 , , ,^ . tan t cos M / V 
 
 sm h = m cos (^ - M), tan A = ^^^ ^^ _^y (2; 
 
 cos h cos A ^m sin (<f> — M), . ^ __ cos A ^ 
 
 cos hsin A=z cos 8 sin t, tan (^ — M)' 
 
 A > 180° when / > 180° and A < 180° when t < 180°. 
 
 For the computation of A and z separately, the following formulas are useful : 
 
 sin / 
 
 tan A =^ ■ 
 
 cos <^ tan 8 (i — tan <j> cot 8 cos t) 
 
 (3) 
 a sin f 
 
 1—6 cos t' 
 where 
 
 a = sec fj) cot 8, 6 ^ tan <^ cot 8. 
 
 Formulas (3) are especially appropriate for the computation of a series of 
 azimuths of close circumpolar stars, since a and l> will be constant for a given 
 place and date. 
 
 cos z = cos (^ — 8) — 2 cos <^ cos 8 sin" ^ /, 
 
 sin" i z = sin" i (<^ ~ 8) + cos <^ cos 8 sin" ^ f, , \ 
 
 (.^ ~ S) = <^ — 8, for^ >8 ^^ 
 
 = 8 — ^, for ^< 8. 
 
ASTRONOMY. Ixix 
 
 For logarithmic application of (4) we may write 
 
 nt^ = cos (^ cos 8, ti^ = sin^ \(^ ~ 8), 
 
 tan iV= — sin \ i, (5) 
 
 n 
 
 ^^"^^ = c-5FiV^=iIO^^i"i^- 
 
 c. Declination and hour angle in terms of altitude and azimuth. 
 
 The fundamental relations for this case are 
 
 sin 8 = sin (^ sin h. — cos ^ cos h cos A, 
 cos S cos t = cos (^ sin ^ -|- sin <^ cos h cos A, (i) 
 
 cos 8 sin / = cos h sin A. 
 
 For logarithmic computation by means of an auxiliary angle M one may write 
 
 m sin M=:. cos /^ cos A, tanM^ cot /^ cos A, 
 
 m cos Jf = sin h, 
 
 ■ 5. • / 1 Tiir\ i J tan A sin Jlf / n 
 
 sm 8 = »z sm (d> — M), tan / := -. ^-^, (2) 
 
 ^^ ^' cos (^ — il/) ^ ^ 
 
 cos S cos / = »z cos (^ — ilf), 
 
 cos S sin / ^ cos A sin A, tan 8 =: tan (<^ — M) cos /. 
 
 d. Hour angle and azimuth in terms of zenith distance. 
 
 cos z — sin (/> sin 8 
 
 cos i ■■ 
 
 cos <j> cos 8 
 
 ^^^, sjn0r-^)_cos^c^)^ ^=1(^ + 8 + ^). 
 
 cos o- cos ((T — z) 
 
 . sin <i cos — sin 8 
 
 cos A = ; . 
 
 cos ^ sin z 
 
 tanH^= '^"^''~'^^T^%7^^ ' <7 = H-^+S + ^)- 
 cos o- sm (a- — 0; 
 
 e. Formulas for parallactic angle. 
 
 cos z = sin 8 sin (^ -{- cos 8 cos <j) cos (, 
 sin cos ^ = cos 8 sin <^ — sin 8 cos <^ cos 2^, 
 sin 2 sin ^ = cos <^ sin /, / \ 
 
 sin 8 = cos sin (^ -|- sin z cos <^ cos f, 
 cos 8 cos ^ = sin 2 sin <^ -j- cos 2 cos <^ cos A, 
 cos 8 sin ^ = cos <t> sin ^. 
 
Ixx ASTRONOMY. 
 
 The first three of these are adapted to logarithmic computation as follows : — 
 
 n sin N-=- cos <^ cos /, 
 
 n cos N^ sin <^, 
 
 cos 2 = « sin (8 -|- JST), 
 
 sin z cos q=-n cos (8 -\- N), 
 
 sin z sin q ^ cos 4> sin /,• 
 
 whence 
 
 tan iV^^ cot <^ cos t, 
 
 tan if sin iV" , ^ 
 
 tan.sm^ = ^.^^g_^_^y (2) 
 
 tan cos f = cot (S + ■^)- 
 
 A similar adaptation results for the last three of equations (i) by interchanging 
 S and z. The equations (2) give both z and q in terms of ^, 8, and t, without 
 ambiguity, since tan z is positive for stars above the horizon. 
 
 If A, z, and q are all required from <^, 8, and /, they are best given by the 
 Gaussian relations 
 
 sin J z sin ^{A + ^) = sin ^t cos i(<^ -|- 8), 
 
 sin J z cos ^{A -\- q)^cos^ t sin J(<^ — 8), ,-. 
 
 cos ^ 2 sin i(^ — ^) ^ sin ^ t sin J(<^ -|- S), 
 
 cos J ^ cos ^A — q) = cos J ^ cos i(<^ — 8). 
 
 f. Hour angle, azimuth, and zenith distance of a star at elongation. 
 
 In this case the parallactic angle is 90° and the required quantities are given by 
 the formulas 
 
 tan <^ 
 
 '=°'^=tsr8"' 
 
 . cos 8 
 
 sm^= T' (i) 
 
 cos <^ ^ ' 
 
 sin d) 
 
 When all of the quantities t. A, and z axe to be computed the following formulas 
 are more advantageous : — 
 
 JP = sin (8 4- ^) sin (8 - <^), 
 
 sm i? ^ 1 — ■ — -^> cos A = 71 sm z = -: — s> (2) 
 
 cos ^ sm 8 cos <^ sm 8 ^ ' 
 
 . . -^ . ^ cos 8 ^ 
 
 tan / = -T-— 7-— — -sj ta.nA= — ^^j tan 2:=: 
 
 sinc^cosS >-<i"^— ^' '•''"^— sin^ 
 
 g. Hour angle, zenith distance, and parallactic angle for transit of a 
 star across prime vertical. 
 
 In this case the azimuth angle is 90° and the required quantities are given by 
 the formulas 
 
COS t 
 
 
 tan 8 
 tan ^' 
 
 COS z 
 
 — 
 
 sin 
 sm 
 
 8 
 
 
 
 cos 
 
 -^. 
 
 
 ^■^ 
 
 cos 
 
 8' 
 
 Ixxi 
 
 (0 
 
 (2) 
 
 (3) 
 
 or, if all of them are to be computed, by the formulas 
 
 K^ = sin (</, + 8) sin (^ - 8), 
 
 • , ^ ■ ^ K 
 
 "^'^'=sin<^cosS' "^"^ = iiM.' <=°s? = S5?S' 
 
 . . -^ . K cos ^ 
 
 tan t = T — : — 5' tan z = —. — =? tan a = — j^- 
 
 cos </) sm 8 sm 8 ^ K 
 
 For special accuracy the following group will be preferred : — 
 
 sin U, — 8) 
 
 tanH^^ r^ll^V 
 =« tan \{^ -\- 8) 
 
 tan'-= (45° - i ?) = tan i(^ + 8) tan \{^ - 8). 
 
 h. Hour angle and azimuth of a star Tvhen in the horizon, or at the 
 time of rising or setting. 
 
 In this case the zenith distance of the star is 90°, and the required quantities 
 
 are given by 
 
 cos t ^ — tan <^ tan 8, 
 
 sin 8 
 
 cos A^ — 7 ; 
 
 cos <p ' 
 
 or by 
 
 tan' 1 i ^°" ('^ - ^) 
 tan 5'— cos (^ + 8)' 
 
 tan''i^- ^^"^^9°°~'^ + ^> . 
 tan ^ ^ _ ^^^ j|.^^o _ ^ _ gy 
 
 On account of refraction, the values of t and A given by these formulas are 
 subject to the following corrections, to wit : — 
 
 R , tan <^ ^ 
 
 cos <^ cos o sm t sm yj ' 
 
 where ^ is the refraction in the horizon. Thus the actual values of the hour 
 angle and azimuth at the time of rising or setting of a star are 
 
 / + A^ and ^ + A^. 
 
Ixxii ASTRONOMY. 
 
 i. Differential formulas. 
 
 The general differential relations for the altitude and azimuth and the declina- 
 tion and hour angle systems of coordinates are : — 
 
 tiz z=— cos q dh -\- sin g cos h dt -\- cos A d(f>, , •. 
 
 sin z dA^ sin q dZ -\- cos q cos I dt — cos z sin A d^i. 
 
 ^S = — cos q dz-\- sin q sin z dA A^ cos t dcj>, / . 
 
 cos 8 d(=^ sin q dz -\- cos ^ sin z ^^ + sin 8 sin / (/^. 
 
 The following values derived from (i) are of interest as showing the dependence 
 of z and ^ on ^ in special cases : — 
 
 m (^) 
 
 cos 8 
 
 For a star in the meridian = o, = rir-i' 
 
 bin z 
 
 For a star in the prime vertical = cos ^, = sin <^, 
 For a star at elongation = cos 8, = o. 
 
 3. Relations of Different Kinds of Time used in Astronomy. 
 a. The sidereal and solar days. 
 
 The sidereal day is the interval between two successive transits of the vernal 
 equinox over the same meridian. The sidereal time at any instant is the hour 
 angle of the vernal equinox reckoned from the meridian towards the west from o 
 to 24 hours. The sidereal time at any place is o when the vernal equinox is in 
 the meridian of that place. 
 
 The solar day is the interval between two successive transits of the sun across 
 any meridian ; and the solar time at any instant is the hour angle of the sun at , 
 that instant. The solar day begins at any place when the sun is in the meridian 
 of that place. 
 
 The mean solar day is the interval between two successive transits over the 
 same meridian of a fictitious sun, called the mean sun, which is assumed to move 
 uniformly in the equator at such a rate that it returns to the vernal equinox at 
 the same instant with the actual sun. 
 
 Time reckoned with respect to the actual sun is called apparent time, while 
 that reckoned with respect to the mean sun is called mean time. The difference 
 between apparent and mean time, which amounts at most to about 16°*, is called 
 the equation of time. This quantity is given for every day in the year in 
 ephemerides. 
 
 The sidereal time when a star or other object crosses the meridian is called the 
 right ascension of the object. The right ascension of the mean sun is also called 
 the sidereal time of mean noon. This time is given for every day in the year in 
 ephemerides for particular meridians, and can be found for any meridian by allow- 
 ing for the difference in longitude. 
 
 The time to which ephemerides and most astronomical calculations are referred 
 
ASTRONOMY. Ixxiii 
 
 is the solar day, beginning at noon, and divided to hours numbered continuously 
 from o* to 24*. This is called astronomical time ; and such a day is called the 
 astronomical day. It begins, therefore, 12 hours later than the civil day. 
 
 b. Relation of apparent and mean time. 
 
 A = apparent time = hour angle of real sun, 
 M = mean time ^ hour angle of mean sun, 
 jE = equation of time. 
 
 M=A-\-£. 
 
 In the use of this relation, £ may be most conveniently derived (by interpola- 
 tion for the place of observation) from an ephemeris. 
 
 c. Relation of sidereal and mean solar intervals of time. 
 
 /= interval of mean solar time, 
 /' = corresponding interval in sidereal time, 
 
 r = the ratio of the tropical year expressed in sidereal days to the tropical 
 year expressed in mean solar days 
 
 366.2422 _ Q 
 
 = ^2 2. — = 1.002738. 
 
 365.2422 
 
 l'=rI—/-\-(r- 1)7=7+0.002738 7 
 7= r-' 7' = 7' - (i - r-') /' = /'- 0.002730 7'. 
 
 Tables for making such calculations are usually given in ephemerides (see, for 
 example, the American Ephemeris). Short tables for this purpose are Tables 
 34 and 35 of this volume. 
 
 Frequent reference is made to the relations 
 
 24* sidereal time = 23* 56" 04.'o9i solar time, 
 24'* mean time = 24* 03"" 56.^555 sidereal time. 
 
 d. Intercohversion of sidereal and mean solar time. 
 
 T„ = mean time at any place, 
 T, = corresponding sidereal time, 
 
 = right ascension of meridian of the place, 
 A = right ascension of mean sun for place and date, 
 
 = sidereal time of mean noon for place and date. 
 
 T,= A -\- Tm expressed in sidereal time. 
 
 7;, = (71 — A) expressed in mean time. 
 
 The quantity A is given in the ephemerides for particular meridians, and can 
 be found by interpolation for any meridian whose longitude with respect to the 
 meridian of the ephemeris is known. The formulas assume that A is taken out 
 of the ephemeris for the next preceding mean noon. 
 
Ixxiv ASTRONOMY. 
 
 e. Relation of sidereal time to the right ascension and hour angle 
 
 of a star. 
 
 T, = sidereal time at any place, 
 
 = right ascension of the meridian of the place, 
 a = right ascension of a star, 
 t := the hour angle of the star at the time 7J. 
 
 T, = a -\- t, f^T, — a. 
 
 4. Determination of Time. 
 a. By meridian transits. 
 
 A determination of time consists in finding the correction to the clock, chro- 
 nometer, or watch used to record time. If 7J denote the true time at any place 
 of an event, T the corresponding observed clock time, and AT' the clock correc- 
 tion. 
 
 To = r+ AT. 
 
 The simplest way to determine the clock correction is to observe the transit of 
 a star, whose right ascension is known, across the meridian. In this case the 
 true time TJ = a, the right ascension of the star ; and if T' is the observed clock 
 time of the transit, 
 
 AT=a — T. 
 
 Meridian transits of stars may be observed by means of a theodolite or transit 
 instrument mounted so that its telescope describes the meridian when rotated 
 about its horizontal axis. The meridian transit instrument is specially designed 
 for this purpose, and affords the most precise method of determining time.* 
 
 Since it is impossible to place the telescope of such an instrument exactly in 
 the meridian, it is essential in precise work to determine certain constants, which 
 define this defect of adjustment, along with the clock correction. These con- 
 stants are the azimuth of the telescope when in the horizon, the inclination of the 
 horizontal axis of the telescope, and the error of collimation of the telescope.t 
 
 Let 
 
 a = azimuth constant, 
 
 6 = inclination or level constant, 
 
 c = collimation constant. 
 
 a is considered plus when the instrument points east of south ; i is plus when 
 the west end of the rotation axis is the higher ; and ( is intrinsically plus when 
 the star observed crosses the thread (or threads) too soon from lack of collima- 
 tion. (The latter constant is generally referred to the clamp or circle on the 
 horizontal axis of the instrument.) 
 
 * The best treatise on the theory and use of this instrument is to be found in Chauvenet's 
 Manual of Spherical and Practical Astronomy, which should be consulted by one desiring to go 
 into the details of the subject. 
 
 t Other equivalent constants may be used, but those given are most commonly employed. 
 
ASTRONOMY, Ixxv 
 
 Also let 
 
 <t> = latitude of the place, 
 8 = declination of star observed, 
 a =: right ascension of star observed, 
 T= observed clock time of star's transit, 
 ^^= the clock correction at an assumed epoch T„, 
 r = the rate of the clock, or other timepiece, 
 
 A sin (d) — 8) 
 
 cos 8 ~ ^^'^ " azimuth factor," 
 
 J, cos ((i — 8) 
 ~ cos 8 ~ *^ " '^^^^ factor," 
 
 C=^ cos^ ^^ ^^ " collimation factor." 
 
 Then, when a, b, e are small (conveniently less than lo" each, and in ordinary 
 practice less than i' each), 
 
 T^ Ar+ Aa -\- Bb ^ Cc -\- r {T - To) = a. 
 
 This is known as Mayer's formula for the computation of time from star transits. 
 The quantity Bb is generally observed directly with a striding level. Assuming 
 it to be known and combined with T, the above equation gives 
 
 AT-\-Aa-{-a-\-r(T-ro) = a-T. (i) 
 
 This equation involves four unknown quantities, AT, a, c, and r; so that in 
 general it will be essential to observe at least four different stars in order to get 
 the objective quantity AT. Where great precision is not needed, the effect of the 
 rate, for short intervals of time, may be ignored, and the collimation c may be 
 rendered insignificant by adjustment. Then the equation (i) is simplified in 
 
 AT-{- Aa = a— T (2) 
 
 This shows that observations of two stars of different declinations will suffice to 
 give AT. Since the factor A is plus for stars south of the zenith (in north lati- 
 tude) and minus for stars north of the zenith, if stars be so chosen as to make the 
 two values of A equal numerically but of opposite signs, AT'will result from the 
 mean of two equations of the form (2). With good instrumental adjustments 
 (b and c small), this simple sort of observation with a theodolite will give AT to 
 the nearest second. 
 
 A still better plan for approximate determination of time is to observe a pair of 
 north and south stars as above, and then reverse the telescope and observe an- 
 other pair similarly situated, since the remaining error of collimation will be partly 
 if not wholly eliminated. Indeed, a well selected and well observed set of four 
 stars will give the error of the timepiece used within a half second or less. This 
 method is especially available to geographers who may desire such an approxi- 
 mate value of the timepiece correction for use in determining azimuth. It will 
 suflOice in the application of the method to set up the instrument (theodolite or tran- 
 sit) in the vertical plane of Polaris, which is always close enough to the meridian. 
 The determination will then proceed according to the following programme : — 
 
Ixxvi ASTRONOMY. 
 
 1. Observe time of transit of a star south of zenith, 
 
 2. Observe time of transit of a star north of zenith. 
 
 Reverse telescope, 
 
 3. Observe time of transit of another star south of zenith, 
 
 4. Observe time of transit of another star north of zenith. 
 
 Each star observation will give an equation of the form (i), and the mean of 
 the four resulting equations is 
 
 ^44 4 4 
 
 Now the coefficient of r in this equation may be always made zero by taking 
 for the epoch 7J the mean of the observed times T. Likewise, ^A and %C may 
 be made small by suitably selected stars, since two of the A's and C's are positive 
 and two negative. The value \ 2(a — T) is thus always a close approximation to 
 AT" for the epoch T(,^\ "SiT, when %A and 2C approximate to zero. But if these 
 suras are not sraall, approximate values of a and c may be found from the four 
 equations of the form (i), neglecting the rate, and these substituted in the above 
 formula will give all needful precision. 
 
 For refined work, as in determining differences of longitude, several groups of 
 stars are observed, half of them with the telescope in one position and half in the 
 reverse position, and the quantities AT, a, c, and r are computed by the method 
 of least squares. In such work it is always advantageous to select the stars with 
 a view to making the sums of the azimuth and collimation coefficients approxi- 
 mate to zero, since this gives the highest precision and entails the simplest com- 
 putations.* 
 
 b. By a single observed altitude of a star. 
 
 An approximate determination of time, often sufficient for the purposes of the 
 geographer, may be had by observing the altitude or zenith distance of a known 
 star. The method requires also a knowledge of the latitude of the place. 
 Let 
 
 Zj = the observed zenith distance of the star, 
 Ji = thje refraction, 
 
 z = the true zenith distance of the star, 
 = z,-\~J?, 
 a, S, = the right ascension and declination of the star, 
 i = hour angle of star at time of observation, 
 T= observed time when Zi is measured, 
 AT:= correction to timepiece, 
 (j} = latitude of place. 
 
 Then the hour angle / may be computed by 
 
 sin (a- — (h) cos (a- — 8) n / 1 1 l^ 1 \ 
 
 tanHif= ^^ / ^ — ^-^. °' = iw + 8 + '2)- 
 
 ^ cos o- cos (o- — z) ^\r I I y 
 
 * -For details of theory and practice in time work done according to this plan see Bulletin 49, 
 'U. S. Geological Survey. 
 
ASTRONOMY. Ixxvii 
 
 Having the hour angle the clock correction A 7" is given by 
 
 A7'=a + /— T, 
 
 in which all terms must be expressed in the same unit ; /. e., in sidereal or in mean 
 time. 
 
 The refraction R may be taken from Table 31. 
 
 The most advantageous position of the star observed, so far as the effect of an 
 error in the measured quantity z-^ is concerned, is in the prime vertical, but stars 
 near the horizon should be avoided on account of uncertainties in refraction. 
 The least favorable position of the star is in the meridian. 
 
 Compared with the preceding method the present method is inferior in preci- 
 sion, but it is often available when the other cannot be applied. 
 
 c. By equal altitudes of a star. 
 
 This method is an obvious extension of the preceding method, and has the 
 advantage of eliminating the effect of constant instrumental errors in the meas- 
 ured altitudes or zenith distances. Thus it is plain that the mean of the times 
 when a (fixed) star has the same altitude east and west of the meridian, whether 
 one can measure that altitude correctly or not, is the time of meridian transit. 
 
 This method may, therefore, give a good approximation to the timepiece 
 correction when nothing better than an engineer's transit, whose telescope can 
 be clamped, is available. When the instrument has a vertical circle (or when a 
 sextant is used) a series of altitudes may be observed before meridian passage of 
 the star, and a similar series in the reverse order with equal altitudes respectively 
 after meridian passage. The half sums of the times of equal altitudes on the two 
 sides of the meridian will give a series of values for the time of meridian transit 
 from which the precision attained may be inferred. 
 
 This method is frequently applied to the sun, observations being made before 
 and after noon. For the theory of the corrections essential in this case on 
 account of the changing position of the sun, on account of inequalities in the 
 observed altitudes, etc., the reader must be referred to special treatises on prac- 
 tical astronomy.* 
 
 5. Determination of Latitude. 
 
 a. By meridian altitudes. 
 
 The readiest method of determining the latitude of a place is to measure the 
 meridian zenith distance or altitude of a known star. When precision is not re- 
 quired this process is a very simple one, since it is only essential to follow a (fixed) 
 star near the meridian until its altitude is greatest, or zenith distance least. Thus, 
 if the observed zenith distance is ^i, the true zenith distance z, and the refrac- 
 tion R, 
 
 z = Zi-^R; 
 
 * The best work of this kind is Chauvenet's Manual of Spherical and Practical Astronomy- It 
 should be consulted by all persons desiring a knowledge of the details of practical astronomy. 
 
Ixxviii ASTRONOMY. 
 
 and if the declination of the star is 8 and the latitude of the place ^, 
 
 according as the star is south or north of the zenith. 
 
 A more accurate application of the same principle is to observe the altitudes 
 of a circumpolar star at upper and lower culmination (above and below the pole). 
 The mean of these altitudes, corrected for refraction, is the latitude of the place. 
 This process, it will be observed, does not require a knowledge of the star's 
 declination. 
 
 b. By the measured altitude of a star at a known time. 
 
 A = measured altitude corrected for refraction, 
 Ts = observed sidereal time, 
 a, 8 ^ right ascension and declination of star, 
 f = hour angle of star, 
 4> = latitude of place. 
 
 Then <f) may be computed by means of the following formulas : — 
 
 ^ = 7] — a, 
 
 , o tan 8 sin ^ sin B 
 
 tan ^ = ■ -, cos y = . . ^ , 
 
 cos r sm 8 
 
 <t> = l3±y. 
 
 In the application of these p may be taken numerically less than 90°, and since 
 t may also be taken less than 90°, yS may be taken with the same sign as 8. y is 
 indeterminate as to sign analytically, but whether it should be taken as positive 
 or negative can be decided in general by an approximate knowledge of the lati- 
 tude, which is always had except in localities near the equator. 
 
 The most advantageous position of a star in determining latitude by this 
 method is in the meridian, and the least advantageous in the prime vertical. 
 When a series of observations on the same star is made, they should be equally 
 distributed about the meridian ; and when more than one star is observed it is 
 advantageous to observe equal numbers of them on the north and south of the 
 zenith. 
 
 The application of this method to the pole star is especially well adapted to 
 the means available to the geographer and engineer, namely, a good theodolite 
 and a good timepiece. In this case the following simple formula for the latitude 
 may be used : — 
 
 (j>^= h — p cos / + \p'^ sin i" sin^ t tan h, 
 
 where/ is the polar distance of Polaris in seconds (about 5400"), and the other 
 symbols have the same meaning as defined above. Tables giving the logarithms 
 of/ and \p^ sin i" are published in the American Ephemeris. 
 
ASTRONOMY. Ixxix 
 
 c. By the zenith telescope. 
 
 The zenith telescope furnishes the most precise means known for the deter- 
 mination of the latitude of a place. For the theory of the instrument and method 
 when applied to refined work the reader must be referred to special treatises.* 
 It will suffice here to state the principle of the method, which may sometimes be 
 advantageously applied by the geographer. Let z^ be the meridian zenith distance 
 of a star south of the zenith, and z^ the meridian zenith distance of another star 
 north of the zenith. Let 8, and 8„ denote the declinations of these stars respec- 
 tively. Then 
 
 «« = <)!> — 8« 
 
 K^^K — ^, 
 whence 
 
 •^ = i (8. + S„) + i {z, - ^„). 
 
 It appears, therefore, that this method requires only that the difference {z^ — 2„) 
 be measured. Herein lies the advantage of the method, since that difference 
 may be made small by a suitable selection of pairs of stars. With the zenith 
 telescope the stars are so chosen that the difference (z^ — z„) may be measured by 
 means of a micrometer in the telescope. 
 
 The essential principles and advantages of this method may be realized also 
 with a theodolite, or other telescope, to which a vertical circle is attached, the 
 difference {z, — z„) being measured on the circle ; and a determination of latitude 
 within s'' or less is thus easy with small theodolites of the best class {i. e., with 
 those whose circles read to lo" or less by opposite verniers or microscopes). 
 
 6. Determination of Azimuth. 
 a. By observation of a star at a known time. 
 
 Tg = sidereal time of observation, 
 a, 8 :^ right ascension and declination of star observed, 
 t :=: hour angle of star, 
 
 (j> = latitude of place, 
 
 A = azimuth of the star at the time 7] counted from the south around by the 
 west through 360°. 
 
 The azimuth A may be computed by the formulas 
 
 a = sec <^ cot 8, 3 = tan ^ cot 8, 
 
 a sin / (i) 
 
 tan A =^ — 
 
 I — d cos i 
 
 The angle A will fall in the same semicircle as f, and A is thus determined by its 
 tangent without ambiguity. The quantities a and i will be sensibly constant for 
 
 * Among which Chauvenet's Manual of Spherical and Practical Astronomy is the best. 
 
Ixxx ASTRONOMY, 
 
 a given star and date ; and hence they need be computed but once for a series of 
 observations on the same star on one date. 
 
 The effects of small errors A/, A^, and AS in the assumed time, latitude, and 
 declination are expressed by 
 
 cos S cos ^ ^ ^ . . . . , sin ^ 
 
 : AA — sm A cot z Ad, -: — - AS, 
 
 sin z ' ^' sin a ' 
 
 respectively, where z and q are the zenith distance and parallactic angle of the 
 star. Hence the effect of A/ will vanish for a star at elongation ; the effect of 
 A(^ vanishes for a star in the meridian, and is always small (in middle latitudes) 
 for a close circumpolar star ; the effect of AS vanishes for a star in the meridian. 
 It appears advantageous, therefore, to observe for azimuth (in middle latitudes) 
 close circumpolar stars at elongations, since the effect of the time error is then 
 least, and the effects of errors in the latitude and declination are small and may 
 be eliminated entirely by observing the same star at both elongations. 
 
 The hour angle t^, the azimuth A^ and the altitude h^ of a star at elongation 
 are given by the formulas (2) of section 2,f. Those best suited to the purpose 
 
 are 
 
 K"" = sin (8 + <^) sin (8 - ^, 
 
 K , cos S , sin <i (2\ 
 
 ^^"^' = sin.^cosS ' tan^. = ^^, tan /J. = -^- W 
 
 Knowing the time of elongation of a close circumpolar star, it suffices for many 
 purposes to observe the angle between the star and some reference terrestrial 
 mark at or in the vicinity of that time. 
 
 For precise determinations of azimuth it is customary to observe a star near 
 its elongation repeatedly, thus obtaining a series of results whose mean will be 
 sensibly free from errors of observation and errors due to instrumental defects. 
 
 The computation of the azimuth A may be made accurately in all cases by the 
 formulas (i) ; but when a close circumpolar star is observed near elongation, it 
 may be more convenient to use the following formulas : — 
 
 A/ = (/ — Q, or the interval before or after elongation at the time of 
 observation, 
 l^A ^{A — A^, or the difference in azimuths of the star at the time 
 
 of elongation and at the time of observation, (3) 
 
 _ M^ sin 8 cos S _(x5lL . f 8 cos 8 
 
 2 f sin /„ cos <^ ^ '2 {p y sm 4 tan 4 cos ^ '• ' 
 
 * To the same order of approximation one may write in the first term of this expression 
 ^(..)^ = /'2sinHA.= ^|^'. 
 
 which latter is the most convenient form when tables giving log ^ — 77 — - for the argument &t 
 
 in time are at hand. Such tables are given in Chauvenet's Manual of Spherical and Practical 
 Astronomy (for full title see p. Ixxxii), and in Formeln und HUlfstafeln fUr Geographische Orts- 
 bestimmungen, von Dr. Th. Albrecht. Leipzig: Wilhelm Engelmann, 4to, 2d ed., 1879. 
 
ASTRONOMY. Ixxxi 
 
 This last formula gives AA in seconds of arc when A# is expressed in seconds 
 of time ; At is considered positive in all cases (in the use of the formula), and 
 with this convention the positive sign is used when the star is between either 
 elongation and upper culmination, and the negative sign when the star is between 
 either elongation and lower culmination. For a given star, place, and date the 
 coefficients of (Af^ and (A^)' will be sensibly constant and their logarithms will 
 thus be constant for a series of observations of a star on any date. By reason of 
 the large factors (p" = 206 264."8y and tan 4 in the denominator of the second 
 term, it will be very small unless Ai" is large. Hence this term may often be 
 neglected. Using both terms, the formula will give AA for Polaris to the nearest 
 o."oi when Af < 40'" and when observations are made in middle latitudes. 
 
 By reference to formulas (2) of section 2, _/] it is seen that 
 
 sin 8 cos S sin'' S cos 8 
 
 sin 4 cos <f) ^ ' 
 
 sin 8 cos S sin'' 8 CDs'* 8 sin ^ 
 
 sin 4 tan 4 cos <f> X^ ' 
 
 K^ = sin (8 4- 4,) sin (8 - <^).* 
 
 b. By an observed altitude of a star. 
 
 k = true altitude of star observed ; t. e., the observed altitude less the refrac- 
 tion, 
 ^ = latitude of place, 
 p = polar distance of star, 
 A = azimuth of star. 
 
 tan^ hA = sin ("• - <t>) sin (a- — h) 
 cos <j cos (o- — /) ' 
 
 The most advantageous position of the star, on account of possible error in the 
 .observed value of h, is that in which sin ^ is a maximum. This position is then 
 at elongation for stars which elongate, in the prime vertical for stars which cross 
 this great circle, and in the horizon for a star which neither elongates nor crosses 
 the prime vertical. A star will elongate when p < go° — ^ ; it will cross the 
 prime vertical when/ lies between 90° — <^ and 90° ; and it will neither elongate 
 nor cross the prime vertical when/ >90°, or when the declination (S) of the star 
 is negative. 
 
 0. By equal altitudes of a star. 
 
 By this method, when a fixed star is observed first east of the meridian and 
 then west of the meridian at the same altitude, the direction of the meridian will 
 
 * In precise work the computed azimuth requires the following correction for daily aberration, 
 
 namely : — 
 
 cos ^ 
 A/i = — o."-',z —■ — cos A, 
 •J sm 3 ' 
 
 -where A is to be reckoned from the south by way of the west through 360°. 
 
Ixxxii ASTRONOMY. 
 
 obviously be given by the mean of the azimuth circle readings for the two 
 observed directions. This process will thus give the direction of the meridian 
 free from the effect of any instrumental errors common to the equal altitudes 
 observed. Neither does it require any knowledge of the star's position (right 
 ascension and declination). It is therefore available to one provided with no- 
 thing but an instrument for measuring altitudes and azimuths, and is susceptible 
 of considerable precision when a series of such equal altitudes is carefully referred 
 to a terrestrial mark. 
 
 When the sun is observed, it is essential to take account of its change in 
 declination between the first and the second observation. Let A-^ and A^ be the 
 true azimuths counted from the meridian toward the east and west respectively 
 at the times t^ and t^ of the two observations. Also, let AS be the increase in 
 declination of the sun in the interval (4 — A). Then 
 
 An A-\ 
 
 COS tj) sin ^(/'a — /i) 
 
 Calling the azimuth circle readings for the east and west observations H^ and Ji^ 
 respectively, the resulting azimuths are 
 
 A, = i(J?, - J?,) - i(A, - A,), 
 A, = i(Ji, - Ry) + ^{A, - A,). 
 
 References. 
 
 Many excellent treatises on spherical and practical astronomy are available. 
 Among these the most complete are the following : — 
 
 " A Manual of Spherical and Practical Astronomy," by William Chauvenet. 
 Philadelphia : J. B. Lippincott & Co., 2 vols., 8vo, sth ed., 1887. " A Treatise 
 on Practical Astronomy, as applied to Astronomy and Geodesy," by C. L. Doo- 
 little. New York: John Wiley & Sons, 8vo, 2d ed., 1888. "Lehrbuch der 
 Spharischen Astronomie," von F. Briinnow. Berlin : Fred. Diimler, 8vo, 1851. 
 " Spherical Astronomy," by F. Briinnow. Translated by the author from the 
 second German edition. London : Asher & Co., 8vo, 1865. 
 
THEORY OF ERRORS. 
 
 I. Laws of Error. 
 
 The theory of errors is that branch of mathematical science which considers the 
 nature and extent of errors in derived quantities due to errors in the data on 
 which such quantities depend. A law of error is a relation between the magni- 
 tude of an error and the probability of its occurrence. The simplest case of a 
 law of error is that in which all possible errors (in the system of errors) are 
 equally likely to occur. An example of such a case is had in the errors of 
 tabular logarithms, natural trigonometric functions, etc. ; all errors from zero to 
 a half unit in the last tabular place being equally likely to occur. 
 
 When quantities subject to errors following simple laws are combined in any 
 manner, the law of error of the quantity resulting from the combination is in 
 general more complex than that of either component. 
 
 Let e denote the magnitude of any error in a system of errors whose law of 
 error is defined by <j>(e). Then if e vary continuously the probability of its 
 occurrence will be expressed by <f)(/)de. If £ vary continuously between equal 
 positive and negative limits whose magnitude is a, the sum of all the probabili- 
 ties <^(€)i/e must be unity, or 
 
 + <^ 
 
 <^(e) </e = I. 
 
 /* 
 
 For the case of tabular logarithms, etc., alluded to above, <^(e) = f , a constant 
 whose value is 1/(2 a) = i, since a = 0.5. 
 
 For the case of a logarithm interpolated between two consecutive tabular 
 values, by the formula v ^Vi-{- (v^ — v^ t ^= v^ {1 — t) -\- v^ t, where Vi and 
 V2 are the tabular values, and t the interval between v^ and the derived value 
 V, <^(e) has the following remarkable forms when the extra decimals (practically 
 the first of them) in (v^ — v^ t are retained : — 
 
 </>(e) = , _f\ f fo' values of e between — ^ and — (i — i), 
 
 = for values of e between — (i — ^) and + (i — 0> (i) 
 
 = y-^ — J— for values of e between + (i — ^^^ + h 
 
Ixxxiv THEORY OF ERRORS. 
 
 It thus appears that <^(e) in this case is represented by the upper base and the 
 two sides of a trapezoid. 
 
 When, as is usually the practice, the quantity (z'j — v^ f is rounded to the 
 nearest unit of the last tabular place, ^(c) becomes more complex, but is still 
 represented by a series of straight lines. It is worthy of remark that the latter 
 species of interpolated value is considerably less precise than the former, wherein 
 an additional figure beyond the last tabular place is retained. 
 
 When an infinite number of infinitesimal errors, each subject to the law of con- 
 stant probability and each as likely to be positive as negative, are combined by 
 addition, the law of the resultant error is of remarkable simplicity and generality. 
 It is expressed by 
 
 ■where e is the Napierian base, tt = 3.14159 -|-, and >^ is a constant dependent on 
 the relative magnitude of the errors in the system. This is the law of error of 
 least squares. It is the law followed more or less closely by most species 
 of observational errors. Its general use is justified by experience rather than 
 by mathematical deduction. 
 
 a. Probable, mean, and average errors. 
 
 For the purposes of comparison of different systems of errors following the 
 same law, three different terms are in use. These are Hc^e. probable error* or that 
 error in the system which is as likely to be exceeded as not ; the mean error, or 
 that error which is the square root of the mean of the squares of all errors in the 
 system ; and the average error, which is the average, regardless of sign, of all 
 errors in the system. Denote these errors by e^, e„, £„, respectively. Then in all 
 systems in which positive and negative errors of equal magnitude are equally 
 likely to occur, and in which the limits of error are denoted by — a and -}- a, the 
 analytical definitions of the probable, mean, and average errors are : — 
 
 — fp o -f €p -\-a 
 
 J.^(e) d^ = J</,0 d. = f<l>(e) d. =: j'4,(.) de = i, 
 
 — a — Cp o -{- fp 
 
 _l_ ^ ^^^ 
 
 ^m = j*<^0 e" A €„ = jf.^(£) c d,. 
 
 * The reader should observe that the word probable is here used in a specially technical sense. 
 Thus, the probable error is not " the most probable error," nor " the most probable value of the 
 actual error," etc., as commonly interpreted. 
 
THEORY OF ERRORS. IxXXV 
 
 b. Probable, mean, average, and maximum actual errors of interpo- 
 lated logarithms, trigonometric functions, etc. 
 
 When values of logarithms, etc., are interpolated from numerical tables by means 
 of first differences, as explained above, the probable and other errors depend on 
 the magnitude of the interpolating factor. Thus, the interpolated value is 
 
 V = Vx-\-(V2 — Vi) t 
 
 where v-^ and v^ are consecutive tabular values and t is the interpolating factor. 
 
 For the species of interpolated value wherein the quantity iy^ — v^ t is not 
 rounded to the nearest unit of the last tabular place (or wherein the next figure 
 beyond that place is retained) the maximum possible actual error is 0.5 of a unit 
 of the last tabular place, and formulas (i) and (3) show that the probable, mean, 
 and average errors are given by the following expressions : — 
 
 «j, := J (i — i) for t between o and \, 
 
 ^\ — ]e *^'2.t (i — for i between \ and §, 
 = i / for ^ between f and i. 
 
 [ - (i - 2 /)^ h 
 96 (i - ^) ^ \ 
 
 I _ (l _ 2/)» ^ ^ 
 
 ^'' ^^ (x — {\ 't ""^ ^ between o and \, 
 
 I_(2/_i)8 
 
 = • — -/ Ts-f- for t between \ and i. 
 
 24 (i — t) t ^ 
 
 It thus appears that the probable error of an interpolated value of the species 
 under consideration decreases from 0.25 to 0.15 of a unit of the last tabular place 
 as t increases from o to 0.5. Hence such interpolated values are more precise 
 than tabular values. 
 
 For the species of interpolated values ordinarily used, wherein {v^ — v^ t is 
 rounded to the nearest unit of the last tabular place, the probable, mean, and 
 average errors are greater than the corresponding errors for tabular values. The 
 laws of error for this ordinary species of interpolated value are similar to but in 
 general more complex than those defined by equations (i). It must suffice here 
 to give the practical results which fiow from these laws for special values of the 
 interpolating factor t.* The following table gives the probable, mean, average, 
 and maximum actual error of such interpolated values for /^ i, |^, J, . . . i>jj. It 
 will be observed that / = i corresponds to a tabular value. 
 
 * For the theory of the errors of this species of interpolated values see Annals of Mathematics, 
 vol. ii. pp. 54-59. 
 
IxXJCVi THEORY OF ERRORS. 
 
 Characteristic Errors of Interpolated Logarithms, etc. 
 
 Interpolating 
 factor 
 
 Probable 
 error 
 
 Mean 
 error 
 
 Average 
 error 
 
 Maximum actual 
 
 t 
 
 «P 
 
 «m 
 
 «a 
 
 error 
 
 I 
 
 0.250 
 
 0.289 
 
 0.250 
 
 \ 
 
 \ 
 
 .292 
 
 .408 
 
 •333 
 
 I 
 
 \ 
 
 .256 
 
 •347 
 
 .287 
 
 # 
 
 4 
 
 .276 
 
 .382 
 
 •313 
 
 I 
 
 \ 
 
 .268 
 
 •370 
 
 ■Z'^-i 
 
 ^ 
 
 \ 
 
 .277 
 
 •38s 
 
 ■z-^^ 
 
 I 
 
 \ 
 
 .274 
 
 .380 
 
 •311 
 
 \\ 
 
 \ 
 
 .279 
 
 •389 
 
 .318 
 
 I 
 
 \ 
 
 .278 
 
 .386 
 
 •316 
 
 \\ 
 
 tV 
 
 .281 
 
 •392 
 
 .320 
 
 I 
 
 2. The Method of Least Squares. 
 
 a. General statement of method. 
 
 When the errors to which observed quantities are subject follow the law ex- 
 pressed by 
 
 a unique method results for the computation of the most probable values of the 
 observed quantities and of quantities dependent on the observed quantities. The 
 method requires that the sum of the weighted squares of the corrections to the 
 observed quantities shall be a minimum,* subject to whatever theoretical condi- 
 tions the corrections must satisfy. These conditions are of two kinds, namely, 
 those expressing relations between the corrections only, and those expressing 
 relations between the corrections and other unknown quantities whose values are 
 disposable in determining the minimum. A familiar illustration of the first class 
 of conditions is presented by the case of a triangle each of whpse angles is mea- 
 sured, the condition being that the sum of the corrections is a constant. An 
 equally familiar illustration of the second class of conditions is found in the case 
 where the sum and difference of two unknown quantities are separately observed ; 
 in this case the two unknowns are to be found along with the corrections. 
 
 Mathematically, the general problem of least squares may be stated in two 
 
 * Hence the term least squares. 
 
THEORY OF ERRORS. Ixxxvii 
 
 equations. Thus, let x,y,z,. . . be the observed quantities with weights /, ^, 
 r, . . , . Let the corrections to the observed quantities be denoted by A^t, Aj, 
 Az, . . . ; so that the corrected quantities are x -|- Ax, y -\- ^y, z-\-^z, . . . . Let 
 the disposable quantities whose values are to be determined along with the correc- 
 tions be denoted by ^, i;, ^ Then, the theoretical conditions which must be 
 
 satisfied by « -j" ^^t y 4" ^y> ^ "t~ ^■^j • • • ^"^ by f , 17, 4 • • • may be symbolized 
 by 
 
 ■Pn (i,V,C---x-\- Ax, y -\- Ay, z -\- Az, . . .) = o. (4) 
 
 Subject to the conditions specified by the n equations (4), we must also have 
 
 / (A^)" -|- ^ (Ayy -\- r {Azf + . . . = a minimum (5) 
 
 = u, say. 
 
 Equations (4) and (5) contain the solution of every problem of adjustment by 
 the method of least squares. Two examples may suffice to illustrate their use. 
 
 First, take the case of the observed angles of a triangle alluded to above. 
 Calling the observed angles x, y, z, we have 
 
 x-\-Ax-\-y-\-Ay-\-z-\-Az^ 180° -f- spherical excess, 
 or 
 
 Ax -\- Ay -\- Az'= 180° -|- spherical excess — (x -\- y -\- z) 
 = c, say. 
 
 This is the only condition of the form (4). The problem is completely stated, 
 then, in the two equations 
 
 Ax -\- Ay-\- Az = c 
 p {Aocf -\- q {Ayf -(- r (Azf = a min. =: u. 
 
 To solve this problem the simplest mode of procedure is to eliminate one of the 
 corrections by means of the first equation and then make u a minimum. Thus, 
 eliminating Az, there results 
 
 u =/ (Axf + q (Ayy -\-r(c-Ax- Ayf. 
 
 The conditions for a minimum of u are : — 
 
 3U /^ I N A I A 
 
 
 
 
 9ax ' 
 
 — V 1 ' y —- V 1 ' — ^ 
 
 
 — ") 
 
 
 
 
 
 du 
 9Ay ' 
 
 = rAx-{-(^-{-r)Ay- 
 
 — re 
 
 = 0; 
 
 
 and these 
 
 give. 
 
 in 
 
 connection with the value Az = 
 
 --C- 
 
 ■ Ax- 
 
 ■Ay, 
 
 where 
 
 
 
 Ax = 
 
 :f, Ay=Q, 
 
 P q 
 Q— ' 
 
 Az: 
 
 r' 
 
 
 
 p^ q^ r 
 
 
 When the weights are equal, or when p ^ q = r, the corrections are — 
 
 Ax = Aj/ = A^ = J f. 
 
Ixxxviii THEORY OF ERRORS. 
 
 Secondly, take the case, also alluded to above, of the observed sum and the 
 
 observed difference of two numbers. Denote the numbers by i and rj, the latter 
 
 being the smaller. Let the observed values of the sum (^ -j~ '^) ^^ denoted 
 
 by xi, X2, . . . x^ and their weights pi, A> • • • /m respectively. Likewise, call 
 
 the observed values of the difference {t — rj), y^, y^, . . . y^ and their weights 
 
 qi, q^. . . q^ respectively. Then there will hs. m -\- n equations of the type (4), 
 
 namely : — 
 
 I + 77 — (ari -f Aa;i) = o, 
 
 f -j- 17 — («2 -j- b.x^ ■= o. 
 
 f + 17 — (^».+ AarJ = o, 
 
 i — v — (ji-\- AjCi) = o, 
 ^ - ^ - ( J2 + ^^2) = o. 
 
 (a) 
 
 f - 77 - 0„ + Ay„) = o ; 
 and the minimum equation is 
 
 u = A (A^,)^ + A (\x,y + . . . + ^1 (Ay^y + q, (Ay,y + . . . = a min. (b) 
 The equations of group (a) give 
 
 Axi =:: S -{- rj — Xi, 
 AX2 = $-\-r] — X2, 
 
 (C) 
 Ayi = i-r, - j/j, 
 Ays, = $ - 7j — y^, 
 
 • ' * > 
 
 and these values in (b) give 
 
 « = A G^ + ^ - ^^r + • ■ • + ?i (f - >? - J'l)^ + • - • (d) 
 
 Thus it appears that all conditions will be satisfied if f and 77 are so determined 
 as to make u in (d) a minimum. Hence, using square brackets to denote sum- 
 mation of like quantities, the values of f and rj must be found from 
 
 || = [/ + ^] f+ |> -^] 77 - |>x + ^^] = o, 
 
 n ^'^ 
 
 5^^ = \j-q-\^+[j, + q-\rj-\jx-qy-\ = o. 
 
 Equations (e) give f and rj, and these substituted in (c) will give the corrections 
 to the observed quantities. 
 
 b. Relation of probable, mean, and average errors. 
 
 The introduction of the law of error (2) in equations (3) furnishes the following 
 relations, when it is assumed that the limits of possible error are —00 and 4-0° '■ 
 
 ^ = 0.6745 e„ = 0.8453 €„. (6) 
 
THEORY OF ERRORS. Ixxxix 
 
 c. Case of a single unknown quantity. 
 
 The case of a single unknown quantity whose observed values are of equal or 
 unequal weight is comprised in the following formulas : — 
 
 Xi, X2, . . . Xn^ observed values of unknown quantity, 
 
 A, A> • • • /m ^ the weights of xi, x^, . . . 
 
 Vi, V2, . . . v^^ most probable corrections to Xi, x^, . . • 
 
 X = most probable value of the unknown quantity, 
 m = the number of independent observations. 
 
 Then the conditional equations (4) are 
 
 X — X\ ^^ Z'^j 
 X — X^ ^2» 
 
 X Xjf^ V^ j 
 
 the minimum equation (5) is 
 
 P\V\ -\-piOi + . . . = \Jv^ = lp{x — x^^ = a min., 
 
 where i=i, 2, . . . m, and 
 
 /i^i -\-piX2 -\- . . .p^x^ _ [px'\ 
 
 ^■~ A+A + ---A [/]■ 
 
 When the weights are equal,/]. =^2 = • • • =A« ^^^ 
 
 m 
 
 or the arithmetic mean of the observed values. 
 
 Weight of a; = [/] when the/V are unequal, 
 ^ m when the/'j are equal. 
 
 \pvv\ 
 Mean error of an observed value of weight unity = y ^_ ^ for unequal weights, 
 
 ^= y _ ■ for equal weights. 
 
 / r pvtii 
 
 Mean error of an observed value of weight/ =: y / _ ^.v ^ for unequal weights. 
 Mean error of x = y ,^ _ j\ r^-i for unequal weights, 
 
 _. 4 / — ISJ — ^ for equal weights. 
 V »z (»« — i) 
 
 The corresponding probable errors are found by multiplying these values by 
 0.6745. See equation (6). 
 
XC THEORY OF ERRORS. 
 
 A formula for the average error sometimes useful is 
 
 \_pv'\ 
 Average error =^ i , _ \ T^ for unequal weights. 
 
 M 
 = "7 — / =^ for equal weights. 
 
 In these the residuals v are all taken with the same sign. A sufficient approxi- 
 
 H 
 
 mation in many cases of equal weights is ^^-^ ; but the above formulas dependent 
 
 on the squares of the residuals are in general more precise. 
 
 An important check on the computation of x is \^pv\ = o ; i. e., the sum of the 
 residuals v, each multiplied by its weight, is zero if the computation is correct. 
 
 d. Case of observed function of several unknown quantities f, 17, ([ . . . . 
 
 A case of frequent occurrence, and one which includes the preceding case, is 
 that in which a function of several unknown quantities is observed. Thus, for 
 example, the observed time of passage of a star across the middle thread of a 
 transit instrument is a function of the azimuth and collimation of the transit 
 instrument and the error of the timepiece used. In cases of this kind the con- 
 ditional equations of the type (4) assume the form 
 
 that is, each of them contains but one observed quantity x along with several 
 disposable (disposable in satisfying the minimum equation) quantities $, r], ^ . . . . 
 
 The process of solution in this case consists in eliminating the corrections 
 Axi, Ax2, . . . from the above conditional equations, substituting their values in 
 the minimum equation (5), and then placing the differential coefficients of u with 
 respect to f, »;, ^ . . . separately equal to zero. There will thus result as many 
 independent equations as there are unknown quantities of the class in which ^, 17, 
 t, . . . fall, the remaining unknown quantities Axi, Aarj, . . . , or the corrections to 
 the observed values, are then found from the conditional equations. 
 
 In many applications it happens that the conditional equations 
 
 F{^, 17, C, • • • x-\- ^x)=zo, 
 
 are not of the linear form. But they may be rendered linear in the following 
 manner. First, eliminate the quantities x -\- J^x from the conditional equations. 
 The result of this elimination may be written 
 
 /fe vA- ■ ■) — X — ^.x = o. 
 Secondly, put 
 
 »; = ■70- 
 
 A'7. 
 
 where fo) Vo, ■ ■ • are approximate values of f, 1;, ... , found in any manner, and 
 Af, At/, ... are corrections thereto. Then supposing the approximate values 
 
THEORY OF ERRORS. xci 
 
 fo, Vo, ■ • • SO close that we may neglect the squares, products, and higher powers 
 of A^, At], . . . , Taylor's series gives 
 
 which is linear with respect to the corrections A^, At/, ... , For brevity, and for 
 the sake of conformity with notation generally used, put 
 
 n = x — /(fo, Vo, &•••); 
 
 V = Ax, 
 
 „-^f ,-^ ,-M_ 
 
 X = Af, Jf = A17, z = A^, . . . . 
 
 Then the conditional equations will assume the form 
 ax -\- by -\- cz -\- . . . — n^v\ 
 and if they are tn in number they may be written individually thus : — 
 
 (a) 
 
 a^ + ^2 + <^2 + . . . — «2 = Z'2, 
 dm -\- K + <^m + • • • — «m = Z'm- 
 
 The minimum equation (5) becomes 
 
 u = [/z;2] = [p(ax -\- by -\- cz -{-... — n)']; 
 
 so that placing ^=-, ^, ^:^, . . . separately equal to zero will give as many 
 
 independent equations as there are values of x,^y, z, . . . . The resulting equa- 
 tions are in the usual (Gaussian) notation of least squares : — 
 
 \J>aa\x -\- [pab']y + \_pac] z -{-... — {pan] = o, 
 
 [pab] -^[/>bb] +[pbc] -\-...-[j>bn]^o, (b) 
 
 [pac] -^[J>bc] -\-U>cc] +..._[/ot] = o, 
 
 The equations (a) are sometimes called observation-equations. The absolute 
 term n is called the observed quantity. It is always equal to the observed quan- 
 tity minus the computed quantity/ (fo, 170,^.. •), which latter is assumed to be 
 free from errors of observation. The term v is called the residual. It is some- 
 times, though quite erroneously, replaced by zero in the equations (a). 
 
 The equations (b) are called normal equations. They are usually formed 
 directly from equations (a) by the following process : Multiply each equation by 
 the coefficient of x and by the weight / of the v in the same equation, and add 
 the products. The result is the first equation of (b), or the normal equation in x. 
 The normal equations in y, z, . . . are found in a similar manner. 
 
XCii THEORY OF ERRORS. 
 
 A noteworthy peculiarity of the normal equations is their symmetry. Hence in 
 forming equations (b) from (a) it is not essential to compute all the coefficients of 
 X, y,z, . . . except in the first equation. 
 
 Checks on the computed values of the numerical terms in the normal equations 
 are found thus : Add the coefficients a, b, c, . . . of x, y, z, . . . in (a) and put 
 
 «i + ^1 + i^i + ■ • ■ = ■fi. 
 
 «2 + ^2 + <^2 + • • • = •S'ZI 
 
 Multiply each of these, first, by its pa ; secondly, by itspb, etc., and then add the 
 products. The results are 
 
 \_paa\ -\- [pad] -|- [pac] -)-... = [pas] 
 [pad] + [pib] + [pic] + . . . = [pl>s] 
 
 These will check the coefficients of x, y, z, . . . in (b). To check the absolute 
 
 terms, multiply each of the above sums by its np, and add the products. The 
 
 result is 
 
 [pan] 4- [J>6n] + [pen] + . . . = [psn], 
 
 which must be satisfied if the absolute terms are correct. 
 
 Checks on the computation of x, y, z, . . . from (b) and of Vi, v^, • . . from (a) 
 are furnished by 
 
 [pav'] = o, [p6v] = o, [pcv] = 0, .... 
 
 To get the unknowns x, y, z, and their weights simultaneously, the best method 
 of procedure is, in general, the following : For brevity replace the absolute terms 
 in (b) by A, B, C, . . . respectively. Then the solution of (b) will be expressed 
 by 
 
 J' = «2 + A + 72 + • • • . (C) 
 
 z—<H + A +73 + • ■ • , 
 
 in which oj, /3i, 71, . . . are numerical quantities ; and 
 
 weight of x = — , 
 
 weight of jc=o> (d) 
 
 Pi 
 
 weight of ^ — > 
 * 73 
 
 To compute mean (and hence probable) errors the following formulas apply : — 
 m = the number of observed quantities n 
 
 = number of equations of condition, 
 ft, = number of the quantities x, y, z, . . . 
 c„ z=^ mean error of an observed quantity («) of weight unity, 
 ip = corresponding probable error = 0.6745 e„,. 
 
THEORY OF ERRORS. XCIU 
 
 \l m — jj. 
 
 4. 
 
 for unequal weights, 
 
 -i — J— for equal weights, 
 
 m — ft. 
 
 Mean error of any observed quantity («) of weight/ = -^i 
 Mean error of x = e„ y/aj, 
 Mean error of .y = e„ y/^^, 
 Mean error of 2 = £„ ^y^^ 
 
 where a^, /ggj 73» • • • are defined by equations (c) and (d) above. 
 
 e. Case of functions of several observed quantities x, y, z 
 
 This case is that in which the conditional equations (4) contain no disposable 
 quantities i, rj, ^, . . . . It is the opposite extreme to that represented by the case 
 of the preceding section.* It finds its most important and extensive application 
 in the adjustment of triangulation, wherein the observed quantities are the angles 
 and bases of the triangulation, and the conditions (4) arise from the geometrical 
 relations which the observed quantities Ji/us their respective corrections must 
 satisfy. 
 
 An outline of the general method of procedure in this case is the following : — 
 The first step consists in stating the conditional equations and in reducing 
 them to the linear form if they are not originally so. The form in which they 
 present themselves is (4) with $, rj, ^, . . . suppressed, or 
 
 i^(Xi + A Xi, ^2 -|- A ^2, *^3 + ^ .^S) • • • ) = °. 
 
 wherein x, y, z, . . . of (4) are replaced by x^, x^, x^ . . . for the purpose of sim- 
 plicity in the sequel. If this equation is not linear, Taylor's series gives 
 
 F(xi, X2,X3...)-\- g^ Axi 4- -g^ Axa = . . . = o, 
 
 since the method supposes that the squares, products, etc., of Axi, Aarj . • . may 
 be neglected. The last equation is then linear with respect to the corrections 
 Axi, Ajc2 • • • which it is desired to find. 
 For brevity put 
 
 F(xx, X2, Xg . . . ) = gi, a. known quantity, 
 
 3F __ 9F_ 3F _ 
 
 dxx — "^^ 9x, — ''" 9xs — '^3, • . . . 
 
 Then the conditional equations will be of the type 
 
 «iAxi -|- '^i^^i + aA^s + • • • + ?i ^= o- 
 
 * The middle ground between these extremes has been little explored ; indeed, most practical 
 applications fall at one or the other of the extremes. 
 
XCIV THEORY OF ERRORS. 
 
 There will be as many equations of this type as there are independent relations 
 which the quantities x^ -\- Axi, x^ -\- Ax^, . . . must satisfy. Suppose there are k 
 such relations, and let the differential coefficients 9J^/9xi, SFjSxi, ... for the sec- 
 ond relation be denoted by bi, h, b^, . . . ; for the third relation by Ci, c^, Cg 
 
 etc. Then all of the conditional equations may be written thus : 
 
 «iAxi + a^!^i + as^Xa + . . . + ^i ^ o, 
 
 bi -\-b2 -\-bi +...+^2 = 0, {a) 
 
 (l -\- C2 -\-C3 + . . . 4- ^3 = O, 
 
 . . . , 
 
 the number of these equations being k. 
 
 Call the weights of the observed quantities x^, x^, . . . pi, p^, . . . . Then, sub- 
 ject to the conditions (a) we must have (in accordance with (5)) 
 
 u =p,{Ax,y + A(A^2y + . . . = [/(A*)^ (b) 
 
 a minimum. 
 
 Equations (a) and (b) contain the solution of all problems falling under the 
 present case. Obviously, the number of conditions (a) must be less than the 
 number of observed quantities x, or less than the number of Ax's in (b) ; in other 
 words, if m denote the number of observed quantities, m > k, ior ii m ^ ^ the 
 minimum equation (b) has no meaning. 
 
 The question presented by (a) and (b) is one of elimination only. Two methods, 
 the one direct and the other indirect, are available. Thus, by the direct method 
 one finds from (a) as many Ax's as there are equations (a), or k such values, and 
 substitutes them in (b). The remaining (m — k) values of Ax in (b) may then be 
 treated as independent and the differential coefficients of u with respect to each 
 of them placed equal to zero. Thus all of the corrections Ax become known. 
 
 By the indirect process, one multiplies the first of equations (a) by a factor Qi, 
 the second by Q2, the third by Qs, . . . and subtracts the differential (with respect 
 to the A^c's) of the sum of these products from half the differential of (b). The 
 result of these operations is 
 
 J ^« = {piAxi — (aiQi -\- biQi -\- CiQi -{- . . .)} liAxi 
 4- {p^Ax^ — (^2^1 + ^262 + AiCa + • . •)} ^^^2 
 + . . . 
 -f {p^Ax^ - (a^Q, + b„,Q, -\-c^Q,-\-.. .)} dAx^ 
 
 Now we may choose the factors Qi, Q2, . . . Qt in such a way as to make k of the 
 coefficients of the differentials in this equation disappear ; and after thus elimi- 
 nating A of these differentials we are at liberty to place the coefficients of the 
 remaining (m — K) differentials equal to zero. Thus all conditions are satisfied 
 by making 
 
 a\Qi + hOs. + c-^Qi + . . . _ /jA^i = o, 
 «2 + *2 -\- c^ + . . . — p^Ax^ = o, 
 
 (c) 
 «m -{- b„ -\- c^ -{-...— /„Ax„, = o ; 
 
 and the values of the corrections will be given by these equations when the fac- 
 tors Qx, Qi, . . . are known. To find the latter it suffices to substitute the values 
 
THEORY OF ERRORS. 
 
 XCV 
 
 of Ax, Ax2, . . . from (c) in (a), whereby there will result /i equations containing 
 the Qi, Qa . . . Qt alone as unknowns. The result of this substitution is 
 
 [f]e.+[f]G.+[f]e.+. 
 
 y] + 
 
 f ] + 
 
 ad ~ 
 71. 
 
 + 
 + 
 
 Vp\ 
 
 <28 + .. 
 
 •+?! = 
 
 °, 
 
 ~ be' 
 
 
 
 
 .P- 
 
 + •■ 
 
 • +?2 = 
 
 o, 
 
 ~ a' 
 
 Vp\ 
 
 + ■ 
 
 ■ -\-qi — 
 
 o, 
 
 {d) 
 
 These equations {d) are derived directly from (<r) in the following manner : multi- 
 ply the first of (c) by x' the second by t-^' etc., sum the products, and compare the 
 
 P\ Pi 
 
 sum with the first of (a). The first of (</) is then evident ; the others are obtained 
 in a similar way. 
 
 The mean error of an observed quantity of weight unity is in this case given by 
 the formula 
 
 _ llp{Axf\ 
 
 where k is the number of conditions (a) ; and the mean error of any observed 
 value of weight/ is 
 
 f. Computation of mean and probable errors of functions of observed 
 
 quantities. 
 
 Let F denote ariy function of one or more independently observed quantities 
 X, y, z, . . .\ that is, let 
 
 V=f{x,y,z..:). 
 
 A question of frequent occurrence with respect to such functions is, What is the 
 mean * error of V in terms of the mean errors oi x, y, z, . . ."i The answer to 
 this question given by the method of least squares assumes that the actual errors 
 (whatever they may be) of x, y, z, . . . are so small that the actual error of Fis a 
 linear function of the errors of x, y, z. In other words, if e„ Cy, e„ . . . denote 
 the actual errors oi x, y, z, . . . , and A Kdenote the corresponding actual error of 
 V, the method assumes that 
 
 sr 
 
 9V 
 
 9V 
 
 ^^=9^'-^9y'''+9^'' + - 
 
 (a) 
 
 wherein the squares, products, etc., of e^, e^, e^, . . . are omitted. 
 
 This condition being fulfilled, let £ denote the mean error of V, and Ej., e-^, ^z ■ ■ • 
 denote those of x, y,z, . . . respectively. Then the law of error of least squares 
 requires that 
 
 * Since the probable error is 0.6745 times the mean error the latter only need be considered. 
 
XCVl THEORY OF ERRORS. 
 
 This equation includes all cases. Its analogy with (a) should be noted, since 
 the step from (a) to {b) is clear when the correct form of {a) is known. Mistakes 
 in the application of {b) are most likely to arise from a lack of knowledge of the 
 independently observed quantities x, y^ z, . . . or from a lack of knowledge of the 
 true form of {a). Hence,* in deriving probable errors of functions of observed 
 quantities attention should be given first to the construction of the expression for 
 the actual error (a). 
 
 A few examples may serve to illustrate the use of (a) and (b). 
 
 (i.) Suppose 
 
 V=f{x, y,z,..) — a{x-y)-\-b{y^z)-\-c{z—\). 
 Then 
 
 
 
 
 A r= ae, + (^ - d)e, + (^ + c)e,. 
 
 (2.) Suppose 
 
 .= = «V + (* - a)\^ + (^ + c)W 
 
 U«-» 
 
 V=f(x,y,z...) = l + by^. 
 
 tien 
 
 3V a 3V b 3V 2 by 
 dx ~ x'' dy — z'' 9z ~ 2? ' 
 
 
 AK— "'e jj e '^^^e 
 
 
 , a'' b\^. ^bY , 
 
 (3.) Suppose 
 
 
 
 
 
 
 
 
 
 V^ a log X -\- 
 
 b sin y 
 
 + ^log 
 
 tan z. 
 
 
 Then 
 
 
 
 
 
 
 
 
 
 3f^ 
 
 a/j-f 
 
 3V ^ 
 
 
 3F 
 
 
 CU. 
 
 
 3x — 
 
 • > 
 
 X 
 
 9y=' 
 
 cosy, 
 
 3z- 
 
 ~ sin z cos z 
 
 and 
 
 
 
 
 
 
 
 
 
 .= = 
 
 =(?! 
 
 )V+(^ 
 
 cos yYi 
 
 ^y'+ii 
 
 ! CfX ■* 
 n -7. s: 
 
 
 (4.) Suppose the case of a single triangle all of whose angles are observed. 
 What is the mean error, ist, of an observed angle; 2d, of the correction to an 
 observed angle ; and 3d, of the corrected or adjusted angle .■' 
 
 Let X, y, z denote the observed angles, /, g, r their weights, and l^x, t^y, Az 
 the corresponding corrections. 
 
 Then, as shown on p. Ixxxvii, 
 
 Aa: + Aj + A2; = (T = 180° -f- sph. excess — {x ■\- y -\- z) 
 = error of closure of triangle, 
 c 
 
 Q = 
 
 - + - + - 
 
 p~ g~ r 
 
 A^ =%, Ly = 5, Az = 2. 
 
 p q r 
 
 * As remarked by Sir George Airy in his T/ieory of Errors. 
 t jit = modulus of common logarithms. 
 
THEORY OF ERRORS. XCVU 
 
 For brevity, put 
 
 g-=- 1 80° -1- spherical excess, h -. 
 
 Then 
 
 p~ q~ r 
 
 Q = h {g — X — y — z) =1 he, 
 
 ^x = -{g-x-y-z), 
 P 
 
 x-\- \x =Ag -x-y-z)-\-x, 
 
 with similar expressions for the other two angles. 
 
 Now by the formula on p. xcv the square of the mean error of an observed 
 angle of weight unity is (since there is but one condition to which Aa;, Aj/, Aa are 
 subject), 
 
 K^xf + g{^yf + r{^zf = |1= k^. 
 
 Hence, the squares of the mean errors of the observed angles x, y, z, their weights 
 being/, q, r respectively, are 
 
 hc^ hf_ h^ 
 
 P ' q' r' 
 
 respectively. 
 
 To get the mean error of a correction, Aa; for example, formula (a) gives 
 A V= A(A^) = -|(.. + ^„ + ^,), 
 
 and the corresponding expressions for the actual errors of t^y and A2 are found 
 from this by replacing p hy q and r respectively. Thus by (p), observing that 
 the mean errors of x, y, z are given above, there result 
 
 Square of mean error of Ax = {hcfff, 
 
 ^y = {hclgy, 
 
 " " " i^z = {hclr)\ 
 
 Likewise, the formula for the actual error oi x -\- ^x is 
 
 A V= A(x + Ax) = (i -^y. 
 
 h _h 
 ' P^" p"" 
 
 and the corresponding expressions for the actual errors oi y -\- Ay and z -\- Az 
 are found by interchange of q and r with/. Thus the squares of the mean errors 
 of the adjusted angles are : — 
 
 for(x + Ax), ^'^i_|y 
 
 for(^ + A^), f-"(^-J). 
 
 for(. + A.), ^\-'^. 
 
XCVUl THEORY OF ERRORS. 
 
 In case the weights are equal, or in case p-=q-=.r, h=z\, and there 
 result, — 
 
 Square of mean error of observed angle = \ c^, 
 
 " " correction to observed angle = \ c\ 
 " " adjusted angle = f r". 
 
 u u 
 
 where c is the error of closure of the triangle ; so that in this case of equal weights 
 the three mean errors are to one another as 4^/3, J, and J^2, 
 
 References. 
 
 The literature of the theory of errors, especially as exemplified by the method 
 of least squares, is very extensive. Amongst the best treatises the following are 
 worthy of special mention : Method of Least Squares, Appendix to vol. ii. of 
 Chauvenet's " Spherical and Practical Astronomy." Philadelphia : J. B. Lippin- 
 cott & Co., 8vo, 5th ed., 1887. " A Treatise on the Adjustment of Observations, 
 with Applications to Geodetic Work and Other Measures of Precision," by T. W. 
 Wright. New York : D. Van Nostrand, 8vo, 1884. " On the Algebraical and 
 Numerical Theory of Errors of Observation and on the Combination of Observa- 
 tions," by Sir George Biddle Airy. London : Macmillan & Co., i2mo, 2d ed., 
 1875. " Die Ausgleichungsrechnung nach der Methode der Kleinsten Quadrate, 
 mit Anwendungen auf die Geodasie und die Theorie der Messinstrumente," von 
 F. R. Helmert. Leipzig : B. G. Teubner, 8vo, 1872. 
 
EXPLANATION OF SOURCE AND USE OF THE 
 
 TABLES. 
 
 Tables i and 2 are copies of tables issued by the Office of Standard Weights 
 and Measures of the United States, edition of November, 1891. 
 
 Table 3 is derived from standard tables giving such data. The arrangement 
 is that given in " Des Ingenieurs Taschenbuch, herausgegeben von dem Verein 
 ' Hiitte '"* (i ith edition, 1877). The numbers have been compared with those 
 given in the latter work, and also with those in Barlow's " Tables." The loga- 
 rithms have been checked by comparison with Vega's 7-place tables. 
 
 Table 4 is abridged from a similar table in the Taschenbuch just referred to. 
 
 Tables 5 and 6 are copies of standard forms for such table. They have 
 been checked by comparison with standard higher-place tables. The mode of 
 using these tables will be evident from the following examples : — 
 
 (i.) To find the logarithm of any number, as 0.06944, we look in Table 5 
 in the column headed N for the first two significant figures of the number, which 
 are in this case 69. In the same horizontal line with 69 we now look for the 
 number in the column headed with the next figure of the given number, which is 
 in the present case 4. We thus find .8414 for the mantissa of the logarithm of 
 the number 694. To get the increase due to the additional figure 4, we look in 
 the same horizontal line under Prop. Parts in the column headed 4 and find the 
 number 2, which is the amount in units of the fourth place to be added to the 
 part of the mantissa previously found. Thus the mantissa of log (0.06944) is 
 .8416. The characteristic for the logarithm in question is —2 =8 — 10. Hence 
 log (0.06944) =8.8416 — 10. 
 
 (2.) To find the number corresponding to any logarithm, as 8.8416— 10, we 
 look in Table 6 in the column headed L for the first two figures of the mantissa, 
 which are in this case 84. In the same horizontal line with 84 we now look for 
 the number in the column headed by the next figure of the mantissa, which is in 
 this case i. We thus find 6394 for the number corresponding to the mantissa 
 8410. Tq get the increase due to the additional figure 6, we look in the same 
 horizontal line under Prop. Parts in the column headed 6 and find 10, which is 
 the amount in units of the fourth place to be added to the number previously 
 found. Thus the significant figures of the number are 6944, and since the char- 
 acteristic of the logarithm is 8— io=— 2, the required number is 0.06944. 
 
 * Berlin : Verlag von Ernst & Korn. This work is an invaluable one to the engineer, archi- 
 tect, geographer, etc. 
 
C EXPLANATION OF SOURCE AND USE OF TABLES. 
 
 Tables 7 and 8 are taken from " Smithsonian Meteorological Tables " (the 
 first volume of this series). Their mode of use will be apparent from the follow- 
 ing example: Required the sine and tangent for 28° 17'. 
 
 sine 28° 10', Table 7 0.4720. Tabular difference = 26. 
 
 Proportional part for 7' (7 X 2.6) . . 18. 
 
 sine 28° 17' 0.4738. 
 
 tangent 28° 10', Table 8 0.5354. Difference for i' = 3.8. 
 
 Increase for 7' (7 X 3-8) 27. 
 
 tangent 28° 17' 0.5381. 
 
 Table 9 is a copy of a similar table published in " Professional Papers, Corps 
 Engineers," U. S. A., No. 12. It has been checked by comparison with other 
 tables in general use. This table is useful in computing latitudes and departures 
 in traverse surveys wherein the bearings of the lines are observed to the nearest 
 quarter of a degree, and in other work where multiples of sines and cosines are 
 required. Thus, if L denote the length and B the bearing from the meridian of 
 any line, the latitude and departure of the line are given by 
 
 LcosB and Zsin^ 
 
 respectively ; the " latitude " being the distance approximately between the paral- 
 lels of latitude at the ends of the line, and the " departure " being the distance 
 approximately between the meridians at the ends of the line. As an example, let 
 it be required to compute the latitude and departure for L ^ 4837, in any unit, 
 and JB = 36° 15'. The computation runs thus : — 
 
 Latitude. Departure. 
 
 For 4000 3225.77 2365-23 
 
 800 645.16 473-05 
 
 30 24.19 17.74 
 
 7 5-63 4-14 
 
 4837 Zcos^ = 3900.77 Zsin^= 2860.16 
 
 Tables 10 and 11 give the logarithms of the principal radii of curvature of the 
 earth's spheroid. They were computed by Mr. B. C. Washington, Jr., and care- 
 fully checked by differences. They depend on the elements of Clarke's spheroid 
 of 1866. The use of these tables is sufficiently explained on p. xlv-xlix. 
 
 Table 12 gives logarithms of radii of curvature of the earth's spheroid in sec- 
 tions inclined to the meridian sections. It is abridged to 5 places from a 6-place 
 table published in the " Report of the U. S. Coast and Geodetic Survey for 
 1876." Its use is explained on pp. Ixi-lxiv. 
 
 Tables 13 and 14 give logarithms of factors needed to compute the spheroidal 
 excess of triangles on the earth's spheroid. No. 13 is constructed for the Eng- 
 lish foot as unit, and No. 14 for the metre. These tables were computed by Mr. 
 
EXPLANATION OF SOURCE AND USE OF TABLES. CI 
 
 Charles H. Kummell. Their use is explained on p. Iviii. The following example 
 will illustrate their use : — 
 
 Latitude of vertex A of triangle 48° 08' 
 
 £ " 47 52 
 
 C " 47 04 
 
 Mean latitude 47 41 
 
 Angle C= 51° 22' 55" log sin C 9.89283 — 10 
 
 log a (feet) 5.64401 
 
 log ^ (feet) 5.58681 
 log factor, Table 13, for 47° 41' 0.37176 
 
 Spheroidal excess = 3i."29o, log 1.49541 
 
 Tables 15 and 16 give logarithms of factors for computing differences of lati- 
 tude, longitude, and azimuth in secondary triangulation whose lines are 12 miles 
 (20 kilometres) or less in length. These tables were computed by Mr. Charles 
 H. Kummell. Table 15 gives factors for the English foot as unit, and Table 16 
 for the metre as unit. The use of these tables is illustrated by a numerical exam- 
 ple given on pp. Ix and Ixi. For lines not exceeding the length mentioned, the 
 tables will give differences of latitude and longitude to the nearest hundredth of 
 a second of arc, using 5-place logarithms of the lengths of the lines. 
 
 Table 17 gives lengths of terrestrial arcs of meridians corresponding to lati- 
 tude intervals of 10", 20", . . . 60", and 10', 20', . . . 60', or lengths corresponding 
 to arcs less than 1°. The unit of length is the English foot. The table was 
 computed by Mr. B. C. Washington, Jr. 
 
 The length corresponding to any latitude interval is the distance along the 
 meridian between parallels whose latitudes are less and greater respectively than 
 the given latitude by half the interval. Thus, for example, the length corre- 
 sponding to the interval 30' and latitude 37" (182047.3 feet) is the distance along 
 the meridian from latitude 36° 45' to latitude 37° 15'. 
 
 By interpolation, we may get from this table the meridional distance corre- 
 sponding to any interval. The following example illustrates this use : Required 
 the distance between latitude 41° 28' 17. "8 and latitude 41° 39' 53."4. The 
 difference of these latitudes is 11' 35."6, and their mean is 41° 34' o5."6. The 
 computation runs thus : — 
 
 
 
 
 Latitude . 
 
 41°. 
 
 Tabular difference. 
 
 10' 
 
 
 
 60724.60 
 
 feet 
 
 
 10.70 feet 
 
 l' 
 
 
 
 6072.46 
 
 «( 
 
 
 1.07 " 
 
 30" 
 
 
 
 3036-23 
 
 K 
 
 
 •54 " 
 
 s" 
 
 
 
 506.04 
 
 U 
 
 
 .09 " 
 
 o."6 
 
 12 
 
 .41 
 
 60.72 
 7-°S 
 
 4( 
 
 sum. 
 
 .01 " 
 
 i4.09 V 
 "60- ^ 
 
 12.41 " 
 
 
 Distance = 
 
 = 70407.10 
 
 a 
 
 
 
 When the degree of precision required is as great as that of the example just 
 
Cii EXPLANATION OF SOURCE AND USE OF TABLES. 
 
 given, it will be more convenient to use formulas (2) on p. xlvi. Thus, in this 
 
 example, — 
 
 log. 
 A<if) = 69S."6 2.8423596 
 
 <l> = 41° 34' os.''6, p„ (Table 10) 7.3196820 
 
 cons't 4-6855749 
 Length ^ 70407.10 feet 4.8476165 
 
 Table 18 gives lengths of terrestrial arcs of parallels corresponding to longi- 
 tude intervals of 10", 20", . . . 60", and 10', 20', . . . 60', or lengths corresponding 
 to arcs less than 1°. The unit is the English foot. This table was computed by 
 Mr. B. C. Washington, Jr. 
 
 The method of using this table is similar to that applicable to Table 17 
 explained above. For the computation of long arcs it will in general be less 
 laborious to use the formulas (i) on p. xlix than to resort to interpolation from 
 Table 18. 
 
 Tables 19-24 give the rectangular co-ordinates for the projection of maps, in 
 accordance with the polyconic system explained on pp. liii-lvi, for the following 
 scales respectively : — 
 
 > unit = English inch. 
 (2 miles to I inch) 
 
 (i mile to I inch) 
 unit = millimetre. 
 
 These tables were computed by Mr. B. C. Washington, Jr. 
 
 The use of these tables and their application in the construction of maps may 
 be best explained by an example. Suppose it is required to draw meridians and 
 parallels for a map of an area of 1° extent in longitude, lying between the paral- 
 lels of 34° and 35°. Let the scale of the map be one mile to the inch, or 1/63360, 
 and let the meridians and parallels be 10' apart respectively. Draw on the pro- 
 jection paper an indefinite straight line AB, Fig. 4, to represent the middle me- 
 ridian of the map. Take any convenient point, as C, on this line for the latitude 
 34°, and lay off from this point the meridional distances CD, C£, CF, . . . CI, 
 given in the second column of Table 22, p. 114.* Through the points £>, E, F, 
 ... I, thus found, draw indefinite straight lines perpendicular to AB. By means 
 of these lines and the tabular co-ordinates, points on the developed parallels and 
 meridians are readily found. Thus, for example, the abscissas for points ten 
 minutes apart on the parallel 34° 20' are 9.53, 19.06, and 28.59 inches. These 
 distances are to be laid off on NN' in both dir.ections from A£. At the points 
 Z, M, N, U, M', N', so determined, erect perpendiculars to NN' equal in 
 length, respectively, to the ordinates corresponding to the longitude intervals 
 
 * The meridional distances and the abscissas of the points on the developed parallels in Fig. 4 
 are one twentieth of the true or tabular values. The ordinates of points on the developed paral- 
 lels are the tabular values. 
 
 abl 
 
 e 19, 
 
 scale 
 
 1 
 
 250000 
 
 
 . 20, 
 
 
 1 
 125000 
 
 
 21, 
 
 
 I 
 126720 
 
 
 22, 
 
 
 63360 
 
 
 23, 
 
 
 200000 
 
 
 24, 
 
 
 1 
 
 800O0 
 
ejSplanation of source and use of tables. 
 
 cm 
 
 lo', 20', 30'. The curved line joining the extremities of these perpendiculars is 
 the parallel required. It may be drawn by means of a flexible ruler. The other 
 parallels are constructed in the same manner. They are all concave towards the 
 north or south according as the map shows a portion of the northern or southern 
 hemisphere. The meridians are drawn in a similar manner through the points 
 {e.g., P, Q, M, R, S,T, C^in Fig. 4) having the same longitude relative to the 
 middle meridian. All meridians are concave towards the middle meridian. 
 
 A test of the graphical work which should always be applied is the approxima- 
 tion to equality of corresponding diagonals in the various quadrilaterals formed. 
 Thus in Fig. 4, FJf should be equal to WY, CNto CN', EVtoEW, etc.* 
 
 
 
 
 X 
 
 1 
 
 
 
 X 
 
 
 
 I 
 
 
 V 
 
 y 
 
 
 
 
 JH 
 
 
 T 
 
 
 
 
 
 G 
 
 
 S 
 
 
 
 
 
 F 
 
 
 M 
 
 
 N 
 
 M 
 
 z 
 
 E 
 
 Z 
 
 M 
 
 JM 
 
 
 
 
 D 
 
 
 Q 
 
 
 W 
 
 
 
 C 
 
 
 F 
 
 V 
 
 35° 
 
 50' 
 
 dor 
 
 3cr 
 
 2d 
 
 l(f 
 
 34° 
 
 Tables 25-29 give areas of quadrilaterals, bounded by meridians and parallels, 
 of the earth's surface. They are taken from " Bulletin 50, U. S. Geological Sur- 
 vey." The unit of length used is the English mile, and the areas are thus given 
 in square miles. The method of using these tables is obvious. 
 
 Table 30 gives data for the computation of heights, from barometric meas- 
 ures, in accordance with the formula of Babinet.f This table is taken from the 
 " Smithsonian Meteorological Tables " (the first volume of this series). The 
 manner of using it is explained in connection with the table. 
 
 * It should be noted that CTVis not equal to EV, A^and Preferring here to points on the 
 developed parallels. 
 
 t Comptes Rendus, Paris, 1850, vol. xxv. p. 309. 
 
civ EXPLANATION OF SOURCE AND USE OF TABLES. 
 
 Table 31 gives the mean astronomical refraction in terms of the apparent alti- 
 tude of a star or other object outside the earth's atmosphere. It is taken from 
 Vega's 7-place table of logarithms. Its use will be evident from the following 
 example : — 
 
 Apparent altitude of star = 34° 17' i2."7 
 
 Refraction^ i' 24."3 — ^ X i-"i = i 24.1 
 
 True altitude of star =34 15 48-6 
 
 Tables 32 and 33 facilitate the interconversion of arc and time. They are 
 taken from the " Smithsonian Meteorological Tables" (the first volume of this, 
 series). The following examples illustrate their use : — 
 
 (i.) To convert 68° 29' 48."8 into time we have from Table 32 — 
 
 68° 
 
 = 
 
 4'' 
 
 32" 
 
 'oo" 
 
 
 29' 
 
 = 
 
 
 I 
 
 56 
 
 
 48" 
 
 = 
 
 
 
 3- 
 
 20 
 
 o."8 
 
 = 
 
 
 
 
 OS 
 
 Equivalent in time = 4 33 59.25 
 (2.) To convert 5"* 43"" 28.'8 into arc we have from Table 33 — 
 
 5>' =75° 00' 00" 
 43" = 10 45 00 
 28' = 7 00 
 
 o.'8 = 12 
 
 Equivalent in arc ^85 52 12 
 
 Tables 34 and 35 facilitate the interconversion of mean solar and sidereal 
 time intervals. They are taken from Vega's 7-place table of logarithms. The 
 mode of using them is explained in the tables themselves. 
 
 Tables 36 and 37 give the lengths of degrees of terrestrial arcs of meridians 
 and parallels expressed in metres,* statute miles (English), and geographic miles 
 (distance corresponding to i' on the earth's equator). These tables are taken 
 from the " Smithsonian Meteorological Tables " (the first volume of this series). 
 
 Table 38 facilitates the interconversion of statute (English) miles and nautical 
 miles. The nautical mile used is that defined by the U. S. Coast and Geodetic 
 Survey, namely : the length of a minute of arc of a great circle of the sphere 
 whose surface equals that of the earth (Clarke's spheroid of 1866). For formula 
 for radius of such sphere see p. lii. This table is taken from the " Smithsonian 
 Meteorological Tables " (the first volume of this series). 
 
 Table 39 gives the English and metric equivalents of other standards of 
 length still in use or obsolescent. It is taken from the " Smithsonian Meteoro- 
 logical Tables " (the first volume of this series). 
 
 Table 40 gives values of the acceleration (g) of gravity, log g; log (1/2^), log v'2 g, 
 
 * It should be observed that the metric values given in these tables depend on Clarke's value 
 of the ratio of the yard to the metre, which is now known to be erroneous by about the i/ loooooth 
 part. 
 
EXPLANATION OF SOURCE AND USE OF TABLES. CV 
 
 and (^/w^) or the length of a seconds pendulum, 'for intervals of 5" of geograph- 
 ical latitude. It was computed by the editor, and is based on the formula for g 
 given by Professor William Harkness in his memoir " On the Solar Parallax and 
 its Related Constants." * 
 
 Table 41 gives the linear expansions of the principal metals. It was compiled 
 by the editor from various sources. The values given for the expansion per 
 degree Centigrade have been rounded (with one exception) to the nearest unit in 
 the millionths place, or to the nearest micron, since different specimens of the 
 same metal vary more or less in the ten-millionths place. 
 
 Table 42 gives the fractional changes in numbers corresponding to changes in 
 the 4th, 5th, . . . 7th place of their logarithms. These fractions are often con- 
 venient in showing the approximate error in a number due to a given error in 
 its logarithm, or the converse. Thus, for example, referring to the remark in a 
 foot-note under explanation of Tables 36 and 37 above, the error in the loga- 
 rithm of Clarke's ratio of the yard to the metre is about 4 units in the sixth place 
 of decimals ; the Table 42 shows, then, that the metric equivalents in Tables 
 36 and 37 are erroneous by about i/ioooooth part. 
 
 * Washington, Government Printing Office, 1891. 
 
GEOGRAPHICAL TABLES 
 
Table 1 . 
 
 FOR CONVERTING U. S. WEIGHTS AND MEASURES. 
 
 CUSTOMARY TO METRIC. 
 
 LINEAR. 
 
 CAPACITY. 
 
 
 
 Inches to 
 milli- 
 metres. 
 
 Feet to 
 metres. 
 
 Yards to 
 metres. 
 
 Miles to 
 kilometres. 
 
 
 Fluid 
 drams to 
 millilitres 
 or cubic 
 
 centi- 
 metres. 
 
 Fluid 
 ounces to 
 milli- 
 litres. 
 
 Quarts to 
 litres. 
 
 Gallons to 
 litres. 
 
 
 2 = 
 
 3 = 
 
 4 = 
 
 8 = 
 
 9 = 
 
 25-4001 
 50-8001 
 76-2002 
 101-6002 
 127-0003 
 1 52-4003 
 177-8004 
 203-2004 
 228-6005 
 
 0-304801 
 0-609601 
 0-914402 
 1-219202 
 1-524003 
 1-828804 
 2-133604 
 2-438405 
 2-743205 
 
 0-914402 
 1-828804 
 2-743205 
 3-657607 
 4-572009 
 5-48641 1 
 6-400813 
 
 7-315215 
 8-229616 
 
 1-60935 
 3-21869 
 4-82804 
 
 •6-43739 
 8-04674 
 9-65608 
 11-26543 
 [2-87478 
 I4"484i2 
 
 2 = 
 
 3 = 
 
 4 = 
 
 8 = 
 
 9 = 
 
 3-70 
 
 7-39 
 11-09 
 
 18-48 
 22-l8 
 25-88 
 29-57 
 33-27 
 
 29-57 
 
 88-72 
 118-29 
 
 147-87 
 177-44 
 207-02 
 
 0-94636 
 1-89272 
 2-83908 
 3-78543 
 4-73179 
 5-67815 
 6-62451 
 7-57087 
 851723 
 
 3-78543 
 7-57087 
 11-35630 
 15-14174 
 18-92717 
 22 71261 
 26-49804 
 30-28348 
 34-06891 
 
 
 SQUARE. 
 
 WEIGHT. 
 
 
 
 Square 
 inches to 
 square 
 centi- 
 metres. 
 
 Square 
 feet to 
 square 
 deci- 
 metres. 
 
 Square 
 yards to 
 square 
 metres. 
 
 Acres to 
 hectares. 
 
 
 Grains to 
 
 milli- 
 grammes. 
 
 Avoirdu- 
 pois 
 ounces to 
 grammes. 
 
 Avoirdu- 
 pois 
 pounds to 
 
 kilo- 
 grammes. 
 
 Troy 
 ounces to 
 grammes. 
 
 
 1 = 
 
 2 = 
 
 3 = 
 
 4 = 
 
 8 = 
 
 9 = 
 
 6-452 
 12-903 
 
 19-355 
 25-807 
 32-258 
 38-710 
 45-161 
 51-613 
 58-065 
 
 18-581 
 27-871 
 37-161 
 46-452 
 55-742 
 65-032 
 
 74-323 
 83-613 
 
 0-836 
 1-672 
 2508 
 
 3344 
 4-181 
 5-017 
 
 ll^ 
 7-525 
 
 0-4047 
 0-8094 
 I-2141 
 I-6187 
 2-0234 
 2-4281 
 2-8328 
 
 3-2375 
 3-6422 
 
 2 = 
 
 3 = 
 
 4 = 
 
 1 = 
 
 7 = 
 
 8 = 
 
 9 = 
 
 64.7989 
 129-5978 
 194-3968 
 259-1957 
 323-9946 
 388-7935 
 453-.5924 
 518-3914 
 583-1903 
 
 283495 
 
 §5-0486 
 1 1 3-398 1 
 141-7476 
 170-0972 
 198-4467 
 226-7962 
 255-1457 
 
 0-45359 
 0-90719 
 
 1-36978 
 1-81437 
 2-26796 
 2-72156 
 3-17515 
 3-62874 
 4-08233 
 
 31-10348 
 62-20696 
 
 93-31044 
 124-41392 
 
 155-51740 
 186-62088 
 
 248-82785 
 279-93133 
 
 
 CUBIC. 
 
 
 
 
 Cubic 
 inches to 
 cubic 
 centi- 
 metres. 
 
 Cubic feet 
 to cubic 
 metres. 
 
 Cubic 
 yards to 
 
 cubic 
 metres. 
 
 Bushels to 
 hectolitres. 
 
 
 2 = 
 
 3 = 
 
 4 = 
 
 7 = 
 
 8 = 
 
 9 = 
 
 16-387 
 32-774 
 49-161 
 
 65-549 
 81-936 
 
 98-323 
 1 14-710 
 131-097 
 147-484 
 
 0-02832 
 0-05663 
 0-08495 
 0-11327 
 O-14158 
 0-16990 
 0-19822 
 0-22654 
 0-25485 
 
 0-765 
 1-529 
 2-294 
 3-058 
 3-823 
 4-587 
 
 6-I16 
 6-881 
 
 0-35239 
 0-70479 
 1-05718 
 1-40957 
 I-76196 
 2- 1 1436 
 2-46675 
 2-81914 
 3-17154 
 
 I G 
 I s< 
 I fa 
 I n 
 I fi 
 I a 
 1 5432-. 
 
 unter's c 
 }. statute 
 thom 
 lutical m 
 ot = 0.3 
 ifoir. pour 
 55639 gra 
 
 lain = 
 mile = 
 
 le = 
 34801 met 
 
 ns = 
 
 20-1168 
 
 259-000 
 
 1-829 
 
 1853-25 
 re, 9-484 
 
 453-5924 
 I kil 
 
 metres, 
 liectares. 
 
 metres. 
 
 metres. 
 0158 log. 
 277 gram, 
 ogramme. 
 
 
 The only authorized material standard of customary length is the Troughton scale belonging to this office, whose 
 length at 59°.(>2 Fahr. conforms to the British standard. The yard in use m the United States is therefore equal to 
 the British yard. 
 
 The only authorized material standard of customary weight is the Troy pound of the Mint. It is of brass of 
 unknown density, and therefore not suitable for a standard of mass. It was derived from the British standard Troy 
 pound of 1758 by direct comparison. The British Avoirdupois pound was also derived from the latter, and contains 
 7,000 grains Troy. 
 
 The grain Troy is therefore the same as the grain Avoirdupois, and the pound Avoirdupois in use in the United 
 States is equal to the British pound Avoirdupois. 
 
 The British gallon = 4.54346 litres. The British bushel = 36.3477 litres. 
 
 The length of the nautical mile given above and adopted by the U. S. Coast and Geodetic Survey many years 
 ago is defined as that of a minute of arc of a great circle of a sphere whose surface equals that of the earth (Clarke's 
 Spheroid of 1866). 
 
 * Issued by U. S. OfBce of Standard Weights and Measures, an4/^ublished here by permilsion of Suoerint^ilient 
 of Coast and Geodetic Survey. 
 
 Tabl 
 
FOR CONVERTING U. S. WEIGHTS AND MEASURES. 
 
 METRIC TO CUSTOMARY. 
 
 Table 2. 
 
 LINEAR. 
 
 CAPACITY. 
 
 
 
 
 
 
 
 Millilitres 
 
 
 
 
 
 
 Metres to 
 inches. 
 
 Metres to 
 feet. 
 
 Metres to 
 yards. 
 
 Kilo- 
 metres to 
 niiJes. 
 
 
 or cubic 
 centi- 
 metres 
 to fluid 
 drams. 
 
 Centi- 
 litres to 
 
 fluid 
 ounces. 
 
 Litres to 
 quarts. 
 
 Deca- 
 litres to 
 gallons. 
 
 Hecto- 
 litres to 
 bushels. 
 
 I — 
 
 39-3700 
 
 3.28083 
 
 I -0936 1 1 
 
 0-62137 
 
 1 =::: 
 
 0-27 
 
 0-338 
 
 1-0567 
 
 2-6417 
 
 2-8377 
 
 2 = 
 
 767400 
 
 6-55.67 
 
 2-187222 
 
 1-24274 
 
 2 = 
 
 0-54 
 0-81 
 
 0-676 
 
 2-1134 
 
 5-2834 
 
 5-6755 
 
 3 = 
 
 llS-lloo 
 
 9-84250 
 
 3-280833 
 
 1-86411 
 
 3 = 
 
 1-014 
 
 3-1700 
 
 7-9251 
 
 8-5132 
 
 4 = 
 
 157-4800 
 
 13-12333 
 
 4-374444 
 
 2-48548 
 
 4 = 
 
 1-08 
 
 •-353 
 
 4-2267 
 
 10-5668 
 
 11-3510 
 
 5 = 
 
 i96'S5oo 
 
 16-40417 
 
 5-468056 
 
 3-10685 
 
 
 1-35 
 
 1-691 
 
 5-2834 
 
 13-2085 
 
 14-1887 
 
 6 = 
 
 236-2200 
 
 19-68500 
 
 6-561667 
 
 3-72822 
 
 6 = 
 
 1-62 
 
 2-029 
 
 6-3401 
 
 15-8502 
 18-4919 
 
 17-0265 
 
 I — 
 
 275-5900 
 
 22-96583 
 
 7-655278 
 
 4-34959 
 
 7 = 
 
 1-89 
 
 2-367 
 
 7-3968 
 
 19-8642 
 
 8 = 
 
 314-9600 
 
 26-24667 
 
 8-748889 
 
 4-97096 
 
 8 = 
 
 2-16 
 
 2-705 
 
 8-4535 
 
 21-1336 
 
 22-7019 
 
 9 = 
 
 354-3300 
 
 29-52750 
 
 9.842500 
 
 5-59233 
 
 9 = 
 
 2-43 
 
 3-043 
 
 9-5101 
 
 23-7753 
 
 25-5397 
 
 SQUARE. 
 
 ■WEIGHT. 
 
 
 Square 
 centi- 
 metres to 
 square 
 inches. 
 
 Square 
 
 metres 
 
 to square 
 
 feet. 
 
 Square 
 
 metres 
 
 to square 
 
 yards. 
 
 Hectares 
 to acres. 
 
 
 Milli- 
 grammes to 
 grains. 
 
 Kilo- 
 grammes to 
 grains. 
 
 Hecto- 
 grammes 
 to ounces 
 avoirdu- 
 pois. 
 
 Kilo- 
 grammes 
 to pounds 
 avoirdu- 
 pois. 
 
 I = 
 
 0-1550 
 
 10-764 
 
 1-195 
 
 2-471 
 
 I = 
 
 0-01543 
 
 15432-36 
 
 3-5274 
 
 2-20462 
 
 2 = 
 
 0-3100 
 
 21-528 
 
 2-392 
 
 4-942 
 
 2 — 
 
 0-03086 
 
 30864-71 
 
 7-0548 
 
 4-40924 
 
 3 = 
 
 0-4650 
 
 32-292 
 
 3-588 
 
 '^'VJ 
 
 3 = 
 
 0-04630 
 
 46297-07 
 
 10-5822 
 
 6-61387 
 
 4 = 
 
 0.6200 
 
 43-055 
 
 4-784 
 
 9-884 
 
 4 = 
 
 0-06173 
 
 61729-43 
 
 14-1096 
 
 8-81849 
 
 
 0-7750 
 
 53-819 
 
 5-980 
 
 12-355 
 
 
 0-07716 
 
 77161-78 
 
 17-6370 
 
 I1-02311 
 
 6 ^ 
 
 0.9300 
 
 64-583 
 
 7-176 
 
 14-826 
 
 6 — 
 
 0-09259 
 
 92594-14 
 
 21-1644 
 
 13-22773 
 
 7 ^^ 
 
 i-o85o 
 
 75-347 
 
 8-372 
 
 17-297 
 
 7 ^ 
 
 0-10803 
 
 108026-49 
 
 24-6918 
 
 15-43236 
 
 8=: 
 
 1-2400 
 
 86111 
 
 9-568 
 
 19-768 
 
 8 = 
 
 0-12346 
 
 123458-85 
 
 28-2192 
 
 17-63698 
 
 9 = 
 
 1-3950 
 
 96-875 
 
 10-764 
 
 22-239 
 
 9 = 
 
 0-13889 
 
 138891-21 
 
 31-7466 
 
 19-84160 
 
 CUBIC. 
 
 WEIGHT — {continued'). 
 
 
 Cubic 
 centi- 
 metres to 
 cubic 
 inches. 
 
 Cubic 
 deci- 
 metres to 
 cubic 
 inches. 
 
 Cubic 
 
 metres 
 
 to cubic 
 
 feet. 
 
 Cubic 
 
 metres to 
 
 cubic 
 
 yards. 
 
 
 Quintals to 
 pounds av. 
 
 Milliers or 
 tonnes to 
 pounds av. 
 
 Kilogrammes 
 
 to ounces 
 
 Troy. 
 
 I — 
 
 o-o5lo 
 
 61-023 
 
 35-314 
 
 1-308 
 
 I = 
 
 220-46 
 
 2204-6 
 
 32-1507 
 
 2 ^ 
 
 0-1220 
 
 122-047 
 
 70-629 
 
 2-616 
 
 2 ^ 
 
 440-92 
 
 4409-2 
 
 64-3015 
 
 3 = 
 
 0-1831 
 
 183-070 
 
 105-943 
 
 3-924 
 
 3 = 
 
 661-39 
 
 6613-9 
 
 96-4522 
 
 4 = 
 
 0-2441 
 
 244-094 
 
 141-258 
 
 5-232 
 
 4 = 
 
 881-85 
 
 8S18-5 
 
 128*6030 
 
 
 0-3051 
 
 305-117 
 
 176-572 
 211-887 
 
 6-540 
 
 5 == 
 
 1102-31 
 
 11023-1 
 
 160-7537 
 
 6 = 
 
 0-3661 
 
 366-140 
 
 7-848 
 
 6 = 
 
 1322-77 
 
 13227-7 
 
 192-9044 
 
 7 = 
 
 0-4272 
 
 427-164 
 
 247-201 
 
 9-156 
 
 7 = 
 
 1543-24 
 
 15432-4 
 
 225-0552 
 
 8 = 
 
 0-4882 
 
 488-187 
 
 282-516 
 317-830 
 
 10-464 
 
 8 = 
 
 1763-70 
 
 17637-0 
 
 257-2059 
 289-3567 
 
 9 = 
 
 0-5492 
 
 549-210 
 
 II-771 
 
 9 = 
 
 1984-16 
 
 1 984 1 -6 
 
 By the concurrent action of the principal governments of the world an International Bureau of Weights and 
 Measures has been established near Paris. Under the direction of the International Committee, two ingots were cast 
 of pure platinum-iridium in the proportion of 9 parts of the former to i of the latter metal. From one of these a cer- 
 tain number of kilogrammes were prepared, from the other a definite number of metre bars. These standards of weight 
 and length were iniercompared, without preference, and certain ones were selected as International prototype stand- 
 ards. The others were distributed by lot, in September, 1889, to the different governments and are called National 
 prototype standards. Those apportioned to the United States were received in 1890 and are in the keeping of this office. 
 
 The metric system was legalized in the United States in 1866. 
 
 The International Standard Metre is derived from the Mfetre des Archives, and its length is defined by the dis- 
 tance between two lines at o'' Centigrade, on a platinum-iridium bar deposited at the International Bureau of Weights 
 and Measures. 
 
 The International Standard Kilogramme is a mass of platinum-iridium deposited at the same place, and its weight 
 in vacuo is the same as that of the Kilogramme des Archives. 
 
 The litre is equal to a cubic decimetre, and it is measured by the quantity of distilled water which, at its maximum 
 density, will counterpoise, the standard kilogramme in a vacuum, the volume of such a quantity of water being, as 
 nearly as has been ascertained, equal to a cubic decimetre. 
 
 SMITMRniUI AIM Xanl E-e 3 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.^ 
 
 »2 
 
 »8 
 
 slu 
 
 p 
 
 log. « 
 
 1 
 
 1000.000 
 
 I 
 
 I 
 
 1. 0000 
 
 1. 0000 
 
 0.00000 
 
 2 
 
 3 
 
 4 
 
 500.000 
 
 333-333 
 250.000 
 
 4 
 
 li 
 
 8 
 27 
 64 
 
 1.4142 
 1.7321 
 2.0000 
 
 1.2599 
 1.4422 
 1-5874 
 
 0.30103 
 0.47712 
 0,60206 
 
 5 
 
 6 
 
 9 
 
 200.000 
 166.667 
 142.857 
 125.000 
 III. Ill 
 
 81 
 
 125 
 216 
 343 
 
 512 
 
 729 
 
 Z.2361 
 
 2-4495 
 2.6458 
 2.8284 
 3.0000 
 
 1.7 100 
 1.8171 
 1.9129 
 
 2.0000 
 2.0801 
 
 0.69897 
 0.77815 
 0.84510 
 0.90309 
 0.95424 
 
 10 
 
 100.000 
 
 100 
 
 1000 
 
 3-1623 
 
 2.1544 
 
 1.00000 
 
 11 
 
 12 
 
 '3 
 14 
 
 90.9091 
 
 833333 
 76.9231 
 71.4286 
 
 121 
 144 
 169 
 196 
 
 1331 
 1728 
 2197 
 2744 
 
 3-3166 
 3.4641 
 3-6056 
 3-7417 
 
 2.2240 
 2.2894 
 
 2.3513 
 2.4101 
 
 1.04139 
 I.079I8 
 
 I.I 1394 
 
 I.I46I3 
 
 15 
 
 i6 
 
 ;^ 
 19 
 
 66.6667 
 62.5000 
 58.8235 
 
 52.6316 
 
 225 
 
 289 
 
 324 
 361 
 
 3375 
 4096 
 
 4913 
 5832 
 6859 
 
 3-8730 
 4.0000 
 4.1231 
 4.2426 
 4-3589 
 
 2.4662 
 2.5198 
 
 2-S7I3 
 2.6207 
 2.6684 
 
 I.I7609 
 I.Z04I2 
 
 1.2304s 
 1.25527 
 1.27875 
 
 20 
 
 21 
 22 
 
 23 
 
 24 
 
 50.0000 
 47.6190 
 45-4545 
 43-4783 
 41.6667 
 
 400 
 441 
 484 
 529 
 576 
 
 8000 
 
 9261 
 
 10648 
 
 12167 
 
 13824 
 
 4.4721 
 4.5826 
 
 4-7958 
 4.8990 
 
 2.7144 
 2.7589 
 2.8020 
 2.8439 
 2.8845 
 
 1-30103 
 1.32222 
 
 1.34242 
 1.36173 
 
 1. 3802 1 
 
 25 
 
 26 
 
 29 
 
 40.0000 
 38.4615 
 37.0370 
 
 3S-7'43 
 34.4828 
 
 625 
 676 
 729 
 
 15625 
 
 19683 
 21952 
 24389 
 
 5.0000 
 5.0990 
 5.1962 
 S-2915 
 S-3852 
 
 2.9240 
 2.9625 
 3.0000 
 3.0366 
 3.0723 
 
 1.39794 
 1.41497 
 
 1.43136 
 1.44716 
 
 1.46240 
 
 30 
 
 31 
 32 
 33 
 34 
 
 33-3333 
 32.2581 
 31.2500 
 
 30.3030 
 29.4118 
 
 900 
 961 
 1024 
 1089 
 1156 
 
 27000 
 29791 
 32768 
 35937 
 39304 
 
 5-4772 
 
 5-6569 
 5-7446 
 5-8310 
 
 3.1072 
 3-1414 
 3-1748 
 3-2075 
 3-2396 
 
 1.47712 
 I.49I36 
 1-50515 
 I.5I85I 
 I.53I48 
 
 35 
 
 36 
 
 ^^ 
 39 
 
 28.5714 
 27.7778 
 27.0270 
 26.3158 
 25.6410 
 
 1225 
 1296 
 1369 
 1444 
 1521 
 
 42875 
 46656 
 
 50653 
 54872 
 59319 
 
 5-9161 
 6.0000 
 6.0828 
 6.1644 
 6.2450 
 
 3.27 1 1 
 
 3-3019 
 3-3322 
 3.3620 
 3-3912 
 
 1.54407 
 1-55630 
 
 1.56820 
 
 1.57978 
 
 1.59106 
 
 40 
 
 41 
 42 
 
 43 
 44 
 
 25.0000 
 24.3902 
 23.809s 
 23.2558 
 22.7273 
 
 1600 
 1681 
 1764 
 1849 
 1936 
 
 64000 
 68921 
 74088 
 
 79507 
 85184 
 
 6.3246 
 6.4031 
 6.4S07 
 
 6-5574 
 6-6332 
 
 3.4200 
 3.4482 
 3.4760 
 3-5034 
 3-5303 
 
 1.60206 
 1. 61 278 
 1.62325 
 1-63347 
 1.64345 
 
 45 
 
 46 
 
 49 
 
 22.2222 
 21.7391 
 21.2766 
 20.8333 
 20.4082 
 
 2025 
 2118 
 2209 
 2304 
 2401 
 
 91125 
 
 97336 
 103823 
 1 10592 
 117649 
 
 6.7082 
 
 6.9282 
 7.0000 
 
 3-6o88 
 3-6342 
 3-6593 
 
 1.65321 
 1.66276 ■ 
 I.672IO 
 I.68I24 
 1.69020 
 
 50 
 
 SI 
 Sz 
 53 
 S4 
 
 20.0000 
 19.6078 
 19.2308 
 18.8679 
 18.5185 
 
 2500 
 2601 
 2704 
 2809 
 2916 
 
 125000 
 132651 
 140608 
 148877 
 157464 
 
 7.0711 
 7.1414 
 7.2111 
 7.2801 
 7-3485 
 
 3.6840 
 3-7084 
 3-7325 
 3-7563 
 3-7798 
 
 1.69897 
 1-70757 
 
 I.7I600 
 1.72428 
 
 1-73239 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 « 
 
 1000.^ 
 
 «2 
 
 «8 
 
 J« 
 
 \n 
 
 log. » 
 
 
 55 
 
 18.1818 
 
 3025 
 
 166375 
 
 7.4162 
 
 3.8030 
 
 1.74036 
 
 
 S6 
 
 17.8571 
 
 3136 
 
 I756I6 
 
 7-4833 
 
 3-8259 
 
 1.74819 
 
 
 57 
 
 17-5439 
 
 3249 
 
 185193 
 
 7-5498 
 
 3-8485 
 
 1-75587 
 
 
 S8 
 
 17.2414 
 
 3364 
 
 I95I12 
 
 7.6158 
 
 3.8709 
 
 1-76343 
 
 
 59 
 
 16.9492 
 
 3481 
 
 205379 
 
 7.6811 
 
 3-8930 
 
 1.77085 
 
 
 60 
 
 16.6667 
 
 3600 
 
 216000 
 
 7.7460 
 
 3-9149 
 
 1-77815 
 
 
 6i 
 
 16.3934 
 
 3721 
 
 226981 
 
 7.8102 
 
 3-9365 
 
 1-78533 
 
 
 62 
 
 16.1290 
 
 3844 
 
 238328 
 
 7.8740 
 
 3-9579 
 
 1.79239 
 
 
 63 
 
 15-8730 
 
 3969 
 
 250047 
 
 7-9373 
 
 3-9791 
 
 1-79934 
 
 
 64 
 
 15.6250 
 
 4096 
 
 262144 
 
 8.0000 
 
 4.0000 
 
 1.80618 
 
 
 65 
 
 15.3846 
 
 4225 
 
 274625 
 
 8.0623 
 
 4.0207 
 
 1.81291 
 
 
 66 
 
 15-1515 
 
 4356 
 
 287496 
 
 8.1240 
 
 4.0412 
 
 1.81954 
 
 
 67 
 
 14.9254 
 
 4489 
 
 300763 
 
 8.1854 
 
 4.0615 
 
 1.82607 
 
 
 68 
 
 14.7059 
 
 4624 
 
 314432 
 
 8.2462 
 
 4.0817 
 
 ^■^W 
 
 
 69 
 
 14.4928 
 
 4761 
 
 328509 
 
 8.3066 
 
 4.1016 
 
 1-83885 
 
 
 70 
 
 14-2857 
 
 4900 
 
 343000 
 
 8.3666 
 
 4.1213 
 
 1.84510 
 
 
 71 
 
 14.0845 
 
 5041 
 
 357911 
 
 8.4261 
 
 4.1408 
 
 1.85126 
 
 
 72 i 
 
 13.8889 
 
 5184 
 
 373248 
 
 8-4853 
 
 4.1602 
 
 1-85733 
 
 
 73 
 
 13.6986 
 
 5329 
 
 389017 
 
 8.5440 
 
 4-1793 
 
 1-86332 
 
 
 74 
 
 13-5135 
 
 5476 
 
 405224 
 
 8.6023 
 
 4.1983 
 
 1.86923 
 
 
 75 
 
 13-3333 
 
 5625 
 
 421875 
 
 8.6603 
 
 4.2172 
 
 1.87506 
 
 
 76 
 
 13-1579 
 
 5776 
 
 438976 
 
 8.7178 
 
 4.2358 
 
 i.88n8i 
 
 
 77 
 
 12.9870 
 
 5929 
 
 456533 
 
 fi^So 
 
 4-2543 
 
 1.88649 
 
 
 78 
 
 1 2.8205 
 
 6084 
 
 474552 
 
 8.8318 
 
 4.2727 
 
 1.89209 
 
 
 79 
 
 12.6582 
 
 6241 
 
 493°39 
 
 8.8882 
 
 4.2908 
 
 1.89763 
 
 
 80 
 
 12.5000 
 
 6400 
 
 512000 
 
 8.9443 
 
 4.3089 
 
 1.90309 
 
 
 81 
 
 12-3457 
 
 656; 
 
 531441 
 
 9.0000 
 
 4.3267 
 
 1.90849 
 
 
 82 
 
 12.1951 
 12.0482 
 
 6724 
 
 551368 
 
 9-°554 
 
 4-3445 
 
 1.91381 
 
 
 83 
 
 6889 
 
 571787 
 
 9.1104 
 
 4-3621 
 
 1. 9 1 908 
 
 
 84 
 
 11.9048 
 
 7056 
 
 592704 
 
 9.1652 
 
 4-3795 
 
 1.92428 
 
 
 85 
 
 11.7647 
 
 7225 
 
 614125 
 
 9.219s 
 
 4.3968 
 
 1.92942 
 
 
 86 
 
 11.6279 
 
 7396 
 
 636056 
 
 9-2736 
 
 4.4140 
 
 1.93450 
 
 
 87 
 
 11.4943 
 
 7569 
 
 658503 
 
 9-3274 
 
 4.4310 
 
 1.93952 
 
 
 88 
 
 11.3636 
 
 7744 
 
 681472 
 
 9.3808 
 
 4.4480 
 
 1.94448 
 
 
 89 
 
 11.2360 
 
 7921 
 
 704969 
 
 9.4340 
 
 4.4647 
 
 1-94939 
 
 
 90 
 
 ii.iili 
 
 8100 
 
 729000 
 
 9.4868 
 
 4.4814 
 
 1-95424 
 
 
 91 
 
 10.9890 
 
 8281 
 
 753571 
 
 9-5394 
 
 4-4979 
 
 1.95904 
 
 
 92 
 
 10.8696 
 
 8464 
 
 778688 
 
 9-5917 
 
 4-5144 
 
 1-96379 
 
 
 93 
 
 10.7527 
 
 8649 
 
 804357 
 830584 
 
 9-6437 
 
 4-5307 
 
 1.96848 
 
 
 94 
 
 10.6383 
 
 8836 
 
 9.6954 
 
 4-5468 
 
 1-97313 
 
 
 95 
 
 10.5263 
 
 9025 
 
 85737s 
 
 9.7468 
 
 4-5629 
 
 1.97772 
 
 
 96 
 
 10.4167 
 
 9216 
 
 884736 
 
 9.7980 
 
 4.5789 
 
 1.98227 
 
 
 97 
 
 10.3093 
 
 9409 
 
 912673 
 
 9.8489 
 
 4-5947 
 
 1.98677 
 
 
 98 
 
 10.2041 
 
 9604 
 
 941192 
 
 9.8995 
 
 4.6104 
 
 1-99123 
 
 
 99 
 
 lO.IOIO 
 
 9801 
 
 970299 
 
 9.9499 
 
 4.6261 
 
 1-99564 
 
 
 100 
 
 10.0000 
 
 lOOOO 
 
 I 000000 
 
 10.0000 
 
 4.6416 
 
 2.00000 
 
 
 1 01 
 
 9.90099 
 
 I020I 
 
 1030301 
 
 10.0499 
 
 4.6570 
 
 2.00432 
 
 
 102 
 
 9.80392 
 
 10404 
 
 1061208 
 
 10.099s 
 
 4.6723 
 
 2.00860 
 
 
 103 
 
 9.70874 
 
 10609 
 
 1092727 
 
 10.1489 
 
 4.6875 
 
 2.01284 
 
 
 104 
 
 9.61538 
 
 I0816 
 
 1124864 
 
 10.1980 
 
 4-7027 
 
 2.01703 
 
 
 105 
 
 9.52381 
 
 11025 
 
 1157625 
 
 10.2470 
 
 4-7177 
 
 2.02119 
 
 
 106 
 
 9-43396 
 
 1 1236 
 
 1191016 
 
 10.2956 
 
 4-7326 
 
 2.02531 
 
 
 107 
 
 9-34579 
 
 11449 
 
 1225043 
 
 10.3441 
 
 4-7475 
 
 2.02938 
 
 
 108 
 
 9.25926 
 
 1 1664 
 
 1259712 
 
 10.3923 
 
 4.7622 
 
 2.03342 
 
 
 109 
 
 9-17431 
 
 11881 
 
 1295029 
 
 10.4403 
 
 4-7769 
 
 2.03743 
 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS. CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.1- 
 
 tfi 
 
 «' 
 
 ^n 
 
 %n 
 
 log. « 
 
 
 110 
 
 9.09091 
 
 I2I00 
 
 I33IOOO 
 
 10.4881 
 
 4.7914 
 
 2.04139 
 
 
 III 
 
 9.00901 
 
 1 2321 
 
 1367631 
 
 10.5357 
 
 4.8059 
 
 2.04532 
 
 
 112 
 
 8.92857 
 
 12544 
 
 1404928 
 
 10.5830 
 
 4.8203 
 
 2.04922 
 
 
 "3 
 
 8.84956 
 
 12769 
 
 1442897 
 
 10.6301 
 
 4.8346 
 
 2.05308 
 
 
 114 
 
 8.77193 
 
 12996 
 
 I481544 
 
 10.6771 
 
 4.8488 
 
 2.05690 
 
 
 115 
 
 8.69565 
 
 13225 
 
 1520875 
 
 10.7238 
 
 4.8629 
 
 2.06070 
 
 
 ii6 
 
 8.62069 
 
 13689 
 
 1560896 
 
 10.7703 
 
 4.8770 
 
 2.06446 
 
 
 117 
 
 8.54701 
 
 1601613 
 
 10.8167 
 
 4.8910 
 
 2.06819 
 
 
 118 
 
 8.47458 
 
 13924 
 
 1643032 
 
 10.8628 
 
 4.9049 
 
 2.07188 
 
 
 119 
 
 8.40336 
 
 I4161 
 
 1685I59 
 
 10.9087 
 
 4.9187 
 
 2.07555 
 
 
 120 
 
 8.33333 
 
 14400 
 
 1728000 
 
 10.9545 
 
 4.9324 
 
 2.07918 
 
 
 121 
 
 8.26446 
 
 I464I 
 
 177 1 561 
 
 11.0000 
 
 4.9461 
 
 Z.08279 
 
 
 122 
 
 8.19672 
 
 14884 
 
 1815848 
 
 11.0454 
 
 4-9597 
 
 2.08636 
 
 
 123 
 
 8.13008 
 
 I5129 
 
 1860867 
 
 11.0905 
 
 4-9732 
 
 2.08991 
 
 
 124 
 
 8.06452 
 
 15376 
 
 1906624 
 
 "•135s 
 
 4.9866 
 
 2.09342 
 
 
 125 
 
 8.00000 
 
 15625 
 
 I953I25 
 
 11.1803 
 
 5.0000 
 
 2.09691 
 
 
 126 
 
 7.93651 
 
 15876 
 
 2000376 
 
 11.2250 
 
 S-0133 
 
 2.10037 
 
 
 127 
 
 7.87402 
 
 16129 
 
 2O4B383 
 
 H.2694 
 
 5.0265 
 
 2.10380 
 
 
 128 
 
 7.81250 
 
 16384 
 
 2097152 
 2146689 
 
 "•3137 
 
 5.0397 
 
 2.10721 
 
 
 129 
 
 7.75194 
 
 16641 
 
 ".3578 
 
 5.0528 
 
 2.11059 
 
 
 130 
 
 7.69231 
 
 16900 
 
 2197000 
 
 11.4018 
 
 5.0658 
 5.0788 
 
 2.11394 
 
 
 131 
 
 7-63359 
 
 I7161 
 
 2248091 
 
 11.4455 
 
 2.1 1727 
 
 
 132 
 
 7.57576 
 
 17424 
 
 2299968 
 
 11.4891 
 
 5.0916 
 
 2.12057 
 
 
 133 
 
 7.51880 
 
 17689 
 
 2352637 
 
 11.5326 
 
 5.1045 
 
 2.12385 
 
 
 134 
 
 7.46269 
 
 17956 
 
 2406104 
 
 11.5758 
 
 5.1172 
 
 2.12710 
 
 
 135 
 
 7.40741 
 
 18225 
 
 2460375 
 
 11.6190 
 
 5.1299 
 
 2.13033 
 
 
 136 
 
 7.35294 
 
 18496 
 
 2515456 
 
 11.6619 
 
 5.1426 
 
 2-13354 
 
 
 '37 
 
 7.29927 
 
 18769 
 
 2571353 
 
 11.7047 
 
 5-1551 
 
 2.13672 
 
 
 138 
 
 7.24638 
 
 19044 
 
 2628072 
 
 "■7473 
 11.7898 
 
 5.1676 
 
 2. 1 3988 
 
 
 139 
 
 7.19424 
 
 19321 
 
 2685619 
 
 5.1801 
 
 2.I430I 
 
 
 140 
 
 7.14286 
 
 19600 
 
 2744000 
 
 11.8322 
 
 5-1925 
 
 2.14613 
 
 
 141 
 
 7.09220 
 
 19881 
 
 2803221 
 
 11.8743 
 
 2.14922 
 
 
 142 
 
 7.04225 
 
 20164 
 
 2863288 
 
 11.9164 
 
 5.2171 
 
 2.15229 
 
 
 143 
 
 6.99301 
 
 20449 
 
 2924207 
 
 11.9583 
 
 5-2293 
 
 2-15534 
 
 
 144 
 
 6.94444 
 
 20736 
 
 2985984 
 
 12.0000 
 
 5-2415 
 
 2.15836 
 
 
 145 
 
 6.89655 
 
 21025 
 
 3048625 
 
 12.0416 
 
 5-2536 
 
 2.16137 
 
 
 146 
 
 6.84932 
 
 21316 
 
 3II2I36 
 
 12.0830 
 
 5.2656 
 
 2.16435 
 
 
 ^''l 
 
 6.80272 
 
 21609 
 
 3176523 
 
 12.1244 
 
 5-2776 
 
 2.16732 
 
 
 148 
 
 6.75676 
 
 21904 
 
 3241792 
 
 12.1655 
 
 5.2896 
 
 2.17026 
 
 
 149 
 
 6.71141 
 
 22201 
 
 3307949 
 
 12.2066 
 
 5-3015 
 
 2.17319 
 
 
 150 
 
 6.66667 
 
 22500 
 22801 
 
 3375000 
 
 12.2474 
 
 S-3133 
 
 2.17609 
 
 
 151 
 
 6.62252 
 
 3442951 
 
 12.2882 
 
 S-3251 
 
 2.17898 
 
 
 152 
 
 6.57893 
 
 23104 
 
 351 1808 
 
 12.3288 
 
 5-3368 
 
 2.18184 
 
 
 153 
 
 6-53595 
 
 23409 
 
 3581577 
 
 12.3693 
 
 5-3485 
 
 2.18469 
 
 
 154 
 
 6.49351 
 
 23716 
 
 3652264 
 
 12.4097 
 
 5-3601 
 
 2.18752 
 
 
 155 
 
 156 
 
 6.45161 
 6.41026 
 
 24025 
 24336 
 
 3723875 
 3796416 
 
 12.4499 
 12.4900 
 
 5-3717 
 5-3832 
 
 2.19033 
 2.19312 
 2.19590 
 2.19866 
 2.20140 
 
 
 157 
 
 6.36943 
 
 24649 
 
 3869893 
 
 12.5300 
 
 5-3947 
 
 
 158 
 
 6.3291 1 
 
 24964 
 
 3944312 
 
 12.5698 
 
 5.4061 
 
 
 159 
 
 6.28931 
 
 25281 
 
 4019679 
 
 12.6095 
 
 S-4175 
 
 
 160 
 
 6.25000 
 
 25600 
 
 4096000 
 
 12.6491 
 
 5.4288 
 
 2.20412 
 
 
 161 
 
 6.21 1 18 
 
 25921 
 
 4173281 
 
 12.6886 
 
 5.4401 
 
 2.20683 
 2.20952 
 
 
 162 
 163 
 
 6.17284 
 
 26244 
 
 4251528 
 
 12.7279 
 
 5.4514 
 
 
 6.13497 
 
 26^96 
 
 4330747 
 
 12.7671 
 
 5.4626 
 
 2.21 2ig 
 2.21484 
 
 
 164 
 
 6.09756 
 
 4410944 
 
 12.8062 
 
 5-4737 
 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 K 
 
 1000.5- 
 
 «2 
 
 „3 
 
 ^« 
 
 ^ 
 
 log. n 
 
 165 
 
 1 66 
 
 leg 
 
 6.06061 
 6.02410 
 S.98802 
 5.95238 
 5.91716 
 
 27225 
 
 278^9 
 28224 
 28561 
 
 4492125 
 4574296 
 4657463 
 4741632 
 4826809 
 
 12.8452 
 12.8841 
 12.9228 
 12.9615 
 13.0000 
 
 5.4848 
 
 5-4959 
 5.5069 
 5-5178 
 5.5288 
 
 2.21748 
 
 2.22011 
 
 2.22272 
 2.22531 
 2.22789 
 
 170 
 
 171 
 
 172 
 
 173 
 174 
 
 5-88235 
 5.84795 
 
 5-81395 
 5-78035 
 
 5-74713 
 
 28900 
 29241 
 29584 
 29929 
 30276 
 
 4913000 
 50002 I I 
 5088448 
 
 S177717 
 526S024 
 
 1 3-0384 ■ 
 13.0767 
 13.1149 
 
 13-1529 
 13.1909 
 
 5-5397 
 5-5505 
 5-5613 
 5-S72I 
 5-5828 
 
 2.23045 
 2.23300 
 2-23553 
 
 2.23805 
 
 2.24055 
 
 175 
 
 176 
 177 
 178 
 179 
 
 5.71429 
 5.68182 
 5.64972 
 5.61798 
 5-58659 
 
 30625 
 30976 
 
 31329 
 31684 
 32041 
 
 5359375 
 5451776 
 5545233 
 5639752 
 5735339 
 
 13.2288 
 13.2665 
 13-3041 
 13-3417 
 13-3791 
 
 5-5934 
 5.6041 
 
 5-6147 
 5.6252 
 
 5-6357 
 
 2.24304 
 2.24551 
 2.24797 
 
 2.25042 
 2.25285 
 
 180 
 
 181 
 182 
 
 184 
 
 5-p4^6 
 
 5-49451 
 5.46448 
 
 5-43478 
 
 32400 
 32761 
 33124 
 33489 
 33856 
 
 5832000 
 
 6028568 
 6128487 
 6229504 
 
 13.4164 
 13-4536 
 13.4907 
 
 13-5277 
 13-5647 
 
 5.6462 
 
 2.25527 
 2.25768 
 
 2.26007 
 2.26245 
 2.26482 
 
 185 
 
 186 
 187 
 :88 
 189 
 
 5-40541 
 5-37634 
 5-34759 
 5-31915 
 5.29101 
 
 34225 
 34596 
 34969 
 
 35344 
 35721 
 
 6331625 
 6434856 
 
 6539203 
 6644672 
 6751269 
 
 13.6015 
 13.6382 
 13.6748 
 13-7113 
 13-7477 
 
 5.6980 
 s-7083 
 5.7185 
 
 5-7388 
 
 2.26717 
 
 2.26951 
 2.27184 
 2.27416 
 2.27646 
 
 190 
 
 191 
 192 
 
 193 
 194 
 
 5.26316 
 5.23560 
 S-20833 
 
 5-18135 
 5.15464 
 
 36100 
 36481 
 36864 
 37249 
 37636 
 
 6859000 
 6967871 
 7077888 
 7189057 
 7301384 
 
 13.7840 
 13.8203 
 13.8564 
 13.8924 
 13.9284 
 
 S-7489 
 5-7590 
 5.7690 
 5.7790 
 5-7890 
 
 2.2787s 
 
 2.28103 
 2.28330 
 2.28556 
 2.28780 
 
 195 
 
 196 
 
 III 
 199 
 
 5.12821 
 5.10204 
 5.07614 
 5-05051 
 5-02513 
 
 38025 
 38416 
 38809 
 39204 
 39601 
 
 7414875 
 7529536 
 7645373 
 7762392 
 7880599 
 
 13.9642 
 14.0000 
 
 14-0357 
 14.0712 
 14.1067 
 
 5;8i86 
 5-8383 
 
 2.29003 
 2.29226 
 
 2.29447 
 2.29667 
 2.29885 
 
 200 
 
 201 
 202 
 203 
 204 
 
 5.00000 
 4.97512 
 4.95050 
 4.9261 1 
 4.90196 
 
 40000 
 40401 
 40804 
 41209 
 41616 
 
 8000000 
 8120601 
 8242408 
 
 n3gS427 
 8489664 
 
 14.1421 
 14.1774 
 14.2127 
 14.2478 
 14.2829 
 
 5.8480 
 5-8578 
 
 5-8675 
 5.8771 
 5.8868 
 
 2.30103 
 
 2.30320 
 
 2-30535 
 2.30750 
 2.30963 
 
 205 
 
 206 
 
 209 
 
 4.87805 
 
 4-85437 
 4.83092 
 4.80769 
 4-78469 
 
 42025 
 
 42436 
 42849 
 43264 
 43681 
 
 8615:25 
 8741816 
 8869743 
 8998912 
 9129329 
 
 14.3178 
 14-3527 
 14-3875 
 14.4222 
 14.4568 
 
 5.8964 
 5-9059 
 5-9155 
 5.9250 
 
 5-9345 
 
 2.31175 
 2-31387 
 2-31597 
 
 2.31806 
 2.32015 
 
 210 
 
 211 
 212 
 
 213 
 214 
 
 4.76190 
 
 4-73934 
 4.71698 
 4.69484 
 4.67290 
 
 44100 
 44521 
 44944 
 45369 
 45796 
 
 9261000 
 
 9393931 
 9528128 
 
 9663597 
 9800344 
 
 14.4914 
 14.5258 
 14.5602 
 
 14-5945 
 14.6287 
 
 5-9439 
 5-9533 
 5.9627 
 
 5-9721 
 5.9814 
 
 2.32222 
 2.32428 
 2.32634 
 2.32838 
 2.33041 
 
 215 
 
 216 
 217 
 218 
 219 
 
 4.65116 
 4.62963 
 4.60829 
 4.58716 
 4.56621 
 
 46225 
 46656 
 47089 
 47524 
 47961 
 
 9938375 
 10077696 
 10218313 
 10360232 
 10503459 
 
 14.6629 
 14.6969 
 14.7309 
 14.7648 
 14.7986 
 
 5-9907 
 6.0000 
 6.0092 
 6.0185 
 6.0277 
 
 2.33244 
 
 2-33445 
 2.33646 
 2.33846 
 2.34044 
 
 Smithsonian Tabues. 
 
Table 3. 
 
 
 
 
 
 
 
 
 VALUES OF RECIPROCALS, 
 ROOTS, AND COMMON 
 
 SQUARES, CUBES, SQUARE ROOTS, CUBE 
 LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.^ 
 
 rfl 
 
 »8 
 
 in 
 
 ^ 
 
 log. n 
 
 
 220 
 
 221 
 222 
 223 
 224 
 
 4-54545 
 4.52489 
 4.50450 
 
 4.48431 
 4.46429 
 
 48400 
 48841 
 
 49284 
 49729 
 50176 
 
 10648000 
 IO793861 
 I 094 I 048 
 I 1089567 
 II239424 
 
 14.8324 
 14.8661 
 14.8997 
 14-9332 
 14-9666 
 
 6.0368 • 
 
 6.0459 
 
 6.0550 
 
 6.0641 
 
 6.0732 
 
 2.34242 
 2.34439 
 2-34635 
 2.34830 
 2.35025 
 
 
 225 
 
 226 
 227 
 228 
 229 
 
 4.44444 
 4.42478 
 4.40529 
 
 4;3668i 
 
 50625 
 51076 
 51529 
 51984 
 52441 
 
 I I 390625 
 1 1 543176 
 1 1 697083 
 
 12008989 
 
 15.0000 
 
 15.0665 
 15.0997 
 15-1327 
 
 6.0822 
 6.0912 
 6.1002 
 6.1091 
 6.1180 
 
 2.35218 
 2.3541 1 
 2.35603 
 
 2-35793 
 2.35984 
 
 
 230 
 
 231 
 232 
 
 233 
 234 
 
 4-34783 
 4.32900 
 
 4-31034 
 4.29185 
 4.27350 
 
 52900 
 53361 
 53824 
 54289 
 
 54756 
 
 I 2 167000 
 12326391 
 
 1 2487 1 68 
 12649337 
 I 28 I 2904 
 
 15.1658 
 15.1987 
 
 i5-23'5 
 15.2643 
 15.2971 
 
 6.1269 
 6.1358 
 6.1446 
 
 6-1534 
 6.1622 
 
 2.36173 
 2.36361 
 
 2.36549 
 2.36736 
 2.36922 
 
 
 235 
 
 236 
 237 
 238 
 239 
 
 4.25532 
 4.23729 
 4.21941 
 4.20168 
 4.18410 
 
 56169 
 56644 
 57121 
 
 12977875 
 13144256 
 '3312053 
 13481272 
 13651919 
 
 15-3297 
 15-3623 
 15.3948 
 15.4272 
 15.4596 
 
 6.1710 
 6.1797 
 6.1885 
 6.1972 
 6.2058 
 
 ■ 2.37107 
 2.37291 
 
 2-37475 
 2.37658 
 2.37840 
 
 
 240 
 
 241 
 
 242 
 
 243 
 244 
 
 4.16667 
 4.14938 
 4.13223 
 
 4- "523 
 4.09836 
 
 57600 
 58081 
 
 58564 
 59049 
 
 59536 
 
 13824000 
 I 3997 52 I 
 14172488 
 14348907 
 14526784 
 
 15.4919 
 15-5242 
 
 15-5563 
 15.5885 
 15.6205 
 
 6.2145 
 6.2231 
 
 6-2317 
 6.2403 
 6.2488 
 
 2.38021 
 2.38202 
 2.38382 
 2.38561 
 2-38739 
 
 
 245 
 
 246 
 247 
 248 
 249 
 
 4.08163 
 4.06504 
 4.04858 
 4.03226 
 4.01606 
 
 60025 
 60516 
 61009 
 61504 
 62001 
 
 14706125 
 14886936 
 15069223 
 15252992 
 15438249 
 
 15.6525 
 15.6844 
 15.7162 
 15.7480 
 15.7797 
 
 6.265S 
 6.2743 
 6.2828 
 6.2912 
 
 2.38917 
 2.39094 
 2.39270 
 
 2.39445 
 2.39620 
 
 
 250 
 
 251 
 252 
 
 253 
 254 
 
 4.00000 
 3.98406 
 3.96825 
 3-95257 
 3-93701 
 
 62500 
 63001 
 
 63504 
 64009 
 64516 
 
 15625000 
 1 58132 51 
 16003008 
 16194277 
 163S7064 
 
 15.8114 
 15.8430 
 
 15-8745 
 15.9060 
 
 15-9374 
 
 6.2996 
 6.3080 
 6.3164 
 6.3247 
 6.3330 
 
 2-39794 
 2.39967 
 2.40140 
 2.40312 
 2.40483 
 
 
 255 
 
 256 
 2S7 
 258 
 259 
 
 3-92157 
 3.90625 
 3.89105 
 
 3-87597 
 3.86100 
 
 65025 
 
 66049 
 66564 
 67081 
 
 16581375 
 16777216 
 
 16974593 
 17173512 
 
 17373979 
 
 15.9687 
 16.0000 
 16.0312 
 16.0624 
 16.0935 
 
 6.3496 
 
 6-3579 
 6.3661 
 
 6-3743 
 
 2.40654 
 2.40824 
 2.40993 
 2.41 162 
 2.41330 
 
 
 260 
 
 261 
 262 
 263 
 264 
 
 3.84615 
 3.83142 
 3.81679 
 3.80228 
 3.78788 
 
 67600 
 
 681 21 
 68644 
 69169 
 69696 
 
 17576000 
 17779581 
 17984728 
 18191447 
 18399744 
 
 16.1245 
 16.155s 
 16.1864 
 16.2173 
 16.2481 
 
 6-3825 
 ^•^907 
 6.3988 
 6.4070 
 6.4151 
 
 2.41497 
 2.41664 
 2.41830 
 2.41996 
 2.42160. 
 
 
 265 
 
 266 
 267 
 268 
 269 
 
 3-77358 
 3-75940 
 3-74532 
 3-73134 
 3-71747 
 
 70225 
 
 70756 
 71289 
 71824 
 72361 
 
 18609625 
 18821096 
 19034163 
 19248832 
 19465109 
 
 16.2788 
 16.3095 
 16.3401 
 16.3707 
 16.4012 
 
 6.4232 
 6.4312 
 6-4393 
 6-4473 
 6-4553 
 
 2.42325 
 2.42488 
 2.42651 
 2.42813 
 2-42975 
 
 
 270 
 
 271 
 272 
 
 273 
 274 
 
 3-70370 
 3.69004 
 3.67647 
 3.66300 
 3.64964 
 
 72900 
 73441 
 73984 
 74529 
 75076 
 
 19683000 
 19902511 
 20123648 
 20346417 
 20570824 
 
 16.4317 
 16.4621 
 16.4924 
 16.5227 
 16.5529 
 
 6-4633 
 6-4713 
 6.4792 
 6.4872 
 6.4951 
 
 2.43136 
 2.43297 
 
 2.43457 
 2.43616 
 
 2-43775 
 
 
 Smithsonian Tables. 
 
Table 3> 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 (/« 
 
 '^» 
 
 log. n 
 
 275 
 
 276 
 
 277 
 278 
 279 
 
 280 
 
 281 
 282 
 283 
 284 
 
 285 
 
 286 
 287 
 288 
 289 
 
 290 
 
 291 
 292 
 
 293 
 294 
 
 295 
 
 296 
 297 
 298 
 299 
 
 300 
 
 301 
 302 
 303 
 304 
 
 305 
 
 306 
 
 307 
 308 
 
 3°9 
 
 310 
 
 3" 
 312 
 
 313 
 314 
 
 315 
 
 316 
 
 318 
 319 
 
 320 
 
 321 
 322 
 323 
 
 324 
 
 325 
 
 326 
 327 
 328 
 329 
 
 3-63636 
 3.62319 
 3.61011 
 3-59712 
 3-58423 
 
 3-57143 
 3-55872 
 3.54610 
 3-53357 
 3-52113 
 
 3-50877 
 3.49650 
 
 3-48432 
 3-47222 
 3.46021 
 
 3.44828 
 
 3-43643 
 3.42466 
 3.41297 
 3.40136 
 
 3-38983 
 3-37838 
 3.36700 
 3-35570 
 3-34448 
 
 3-33333 
 3.32226 
 3.31 1 26 
 
 3-30033 
 3.28947 
 
 3.27869 
 3.26797 
 
 3-25733 
 3.24675 
 3.23625 
 
 3.22581 
 3-21543 
 3-20513 
 3.19489 
 3. 1847 1 
 
 3.17460 
 3.16456 
 3-15457 
 3-14465 
 3.13480 
 
 3.12500 
 3.11527 
 
 3-10559 
 3.09598 
 3.08642 
 
 3.07692 
 3.06748 
 3.05810 
 3.04878 
 3-03951 
 
 75625 
 76176 
 76729 
 77284 
 77841 
 
 78400 
 78961 
 79524 
 
 80656 
 
 81225 
 81796 
 82369 
 82944 
 83521 
 
 84100 
 84681 
 85264 
 
 86436 
 
 87025 
 87616 
 88209 
 
 89401 
 
 goooo 
 90601 
 91204 
 91809 
 92416 
 
 93025 
 93636 
 94249 
 94864 
 95481 
 
 96100 
 96721 
 97344 
 97969 
 98596 
 
 99225 
 
 99856 
 
 100489 
 
 101124 
 
 101761 
 
 102400 
 103041 
 103684 
 104329 
 104976 
 
 105625 
 106276 
 106929 
 107584 
 108241 
 
 20796S75 
 21024570 
 
 21253933 
 21484952 
 21717639 
 
 21952000 
 22188041 
 22425768 
 22665187 
 22906304 
 
 23149125 
 23393656 
 23639903 
 23887872 
 24137569 
 
 24389000 
 246421 7 1 
 24897088 
 
 25153757 
 25412184 
 
 25672375 
 
 25934336 
 26198073 
 26463592 
 26730899 
 
 27000000 
 27270901 
 27543608 
 27818127 
 28094464 
 
 28372625 
 28652616 
 
 28934443 
 29218112 
 29503629 
 
 29791000 
 30080231 
 30371328 
 30664297 
 30959144 
 
 31255875 
 31554496 
 
 31855013 
 32157432 
 32461759 
 
 32768000 
 33076161 
 33386248 
 33698267 
 34012224 
 
 34328125 
 34645976 
 34965783 
 35287552 
 35611289 
 
 16.5831 
 
 6.5030 
 
 16.6132 
 
 6.5108 
 
 16.6433 
 
 6.5187 
 
 16.6733 
 
 6.5265 
 
 16.7033 
 
 6-5343 
 
 16.7332 
 
 6.5421 
 
 16.7631 
 
 6.5499 
 
 16.7929 
 
 6-5577 
 
 16.8226 
 
 6.5654 
 
 16.8523 
 
 6-5731 
 
 16.8819 
 
 6.5808 
 
 16.9115 
 
 6.5885 
 
 16.9411 
 
 6.5962 
 
 16.9706 
 
 6.6039 
 
 17.0000 
 
 6.6115 
 
 17.0294 
 17.0587 
 17.0880 
 17.1172 
 
 17.1464 
 17.1756 
 
 17.2047 
 
 17-2337 
 
 17.2627 
 17.2916 
 
 17.3205 
 
 17-3494 
 17.3781 
 
 17.4069 
 
 17-4356 
 
 17.4642 
 17.4929 
 17.5214 
 17.5499 
 17.5784 
 
 17.6068 
 17.6352 
 
 17.6635 
 17.6918 
 17.7200 
 
 17.7482 
 17.7764 
 17.8045 
 
 17.8326 
 17.8606 
 
 17.8885 
 17.9165 
 17.9444 
 
 17.9722 
 18.0000 
 
 18.0278 
 18.0555 
 
 18.0831 
 18.1108 
 
 18.1384 
 
 6.6191 
 6.6267 
 
 6-6343 
 
 6.6419 
 6.6494 
 
 6.6569 
 6.6644 
 6.6719 
 
 6.6794 
 6.6869 
 
 6.6943 
 6.7018 
 
 6.7092 
 6.7166 
 6.7240 
 
 6-7313 
 6-7387 
 
 6.7460 
 
 6-7533 
 6.7606 
 
 6.7679 
 6.7752 
 6.7824 
 6.7897 
 6.7969 
 
 6.8041 
 6.81 13 
 6.8185 
 6.8256 
 6.8328 
 
 6.8399 
 6.8470 
 6.8541 
 6.8612 
 6.8683 
 
 6-8753 
 6.8824 
 6.8894 
 6.8964 
 6-9034 
 
 2-43933 
 2.44091 
 2.44248 
 2.44404 
 2.44560 
 
 2.44716 
 2.44871 
 2.45025 
 2.45179 
 2.45332 
 
 2.45484 
 2.45637 
 2.45788 
 
 2-45939 
 2.46090 
 
 2.46240 
 2.46389 
 2.46538 
 2.46687 
 2.46835 
 
 2.46982 
 2.47129 
 2.47276 
 2.47422 
 2.47567 
 
 2.47712 
 2.47857 
 2.48001 
 2.48144 
 2.48287 
 
 2.48430 
 2.48572 
 2.48714 
 2.48855 
 2.48996 
 
 2.49136 
 2.49276 
 2.49415 
 
 2.49554 
 2-49693 
 
 2.49831 
 2.49969 
 2.50106 
 2.50243 
 2-50379 
 
 2.50515 
 2.50651 
 2.50786 
 2.50920 
 2.51055 
 
 2.51188 
 2.51322 
 
 2-51455 
 2.51587 
 2.51720 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 » 
 
 1000.^ 
 
 »2 
 
 «8 
 
 in 
 
 \n 
 
 log. « 
 
 330 
 
 3-03030 
 
 108900 
 
 35937000 
 
 18.1659 
 
 6.9104 
 
 2.51851 
 2.51983 
 
 331 
 
 3.02115 
 
 IO9561 
 
 36264691 
 
 18.1934 
 
 6.9174 
 
 332 
 
 3.01205 
 
 I 10224 
 
 36594368 
 
 18.2209 
 
 6.9244 
 
 2.52114 
 
 333 
 
 3.00300 
 
 I 10889 
 
 36926037 
 
 18.2483 
 
 6.9313 
 
 2.52244 
 
 334 
 
 2.99401 
 
 "I556 
 
 37259704 
 
 18.2757 
 
 6.9382 
 
 2.52375 
 
 335 
 
 2.98507 
 
 II2225 
 
 37595375 
 
 18.3030 
 
 6.9451 
 
 2.52504 
 
 336 
 
 2.97619 
 
 I I 2896 
 
 37933056 
 
 18.3303 
 
 6.9521 
 
 2.52634 
 
 337 
 
 2.96736 
 
 "3569 
 
 38272753 
 
 '§•3576 
 
 6.9589 
 
 2.52763 
 
 338 
 
 2.95858 
 2.94985 
 
 II4244 
 
 38614472 
 
 18.384S 
 
 6.9658 
 
 2.52892 
 
 339 
 
 II492I 
 
 38958219 
 
 18.4120 
 
 6.9727 
 
 2.53020 
 
 340 
 
 2.941 18 
 
 II560O 
 
 39304000 
 
 18.4391 
 
 6.979s 
 
 2.5314S 
 
 341 
 
 2-93255 
 2.92398 
 
 I16281 
 
 39651821 
 
 18.4662 
 
 6.9864 
 
 2-53275 
 
 342 
 
 1 1 6964 
 
 40001688 
 
 18.4932 
 
 6.9932 
 
 2-53403 
 
 343 
 
 2.91545 
 
 I 17649 
 
 40353607 
 
 18.5203 
 
 7.0000 
 
 2-53529 
 
 344 
 
 2.90698 
 
 I 18336 
 
 40707584 
 
 18.5472 
 
 7.0068 
 
 2.53656 
 
 345 
 
 2.89855 
 
 I I 9025 
 
 41063625 
 
 18.5742 
 
 7.0136 
 
 2.53782 
 
 346 
 
 ^•oi°i7 
 
 II9716 
 
 41421736 
 
 18.6011 
 
 7-0203 
 
 2.53908 
 
 347 
 
 2.88184 
 
 120409 
 
 41781923 
 
 18.6279 
 
 7.0271 
 
 2-54033 
 2.54158 
 2.54283 
 
 348 
 
 2.87356 
 
 I2II04 
 
 42144192 
 
 18.6548 
 18.6815 
 
 7-0338 
 
 349 
 
 2.86533 
 
 I2180I 
 
 42508549 
 
 7.0406 
 
 350 
 
 2.85714 
 
 122500 
 
 42875000 
 
 18.7083 
 
 7-0473 
 
 2.54407 
 
 351 
 
 2.84900 
 
 I232OI 
 
 43243551 
 
 18.7350 
 
 7-0540 
 
 2-54531 
 
 352 
 
 2.84091 
 
 123904 
 
 43614208 
 
 18.7617 
 
 7.0607 
 
 2-54654 
 
 353 
 
 2.83286 
 
 124609 
 
 43986977 
 
 18.7883 
 
 7.0674 
 
 2.54777 
 
 354 
 
 2.82486 
 
 I25316 
 
 44361864 
 
 18.8149 
 
 7-0740 
 
 2.54900 
 
 355 
 
 2.8i6go 
 
 126025 
 
 44738875 
 
 18.8414 
 
 7.0807 
 
 2.55023 
 
 356 
 
 2.80899 
 
 126736 
 
 45118016 
 
 18.8680 
 
 7-0873 
 
 2-55145 
 
 357 
 
 2.801 1 2 
 
 127449 
 
 45499293 
 
 18.8944 
 
 7.0940 
 
 2.55267 
 
 358 
 
 2.79330 
 
 128164 
 
 45882712 
 
 18.9209 
 
 7.1006 
 
 2.55388 
 
 359 
 
 2.78552 
 
 128881 
 
 46268279 
 
 18.9473 
 
 7.1072 
 
 2.55509 
 
 360 
 
 2.77778 
 
 129600 
 
 46656000 
 
 18.9737 
 
 7-1138 
 
 2.55630 
 
 361 
 
 2.77008 
 
 I3032I 
 
 47045881 
 
 19.0000 
 
 7.1204 
 
 2-55751 
 
 362 
 
 2.76243 
 
 I3IO44 
 
 47437928 
 
 19.0263 
 
 7.1269 
 
 2-55871 
 
 363 
 
 2.75482 
 
 I3I769 
 
 47832147 
 
 19.0526 
 
 7-1335 
 
 2-55991 
 
 364 
 
 2.74725 
 
 132496 
 
 48228544 
 
 19.0788 
 
 7.1400 
 
 2.56110 
 
 365 
 
 2-73973 
 
 133225 
 
 48627125 
 
 19.1050 
 
 7.1466 
 
 2.56229 
 
 366 
 
 2.73224 
 
 133956 
 134689 
 
 49027896 
 
 19.1311 
 
 7-1531 
 
 2.56348 
 
 ^% 
 
 2.72480 
 
 49430863 
 
 19-1572 
 I91833 
 
 
 2.56467 
 
 368 
 
 2.71739 
 
 135424 
 
 49836032 
 
 7.1661 
 
 2.56585 
 
 369 
 
 2.71003 
 
 I36161 
 
 50243409 
 
 19.2094 
 
 7.1726 
 
 2.56703 
 
 370 
 
 2.70270 
 
 136900 
 
 50653000 
 
 19.2354 
 
 7.1791 
 
 2.56820 
 
 371 
 
 2.69542 
 2.68817 
 
 137641 
 
 51064811 
 
 19.2614 
 
 7.1855 
 
 2.56937 
 
 372 
 
 138384 
 
 51478848 
 
 19.2873 
 
 7.1920 
 
 2.57054 
 
 373 
 
 2.68097 
 
 I39I29 
 
 51895117 
 
 19-3132 
 
 7-1984 
 
 2.57171 
 
 374 
 
 2.67380 
 
 139876 
 
 52313624 
 
 19-3391 
 
 7.2048 
 
 2.57287 
 
 375 
 
 2.66667 
 
 140625 
 
 52734375 
 
 19.3649 
 
 7.2112 
 
 2. i;740'j 
 
 376 
 
 2-65957 
 
 141376 
 
 53157376 
 
 19-3907 
 
 7.2177 
 
 2-57519 
 
 377 
 
 2.65252 
 
 I42129 
 
 53582633 
 
 19.4165 
 
 7.2240 
 
 2.57634 
 
 378 
 
 2.64550 
 2.63852 
 
 142884 
 
 54010152 
 
 19.4422 
 
 7.2304 
 
 2.57749 
 
 379 
 
 143641 
 
 54439939 
 
 19.4679 
 
 7.2368 
 
 2.57864 
 
 380 
 
 2.63158 
 
 144400 
 
 54872000 
 
 19-4936 
 
 7.2432 
 
 2.57978 
 
 ^i' 
 
 2.62467 
 
 I45161 
 
 55306341 
 
 19.5192 
 
 7.2495 
 
 2.1:8002 
 
 382 
 
 2.61780 
 
 145924 
 
 55742968 
 
 19.5448 
 
 7-2558 
 
 2.58206 
 2.58320 
 2-58433 
 
 383 
 
 2.61097 
 
 146689 
 
 56181887 
 
 19.5704 
 
 7.2622 
 
 384 
 
 2.60417 
 
 147456 
 
 56623104 
 
 19-5959 
 
 7.2685 
 
 Smithsonian 
 
 Tables. 
 
 
 
 
 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.^ 
 
 «2 
 
 «» 
 
 in 
 
 %n 
 
 log. « 
 
 3R5 
 
 386 
 
 389 
 
 2.59740 
 
 2.58398 
 2.57732 
 2.57069 
 
 148225 
 148996 
 
 149769 
 150544 
 
 I5I32I 
 
 57066625 
 
 57512456 
 57960603 
 58411072 
 58863869 
 
 19.6214 
 19.6469 
 19.6723 
 19.6977 
 19.7231 
 
 7.2748 
 7.2811 
 7.2874 
 7.2936 
 7.2999 
 
 2.58546 
 2.58659 
 2.58771 
 2.58883 
 2.58995 
 
 390 
 
 391 
 392 
 393 
 394 
 
 2.56410 
 
 2-S5754 
 2.55102 
 
 2.54453 
 2.53807 
 
 1 52100 
 152881 
 153664 
 
 154449 
 155236 
 
 59319000 
 
 59776471 
 60236288 
 60698457 
 61 162984 
 
 19.7484 
 19-7737 
 19.7990 
 19.8242 
 19.8494 
 
 7.3061 
 7-3124 
 7.3186 
 7.3248 
 7-3310 
 
 2.59106 
 2.59218 
 2.59329 
 
 2-59439 
 2.59550 
 
 395 
 
 396 
 
 398 
 399 
 
 2-53165 
 2.52C25 
 2.51889 
 2.51256 
 2.50627 
 
 156025 
 156816 
 157609 
 158404 
 159201 
 
 6162987s 
 62099136 
 62570773 
 63044792 
 63521199 
 
 19.8746 
 19.8997 
 19.9249 
 19.9499 
 19.9750 
 
 7-3372 
 7-3434 
 7-3496 
 7-3558 
 7-3619 
 
 2.59660 
 2.59770 
 2.59879 
 2.59988 
 2.60097 
 
 400 
 
 401 
 402 
 403 
 404 
 
 2.50000 
 
 2-49377 
 2.48756 
 2.48139 
 2-47525 
 
 160000 
 16080I 
 161604 
 162409 
 163216 
 
 64000000 
 64481201 
 64964808 
 65450827 
 65939264 
 
 20.0000 
 20.0250 
 20.0499 
 20.0749 
 20.0998 
 
 7-3681 
 7-3742 
 7-3803 
 7.3864 
 7-3925 
 
 2.60206 
 2.60314 
 2.60423 
 2.60531 
 2.60638 
 
 405 
 
 406 
 
 409 
 
 2.46914 
 2.46305 
 2.45700 
 2.45098 
 2-44499 
 
 164025 
 164836 
 165649 
 166464 
 167281 
 
 66430125 
 66923416 
 
 67419I43 
 679I73I2 
 68417929 
 
 20.1246 
 20.1494 
 20.1742 
 20.1990 
 20.2237 
 
 7-3986 
 7.4047 
 7.4108 
 7.4169 
 7.4229 
 
 2.60746 
 2.60853 
 
 2^61066 
 2.61172 
 
 410 
 
 411 
 412 
 
 413 
 414 
 
 2.43902 
 
 2.43309 
 2.42718 
 2.42 1 31 
 2.41546 
 
 168100 
 168921 
 
 169744 
 170569 
 171396 
 
 68921000 
 69426531 
 69934528 
 70444997 
 70957944 
 
 20.2485 
 20.2731 
 20.2978 
 20.3224 
 20.3470 
 
 7.4290 
 
 7-4350 
 7.4410 
 7.4470 
 7-4530 
 
 2.61278 
 2.61384 . 
 2.61490 
 2.6159s 
 2.61700 
 
 415 
 
 416 
 
 417 
 418 
 419 
 
 2.40964 
 2.40385 
 2.39808 
 
 172225 
 
 1738^9 
 174724 
 I75561 
 
 71473375 
 71991296 
 
 725II713 
 73034632 
 73560059 
 
 20.3715 
 20.3961 
 20.4206 
 20.4450 
 20.4695 
 
 7-4590 
 7.4650 
 7.4710 
 7.4770 
 7-4829 
 
 2.61805 
 2.61909 
 2.62014 
 2.621 1 8 
 2.62221 
 
 420 
 
 421 
 422 
 
 423 
 424 
 
 2.38095 
 2.37530 
 
 2.36407 
 2.35849 
 
 176400 
 177241 
 178084 
 178929 
 179776 
 
 74088000 
 74618461 
 
 76225024 
 
 20.4939 
 20.5183 
 20.5426 
 20.5670 
 20.5913 
 
 7-4889 
 7.4948 
 
 7-5007 
 7.5067 
 7.5126 
 
 2.62325 
 2.62428 
 2.62531 
 2.62634 
 2.62737 
 
 425 
 
 426 
 427 
 428 
 429 
 
 2.35294 
 2.34742 
 2.34192 
 
 2.3364s 
 2.33100 
 
 180625 
 181476 
 182329 
 183184 
 184041 
 
 76765625 
 77308776 
 
 77854483 
 78402752 
 
 78953589 
 
 20.6155 
 20.6398 
 20.6640 
 20.6882 
 20.7123 
 
 7-5185 
 7.5244 
 
 7-5302 
 7-5361 
 7.5420 
 
 2.62839 
 2.62941 
 2.63043 
 2.63144 
 2.63246 
 
 430 
 
 431 
 432 
 433 
 434 
 
 2.32558 
 2.32019 
 2.31481 
 
 2.30947 
 2.30415 
 
 184900 
 185761 
 186624 
 187489 
 188356 
 
 79507000 
 80062991 
 80621568 
 81182737 
 81746504 
 
 20.7364 
 20.7605 
 20.7846 
 20.8087 
 20.8327 
 
 7-5478 
 7-5537 
 7-5S9S 
 7-5654 
 7.5712 
 
 2-63347 
 2-63448 
 2.63548 
 2.63649 
 2-63749 
 
 435 
 
 436 
 437 
 438 
 439 
 
 2.29885 
 2.29358 
 2.28833 
 2.28311 
 
 2.27790 
 
 189225 
 190096 
 190969 
 191844 
 192721 
 
 82312875 
 82881856 
 
 83453453 
 84027672 
 84604519 
 
 20.8567 
 20.8806 
 20.9045 
 20.9284 
 20.9523 
 
 7-5770 
 7.5828 
 7.5886 
 
 7-5944 
 7.6001 
 
 2.63849 
 
 2-63949 
 2.64048 
 2.64147 
 2.64246 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, C 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS 
 
 CUBE 
 
 n 
 
 1000.^- 
 
 «2 
 
 «8 
 
 ^n 
 
 %n 
 
 log. n 
 
 440 
 
 2.27273 
 
 193600 
 
 85184000 
 
 20.9762 
 
 7.6059 
 
 2-64345 
 
 441 
 
 2.26757 
 
 194481 
 
 85766121 
 
 21.0000 
 
 7.6117 
 
 2.64444 
 
 442 
 
 Z.26244 
 
 195364 
 
 86350888 
 
 21.0238 
 
 7.6174 
 
 2.64542 
 
 443 
 
 2.25734 
 
 196249 
 
 86938307 
 
 21.0476 
 
 7.6232 
 7.6289 
 
 2.64640 
 
 444 
 
 2.25225 
 
 I97I36 
 
 87528384 
 
 21.0713 
 
 2.64738 
 
 445 
 
 2.24719 
 
 198025 
 
 88121125 
 
 21.0950 
 21.1187 
 
 7.6346 
 
 2.64836 
 
 446 
 
 2.24215 
 
 I 9891 6 
 
 88716536 
 
 7.6403 
 
 2.64933 
 
 447 
 
 2.23714 
 
 199809 
 
 89314623 
 
 21.1424 
 
 7.6460 
 
 2.65031 
 
 448 
 
 2.23214 
 
 200704 
 
 89915392 
 90518849 
 
 21.1660 
 
 7-6517 
 
 2.65128 
 
 449 
 
 2.22717 
 
 201601 
 
 21.1896 
 
 7-6574 
 
 2.6522s 
 
 450 
 
 2.22222 
 
 202500 
 
 91125000 
 
 21.2132 
 
 7-6631 
 
 2.65321 
 
 451 
 
 2.21730 
 
 203401 
 
 9173385I 
 
 21.2368 
 
 7.6688 
 
 2.65418 
 
 452 
 
 2.21239 
 
 204304 
 
 92345408 
 
 21.2603 
 21.2838 
 
 7.6744 
 
 2.65514 
 
 453 
 
 2.20751 
 
 205209 
 
 92959677 
 
 7.6801 
 
 2.65610 
 
 454 
 
 2.20264 
 
 2061 16 
 
 93576664 
 
 21.3073 
 
 7.6857 
 
 2.65706 
 
 455 
 
 2.19780 
 
 207025 
 
 9419637s 
 
 21.3307 
 
 7-6914 
 
 2.65801 
 
 456 
 
 2.19298 
 
 207936 
 
 94818816 
 
 21.3542 
 
 7.6970 
 
 2.65896 
 
 ^^57 
 
 2.I8818 
 
 208849 
 
 95443993 
 
 21.3776 
 
 7.7026 
 
 2.65992 
 
 458 
 
 2.18341 
 
 209764 
 
 96071912 
 
 21.4009 
 
 7.7082 
 
 2.66087 
 
 459 
 
 2.I7S65 
 
 210681 
 
 96702579 
 
 21.4243 
 
 7-7138 
 
 2.66181 
 
 460 
 
 2.I739I 
 
 211600 
 
 97336000 
 
 21.4476 
 
 7-7194 
 
 2.66276 
 
 461 
 
 2.16920 
 
 212521 
 
 97972181 
 
 21.4709 
 
 7.7250 
 
 2.66370 
 
 462 
 
 2.16450 
 
 213444 
 
 98611128 
 
 21.4942 
 
 7.7306 
 
 2.66464 
 
 463 
 
 2.15983 
 
 214369 
 
 99252847 
 
 21.5174 
 
 7.7362 
 
 2.66558 
 
 464 
 
 2-15517 
 
 215296 
 
 99897344 
 
 21.5407 
 
 7.7418 
 
 2.66652 
 
 465 
 
 2.15054 
 
 216225 
 
 100544625 
 
 21.5639 
 
 7-7473 
 
 2.6674s 
 
 466 
 
 2.14592 
 
 217156 
 
 101194696 
 
 21.5870 
 
 7-7529 
 
 2.66839 
 
 467 
 
 2-14133 
 
 218089 
 
 101847563 
 
 21.6102 
 
 7-7584 
 
 2.66932 
 
 468 
 
 2.13675 
 
 219024 
 
 J02503232 
 
 21.6333 
 
 7-7639 
 
 2.67025 
 
 469 
 
 2.13220 
 
 219961 
 
 103161709 
 
 21.6564 
 
 7.769s 
 
 2.67117 
 
 470 
 
 2.12766 
 
 220900 
 221841 
 
 103823000 
 
 21.679s 
 
 7-7750 
 
 2.67210 
 
 471 
 
 2.I23I4 
 
 104487111 
 
 21.7025 
 
 7.7805 
 
 2.67302 
 
 472 
 
 2.1 1864 
 
 222784 
 
 105154048 
 
 21.7256 
 
 7.7860 
 
 2.67486 
 
 473 
 
 2.11416 
 
 223729 
 
 105823817 
 
 21.7486 
 
 7-7915 
 
 474 
 
 2.10970 
 
 224677 
 
 106496424 
 
 21.77J5 
 
 7.7970 
 
 2.67578 
 
 475 
 
 2.10526 
 
 225625 
 
 107171875 
 
 21.7945 
 
 7-8025 
 
 2.67669 
 
 476 
 
 2.10084 
 
 226576 
 
 107850176 
 
 21.8174 
 
 7.8079 
 
 2.67761 
 2.67852 
 
 477 
 
 2.09644 
 
 227529 
 
 108531333 
 
 21.8403 
 
 7-8134 
 
 478 
 
 2.09205 
 2.08768 
 
 228484 
 
 109215352 
 
 21.8632 
 
 7.8188 
 
 2.67943 
 
 479 
 
 229441 
 
 109902239 
 
 21.8861 
 
 7.8243 
 
 2.68034 
 
 480 
 
 2.08333 
 
 230400 
 
 110592000 
 
 21.9089 
 
 7-8297 
 
 2.68124 
 
 481 
 
 2.07900 
 
 231361 
 
 11 1284641 
 
 21.9317 
 
 7-8352 
 
 2.6821s 
 
 482 
 
 2.07469 
 
 232324 
 
 111980168 
 
 21.9545 
 
 7.8406 
 
 2.68305 
 
 483 
 
 2.07039 
 
 233289 
 
 112678587 
 
 21.9773 
 
 7.8460 
 
 2.6839s 
 
 484 
 
 2.0661 2 
 
 234256 
 
 I 13379904 
 
 22.0000 
 
 7.8514 
 
 2.6848s 
 
 485 
 
 2.06186 
 
 235225 
 
 114084125 
 
 22.0227 
 
 7.8568 
 
 2.68574 
 2.68664 
 
 486 
 
 2.05761 
 
 236196 
 
 114791256 
 
 22.0454 
 22.0681 
 
 7.8622 
 
 487 
 
 2-0533? 
 
 237169 
 
 I 15501303 
 
 7-8676 
 
 2:i^li 
 
 488 
 
 2.04918 
 
 238144 
 
 116214272 
 
 22.0907 
 
 7-87^4 
 
 489 
 
 2.04499 
 
 2391 2 I 
 
 116930169 
 
 22.1133 
 
 2.68931 
 
 490 
 
 491 
 
 2.04082 
 2.03666 
 
 240100 
 241081 
 
 I 17649000 
 118370771 
 
 22.1359 
 22.1^85 
 
 22.l8ll 
 
 7-^f37 
 7.8891 
 
 2.69020 
 2.69108 
 2.69197 
 2.69285 
 2.69373 
 
 492 
 
 2.03252 
 
 242064 
 
 119095488 
 
 7.8944 
 
 493 
 
 2.02840 
 
 243049 
 
 119823157 
 
 22.2036 
 
 7.8998 
 
 494 
 
 2.02429 
 
 244036 
 
 120553784 
 
 22.2261 
 
 7-9051 
 
 Smithsonian 
 
 Tables. 
 
 1 
 
 
 
 
 
 12 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.'- 
 
 rfi 
 
 »8 
 
 V« 
 
 ^n 
 
 log. n 
 
 
 495 
 
 496 
 
 499 
 
 2.02020 
 2.01613 
 2.01207 
 2.00803 
 2.00401 
 
 245025 
 246016 
 247009 
 248004 
 249001 
 
 121287375 
 122023936 
 122763473 
 123505992 
 124251499 
 
 22.2486 
 22.2711 
 22.2935 
 
 22.3159 
 22.3383 
 
 7-9105 
 7.9158 
 7.9211 
 7.9264 
 7-9317 
 
 2.69461 
 2.69548 
 2.69636 
 2.69723 
 2.69810 
 
 
 500 
 
 SOI 
 S02 
 
 503 
 504 
 
 2.00000 
 1.99601 
 
 ;« 
 1.98413 
 
 250000 
 251001 
 252004 
 253009 
 254016 
 
 125000000 
 125751501 
 126506008 
 127263527 
 128024064 
 
 22.3607 
 22.3830 
 22.4054 
 22.4277 
 22.4499 
 
 7-9370 
 7.9420 
 7.9476 
 7.9528 
 7.9581 
 
 2.69897 
 2.69984 
 2.70070 
 2.70157 
 2.70243 
 
 
 505 
 
 506 
 507 
 S08 
 509 
 
 1.98020 
 1.97628 
 1.97239 
 1.96850 
 1.96464 
 
 256036 
 257049 
 258064 
 259081 
 
 128787625 
 129554216 
 130323843 
 I3IO965I2 
 131872229 
 
 22.4722 
 22.4944 
 22.5167 
 22.5389 
 22.5610 
 
 719686 
 7-9739 
 7-9791 
 7-9843 
 
 2.70329 
 2.70415 
 2.70501 
 2.70586 
 2.70672 
 
 
 510 
 
 SI I 
 
 512 
 
 513 
 514 
 
 1.96078 
 1.95695 
 1.95312 
 1.94932 
 
 1-94553 
 
 260100 
 261121 
 262144 
 263169 
 264196 
 
 132651000 
 133432831 
 134217728 
 135005697 
 135796744 
 
 22.5832 
 22.6053 
 22.6274 
 22.6495 
 22.6716 
 
 7.9896 
 7.9948 
 8.0000 
 8.0052 
 8.0104 
 
 2.70757 
 2.70842 
 2.70927 
 2.71012 
 2.71096 
 
 
 515 
 
 S16 
 
 S18 
 519 
 
 1-94175 
 1-93798 
 1.93424 
 1.93050 
 1.92678 
 
 266256 
 267289 
 268324 
 269361 
 
 136590875 
 137388096 
 138188413 
 138991832 
 139798359 
 
 22.6936 
 22.7 1 56 
 22.7376 
 22.7596 
 22.7816 
 
 8.0156 
 8.0208 
 8.0260 
 8.0311 
 8.0363 
 
 2.71181 
 2.7126s 
 2.71349 
 
 2.71433 
 2-71517 
 
 
 520 
 
 521 
 522 
 
 523 
 524 
 
 1.92308 
 1.91939 
 1.91571 
 1. 9 1 205 
 1.90840 
 
 270400 
 271441 
 272484 
 273529 
 274576 
 
 140608000 
 141420761 
 142236648 
 143055667 
 143877824 
 
 22.8035 
 22.8254 
 22.8473 
 22.8692 
 22.8910 
 
 8^0466 
 8.0517 
 8.0569 
 8.0620 
 
 2.71600 
 2.71684 
 2.71767 
 2.71850 
 2.71933 
 
 
 525 
 
 526 
 527 
 528 
 529 
 
 1.90476 
 1.90114 
 
 1-89753 
 1.89394 
 1.89036 
 
 275625 
 276676 
 277729 
 ■ 278784 
 279841 
 
 144703125 
 
 146363183 
 
 I47197952 
 148035889 
 
 22.9129 
 
 22.9347 
 22.9565 
 22.9783 
 23.0000 
 
 8.0671 
 8.0723 
 8.0774 
 8.0825 
 8.0876 
 
 2.72016 
 2.72099 
 2.72181 
 2.72263 
 2.72346 
 
 
 530 
 
 531 
 532 
 533 
 534 
 
 1.88679 
 1.88324 
 1.87970 
 1.87617 
 r.87266 
 
 280900 
 281961 
 283024 
 284089 
 285156 
 
 148877000 
 149721291 
 150568768 
 
 I514I9437 
 152273304 
 
 23.0217 
 
 23-0434 
 23.0651 
 23.0868 
 23.1084 
 
 8.0927 
 8.0978 
 8.1028 
 8.1079 
 8.1130 
 
 2.72428 
 2.72509 
 2.72591 
 2.72673 
 2.72754 
 
 
 535 
 
 536 
 537 
 538 
 539 
 
 1.86916 
 1.86567 
 1.86220 
 1.85874 
 1.85529 
 
 286225 
 287296 
 288369 
 
 289444 
 290521 
 
 I53I30375 
 153990656 
 
 1 548541 53 
 155720872 
 156590819 
 
 23.1301 
 23.1517 
 
 23-1733 
 23.1948 
 23.2164 
 
 8.1180 
 8.1231 
 8.1281 
 
 2-72835 
 2.72916 
 
 2-72997 
 2.73078 
 
 2.73159 
 
 
 540 
 
 541 
 542 
 543 
 544 
 
 1.85185 
 1.84843 
 1.84502 
 1.84162 
 1.83824 
 
 291600 
 292681 
 293764 
 294849 
 295936 
 
 157464000 
 158 340421 
 159220088 
 160103007 
 160989184 
 
 23-2379 
 23.2594 
 23.2809 
 23.3024 
 23-3238 
 
 8.1483 
 
 8.1583 
 8.1633 
 
 2.73239 
 2.73320 
 2.73400 
 2.73480 
 2.73560 
 
 
 545 
 
 546 
 
 548 
 549 
 
 1.83486 
 1.83150 
 1. 828 1 5 
 1.82482 
 1.82149 
 
 297025 
 298116 
 299209 
 300304 
 301401 
 
 161878625 
 162771336 
 163667323 
 164566592 
 165469149 
 
 23-3452 
 23.3666 
 23.3880 
 23.4094 
 23-4307 
 
 8.1683 
 8.1733 
 ^■'783 
 8.1833 
 8.1882 
 
 2.73640 
 2.73719 
 
 2.73799 
 2.73878 
 
 2.73957 
 
 
 Smithsonian Tables. 
 
 13 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 « 
 
 1000.^ 
 
 «2 
 
 «3 
 
 V« 
 
 v« 
 
 log. » 
 
 
 550 
 
 552 
 553 
 554 
 
 1.81818 
 ; 1.81488 
 1.81159 
 1.80832 
 1.80505 
 
 302500 
 303601 
 304704 
 305809 
 306916 
 
 166375000 
 16728415I 
 
 169112377 
 I7OO3I464 
 
 23.4521 
 23-4734 
 23-4947 
 23.5160 
 
 23-5372 
 
 8.1932 
 8.1982 
 8.2031 
 8.2081 
 8.2130 
 
 2.74036 
 2.74115 
 2.74194 
 2.74273 
 
 2.74351 
 
 
 555 
 
 556 
 557 
 558 
 559 
 
 1.80180 
 1.79856 
 
 1-79533 
 1.79211 
 1.78891 
 
 308025 
 309136 
 310249 
 311364 
 312481 
 
 170953875 
 171879616 
 172808693 
 173741112 
 174676879 
 
 23.5584 
 
 23-5797 
 23.6008 
 23.6220 
 23.6432 
 
 8.2180 
 8.2229 
 8.2278 
 8.2327 
 8-2377 
 
 2.74429 
 2.74507 
 2.74586 
 2.74663 
 2.74741 
 
 
 560 
 
 562 
 563 
 564 
 
 1-78571 
 1.78253 
 1.77936 
 1.77620 
 1-77305 
 
 313600 
 314721 
 315844 
 316969 
 318096 
 
 I756160OO 
 176558481 
 177504328 
 178453547 
 I 794061 44 
 
 23-6643 
 23.6854 
 23.7065 
 23.7276 
 23.7487 
 
 8.2426 
 
 8.2475 
 8.2524 
 
 8-2573 
 8.2621 
 
 2.74819 
 2.74896 
 2.74974 
 2.75051 
 2.75128 
 
 
 565 
 
 566 
 
 567 
 568 
 
 569 
 
 1. 76991 
 1.76678 
 1.76367 
 1.76056 
 1-75747 
 
 319225 
 320356 
 321489 
 322624 
 323761 
 
 180362125 
 181321496 
 182284263 
 183250432 
 184220009 
 
 23.7697 
 23.7908 
 23.8118 
 23.8328 
 23-8537 
 
 8.2670 
 8.2719 
 8.2768 
 8.2816 
 8.2865 
 
 2.75205 
 2.75282 
 2.75358 
 2-75435 
 2.7551 1 
 
 
 570 
 
 571 
 572 
 573 
 574 
 
 1-75439 
 1-75131 
 1.74825 
 1.74520 
 1.74216 
 
 324900 
 326041 
 327184 
 328329 
 329476 
 
 185193000 
 1861694II 
 187149248 
 188132517 
 1891 19224 
 
 23-8747 
 23.8956 
 23.9165 
 
 23-9374 
 23-9583 
 
 8.2913 
 8.2962 
 8.3010 
 8.3059 
 8.3107 
 
 2.75664 
 2.75740 
 
 2.75815 
 2.75891 
 
 
 575 
 
 576 
 577 
 578 
 579 
 
 1-73913 
 1-736" 
 1-73310 
 1.73010 
 1.72712 
 
 330625 
 331776 
 332929 
 334084 
 335241 
 
 190109375 
 191102976 
 192100033 
 193100552 
 194104539 
 
 23.9792 
 24.0000 
 24.0208 
 24.0416 
 24.0624 
 
 8.3155 
 8.3203 
 8.3251 
 8.3300 
 8-3348 
 
 2.75967 
 2.76042 
 2.76118 
 
 1-& 
 
 
 580 
 
 581 
 582 
 
 584 
 
 1.72414 
 1.72117 
 1.71821 
 1-71527 
 1-71233 
 
 336400 
 337561 
 338724 
 339889 
 341056 
 
 195112000 
 196122941 
 197137368 
 198155287 
 199176704 
 
 24.0832 
 24-1039 
 24.1247 
 24.1454 
 24.1661 
 
 8.3396 
 
 8-3443 
 8.3491 
 
 §•3539 
 8-3587 
 
 2.76343 
 2.76418 
 2.76492 
 2.76567 
 2.76641 
 
 
 585 
 
 586 
 
 587 
 588 
 
 589 
 
 1.70940 
 1.70648 
 1.70358 
 1.70068 
 1.69779 
 
 342225 
 343396 
 344569 
 
 345744 
 346921 
 
 200201625 
 201230056 
 202262003 
 203297472 
 204336469 
 
 24.1868 
 24.2074 
 24.2281 
 24.2487 
 24.2693 
 
 8.3730 
 8-3777 
 8.3825 
 
 2.76716 
 
 2.76864 
 2.76938 
 2.77012 
 
 
 590 
 
 591 
 592 
 
 593 
 
 594 
 
 1.69492 
 1.69205 
 1.68919 
 1.68634 
 1.68350 
 
 348100 
 349281 
 350464 
 351649 
 352836 
 
 205379000 
 206425071 
 207474688 
 208527857 
 209584584 
 
 24.2899 
 24.3105 
 
 24-3311 
 24.3516 
 24.3721 
 
 8.3872 
 8.3919 
 8.3967 
 8.4014 
 8.4061 
 
 2.77085 
 
 2-77159 
 2.77232 
 2.77305 
 2-77379 
 
 
 595 
 
 596 
 597 
 598 
 599 
 
 1.68067 
 1.67785 
 1.67504 
 1.67224 
 1.66945 
 
 354025 
 355216 
 356409 
 357604 
 358801 
 
 210644875 
 211708736 
 212776173 
 
 213847192 
 214921799 
 
 24.3926 
 24.4131 
 
 24-4336 
 24.4540 
 
 24-4745 
 
 8.4108 
 
 8-4155 
 8.4202 
 
 8.4249 
 8.4296 
 
 2.77452 
 2.77525 
 
 2.77597 
 2.77670 
 
 2.77743 
 
 
 600 
 
 601 
 602 
 603 
 
 604 
 
 1.66667 
 1.66389 
 1.66113 
 1-65837 
 1-65563 
 
 360000 
 361 201 
 362404 
 363609 
 364816 
 
 216000000 
 217081801 
 218167208 
 219256227 
 220348864 
 
 24-4949 
 24-5153 
 24-5357 
 24.5561 
 24.5764 
 
 8-4343 
 8.4390 
 
 8.4437 
 8.4484 
 8.4530 
 
 2.77815 
 2.77887 
 2.77960 
 2.78032 
 2.78104 
 
 
 Smithsonian 
 
 Tables. 
 
 
 
 
 
 
 
 14 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 » 
 
 1000.J 
 
 «2 
 
 k3 
 
 v« 
 
 v« 
 
 log. K 
 
 
 605 
 
 606 
 
 609 
 
 1.65289 
 1.65017 
 1.64745 
 1.64474 
 1.64204 
 
 366025 
 367236 
 368449 
 369664 
 370881 
 
 221445125 
 222545016 
 223648543 
 224755712 
 225866529 
 
 24.5967 
 24.6171 
 24.6374 
 24.6577 
 24.6779 
 
 8.4577 
 8.4623 
 
 8.4670 
 
 8.4716 
 8.4763 
 
 2.78176 
 2.78247 
 2.78319 
 2.78390 
 2.78462 
 
 
 610 
 
 611 
 612 
 613 
 614 
 
 1.63666 
 1-63399 
 
 i!62866 
 
 372100 
 373321 
 
 374544 
 375769 
 376996 
 
 226981000 
 228099131 
 229220928 
 230346397 
 231475544 
 
 24.6982 
 
 24.7184 
 24.7386 
 24.7588 
 24.7790 
 
 8.4809 
 8.4856 
 8.4902 
 8.4948 
 8.4994 
 
 2.78533 
 2.78604 
 2.7867s 
 2.78746 
 2.78817 
 
 
 615 
 
 616 
 617 
 618 
 619 
 
 1.62602 
 1.62338 
 1.6207 s 
 1.61812 
 1.61551 
 
 378225 
 
 380689 
 381924 
 383161 
 
 232608375 
 233744896 
 234885113 
 236029032 
 237176659 
 
 24.7992 
 24.8193 
 
 24-8395 
 24.8596 
 24.8797 
 
 8.5040 
 
 8.5086 
 8.5132 
 8.5178 
 
 8.5224 
 
 2.78888 
 2.78958 
 2.79029 
 2.79099 
 2.79169 
 
 
 620 
 
 621 
 622 
 623 
 624 
 
 1. 61 290 
 1.61031 
 1.60772 
 1.60514 
 1.60256 
 
 384400 
 
 at 
 388129 
 389376 
 
 238328000 
 239483061 
 240641848 
 241804367 
 242970624 
 
 24.8998 
 24.9199 
 24-9399 
 
 24.9600 
 24.9800 
 
 8.5270 
 8.5316 
 8.5362 
 8.5408 
 
 8-5453 
 
 2.79239 
 2.79309 
 2.79379 
 2.79449 
 2.79518 
 
 
 625 
 
 626 
 
 628 
 629 
 
 1.60000 
 
 I-S9744 
 1.59490 
 1.59236 
 
 390625 
 391876 
 393129 
 394384 
 395641 
 
 244140625 
 245314376 
 246491883 
 247673152 
 248858189 
 
 25.0000 
 25.0200 
 25.0400 
 
 25.0599 
 25.0799 
 
 8.5499 
 8.5544 
 8.5590 
 
 8-5635 
 8.5681 
 
 2-79934 
 2.79657 
 2.79727 
 2.79796 
 2.7986s 
 
 
 630 
 
 631 
 632 
 633 
 634 
 
 1-58730 
 1.58479 
 1.58228 
 I-S7978 
 1.57729 
 
 396900 
 398161 
 
 399424 
 400689 
 401956 
 
 250047000 
 251239591 
 252435968 
 253636137 
 254840104 
 
 25.0998 
 25.1197 
 25.1396 
 25.1595 
 25.1794 
 
 8.5726 
 8.5772 
 8.5817 
 8.5862 
 8.5907 
 
 2-79934 
 2.8nnn3 
 2.80072 
 2.80140 
 2.80209 
 
 
 635 
 
 636 
 
 638 
 639 
 
 1.57480 
 
 \m 
 1.56740 
 1.5649s 
 
 403225 
 404496 
 405769 
 407044 
 408321 
 
 256047875 
 257259456 
 
 258474853 
 259694072 
 260917119 
 
 25.1992 
 
 25.2190 
 25.2389 
 
 25.2587 
 25.2784 
 
 8.5952 
 8-5997 
 
 8!6o88 
 8.6132 
 
 2.80277 
 2.80346 
 2.80414 
 2.80482 
 2.80550 
 
 
 640 
 
 641 
 642 
 
 643 
 644 
 
 1.56250 
 1.56006 
 1-55763 
 
 1-555" 
 1.55280 
 
 409600 
 410881 
 412164 
 413449 
 414736 
 
 262144000 
 263374721 
 264609288 
 
 265847707 
 267089984 
 
 25.2982 
 25.3180 
 
 25-3377 
 25-3574 
 25.3772 
 
 8.6177 
 8.6222 
 8.6267 
 8.6312 
 8.6357 
 
 2.80618 
 2.80686 
 2.80754 
 2.80821 
 2.80889 
 
 
 645 
 
 646 
 647 
 648 
 649 
 
 1-55039 
 1-54799 
 1.54560 
 1.54321 
 1.54083 
 
 416025 
 417316 
 418609 
 419904 
 421201 
 
 268336125 
 269586136 
 270840023 
 272097792 
 273359449 
 
 25.3969 
 25.4165 
 25.4362 
 25.4558 
 
 25-4755 
 
 8.6401 
 8.6446 
 8.6490 
 8-6535 
 8-6579 
 
 2.80956 
 2.81023 
 2.81090 
 2.81158 
 2.81224 
 
 
 650 
 
 652 
 
 ^" 
 654 
 
 1.53846 
 1.53610 
 1-53374 
 1-53139 
 1.52905 
 
 422500 
 423801 
 425104 
 426409 
 427716 
 
 274625000 
 275894451 
 277167808 
 278445077 
 279726264 
 
 25-4951 
 25-5147 
 25-5343 
 25-5539 
 25-5734 
 
 8.6624 
 8.6668 
 8.6713 
 
 8.6801 
 
 2.81291 
 2.81358 
 2.81425 
 2.81491 
 2.81558 
 
 
 655 
 
 656 
 
 659 
 
 1.52672 
 
 1-52439 
 1.52207 
 1.51976 
 1-51745 
 
 429025 
 430336 
 431649 
 432964 
 434281 
 
 281011375 
 282300416 
 
 283593393 
 284890312 
 286191179 
 
 25.5930 
 25.6125 
 25.6320 
 25.6515 
 25.6710 
 
 8.6845 
 8.6890 
 8.6934 
 8.6978 
 8.7022 
 
 2.81624 
 2.81690 
 
 2.81757 
 2.81823 
 2.81889 
 
 
 Smithsonian Tables. 
 
 IS 
 
Table 3. 
 
 
 
 
 
 
 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 « 
 
 1000.^ 
 
 »2 
 
 »3 
 
 ^n 
 
 %n 
 
 log. « 
 
 
 660 
 
 66i 
 662 
 663 
 664 
 
 1-51515 
 1.51286 
 
 1-51057 
 1.50830 
 1.50602 
 
 435600 
 436921 
 438244 
 
 439569 
 440896 
 
 287496000 
 288804781 
 290117528 
 
 291434247 
 292754944 
 
 25.6905 
 25.7099 
 25.7294 
 25.7488 
 25.7682 
 
 8.7066 
 8.7110 
 
 f-7'S4 
 8.7198 
 8.7241 
 
 2.81954 
 2.82020 
 2.82086 
 2.82151 
 2.82217 
 
 
 665 
 
 666 
 667 
 668 
 669 
 
 1.50376 
 1. 50150 
 1.49925 
 1. 49701 
 1-49477 
 
 442225 
 
 443556 
 444889 
 446224 
 447561 
 
 294079625 
 295408296 
 296740963 
 298077632 
 299418309 
 
 25.7876 
 25.8070 
 25.8263 
 25.8457 
 25.8650 
 
 8.7285 
 8.7329 
 8-7373 
 8.7416 
 8.7460 
 
 2.82282 
 ■ 2.82347 
 2.82413 
 2.82478 
 2.82543 
 
 
 670 
 
 671 
 672 
 
 673 
 674 
 
 1.49254 
 1.49031 
 1. 488 10 
 1.48588 
 1.48368 
 
 448900 
 450241 
 
 451584 
 452929 
 454276 
 
 300763000 
 3021 117 11 
 303464448 
 304821217 
 306182024 
 
 25.8844 
 25.9037 
 25.9230 
 25.9422 
 25.9615 
 
 8.7503 
 8.7547 
 8.7590 
 8.7634 
 8.7677 
 
 2.82607 
 2.82672 
 2.82737 
 2.82802 
 2.82866 
 
 
 675 
 
 676 
 677 
 678 
 679 
 
 1.48148 
 1.47929 
 1.47710 
 
 1-47493 
 1.47275 
 
 455625 
 456976 
 458329 
 
 46104I 
 
 307546875 
 308915776 
 310288733 
 311665752 
 313046839 
 
 25.9808 
 26.0000 
 26.0192 
 26.0384 
 26.0576 
 
 8.7721 
 8.7764 
 8.7807 
 8.7850 
 8.7893 
 
 2.82930 
 Z.82995 
 2.83059 
 2.83123 
 2.83187 
 
 
 680 
 
 681 
 682 
 683 
 684 
 
 1.47059 
 1.46843 
 1.46628 
 1.46413 
 1.46199 
 
 462400 
 463761 
 
 466489 
 467856 
 
 314432000 
 315821241 
 317214568 
 318611987 
 320013504 
 
 26.0768 
 26.0960 
 26.1151 
 26.1343 
 26.1534 
 
 5-7937 
 8.7980 
 8.8023 
 8.8066 
 8.8108 
 
 2.83251 
 2.83315 
 2.83378 
 2.83442 
 2.83506 
 
 
 685 
 
 686 
 687 
 688 
 689 
 
 1.4598s 
 1-45773 
 1.45560 
 
 1-45349 
 1-45138 
 
 469225 
 470596 
 471969 
 
 473344 
 474721 
 
 321419125 
 322828856 
 324242703 
 325660672 
 327082769 
 
 26.1725 
 26.1916 
 26.2107 
 26.2298 
 26.2488 
 
 8.8152 
 8.8194 
 8.8237 
 8.8280 
 8.8323 
 
 2.83569 
 2.83632 
 2.83696 
 
 2.83759 
 2.83822 
 
 
 690 
 
 691 
 692 
 
 693 
 694 
 
 1.44928 
 1.44718 
 1.44509 
 1.44300 
 1.44092 
 
 476100 
 477481 
 478864 
 480249 
 481636 
 
 328509000 
 
 329939371 
 331373888 
 332812557 
 334255384 
 
 26.2679 
 26.2869 
 26.3059 
 26.3249 
 26.3439 
 
 8.8366 
 8.8408 
 8.8451 
 
 8.8493 
 8.8536 
 
 2.83885 
 2.83948 
 2.84011 
 2.84073 
 2.84136 
 
 
 695 
 
 696 
 697 
 698 
 699 
 
 1-4388S 
 1.43678 
 1.43472 
 1.43266 
 1.43062 
 
 483025 
 
 485809 
 487204 
 488601 
 
 335702375 
 
 338608873 
 34OO6S392 
 341532099 
 
 26.3629 
 26.3818 
 26.4008 
 26.4197 
 26.4386 
 
 8.8578 
 8.8621 
 8.8663 
 8.8706 
 8.8748 
 
 2.84198 
 2.84261 
 
 2.84386 
 2.84448 
 
 
 700 
 
 701 
 702 
 
 703 
 704 
 
 1.42857 
 1.42653 
 1.42450 
 1.42248 
 1.42045 
 
 490000 
 491401 
 492804 
 494209 
 495616 
 
 343000000 
 344472IOI 
 345948408 
 347428927 
 348913664 
 
 26.4575 
 26.4764 
 26.4953 
 26.5141 
 26.533° 
 
 8.8790 
 8.8833 
 8.8875 
 8.8917 
 8.8959 
 
 2.84510 
 2.84572 
 2.84634 
 2.84696 
 2.84757 
 
 
 705 
 
 706 
 707 
 708 
 709 
 
 1.41844 
 1.41643 
 1-41443 
 1-41243 
 1.41044 
 
 497025 
 498436 
 499849 
 501264 
 502681 
 
 350402625 
 351895816 
 
 353393243 
 354894912 
 356400829 
 
 26.5518 
 26.5707 
 26.5895 
 26.6083 
 26.6271 
 
 8.9001 
 
 8.9043 
 8.908s 
 8.9127 
 8.9169 
 
 2.84819 
 2.84880 
 2.84942 
 2.85003 
 2.85065 
 
 
 710 
 
 711 
 712 
 713 
 714 
 
 1.40845 
 1.40647 
 1.40449 
 1.40252 
 1.40056 
 
 504100 
 505521 
 
 &t 
 509796 
 
 357911000 
 
 359425431 
 360944128 
 362467097 
 363994344 
 
 26.6458 
 26.6646 
 26.6833 
 26.7021 
 26.7208 
 
 8.9211 
 
 8-9253 
 8.9295 
 
 8-9337 
 8.9378 
 
 2.85126 
 2.85187 
 2.85248 
 2.85309 
 2.85370 ■ 
 
 
 16 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 « 
 
 1000.*- 
 
 »2 
 
 «8 
 
 ^« 
 
 \n 
 
 log. K 
 
 715 
 
 716 
 717 
 718 
 719 
 
 1.39860 
 1.3966s 
 1.39470 
 1.39276 
 1.39082 
 
 5II225 
 512656 
 514089 
 
 515524 
 516961 
 
 365525875 
 367061696 
 368601813 
 370146232 
 371694959 
 
 26.7395 
 26.7582 
 26.7769 
 
 26.7955 
 26.8142 
 
 8.9420 
 8.9462 
 8.9503 
 
 8.9545 
 8.9587 
 
 2.85431 
 2.85491 
 2.85552 
 2.85612 
 2.85673 
 
 720 
 
 721 
 722 
 
 723 
 724 
 
 1.38889 
 1.38696 
 1.38504 
 
 1-38313 
 1.38122 
 
 518400 
 
 519841 
 521284 
 
 522729 
 524176 
 
 373248000 
 374805361 
 376367048 
 377933067 
 379503424 
 
 26.8328 
 26.8514 
 26.8701 
 26.8887 
 26.9072 
 
 8.9628 
 8.9670 
 
 8-97" 
 8.9752 
 8.9794 
 
 2-85733 
 2.85794 
 
 2.85854 
 2.85914 
 2.85974 
 
 725 
 
 726 
 727 
 728 
 729 
 
 I-3793I 
 I-3774I 
 I -37 552 
 1-37363 
 1-37174 
 
 525625 
 527076 
 528529 
 529984 
 53I44I 
 
 381078125 
 382657176 
 
 384240583 
 385828352 
 387420489 
 
 26.9258 
 26.9444 
 26.9629 
 26.9815 
 27.0000 
 
 f-9^3S 
 8.9876 
 8.9918 
 8.9959 
 g.oooo 
 
 2.86034 
 2.86094 
 2.86153 
 2.86213 
 2.86273 
 
 730 
 
 731 
 732 
 733 
 734 
 
 1.36986 
 1.36799 
 1.36612 
 1.36426 
 1.36240 
 
 532900 
 534361 
 535824 
 537289 
 538756 
 
 389017000 
 39061789I 
 392223168 
 393832837 
 395446904 
 
 27.0185 
 27.0370 
 
 27-0555 
 27.0740 
 27.0924 
 
 9.0041 
 9.0082 
 9.0123 
 9.0164 
 9.0205 
 
 2.86332 
 2.86392 
 2.86451 
 2.86510 
 2.86570 
 
 735 
 
 736 
 
 738 
 739 
 
 1.36054 
 1.35870 
 1.35685 
 I-3550I 
 1-35318 
 
 540225 
 541696 
 543169 
 544644 
 54612I 
 
 398688256 
 400315553 
 401947272 
 403583419 
 
 27.1109 
 27.1293 
 27.1477 
 27.1662 
 27.1846 
 
 9.0246 
 9.0287 
 9.0328 
 9.0369 
 9.0410 
 
 2.86629 
 
 2.86688 
 2.86747 
 2.86806 
 2.86864 
 
 740 
 
 741 
 742 
 743 
 744 
 
 1-35135 
 1-34953 
 I-3477I 
 1.34590 
 1.34409 
 
 547600 
 549081 
 550564 
 552049 
 553536 
 
 405224000 
 406869021 
 408518488 
 410172407 
 411830784 
 
 27.2029 
 27.2213 
 27.2397 
 27.2580 
 27.2764 
 
 9-0450 
 9.0491 
 
 9-0532 
 9.0572 
 9.0613 
 
 2.86923 
 2.86982 
 2.87040 
 2.87099 
 2.87157 
 
 745 
 
 746 
 747 
 748 
 749 
 
 1.34228 
 1.34048 
 1.33869 
 1.33690 
 1-335" 
 
 556516 
 558009 
 
 559504 
 561001 
 
 413493625 
 415160936 
 416832723 
 418508992 
 420189749 
 
 27.2947 
 27.3130 
 
 27-3313 
 27.3496 
 
 27-3679 
 
 9.0654 
 9.0694 
 90735 
 9-0775 
 9.0816 
 
 2.87216 
 2.87274 
 2-87332 
 2.87390 
 2.87448 
 
 750 
 
 751 
 752 
 7S3 
 
 754 
 
 1-33333 
 1-33156 
 1.32979 
 1.32802 
 1.32626 
 
 562500 
 564001 
 
 565504 
 567009 
 568516 
 
 421875000 
 423564751 
 425259008 
 426957777 
 428661064 
 
 27.3861 
 27.4044 
 27.4226 
 27.4408 
 27.4591 
 
 9.0856 
 9.0896 
 
 9-0937 
 9.0977 
 9.1017 
 
 2.87506 
 2.87564 
 2.87622 
 2.87679 
 2.87737 
 
 755 
 
 756 
 757 
 758 
 759 
 
 1.32450 
 1.3227 s 
 1. 32100 
 1.31926 
 1-31752 
 
 570025 
 571536 
 573049 
 574564 
 576081 
 
 430368875 
 432081 21 6 
 
 433798093 
 4355I95I2 
 437245479 
 
 27-4773 
 27-4955 
 27.5136 
 27-5318 
 27.5500 
 
 9.1057 
 9.1098 
 9.1138 
 9.1178 
 9.1218 
 
 2.87795 
 2.87852 
 2.87910 
 2.87967 
 2.88024 
 
 760 
 
 761 
 762 
 
 763 
 764 
 
 1-31579 
 1.31406 
 
 1-31234 
 1.31062 
 1.30890 
 
 577600 
 
 579I2I 
 580644 
 582169 
 583696 
 
 438976000 
 
 440711081 
 
 442450728 
 444194947 
 
 445943744 
 
 27.5681 
 27.5862 
 
 27-6043 
 27.6225 
 27.6405 
 
 9.1258 
 9.1298 
 9-1338 
 9-1378 
 9-1418 
 
 2.88081 
 2.88138 
 2.8819s 
 2.88252 
 2.88309 
 
 765 
 
 766 
 767 
 768 
 769 
 
 1.30719 
 1.30548 
 1.30378 
 1.30208 
 1.30039 
 
 5882I9 
 589824 
 591361 
 
 447697125 
 449455096 
 451217663 
 452984832 
 454756609 
 
 27.6586 
 27.6767 
 27.6948 
 27.7128 
 27.7308 
 
 9-1458 
 9.1498 
 
 9-1537 
 9-1577 
 9.1617 
 
 2.88366 
 2.88423 
 2.88480 
 2.88536 
 2.88593 
 
 Smithsonian Tables. 
 
 17 
 
Table 3. 
 
 
 
 
 
 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.^ 
 
 k2 
 
 »s 
 
 in 
 
 v« 
 
 log. « 
 
 770 
 
 771 
 772 
 773 
 774 
 
 1.29870 
 1.29702 
 
 1-29534 
 1.29366 
 1.29199 
 
 592900 
 
 594441 
 595984 
 
 597529 
 599076 
 
 456533000 
 45831401 I 
 
 461889917 
 463684824 
 
 27.7489 
 27.7669 
 27.7849 
 27.8029 
 27.8209 
 
 9.1696 
 9.1736 
 
 9-I77S 
 9.1815 
 
 2.88649 
 2.88705 
 2.88762 
 2.88818 
 2.88874 
 
 775 
 
 776 
 777 
 778 
 779 
 
 1.29032 
 1.28866 
 1.28700 
 1.2853s 
 1.28370 
 
 600625 
 602176 
 603729 
 605284 
 606841 
 
 465484375 
 467288576 
 469097433 
 470910952 
 472729139 
 
 27.8388 
 27.8568 
 27.8747 
 27.8927 
 27.9106 
 
 9.1855 
 9.1894 
 9-1933 
 9-1973 
 9.2012 
 
 2.88930 
 2.88986 
 2.89042 
 2.89098 
 2.89154 
 
 780 
 
 781 
 
 782 
 
 783 
 784 
 
 1.28205 
 1.28041 
 1.27877 
 1.27714 
 1-27551 
 
 608400 
 609961 
 61 1 524 
 613089 
 614656 
 
 474552000 
 
 476379541 
 478211768 
 480048687 
 481890304 
 
 27.9285 
 27.9464 
 27.9643 
 27.9821 
 28.0000 
 
 9.2052 
 9.2091 
 9.2130 
 9.2170 
 9-2209 
 
 2.89209 
 2.89265 
 2.89321 
 2.89376 
 2.89432 
 
 785 
 
 786 
 
 788 
 789 
 
 1.27389 
 1.27226 
 1.27065 
 1.26904 
 1.26743 
 
 616225 
 617796 
 619369 
 620944 
 622521 
 
 483736625 
 485587656 
 487443403 
 489303872 
 491169069 
 
 28.0179 
 28.0357 
 28.0535 
 28.0713 
 28.0891 
 
 9.2248 
 9.2287 
 9.2326 
 
 9-2365 
 9.2404 
 
 2.89487 
 2.89542 
 2.89597 
 2.89653 
 2.89708 
 
 790 
 
 791 
 792 
 793 
 794 
 
 1.26582 
 1.26422 
 1.26263 
 1.26103 
 1.25945 
 
 624100 
 625681 
 627264 
 628849 
 630436 
 
 493039000 
 494913671 
 496793088 
 498677257 
 500566184 
 
 28.1069 
 28.1247 
 28.1425 
 28.1603 
 28.1780 
 
 9-2443 
 9.2482 
 9.2521 
 9.2560 
 9.2599 
 
 2.89763 
 2.89818 
 2.89873 
 2.89927 
 2.89982 
 
 795 
 
 796 
 797 
 798 
 799 
 
 1.25786 
 1.25628 
 1-25471 
 1-25313 
 1.25156 
 
 632025 
 633616 
 635209 
 636804 
 638401 
 
 502459875 
 
 504358336 
 506261573 
 508169592 
 510082399 
 
 28.1957 
 28.2135 
 28.2312 
 28.2489 
 28.2666 
 
 9.2638 
 9.2677 
 9.2716 
 9.2754 
 
 9-2793 
 
 2.90037 
 2.90091 
 2.90146 
 2.90200 
 2.9025s 
 
 800 
 
 801 
 802 
 803 
 804 
 
 1.25000 
 1.24844 
 1.24688 
 
 1-24533 
 1.24378 
 
 640000 
 641601 
 643204 
 644809 
 646416 
 
 512000000 
 513922401 
 515849608 
 517781627 
 519718464 
 
 28.2843 
 28.3019 
 28.3196 
 
 28.3373 
 28.3549 
 
 9.2832 
 9.2870 
 
 2.90309 
 2.90363 
 2.90417 
 2.90472 
 2.90526 
 
 805 
 
 806 
 
 809 
 
 1.24224 
 1.24069 
 1.23916 
 1.23762 
 1.23609 
 
 648025 
 649636 
 651249 
 652864 
 654481 
 
 521660125 
 523606616 
 
 525557943 
 527514112 
 529475129 
 
 28.3725 
 28.3901 
 28.4077 
 28.4253 
 28.4429 
 
 9.3025 
 
 9-3063 
 9.3102 
 9.3140 
 9-3179 
 
 2.90580 
 
 2.90741 
 2.90795 
 
 810 
 
 8n 
 
 812 
 
 814 
 
 1-23457 
 1.23305 
 
 1-23153 
 1.23001 
 1.22850 
 
 656100 
 657721 
 
 662596 
 
 531441000 
 533411731 
 535387328 
 537367797 
 539353144 
 
 28.4605 
 28.4781 
 28.4956 
 28.5132 
 28.5307 
 
 9-3217 
 9-3255 
 9.3294 
 
 9-3332 
 9-3370 
 
 2.90849 
 2.90902 
 2.90956 
 2.91009 
 2.91062 
 
 815 
 
 816 
 817 
 818 
 819 
 
 1.22699 
 1.22549 
 1.22399 
 1.22249 
 1.22100 
 
 664225 
 665856 
 667489 
 669124 
 670761 
 
 541343375 
 543338496 
 545338513 
 547343432 
 549353259 
 
 28.5482 
 28.5657 
 28.5832 
 28.6007 
 28.6182 
 
 9.3408 
 9-3447 
 9-3485 
 9-3523 
 9-3561 
 
 2.91 1 16 
 2.91169 
 2.91222 
 2.9127s 
 2.91328 
 
 820 
 
 821 
 822 
 823 
 824 
 
 1.21951 
 1. 2 1 803 
 1.21655 
 1.21507 
 I.2I359 
 
 672400 
 674041 
 675684 
 677329 
 678976 
 
 551368000 
 553387661 
 555412248 
 557441767 
 559476224 
 
 28.6356 
 28.6531 
 28.6705 
 28.6880 
 28.7054 
 
 9-3599 
 9-3637 
 9-3675 
 9-3713 
 9-3751 
 
 2.91381 
 2.91434 
 2.91487 
 2.91540 
 2.91593 
 
 Smithsonian Tables. 
 
 18 
 
VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS. C 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS 
 
 Tablg 3. 
 UARE ROOTS, CUBE 
 
 n 
 
 1000.^ 
 
 tfl 
 
 «= 
 
 ^n 
 
 V« 
 
 log. « 
 
 825 
 
 826 
 827 
 828 
 829 
 
 I.2Z2I2 
 1.21065 
 1. 2091 9 
 1.20773 
 1.20627 
 
 680625 
 682276 
 683929 
 685584 
 687241 
 
 561515625 
 563559976 
 565609283 
 567663552 
 569722789 
 
 28.7228 
 28.7402 
 28.7576 
 28.7750 
 28.7924 
 
 9-3789 
 9-3827 
 9-3865 
 9-3902 
 9-3940 
 
 2.91645 
 2.91698 
 2.91751 
 2.91803 
 2-91855 
 
 830 
 
 832 
 833 
 834 
 
 1.20482 
 1.20337 
 1. 20192 
 1.20048 
 I.I9904 
 
 688900 
 690561 
 692224 
 693889 
 695556 
 
 571787000 
 573856191 
 575930368 
 578009537 
 580093704 
 
 28.8097 
 28.8271 
 28.8444 
 28.8617 
 28.8791 
 
 9-3978 
 9.4016 
 
 9-4053 
 9.4091 
 9.4129 
 
 2.91908 
 2.91960 
 2.92012 
 2.92065 
 2.921 17 
 
 835 
 
 836 
 
 838 
 839 
 
 I. 197 60 
 I.I9617 
 
 I.19474 
 I.I9332 
 I.I919O 
 
 697225 
 698896 
 700569 
 702244 
 703921 
 
 58218287s 
 584277056 
 
 588480472 
 590589719 
 
 28.8964 
 
 28.9137 
 28.9310 
 28.9482 
 28.965s 
 
 9.4166 
 9.4204 
 9.4241 
 9.4279 
 9.4316 
 
 2.92169 
 2.92221 
 2.92273 
 2.92324 
 2.92376 
 
 840 
 
 841 
 842 
 
 843 
 844 
 
 1.19048 
 1.18906 
 1.1876s 
 1. 1 8624 
 I.18483 
 
 705600 
 707281 
 708964 
 710649 
 712336 
 
 592704000 
 594823321 
 596947688 
 599077107 
 601211584 
 
 28.9828 
 29.0000 
 29.0172 
 
 29-0345 
 29.0517 
 
 9-4354 
 9-4391 
 9-4429 
 9.4466 
 
 9-4503 
 
 2.92428 
 2.92480 
 2.92531 
 2.92583 
 2.92634 
 
 845 
 
 846 
 847 
 848 
 849 
 
 1-18343 
 1. 1 8203 
 1. 18064 
 1.1792s 
 I.17786 
 
 714025 
 715716 
 717409 
 
 719104 
 720801 
 
 603351 I 25 
 
 605495736 
 607645423 
 609800192 
 61 1960049 
 
 29.0689 
 29.0861 
 29.1033 
 29.1204 
 29.1376 
 
 9-4541 
 9.4578 
 
 9-4615 
 9.4652 
 9.4690 
 
 2.92686 
 2.92737 
 2.92788 
 2.92840 
 2.92891 
 
 850 
 
 852 
 
 853 
 
 854 
 
 I.I7647 
 1.17509 
 1-17371 
 1-17233 
 I.I7096 
 
 722500 
 724201 
 725904 
 727609 
 729316 
 
 614125000 
 616295051 
 618470208 
 620650477 
 622835864 
 
 29.1548 
 29.1719 
 29.1890 
 29.2062 
 29.2233 
 
 9-4727 
 9.4764 
 9.4801 
 9.4838 
 9.4875 
 
 2.92942 
 2.92993 
 2.93044 
 2.93095 
 2.93146 
 
 855 
 
 856 
 
 1% 
 858 
 
 859 
 
 1.16959 
 1. 16822 
 1. 16686 
 1. 16550 
 I.16414 
 
 731025 
 732736 
 
 734449 
 736164 
 737881 
 
 625026375 
 627222016 
 629422793 
 63 1 6287 1 2 
 633839779 
 
 29.2404 
 29.2575 
 29.2746 
 29.2916 
 29.3087 
 
 9.4912 
 
 9-4949 
 9.4986 
 9.5023 
 9.5060 
 
 2.93197 
 2.93247 
 2.93298 
 2-93349 
 2-93399 
 
 860 
 
 861 
 862 
 863 
 864 
 
 1.16279 
 I.16144 
 1. 16009 
 1-15875 
 1-15741 
 
 739600 
 741321 
 743044 
 744769 
 746496 
 
 636056000 
 638277381 
 640503928 
 642735647 
 644972544 
 
 29.3258 
 29.3428 
 29.3598 
 29.3769 
 29-3939 
 
 9-5097 
 9-5134 
 9.5171 
 
 9-5207 
 9.5244 
 
 2.93450 
 2.93500 
 
 2-93551 
 2.93601 
 
 2-93651 
 
 865 
 
 866 
 867 
 868 
 869 
 
 1. 1 5607 
 1-15473 
 1-15340 
 1.15207 
 1.15075 
 
 748225 
 
 749956 
 751689 
 
 753424 
 755161 
 
 647214625 
 649461896 
 
 651714363 
 653972032 
 656234909 
 
 29.4109 
 29.4279 
 29.4449 
 29.4618 
 29.4788 
 
 9-5281 
 9-5317 
 9-5354 
 9-5391 
 9-5427 
 
 2.93702 
 2.93752 
 2.93802 
 2.93852 
 2.93902 
 
 870 
 
 872 
 874 
 
 1-14943 
 1.14811 
 
 1.14679 
 1. 14548 
 1.14416 
 
 756900 
 
 762129 
 763876 
 
 658503000 
 6607763U 
 663054848 
 665338617 
 667627624 
 
 29.4958 
 295127 
 29.5296 
 29.5466 
 29-5635 
 
 9.5464 
 9-5501 
 9-5537 
 9-5574 
 9.5610 
 
 2.93952 
 2.94002 
 2.94052 
 2.94101 
 2.94151 
 
 875 
 
 876 
 
 878 
 879 
 
 1. 1 4286 
 
 1-14155 
 1. 14025 
 
 76562s 
 767376 
 769129 
 770884 
 772641 
 
 66992187s 
 672221376 
 674526133 
 676836152 
 6791 51439 
 
 29.5804 
 
 29-5973 
 29.6142 
 29.6311 
 29.6479 
 
 9.5647 
 9-5683 
 9-5719 
 9-5756 
 9.5792 
 
 2.94201 
 2.94250 
 2.94300 
 
 2.94349 
 2.94399 
 
 Smithsonian Tables. 
 
 19 
 
Table 3. 
 
 
 
 
 
 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 n 
 
 1000.^ 
 
 rfl 
 
 «8 
 
 in 
 
 > 
 
 log. n 
 
 880 
 
 88i 
 882 
 883 
 884 
 
 1.13636 
 1-13507 
 '•13379 
 1.13250 
 1.13122 
 
 774400 
 776161 
 777924 
 779689 
 781456 
 
 681472000 
 683797841 
 686128968 
 688465387 
 690807104 
 
 29.6648 
 29.6816 
 29.6985 
 
 29-7153 
 29.7321 
 
 9.5828 
 9.586s 
 9-5901 
 9-5937 
 9-5973 
 
 2.94448 
 2.94498 
 2.94547 
 2.94596 
 2.94645 
 
 885 
 
 886 
 887 
 888 
 889 
 
 ;:;^8%^ 
 1. 12740 
 1.12613 
 1.12486 
 
 783225 
 
 788544 
 790321 
 
 693154125 
 695506456 
 697864103 
 700227072 
 702595369 
 
 29-7489 
 29.7658 
 29.7825 
 29.7993 
 29.8161 
 
 9.6010 
 9.6046 
 9.6082 
 9.6118 
 9.6154 
 
 2.94694 
 
 2-94743 
 2,94792 
 2.94841 
 2.94890 
 
 890 
 
 891 
 892 
 
 893 
 894 
 
 1.12360 
 1.12233 
 1.12108 
 1.11982 
 1.11857 
 
 792100 
 793881 
 795664 
 
 797449 
 799236 
 
 704969000 
 
 707347971 
 709732288 
 712121957 
 714516984 
 
 29-8329 
 29.8496 
 29.8664 
 29.8831 
 29.8998 
 
 9.6190 
 9.6226 
 9.6262 
 9.6298 
 9-6334 
 
 2.94939 
 
 2.95036 
 2.95085 
 2.95134 
 
 895 
 
 896 
 
 899 
 
 1.11732 
 I.I 1607 
 1.11483 
 
 i-"3S9 
 1.1123s 
 
 801025 
 802816 
 804609 
 806404 
 808201 
 
 716917375 
 719323136 
 721734273 
 724150792 
 726572699 
 
 29.9166 
 
 29-9333 
 29.9500 
 29.9666 
 29-9833 
 
 9.6370 
 9.6406 
 9,6442 
 9.6477 
 9-6513 
 
 2.95182 
 2.95231 
 2.95279 
 2.95328 
 2.95376 
 
 900 
 
 901 
 902 
 
 903 
 904 
 
 I. mil 
 1. 10988 
 1. 10865 
 1. 10742 
 1.10619 
 
 810000 
 811801 
 813604 
 815409 
 817216 
 
 729000000 
 731432701 
 733870808 
 
 736314327 
 738763264 
 
 30.0000 
 30.0167 
 
 30-0333 
 30.0500 
 30.0666 
 
 9-6549 
 9-6585 
 9.6620 
 9.6656 
 9.6692 
 
 2.95424 
 2.95472 
 2.95521 
 2.95569 
 2.95617 
 
 905 
 
 906 
 
 909 
 
 1.10497 
 
 1-I037S 
 1. 102 54 
 1.10132 
 
 I.IOOII 
 
 819025 
 820836 
 822649 
 824464 
 826281 
 
 741 2 17625 
 743677416 
 746142643 
 748613312 
 751089429 
 
 30.0832 
 30.0998 
 30.1164 
 
 30-1330 
 30.1496 
 
 9.6727 
 9.6763 
 9.6799 
 9.6834 
 9.6870 
 
 2.95665 
 2.95713 
 2-95761 
 2.95809 
 2.95856 
 
 910 
 
 911 
 912 
 
 913 
 914 
 
 1.09890 
 1.09769 
 1.09649 
 
 1.09529 
 
 1.09409 
 
 828100 
 829921 
 831744 
 833569 
 835396 
 
 753571000 
 756058031 
 758550528 
 761048497 
 763551944 
 
 30.1662 
 30.1828 
 
 30-1993 
 30.2159 
 30.2324 
 
 9.6905 
 9.6941 
 9.6976 
 9.7012 
 9.7047 
 
 2.95904 
 2.95952 
 2.95999 
 2.96047 
 2.9609s 
 
 915 
 
 916 
 917 
 918 
 919 
 
 1.09290 
 I.09I70 
 
 I. 09051 
 1.08932 
 I.088I4 
 
 837225 
 839056 
 840889 
 842724 
 844561 
 
 766060875 
 768575296 
 771095213 
 773620632 
 776151559 
 
 30.2490 
 30.2655 
 30.2820 
 30.2985 
 30-3150 
 
 9.7082 
 9.7118 
 
 9-7 '53 
 
 9.7188 
 9.7224 
 
 2.96142 
 2.96190 
 2.96277 
 2.96284 
 2.96332 
 
 920 
 
 921 
 
 922 
 
 923 
 924 
 
 1.08696 
 1.08578 
 1.08460 
 1.08342 
 1.08225 
 
 846400 
 848241 
 850084 
 851929 
 853776 
 
 778688000 
 781229961 
 783777448 
 786330467 
 788889024 
 
 30-3315 
 30.3480 
 
 30-3645 
 30.3809 
 
 30-3974 
 
 9-7259 
 9.7294 
 
 9-7329 
 9-7364 
 9.7400 
 
 2.96379 
 2.96426 
 2.96473 
 2.96520 
 2.96567 
 
 925 
 
 926 
 927 
 928 
 929 
 
 1.08108 
 1.07991 
 1.07875 
 1.07759 
 1.07643 
 
 855625 
 857476 
 
 859329 
 861184 
 863041 
 
 791453125 
 794022776 
 796597983 
 799178752 
 801765089 
 
 30.4138 
 30.4302 
 
 30-4467 
 30.4631 
 
 30-4795 
 
 9-7435 
 9-7470 
 9-7505 
 9-7540 
 9-7575 
 
 2.96614 
 2.96661 
 2.96708 
 2.96755 
 2.96802 
 
 930 
 
 931 
 932 
 933 
 934 
 
 1.07527 
 
 1.07411 
 1.07296 
 1.07181 
 1.07066 
 
 864900 
 866761 
 868624 
 870489 
 872356 
 
 804357000 
 806954491 
 809557568 
 812166237 
 814780504 
 
 30-4959 
 30-5123 
 30.5287 
 30.5450 
 30.5614 
 
 9.7610 
 9.764s 
 9.7680 
 9-7715 
 9-775° 
 
 2.96848 
 2.9689s 
 2.96942 
 2.96988 
 2-97035 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 » 
 
 lOOO.j- 
 
 tfi 
 
 «8 
 
 in 
 
 %n 
 
 log. n 
 
 935 
 
 936 
 937 
 938 
 939 
 
 1.06952 
 
 1.06838 
 
 1.06724 
 I.066IO 
 1.06496 
 
 874225 
 876096 
 S77969 
 879844 
 881721 
 
 817400375 
 820025856 
 822656953 
 825293672 
 827936019 
 
 30.5778 
 30.5941 
 30.6105 
 30.6268 
 30.6431 
 
 9-7785 
 9.7819 
 
 9-7854 
 9.7889 
 9.7924 
 
 ■Z.Oi-JQ&Y 
 
 2.97128 
 
 2.97174 
 
 2.97220 
 
 2.97267 
 
 940 
 
 941 
 942 
 943 
 944 
 
 1.06383 
 1.06270 
 1. 06 1 57 
 1.06045 
 1.05932 
 
 883600 
 885481 
 .887364 
 889249 
 891 136 
 
 830584000 
 833237621 
 835896888 
 838561807 
 841232384 
 
 30-6594 
 30-6757 
 30.6920 
 30.7083 
 30.7246 
 
 9-7959 
 9-7993 
 9.8028 
 9.8063 
 9.8097 
 
 2-97313 
 
 2-97359 
 2.97405 
 2.97451 
 2.97497 
 
 945 
 
 946 
 
 947 
 948 
 
 949 
 
 1.05820 
 1.05708 
 1.05597 
 1.05485 
 1-05374 
 
 893025 
 894916 
 896809 
 898704 
 900601 
 
 843908625 
 846590536 
 849278123 
 
 851971392 
 854670349 
 
 30.7409 
 30-7571 
 30-7734 
 30.7896 
 30.8058 
 
 9.8132 
 9.8167 
 9.8201 • 
 9.8236 
 9.8270 
 
 2-97543 
 2.97589 
 
 2-97635 
 2.97681 
 2.97727 
 
 950 
 
 951 
 952 
 9S3 
 954 
 
 1.05263 
 1.05152 
 1.05042 
 1.04932 
 1.04822 
 
 902500 
 904401 
 906304 
 908209 
 910116 
 
 857375000 
 860085351 
 862801408 
 865523177 
 868250664 
 
 30.8221 
 
 30-8383 
 30.8545 
 30.8707 
 30.8869 
 
 9.8305 
 9-8339 
 
 9.8408 
 9.8443 
 
 2.97772 
 2.97818 
 2.97864 
 2.97909 
 2.97955 
 
 955 
 
 956 
 
 958 
 959 
 
 1.047 1 2 
 1.04603 
 1.04493 
 1.04384 
 1.04275 
 
 912025 
 
 913936 
 915849 
 917764 
 919681 
 
 870983875 
 873722816 
 876467493 
 879217912 
 881974079 
 
 30.9031 
 30.9192 
 
 30-9354 
 30.9516 
 30.9677 
 
 9-8477 
 9.8511 
 9.8546 
 9.8580 
 9.8614 
 
 2.98000 
 2.98046 
 2.98091 
 2.98137 
 2.98182 
 
 960 
 
 961 
 962 
 
 963 
 964 
 
 1. 041 67 
 1.04058 
 1.03950 
 1.03832 
 1-03734 
 
 921600 
 923521 
 925444 
 927369 
 929296 
 
 884736000 
 887503681 
 890277128 
 893056347 
 895841344 
 
 30-9839 
 31.0000 
 31.0161 
 31.0322 
 31.0483 
 
 9.8648 
 9.8683 
 9.8717 
 
 9.8785 
 
 2.98227 
 2.98272 
 2.98318 
 
 2.98408 
 
 965 
 
 966 
 
 969 
 
 1.03627 
 1.03520 
 
 1-03413 
 1.03306 
 1.03199 
 
 931225 
 
 933156 
 935089 
 937024 
 938961 
 
 898632125 
 901428696 
 904231063 
 907039232 
 909853209 
 
 31.0644 
 31.0805 
 31.0966 
 31.1127 
 31.1288 
 
 9.8819 
 
 9.8854 
 9.8888 
 9.8922 
 9.8956 
 
 2.98453 
 2.98498 
 2.98543 
 2.98588 
 2.98632 
 
 970 
 
 971 
 972 
 
 973 
 974 
 
 1.03093 
 1.02987 
 1. 02881 
 1.02775 
 1.02669 
 
 940900 
 942841 
 944784 
 946729 
 948676 
 
 912673000 
 91 5498611 
 918330048 
 92I167317 
 924010424 
 
 31.1448 
 31.1609 
 31.1769 
 31.1929 
 31.2090 
 
 9.8990 
 9.9024 
 9.9058 
 9.9092 
 9.9126 
 
 2.98677 
 2.98722 
 2.98767 
 2.98811 
 2.98856 
 
 975 
 
 976 
 
 977 
 978 
 
 979 
 
 1.02564 
 1.02459 
 1.02354 
 1.02249 
 1.02145 
 
 950625 
 952576 
 
 954529 
 956484 
 958441 
 
 926859375 
 9297 I 41 76 
 932574833 
 935441352 
 938313739 
 
 31.2250 
 31.2410 
 31.2570 
 31-2730 
 31.2890 
 
 9.9160 
 9.9194 
 9.9227 
 9.9261 
 9-9295 
 
 2.98900 
 
 2.99034 
 2.99078 
 
 980 
 
 981 
 982 
 
 983 
 984 
 
 1. 02041 
 I.OI937 
 1.01833 
 1.01729 
 1. 01 626 
 
 960400 
 962361 
 964324 
 966289 
 968256 
 
 941 192000 
 944076141 
 946966168 
 949862087 
 952763904 
 
 31.3050 
 31.3209 
 31-3369 
 
 31-3688 
 
 9-9329 
 99363 
 9.9396 
 9.9430 
 9-9464 
 
 2.99123 
 2.99167 
 2.99211 
 2.99255 
 2.99300 
 
 985 
 
 986 
 
 1 
 
 989 
 
 1-01523 
 1. 01420 
 
 1-01317 
 1.01215 
 1.01112 
 
 970225 
 972196 
 974169 
 976144 
 978121 
 
 955671625 
 958585256 
 961 504803 
 964430272 
 967361669 
 
 31-3847 
 31.4006 
 31.4166 
 
 31-4325 
 31.4484 
 
 9-9497 
 9-9531 
 9-9565 
 9.9598 
 9.9632 
 
 2.99344 
 2.99388 
 2.99432 
 2.99476 
 2.99520 
 
 Smithsonian Tables. 
 
Table 3. 
 
 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE 
 ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 
 
 » 
 
 1000.^ 
 
 «2 
 
 «' . 
 
 */« 
 
 ^ 
 
 log. » 
 
 990 
 
 991 
 992 
 993 
 994 
 
 995 
 
 996 
 997 
 998 
 999 
 
 1000 
 
 I.OIOIO 
 
 1.00908 
 1.00806 
 1.00705 
 1.00604 
 
 1.00503 
 
 1.00402 
 1.00301 
 
 1.00200 
 1. 00 1 00 
 
 I. OOOOO 
 
 980100 
 982081 
 984064 
 986049 
 988036 
 
 990025 
 
 992016 
 994009 
 996004 
 998001 
 
 I 000000 
 
 970299000 
 
 973242271 
 
 976I9I488 
 
 979146657 
 
 982107784 
 
 985074875 
 988047936 
 991026973 
 9940H992 
 997002999 
 
 lOOOOOOOOO 
 
 31-4643 
 31.4802 
 31.4960 
 
 31-51 19 
 31.5278 
 
 31-5436 
 
 31-5595 
 31-5753 
 31-59" 
 31.6070 
 
 31.6228 
 
 9.9666 
 9.9699 
 
 9-9733 
 9.9766 
 g.9800 
 
 9-9833 
 9.9865 
 9.9900 
 
 9-9933 
 9.9967 
 
 10.0000 
 
 2.99564 
 2.99607 
 2.99651 
 2-99695 
 2-99739 
 
 2.99782 
 2.99826 
 2.99870 
 2.99913 
 2.99957 
 
 3.00000 
 
 Smithsonian Tables. 
 
CIRCUMFERENCE 
 
 AND AREA OF CIRCLE 
 DIAMETER d. 
 
 IN 
 
 Table 4. 
 TERMS OF 
 
 ird 
 
 ^ird^ 
 
 i^d^ 
 
 ird 
 
 10 
 II 
 
 12 
 
 13 
 14 
 15 
 
 16 
 
 17 
 18 
 
 19 
 20 
 
 21 
 22 
 24 
 
 2S 
 26 
 
 27 
 
 28 
 29 
 30 
 
 31 
 32 
 33 
 
 34 
 36 
 
 37 
 38 
 39 
 
 31.416 
 
 34-558 
 37.699 
 
 40.841 
 43.982 
 47.124 
 
 50.265 
 53-407 
 56-549 
 
 59.690 
 62.832 
 65-973 
 
 69.115 
 72.257 
 75-398 
 
 78.540 
 8i.68i 
 84.823 
 
 87.965 
 91.106 
 94.248 
 
 97-389 
 100.53 
 103.67 
 
 106.81 
 iog.96 
 113.10 
 
 116.24. 
 119.38 
 122.52 
 
 78.5398 
 9S-°332 
 113.097 
 
 132.732 
 
 153-938 
 176.715 
 
 201.062 
 226.980 
 254.469 
 
 283.529 
 
 314-159 
 346.361 
 
 380.133 
 415.476 
 
 452-389 
 
 490.874 
 530-929 
 572-555 
 
 615.752 
 660.520 
 706.858 
 
 754.768 
 804.248 
 855.299 
 
 907.920 
 962.113 
 1017.88 
 
 1075.21 
 1134.11 
 1194.59 
 
 40 
 
 41 
 42 
 
 43 
 44 
 45 
 
 46 
 47 
 48 
 
 49 
 50 
 51 
 
 52 
 53 
 54 
 
 57 
 
 1^ 
 
 61 
 62 
 63 
 
 64 
 
 65 
 66 
 
 67 
 68 
 
 125.66 
 128.81 
 131-95 
 
 135-09 
 138-23 
 141-37 
 
 144-51 
 147-65 
 150.80 
 
 153-94 
 157.08 
 160.22 
 
 163.36 
 166.50 
 169.65 
 
 172.79 
 
 175-93 
 179.07 
 
 182.21 
 
 185-35 
 188.50 
 
 191.64 
 194.78 
 197.92 
 
 201.06 
 204.20 
 207-35 
 
 210.49 
 213.63 
 216.77 
 
 1256.64 
 1320.25 
 1385-44 
 
 1452.20 
 
 1520-53 
 1590.43 
 
 1661.90 
 
 1734-94 
 1809.56 
 
 1885.74 
 1963.50 
 2042.82 
 
 2123.72 
 2206.18 
 2290.22 
 
 2375-83 
 2463.01 
 2551.76 
 
 2642.08 
 
 2733-97 
 2827.43 
 
 2922.47 
 3019.07 
 3"7-25 
 
 3216.99 
 3318.31 
 3421.19 
 
 3525-65 
 3631.68 
 
 3739-28 
 
 70 
 
 71 
 72 
 
 73 
 74 
 75 
 
 76 
 77 
 78 
 
 80 
 81 
 
 82 
 83 
 
 85 
 86 
 
 87 
 
 88 
 
 89 
 90 
 
 91 
 92 
 93 
 
 94 
 
 96 
 
 97 
 98 
 99 
 
 219.91 
 223.05 
 226.19 
 
 229-34 
 232.48 
 235-62 
 
 238.76 
 241.90 
 245.04 
 
 248.19 
 251-33 
 254-47 
 
 257.61 
 260.75 
 263.89 
 
 267.04 
 270.18 
 273-32 
 
 276.46 
 279.60 
 282.74 
 
 285.88 
 289.03 
 292.17 
 
 295-31 
 298.45 
 
 301-59 
 
 304-73 
 307.88 
 311.02 
 
 3848-45 
 3959-19 
 4071.50 
 
 4185.39 
 4300-84 
 4417.86 
 
 4536-46 
 4656.63 
 4778.36 
 
 4901.67 
 5026.55 
 5153-00 
 
 5281.02 
 5410.61 
 5541-77 
 
 5674.50 
 5808.80 
 5944.68 
 
 6082.12 
 6221.14 
 6361-73 
 
 6503.88 
 6647.61 
 6792.91 
 
 6939-78 
 7088.22 
 7238.23 
 
 7389.81 
 7542.96 
 7697.69 
 
 Smithsonian Tables. 
 
 23 
 
Table 5. 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 0000 0043 0086 0128 0170 
 0414 04S3 0492 0531 0569 
 0792 0828 0864 0899 0934 
 1139 1173 1206 1239 1271 
 1461 1492 1523 1553 1584 
 
 1761 1790 1818 1847 1875 
 2041 2068 2095 2122 2148 
 2304 2330 2355 2380 2405 
 2553 2577 2601 2625 2648 
 2788 2810 2833 2856 2878 
 
 3010 3032 3054 3075 3096 
 3222 3243 3263 3284 3304 
 3424 3444 3464 3483 3502 
 3617 3636 3655 3674 3692 
 3802 3820 3838 3856 3874 
 
 3979 3997 4°i4 4031 4048 
 4150 4166 4183 4200 4216 
 43H 433° 4346 4362 4378 
 4472 4487 4S°2 4518 4533 
 4624 4639 4654 4669 4683 
 
 4771 4786 4800 4814 4829 
 4914 4928 4942 49SS 4969 
 5051 5065 5079 5092 5105 
 518s 5198 521 I 5224 5237 
 5315 5328 5340 S3S3 5366 
 
 5441 54S3 5465 5478 5490 
 5563 S575 5587 5599 56" 
 5682 5694 5705 5717 5729 
 5798 5809 5821 5832 5843 
 59" 5922 5933 5944 5955 
 
 6021 6031 6042 6053 6064 
 6128 6138 6149 6160 6170 
 6232 6243 6253 6263 6274 
 
 6335 634s 6355 6365 6375 
 6435 6444 6454 6464 6474 
 
 6532 6542 6551 6561 6571 
 
 6628 6637 6646 6656 6665 
 
 6721 6730 6739 6749 6758 
 
 6812 6821 6830 6839 6848 
 
 6902 691 I 6920 6928 6937 
 
 6990 6998 7007 7016 7024 
 7076 7084 7093 7101 71 10 
 7160 7168 7177 7185 7193 
 7243 7251 7259 7267 7275 
 7324 7332 7340 7348 7356 
 
 0212 0253 0294 0334 0374 
 0607 0645 0682 0719 07S5 
 0969 1004 1038 1072 I 106 
 1303 1335 1367 1399 143° 
 1614 1644 1673 1703 1732 
 
 1903 1931 1959 1987 2014 
 2175 2201 2227 2253 2279 
 2430 24SS 2480 2504 2529 
 2672 269s 2718 2742 2765 
 2900 2923 2945 2967 2989 
 
 3118 3139 3160 3181 3201 
 3324 3345 3365 3385 3404 
 3522 3541 3560 3579 3598 
 37" 3729 3747 3766 3784 
 3892 3909 3927 3945 3962 
 
 4065 4082 4099 41 16 4133 
 4232 4249 4265 4281 4298 
 
 4393 4409 4425 4440 4456 
 4548 4564 4579 4594 4609 
 4698 4713 4728 4742 4757 
 
 4843 4857 4871 4886 4900 
 
 4983 4997 50" 5024 5038 
 5119 5132 5145 5159 5172 
 5250 5263 5276 5289 5302 
 5378 5391 5403 5416 5428 
 
 5502 SSH 5527 5539 5551 
 5623 563s 5647 5658 5670 
 5740 5752 5763 5775 5786 
 5855 5866 5877 5888 5899 
 5966 5977 5988 5999 6010 
 
 6075 6085 6096 6107 61 17 
 6180 6191 6201 6212 6222 
 6284 6294 6304 6314 6325 
 6385 6395 6405 6415 6425 
 6484 6493 6503 6513 6522 
 
 6580 6590 6599 6609 6618 
 6675 6684 6693 6702 6712 
 6767 6776 6785 6794 6803 
 6857 6866 6875 6884 6893 
 6946 6955 6964 6972 6981 
 
 7033 7042 7050 7059 7067 
 7118 7126 7135 7143 7152 
 7202 7210 7218 7226 7235 
 7284 7292 7300 7308 7316 
 7364 7372 7380 7388 7396 
 
 Prop. Parts. 
 
 12 3 
 
 4 8 12 
 4 8 II 
 3 7 10 
 3 6 10 
 369 
 
 246 
 246 
 246 
 246 
 2 4 5 
 
 7 8 9 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 2 
 
 2 2 
 
 2 2 
 
 17 21 25 
 
 29 33 37 
 
 15 19 23 
 
 26 30 34 
 
 14 17 21 
 
 24 28 31 
 
 13 16 19 
 
 23 26 29 
 
 12 15 18 
 
 21 24 27 
 
 " 14 17 
 
 20 22 25 
 
 II 13 16 
 
 18 21 24 
 
 10 12 15 
 
 17 20 22 
 
 912 14 
 
 10 19 21 
 
 911 13 
 
 16 18 20 
 
 8 II 13 
 8 10 12 
 8 10 12 
 7 9" 
 7 9" 
 
 15 17 19 
 14 16 18 
 
 141517 
 131517 
 12 14 16 
 
 7 9 10 12 14 15 
 
 7 8 10 II 13 15 
 
 689 II 13 14 
 
 689 II 12 14 
 
 679 10 12 13 
 
 6 7 9 
 6 7 8 
 5 7 8 
 568 
 S 6 8 
 
 10 II 13 
 
 10 II 12 
 
 9 II 12 
 
 9 10 12 
 
 9 10 II 
 
 9 10 II 
 8 10 II 
 8 9 10 
 8 9 10 
 8 9 10 
 
 8 9 10 
 
 7 8 9 
 
 7 8 9 
 
 7 8 9 
 
 3 4 5 
 
 3 4 5 
 
 3 4 5 
 
 3 4 5 
 
 3 4 5 
 
 7 8 
 
 7 8 
 
 7 7 
 
 6 7 
 
 6 7 
 
 N. 
 
 123 456 789 
 
 Smithsonian Tables. 
 
 24 
 
LOGARITHMS OF NUMBERS. 
 
 Table 5. 
 
 N. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 Prop. Parts. 
 123 456 789 
 
 
 
 
 55 
 
 59 
 
 7404 7412 7419 7427 7435 
 7482 7490 7497 7505 7513 
 
 7559 7566 7574 7582 7589 
 7634 7642 7649 7657 7664 
 7709 7716 7723 7731 7738 
 
 7443 7451 7459 7466 7474 
 7520 7528 7536 7S43 7551 
 7597 7604 7612 7619 7627 
 7672 7679 7686 7694 770I 
 7745 7752 7760 7767 7774 
 
 122 
 1 2 2 
 I 2 2 
 112 
 I I 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 
 4 
 4 
 4 
 4 
 
 5 
 5 
 5 
 4 
 4 
 
 5 6 7 
 
 5 f 7 
 5 6 7 
 5 6 7 
 567 
 
 60 
 
 6i 
 62 
 
 P 
 64 
 
 7782 7789 7796 7803 7810 
 7853 7860 7868 7875 7882 
 7924 7931 7938 794S 7952 
 7993 8000 8007 8014 8021 
 8062 8069 8075 8082 8089 
 
 7818 7825 7832 7839 7846 
 7889 7896 7903 7910 7917 
 7959 7966 7973 7980 7987 
 8028 8035 8041 8048 8055 
 8096 8102 8109 8n6 8122 
 
 112 
 112 
 112 
 I I 2 
 I I 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 4 
 3 
 3 
 3 
 
 4 
 4 
 4 
 4 
 4 
 
 566 
 5 6 6 
 5 6 6 
 
 ^ 5 t 
 5 5 6 
 
 65 
 
 66 
 
 67 
 68 
 69 
 
 8129 8136 8142 8149 8156 
 8195 8202 8209 8215 8222 
 8261 8267 8274 8280 8287 
 P^l f33i 8338 8344 8351 
 8388 8395 8401 8407 8414 
 
 8162 8169 8176 8182 8189 
 8228 8235 8241 8248 8254 
 8293 8299 8306 8312 8319 
 8357 8363 8370 8376 8382 
 8420 8426 8432 8439 8445 
 
 112 
 I I 2 
 112 
 112 
 I I 2 
 
 3 
 3 
 3 
 3 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 4 
 4 
 4 
 4 
 
 5 5 6 
 5 5 6 
 5 5 6 
 456 
 456 
 
 70 
 
 71 
 72 
 
 73 
 74 
 
 8451 8457 8463 8470 8476 
 8513 8519 8525 8531 8S37 
 8573 8579 8585 8591 8S97 
 8633 8639 8645 8651 8657 
 8692 8698 8704 8710 8716 
 
 8482 8488 8494 8500 8506 
 8543 8549 8555 8561 8567 
 8603 8609 8615 8621 8627 
 8663 8669 867s 8681 8686 
 8722 8727 8733 8739 8745 
 
 112 
 112 
 112 
 112 
 112 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 
 4 
 4 
 4 
 4 
 
 4 5 6 
 4 5 5 
 4 5 5 
 4 5 5 
 4 5 5 
 
 75 
 
 76 
 77 
 78 
 79 
 
 8751 8756 8762 8768 8774 
 8808 8814 8820 8825 8831 
 8865 8871 8876 8882 8887 
 8921 8927 8932 8938 8943 
 8976 8982 8987 8993 8998 
 
 8779 8785 8791 8797 8802 
 8837 8842 8848 8854 8859 
 8893 8899 8904 8910 891 s 
 8949 8954 8960 8965 8971 
 9004 9009 9015 9020 9025 
 
 112 
 112 
 
 112 
 112 
 112 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 4 5 5 
 4 5 5 
 4 4 5 
 4 4 5 
 4 4 5 
 
 80 
 
 81 
 82 
 
 f3 
 84 
 
 9031 9036 9042 9047 9053 
 9085 9090 9096 9101 9106 
 9138 9143 9149 9154 9159 
 9191 9196 9201 9206 9212 
 9243 9248 9-253 9258 9263 
 
 9058 9063 9069 9074 9079 
 9112 9117 9122 9128 9133 
 9165 9170 917s 9180 9186 
 9217 9222 9227 9232 9238 
 9269 9274 9279 9284 9289 
 
 I I 2 
 112 
 112 
 112 
 112 
 
 2 
 2 
 2 
 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 3 
 
 4 4 5 
 4 4 5 
 4 4 5 
 4 4 5 
 4 4 5 
 
 85 
 
 86 
 
 87 
 88 
 89 
 
 9294 9299 9304 9309 9315 
 9345 9350 9355 9360 9365 
 9395 9400 9405 9410 941 s 
 9445 9450 9455 946o 9465 
 9494 9499 9504 9509 95i3 
 
 9320 9325 9330 933S 9340 
 9370 9375 9380 9385 9390 
 9420 9425 9430 9435 9440 
 9469 9474 9479 9484 9489 
 9518 9523 9528 9533 9538 
 
 I I 2 
 I I 2 
 Oil 
 Oil 
 Oil 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 4 5 
 4 4 5 
 3 4 4 
 3 4 4 
 3 4 4 
 
 90 
 
 91 
 92 
 
 93 
 94 
 
 9542 9547 9552 9557 9562 
 9590 9595 9600 9605 9609 
 9638 9643 9647 9652 9657 
 968s 9689 9694 9699 9703 
 9731 9736 9741 9745 97SO 
 
 9566 9571 9576 9581 9586 
 9614 9619 9624 9628 9633 
 9661 9666 9671 9675 9680 
 9708 9713 9717 9722 9727 
 9754 9759 9763 9768 9773 
 
 Oil 
 Oil 
 Oil 
 Oil 
 Oil 
 
 2 
 2 
 2 
 2 
 
 2 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 
 95 
 
 96 
 
 99 
 
 9777 9782 9786 9791 9795 
 9823 9827 9832 9836 9841 
 9868 9872 9877 9881 9886 
 9912 9917 9921 9926 9930 
 9956 9961 9965 9969 9974 
 
 9800 9805 9809 9814 9818 
 9845 9850 9854 9859 9863 
 9890 9894 9899 9903 9908 
 9934 9939 9943 9948 9952 
 9978 9983 9987 9991 9996 
 
 Oil 
 Oil 
 Oil 
 Oil 
 Oil 
 
 2 
 2 
 2 
 2 
 2 
 
 2 
 2 
 2 
 2 
 2 
 
 3 
 3 
 3 
 3 
 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 3 4 
 
 N. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 12 3 
 
 4 
 
 5 
 
 6 
 
 7 8 9 
 
 Smithsonian Tables. 
 
 25 
 
Table 6. 
 
 ANTILOCARITHMS. 
 
 L. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 
 Prop. Parts. 
 
 
 
 
 
 12 3 t a V 
 
 f 
 
 
 .00 
 
 1000 1002 1005 1007 1009 
 1023 1026 1028 1030 1033 
 
 1012 1014 1016 1019 1021 
 
 ] 
 
 I I I 
 
 222 
 
 
 .01 
 
 1035 1038 1040 1042 1045 
 
 1 
 
 I I I 
 
 222 
 
 
 .02 
 
 1047 1050 1052 1054 I0S7 
 
 1059 1062 1064 1067 1069 
 1084 1086 1089 1091 1094 
 
 1 
 
 I I I 
 
 222 
 
 
 •°3 
 
 1072 1074 1076 1079 lOSI 
 
 ) 
 
 III 
 
 222 
 
 
 .04 
 
 1096 1099 1102 1104 1107 
 
 1109 1112 IH4 1117 1H9 
 
 I I 
 
 112 
 
 222 
 
 
 .05 
 
 1122 1125 1127 1130 1132 
 
 H35 1 138 1 140 1 143 1 146 
 
 I 
 
 112 
 
 222 
 
 
 .06 
 
 1148 1151 1153 1156 1159 
 1175 1178 1180 1183 1186 
 
 1161 1164 1167 1169 1172 
 
 I 
 
 I I 2 
 
 222 
 
 
 .07 
 
 1189 iigi 1194 1197 1199 
 
 I 1 
 
 112 
 
 222 
 
 
 .08 
 
 1202 1205 1208 1211 1213 
 
 1216 1219 1222 1225 1227 
 
 I 
 
 112 
 
 2 2 3 
 
 
 .09 
 
 1230 1233 1236 1239 1242 
 
 1245 1247 1250 1253 1256 
 
 I 
 
 t 112 
 
 223 
 
 
 .10 
 
 1259 1262 1265 1268 1271 
 1288 1291 1294 1297 1300 
 
 1274 1276 1279 1282 1285 
 
 I 
 
 [ 112 
 
 223 
 
 
 .11 
 
 1303 1306 1309 1312 131 5 
 
 I 
 
 [ 12 2 
 
 2 2 3 
 
 
 .12 
 
 1318 1321 1324 1327 1330 
 
 1334 1337 1340 1343 1346 
 
 I 
 
 [ 12 2 
 
 223 
 
 
 •13 
 
 1349 1352 1355 1358 1361 
 1380 1384 1387 139° 1393 
 
 1365 1368 1371 1374 1377 
 
 I 
 
 I 12 2 
 
 2 3 3 
 
 
 .14 
 
 1396 1400 1403 1406 1409 
 
 I 
 
 [ I :i 2 
 
 2 3 3 
 
 
 .15 
 
 1413 1416 1419 1422 1426 
 
 1429 1432 1435 1439 1442 
 
 I 
 
 [ 12 2 
 
 2 3 3 
 
 
 .16 
 
 I44S 1449 "452 I4S5 1459 
 
 1462 1466 1469 1472 1476 
 
 I 
 
 t 12 2 
 
 2 3 3 
 
 
 •17 
 
 1479 1483 i486 1489 1493 
 
 1496 1500 1503 1507 1 510 
 
 I 
 
 12 2 
 
 233 
 
 
 .18 
 
 1514 1517 1521 1524 1528 
 
 1531 IS3S 1538 1542 1545 
 
 I 
 
 C 12 2 
 
 2 3 3 
 
 
 .19 
 
 1549 1552 1556 1560 1563 
 
 1567 1570 1574 1578 1 581 
 
 I 
 
 I 12 2 
 
 3 3 3 
 
 
 .20 
 
 1585 1589 1592 1596 1600 
 
 1603 1607 1611 1614 1618 
 
 I 
 
 [ 12 2 
 
 3 3 3 
 
 
 .21 
 
 1622 1626 1629 1633 1637 
 
 1641 1644 1648 1652 1656 
 
 I 
 
 I 2 2 2 
 
 3 3 3 
 
 
 .22 
 
 1660 1663 1667 1671 1675 
 
 1679 1683 1687 1690 1694 
 
 I 
 
 [ 2 2 2 
 
 3 3 3 
 
 
 •23 
 
 1698 1702 1706 1710 1714 
 
 1718 1722 1726 1730 1734 
 
 I 
 
 I 2 2 2 
 
 3 3 4 
 
 
 .24 
 
 1738 1742 1746 1750 1754 
 
 1758 1762 1766 1770 1774 
 
 I 
 
 [ 2 2 2 
 
 3 3 4 
 
 
 .25 
 
 1778 1782 1786 1791 1795 
 
 1799 1803 1807 181 I 1816 
 
 I 
 
 [ 2 2 2 
 
 3 3 4 
 
 
 .26 
 
 1820 1824 1828 1832 1837 
 
 1841 1845 1849 1854 1858 
 1884 1888 1892 1897 1901 
 
 I 
 
 t 223 
 
 3 3 4 
 
 
 .27 
 
 1862 1866 1871 187s 1879 
 
 I 
 
 [ 223 
 
 3 3 4 
 
 
 .28 
 
 1905 1910 1914 1919 1923 
 
 1928 1932 1936 1941 1945 
 1972 1977 1982 1986 1991 
 
 I 
 
 [ 223 
 
 3 4 4 
 
 
 .29 
 
 1950 1954 1959 1963 1968 
 
 I 
 
 I 223 
 
 3 4 4 
 
 
 .30 
 
 1995 2000 2004 2009 2014 
 
 2018 2023 2028 2032 2037 
 2065 2070 2075 2080 2084 
 
 I 
 
 I 223 
 
 3 4 4 
 
 
 •31 
 
 2042 2046 2051 2056 2061 
 
 I 
 
 I 223 
 
 3 4 4 
 
 
 •32 
 
 2089 2094 2099 2104 2 1 09 
 2138 2143 2148 2153 2158 
 
 2113 2118 2123 2128 2133 
 2163 2168 2173 2178 2183 
 
 I 
 
 I 223 
 
 3 4 4 
 
 
 •33 
 
 I 
 
 I 223 
 
 3 4 4 
 
 
 ■34 
 
 2188 2193 2198 2203 2208 
 
 2213 2218 2223 2228 2234 
 
 I I 
 
 2 233 
 
 4 4 S 
 
 
 .35 
 
 2239 2244 2249 2254 2259 
 
 2265 2270 2275 2280 2286 
 
 I I 
 
 2 233 
 
 4 4 S 
 
 
 •36 
 
 2291 2296 2301 2307 2312 
 
 2317 2323 2328 2333 2339 
 2371 2377 2382 2388 2393 
 
 I I 
 
 2233 
 
 4 4 5 
 
 
 •37 
 
 2344 2350 235s 2360 2366 
 
 I 1 
 
 2 233 
 
 4 4 5 
 
 
 •38 
 
 2399 2404 2410 2415 2421 
 
 2427 2432 2438 2443 2449 
 
 I I 
 
 2 233 
 
 4 4 5 
 
 
 •39 
 
 2455 2460 2466 2472 2477 
 
 2483 2489 2495 2500 2506 
 
 I I 
 
 2 233 
 
 4 S S 
 
 
 .40 
 
 2512 2518 2523 2529 2535 
 
 2541 2547 2553 25S9 2564 
 2600 2606 2612 2618 2624 
 
 I I 
 
 2 234 
 
 4 5 5 
 
 
 .41 
 
 2570 2576 2582 2588 2594 
 
 I I 
 
 2 234 
 
 4 S 5 
 4 S 6 
 
 
 .42 
 
 2630 2636 2642 2649 2655 
 
 2661 2667 2673 2679 2685 
 
 I I 
 
 2 234 
 
 
 •43 
 
 2692 2698 2704 2710 2716 
 
 2723 2729 273s 2742 2748 
 
 I I 
 
 2334 
 
 4 S 6 
 
 
 •44 
 
 2754 2761 2767 2773 2780 
 
 2786 2793 2799 2805 2812 
 
 I I 
 
 2 3 3 4 
 
 456 
 
 
 .45 
 
 2818 2825 2831 2838 2844 
 
 2851 2858 2864 2871 2877 
 
 1 I 
 
 2 3 3 4 
 
 5 5 6 
 
 
 .46 
 
 2884 2891 2897 2904 291 I 
 
 2917 2924 2931 2938 2944 
 
 I 1 
 
 2 3 3 4 
 
 S S 6 
 
 
 •47 
 
 2951 2958 2965 2972 2979 
 
 2985 2992 2999 3006 3013 
 
 I I 
 
 2 3 3 4 
 
 5 1 ^ 
 S 6 6 
 
 
 .48 
 
 3020 3027 3034 3041 3048 
 
 3055 3062 3069 3076 3083 
 
 I I 
 
 2 3 4 4 
 
 
 ■49 
 
 3090 3097 3105 31 1 2 31 19 
 
 3126 3133 3141 3148 31SS 
 
 I I 
 
 2 3 4 4 
 
 566 
 
 
 L. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 1 2 
 
 3 4 5 6 
 
 7 8 9 
 
 
 Smithsonian Tables. 
 
 26 
 
ANTILOCARITHMS. 
 
 Table 6. 
 
 L. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 1 2 
 
 Prop. Parts. 
 3 4 5 6 7 8 9 
 
 
 
 
 .50 
 
 ■5' 
 
 .52 
 
 •53 
 ■54 
 
 3162 3170 3177 3184 3192 
 3236 3243 3251 3258 3266 
 33" 3319 3327 3334 3342 
 3388 3396 3404 3412 3420 
 3467 3475 3483 3491 3499 
 
 3199 3206 3214 3221 3228 
 3273 32S1 3289 3296 3304 
 
 3350 3357 3365 3373 338" 
 3428 3436 3443 3451 3459 
 3508 3516 3524 3532 3540 
 
 I I 
 
 I 2 
 I 2 
 I 2 
 I 2 
 
 2 
 2 
 2 
 2 
 2 
 
 3 4 
 3 4 
 3 4 
 3 4 
 3 4 
 
 4 
 
 5 
 5 
 5 
 5 
 
 5 6 7 
 5 6 7 
 
 5 6 7 
 
 6 6 7 
 667 
 
 .55 
 
 .56 
 
 ■11 
 ■59 
 
 3548 3556 3565 3573 3581 
 3631 3639 3648 3656 3664 
 3715 3724 3733 3741 3750 
 3802 381 I 3819 3828 3837 
 3890 3899 3908 3917 3926 
 
 3589 3597 3606 3614 3622 
 3673 3681 3690 3698 3707 
 3758 3767 3776 3784 3793 
 3846 3855 3864 3873 3882 
 3936 3945 3954 3963 3972 
 
 I 2 
 I 2 
 I 2 
 I 2 
 I 2 
 
 2 
 3 
 3 
 3 
 
 3 
 
 3 4 
 3 4 
 
 3 4 
 
 4 4 
 4 5 
 
 5 
 5 
 5 
 5 
 
 5 
 
 6 7 7 
 6 7 8 
 6 7 8 
 678 
 678 
 
 .60 
 
 .6i 
 .62 
 
 ■P 
 .64 
 
 3981 3990 3999 4009 4018 
 4074 4083 4093 4102 41 1 1 
 4169 4178 4188 4198 4207 
 4266 4276 4285 429s 4305 
 4365 4375 4385 4395 44o6 
 
 4027 4036 4046 4055 4064 
 4121 4130 4140 4150 4159 
 4217 4227 4236 4246 4256 
 
 431 5 4325 4335 4345 4355 
 4416 4426 4436 4446 4457 
 
 I 2 
 I 2 
 I 2 
 I 2 
 I 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 S 
 4 5 
 4 5 
 4 5 
 4 5 
 
 6 
 6 
 6 
 6 
 6 
 
 6 7 8 
 
 7 8 9 
 7 8 9 
 7 8 9 
 7 8 9 
 
 .65 
 
 .66 
 
 :69 
 
 4467 4477 4487 4498 4508 
 4571 4581 4592 4603 4613 
 4677 4688 4699 4710 4721 
 4786 4797 4808 4819 4831 
 4898 4909 4920 4932 4943 
 
 4519 4529 4539 4550 4560 
 4624 4634 4645 4656 4667 
 4732 4742 4753 4764 4775 
 4842 4853 4864 4875 4887 
 4955 4966 4977 4989 5°°° 
 
 I 2 
 I 2 
 I 2 
 I 2 
 I 2 
 
 3 
 3 
 3 
 3 
 3 
 
 4 5 
 
 4 5 
 
 5 6 
 
 6 
 6 
 
 7 
 7 
 7 
 
 7 8 9 
 
 7 9 10 
 
 8 9 10 
 8 9 10 
 8 9 10 
 
 .70 
 
 ■71 
 .72 
 
 ■73 
 ■74 
 
 5012 5023 5035 5047 5058 
 5129 5140 5152 5164 5176 
 5248 5260 5272 5284 5297 
 5370 5383 5395 5408 5420 
 5495 5508 5521 5534 5546 
 
 5070 5082 5093 5105 5117 
 5188 5200 5212 5224 5236 
 5309 5321 5333 5346 5358 
 5433 5445 5458 5470 5483 
 5559 5572 5585 5598 5610 
 
 I 2 
 I 2 
 I 2 
 I 3 
 I 3 
 
 4 
 4 
 4 
 4 
 4 
 
 5 i 
 5 6 
 
 7 
 7 
 
 8 
 
 8 911 
 
 8 10 II 
 
 9 10 II 
 9 10 II 
 9 10 12 
 
 .75 
 
 .76 
 
 ■77 
 .78 
 
 79 
 
 5623 5636 5649 5662 5675 
 5754 5768 5781 5794 5808 
 5888 5902 5916 5929 5943 
 6026 6039 6053 6067 6081 
 6166 6180 6194 6209 6223 
 
 5689 5702 5715 5728 5741 
 5821 5834 5848 5861 5875 
 5957 5970 5984 5998 6012 
 6095 6109 6124 6138 6152 
 6237 6252 6266 6281 6295 
 
 I 3 
 I 3 
 I 3 
 I 3 
 I 3 
 
 4 
 4 
 4 
 4 
 4 
 
 5 7 
 5 7 
 
 5 7 
 
 6 7^ 
 
 8 
 8 
 8 
 8 
 9 
 
 9 10 12 
 
 9 II 12 
 
 10 II 12 
 
 10 II 13 
 
 10 II 13 
 
 .80 
 
 .81 
 .82 
 
 .84 
 
 6310 6324 6339 6353 6368 
 6457 6471 6486 6501 6516 
 6607 6622 6637 6653 6668 
 6761 6776 6792 6808 6823 
 6918 6934 6950 6966 6982 
 
 6383 6397 6412 6427 6442 
 6531 6546 6561 6577 6592 
 6683 6699 6714 6730 6745 
 6839 6855 6871 6887 6902 
 6998 7015 7031 7047 7063 
 
 1 3 
 
 2 3 
 2 3 
 2 3 
 2 3 
 
 4 
 5 
 5 
 5 
 
 5 
 
 6 7 
 6 8 
 6 8 
 6 8 
 6 8 
 
 9 
 9 
 9 
 9 
 10 
 
 10 12 13 
 
 11 12 14 
 II 12 14 
 II 13 14 
 II 13 15 
 
 .85 
 
 .86 
 ■87 
 .88 
 .89 
 
 7079 7096 7112 7129 7145 
 7244 7261 7278 7295 731 1 
 
 7413 743° 7447 7464 7482 
 7586 7603 7621 7638 7656 
 7762 7780 7798 7816 7834 
 
 7161 7178 7194 7211 7228 
 7328 7345 7362 7379 7396 
 7499 7516 7534 755' 7568 
 7674 7691 7709 7727 7745 
 7852 7870 7889 7907 7925 
 
 2 3 
 2 3 
 2 3 
 
 2 4 
 
 2 4 
 
 5 
 5 
 5 
 5 
 5 
 
 7 8 
 7 8 
 7 9 
 7 9 
 7 9 
 
 10 
 10 
 10 
 II 
 II 
 
 12 13 15 
 12 13 15 
 12 14 16 
 
 12 14 16 
 
 13 14 16 
 
 .90 
 
 .91 
 .92 
 ■93 
 •94 
 
 7943 7962 7980 7998 8017 
 8128 8147 8166 8185 8204 
 8318 8337 8356 8375 8395 
 851 I 8531 8551 8570 8590 
 8710 8730 8750 8770 8790 
 
 803s 8054 8072 8091 81 10 
 8222 8241 8260 8279 8299 
 8414 8433 8453 8472 8492 
 8610 8630 8650 8670 8690 
 8810 8831 8851 8872 8892 
 
 2 4 
 2 4 
 2 4 
 2 4 
 2 4 
 
 6 
 6 
 6 
 6 
 6 
 
 7 9 
 
 8 10 
 8 10 
 8 10 
 
 II 
 II 
 
 12 
 12 
 12 
 
 13 15 17 
 
 13 15 17 
 
 14 15 17 
 14 16 18 
 14 16 18 
 
 .95 
 
 .96 
 
 •99 
 
 8913 8933 8954 8974 8995 
 9120 9141 9162 9183 9204 
 9333 9354 9376 9397 9419 
 955° 9572 9594 9616 9638 
 9772 9795 9817 9840 9863 
 
 9016 9036 9057 9078 9099 
 9226 9247 9268 9290 931 I 
 9441 9462 9484 9506 9528 
 
 9661 9683 9705 9727 975° 
 9886 9908 9931 9954 9977 
 
 2 4 
 2 4 
 
 2 4 
 2 4 
 
 2 5 
 
 6 
 6 
 
 7 
 7 
 7 
 
 8 10 
 811 
 911 
 911 
 911 
 
 12 
 13 
 13 
 13 
 14 
 
 15 17 19 
 15 17 19 
 
 15 17 20 
 
 16 18 20 
 16 18 20 
 
 L. 
 
 12 3 4 
 
 5 6 7 8 9 
 
 1 2 
 
 3 
 
 4 5 
 
 6 
 
 7 8 9 
 
 Smithsonian Tables. 
 
 27 
 
Table 
 
 7. 
 
 NATURAL SINES 
 
 AND COSINES. 
 
 
 
 
 
 
 
 
 Natural Sines. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Prop. 
 
 Angle. 
 
 0' 
 
 10' 
 
 20' 
 
 SC 
 
 40' 
 
 SC 
 
 60' 
 
 Angle. 
 
 Faits 
 lor 1'. 
 
 0° 
 
 .0000 00 
 
 .0029 09 
 
 .0058 18 
 
 .0087 27 
 
 .011635 
 .0290 8 
 
 .014544 
 
 .0174 52 
 
 89° 
 
 2.9 
 
 I 
 
 •0174 52 
 
 .0203 6 
 
 .0232 7 
 
 .0261 8 
 
 .03199 
 
 .03490 
 
 88 
 
 2.9 
 
 2 
 
 .0349 
 
 •0378 I 
 
 .0407 I 
 
 .0436 2 
 
 .0465 3 
 
 .04943 
 
 .05234 
 
 87 
 
 2.9 
 
 3 
 
 .0523 4 
 
 •0552 4 
 
 .0581 4 
 
 .0610 5 
 
 •0639 S 
 
 .06685 
 
 .0697 6 
 
 86 
 
 2.9 
 
 4 
 
 .0697 6 
 
 .0726 6 
 
 .0755 6 
 
 .0784 6 
 
 .08136 
 
 .0842 5 
 
 .08716 
 
 85 
 
 2.9 
 
 5 
 
 .0871 6 
 
 .0900 5 
 
 .0929 5 
 
 .0958 5 
 
 .0987 4 
 
 .10164 
 
 •1045 3 
 
 84 
 
 2.9 
 
 6 
 
 •10453 
 
 .10742 
 
 .11031 
 
 .11320 
 
 .11609 
 
 .11898 
 
 .12187 
 
 P 
 
 2.9 
 
 7 
 
 .12187 
 
 .1247 6 
 
 .12764 
 
 ■130S 3 
 
 •1334 
 
 •1363 
 
 •1392 
 
 82 
 
 2.9 
 
 8 
 
 .1392 
 
 .1421 
 
 .1449 
 
 .1478 
 
 •1507 
 
 .1536 
 
 .1564 
 
 81 
 
 2.9 
 
 9 
 
 .1564 
 
 •1593 
 
 .1622 
 
 .1650 
 
 .1679 
 
 .1708 
 
 •1736 
 
 80 
 
 2.9 
 
 10 
 
 •1736 
 
 .1765 
 
 .1794 
 
 .1822 
 
 .1851 
 
 .1880 
 
 .1908 
 
 79 
 
 2.9 
 
 II 
 
 .1908 
 
 •1937 
 
 .1965 
 
 .1994 
 
 .2022 
 
 .2051 
 
 .2079 
 
 78 
 
 ^t 
 
 12 
 
 .2079 
 
 .2108 
 
 .2136 
 
 .2164 
 
 .2193 
 
 .2221 
 
 .2250 
 
 77 
 
 2.8 
 
 13 
 
 .2250 
 
 .2278 
 
 .2306 
 
 •2334 
 
 •2363 
 
 .2391 
 
 •^4J9 
 
 76 
 
 2.8 
 
 14 
 
 .2419 
 
 .2447 
 
 .2476 
 
 .2504 
 
 •2532 
 
 .2560 
 
 .2588 
 
 75 
 
 2.8 
 
 15 
 
 .2588 
 
 .2616 
 
 .2644 
 
 .2672 
 
 .2700 
 
 .2728 
 
 .2756 
 
 74 
 
 2.8 
 
 16 
 
 .2756 
 
 .2784 
 
 .2812 
 
 .2840 
 
 .2868 
 
 .2896 
 
 .2924 
 
 73 
 
 2.8 
 
 17 
 
 .2924 
 
 •2952 
 
 .2979 
 
 .3007 
 
 •3035 
 
 .3062 
 
 .3090 
 
 72 
 
 2.8 
 
 18 
 
 .3090 
 
 .3118 
 
 •3145 
 
 •3173 
 
 .3201 
 
 .3228 
 
 •3256 
 
 71 
 
 2.8 
 
 19 
 
 .3256 
 
 ■3283 
 
 ■331 1 
 
 •3338 
 
 •3365 
 
 •3393 
 
 .3420 
 
 70 
 
 2.7 
 
 20 
 
 .3420 
 
 •3448 
 
 •3475 
 
 •3502 
 
 •3529 
 
 •3557 
 
 •3584 
 
 69 
 
 2.7 
 
 21 
 
 ■3584 
 
 .3611 
 
 •3638 
 
 •366s 
 
 .3692 
 
 •3719 
 
 •3746 
 
 68 
 
 2.7 
 
 22 
 
 •3746 
 
 •3773 
 
 .3800 
 
 .3827 
 
 •3854 
 
 .3881 
 
 •3907 
 
 ^l 
 
 2.7 
 
 23 
 
 •3907 
 
 •3934 
 
 .3961 
 
 ■3987 
 
 .4014 
 
 .4041 
 
 .4067 
 
 66 
 
 2.7 
 
 24 
 
 .4067 
 
 .4094 
 
 .4120 
 
 .4147 
 
 ■4173 
 
 .4200 
 
 .4226 
 
 65 
 
 2.7 
 
 25 
 
 .4226 
 
 •4253 
 
 .4279 
 
 ■4305 
 
 •4331 
 
 •4358 
 
 .4384 
 
 64 
 
 2.6 
 
 26 
 
 •4384 
 
 .4410 
 
 •4436 
 
 .4462 
 
 
 •4514 
 
 .4540 
 
 ^3 
 
 2.6 
 
 27 
 
 •4S40 
 
 .4566 
 
 •4592 
 
 .4617 
 
 •4643 
 
 .4669 
 
 14848 
 
 62 
 
 2.6 
 
 28 
 
 .4695 
 
 .4720 
 
 .4746 
 
 •4772 
 
 •4797 
 
 .4823 
 
 61 
 
 2.6 
 
 29 
 
 .4848 
 
 .4874 
 
 .4899 
 
 .4924 
 
 .4950 
 
 •4975 
 
 .5000 
 
 60 
 
 2.5 
 
 30 
 
 .5000 
 
 .5025 
 
 .5050 
 
 •507s 
 
 .5100 
 
 •5125 
 
 .5150 
 
 59 
 
 2.5 
 
 31 
 
 .5150 
 
 •517s 
 
 .5200 
 
 .5225 
 
 .5250 
 
 •527s 
 
 •5299 
 
 58 
 
 2.5 
 
 32 
 
 .5299 
 
 •5324 
 
 •5348 
 
 •5373 
 
 •5398 
 
 .5422 
 
 .5446 
 
 5? 
 
 2.5 
 
 33 
 
 •5446 
 
 •5471 
 
 •5495 
 
 •5519 
 
 •5544 
 
 •5568 
 
 •5592 
 
 56 
 
 2.4 
 
 34 
 
 •5592 
 
 .5616 
 
 .5640 
 
 
 .5688 
 
 •5712 
 
 •5736 
 
 55 
 
 2.4 
 
 3S 
 
 •5736 
 
 .5760 
 
 •5783 
 
 .5807 
 
 •5831 
 
 •5854 
 
 .5878 
 
 54 
 
 2.4 
 
 36 
 
 .5878 
 
 .5901 
 
 .5925 
 
 .5948 
 
 •5972 
 
 •5995 
 
 .6018 
 
 53 
 
 2-3 
 
 
 .6018 
 
 
 .6065 
 
 .6088 
 
 .6111 
 
 .6134 
 
 •6157 
 
 52 
 
 2-3 
 
 38 
 
 •6157 
 
 ".6i8o 
 
 .6202 
 
 .6225 
 
 .6248 
 
 .6271 
 
 .629; 
 
 51 
 
 2^3 
 
 39 
 
 .6293 
 
 .6316 
 
 •6338 
 
 .6361 
 
 •6383 
 
 .6406 
 
 .6428 
 
 50 
 
 2.3 
 
 40 
 
 .6428 
 
 .6450 
 
 .6472 
 
 •6494 
 
 •6517 
 
 •6539 
 
 .6561 
 
 49 
 
 2.2 
 
 41 
 
 .6561 
 
 •6583 
 
 .6604 
 
 .6626 
 
 .6648 
 
 .6670 
 
 .6691 
 
 48 
 
 2.2 
 
 42 
 
 .6691 
 
 •6713 
 
 .6734 
 
 .6756 
 
 .6777 
 
 .6799 
 
 .6820 
 
 47 
 
 2.2 
 
 43 
 
 .6820 
 
 .6841 
 
 .6862 
 
 .6884 
 
 .6905 
 
 .6926 
 
 .6947 
 
 46 
 
 2.1 
 
 44 
 
 .6947 
 
 .6967 
 
 .6988 
 
 .7009 
 
 .7030 
 
 •7050 
 
 .7071 
 
 45 
 
 2.1 
 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Angle. 
 
 
 Smiths 
 
 9NIAN T«B 
 
 LES. 
 
 
 Natural Cosines. 
 
 
 
 
 
 
 
 
 
 2i 
 
 $ 
 
 
 
 
 
Table 7. 
 
 NATURAL SINES AND COSINES. 
 
 Natural Sines. 
 
 
 
 
 
 
 
 
 
 
 Piop. 
 
 
 AjiEle. 
 
 C 
 
 10' 
 
 20' 
 
 30- 
 
 40' 
 
 50' 
 
 60' 
 
 Angle. 
 
 Parts 
 for 1', 
 
 
 45° 
 
 .7071 
 
 .7092 
 
 .7112 
 
 ■7133 
 
 •7153 
 
 ■7173 
 
 •7193 
 
 44° 
 
 2.0 
 
 
 46 
 
 •7193 
 
 .7214 
 
 ■7234 
 
 ■7254 
 
 ■7274 
 
 .7294 
 
 ■7314 
 
 43 
 
 2.0 
 
 
 "^l 
 
 ■73H 
 
 ■7333 
 
 ■7353 
 
 ■7373 
 
 •7392 
 
 .7412 
 
 •7431 
 
 42 
 
 2.0 
 
 
 48 
 
 •7431 
 
 ■7451 
 
 .7470 
 
 .7490 
 
 ■7509 
 
 ■7528 
 
 •7547 
 
 41 
 
 19 
 
 
 49 
 
 •7547 
 
 ■7566 
 
 •7585 
 
 .7604 
 
 .7623 
 
 .7642 
 
 .7660 
 
 40 
 
 1.9 
 
 
 50 
 
 .7660 
 
 .7679 
 
 .7698 
 
 .7716 
 
 ■7735 
 
 •7753 
 
 .7771 
 
 39 
 
 1.9 
 
 
 51 
 
 •7771 
 
 .7790 
 
 .7808 
 
 .7826 
 
 .7844 
 
 .7862 
 
 .7880 
 
 38 
 
 1.8 
 
 
 S2 
 
 .7880 
 
 .7898 
 
 .7916 
 
 ■7934 
 
 
 .7969 
 
 .7986 
 
 37 
 
 1.8 
 
 
 S3 
 
 .7986 
 
 .8004 
 
 .8021 
 
 .8039 
 
 :8o56 
 
 ■8073 
 
 .8090 
 
 36 
 
 1-7 
 
 
 54 
 
 .8090 
 
 .8107 
 
 .8124 
 
 .8141 
 
 .8158 
 
 .8175 
 
 .8192 
 
 35 
 
 '•7 
 
 
 55 
 
 .8192 
 
 .8208 
 
 .8225 
 
 .8241 
 
 .8258 
 
 .8274 
 
 .8290 
 
 34 
 
 1.6 
 
 
 S6 
 
 .8290 
 
 .8307 
 
 .8323 
 
 ■8339 
 
 •8355 
 
 ■8371 
 
 .8387 
 
 33 
 
 1.6 
 
 
 
 ■^387 
 
 ■8403 
 
 .8418 
 
 ■8434 
 
 .8450 
 
 .8465 
 
 .8480 
 
 32 
 
 1.6 
 
 
 58 
 
 .8480 
 
 .8496 
 
 .8511 
 
 .8526 
 
 ■8542 
 
 ■8557 
 
 •8572 
 
 31 
 
 i-S 
 
 
 59 
 
 •8572 
 
 ■8587 
 
 .8601 
 
 .8616 
 
 .8631 
 
 .8646 
 
 .8660 
 
 30 
 
 15 
 
 
 60 
 
 .8660 
 
 ■8675 
 
 .8689 
 
 .8704 
 
 .8718 
 
 ■8732 
 
 .8746 
 
 29 
 
 1.4 
 
 
 61 
 
 .8746 
 
 .8760 
 
 ■8774 
 
 .8788 
 
 .8802 
 
 .8816 
 
 .8829 
 
 28 
 
 1.4 
 
 
 62 
 
 .8829 
 
 .8843 
 
 ■8857 
 
 .8870 
 
 .8884 
 
 .8897 
 
 .8910 
 
 27 
 
 1.4 
 
 
 63 
 
 .8910 
 
 .8923 
 
 .8936 
 
 ■8949 
 
 .8962 
 
 ■8975 
 
 .8988 
 
 26 
 
 13 
 
 
 64 
 
 .8988 
 
 .9001 
 
 .9013 
 
 .9026 
 
 .9038 
 
 .9051 
 
 .9063 
 
 25 
 
 1-3 
 
 
 65 
 
 .9063 
 
 ■907s 
 
 .9088 
 
 .9100 
 
 .9112 
 
 .9124 
 
 •9135 
 
 24 
 
 1.2 
 
 
 66 
 
 ■9135 
 
 .9147 
 
 ■9159 
 
 .9171 
 
 .9182 
 
 .9194 
 
 .9205 
 
 23 
 
 1.2 
 
 
 67 
 
 .920s 
 
 .9216 
 
 .9228 
 
 ■9239 
 
 .9250 
 
 .9261 
 
 .9272 
 
 22 
 
 I.I 
 
 
 68 
 
 .9272 
 
 .9283 
 
 ■9293 
 
 ■9304 
 
 ■9315 
 
 ■9325 
 
 •9336 
 
 21 
 
 I.I 
 
 
 69 
 
 •9336 
 
 ■9346 
 
 •9356 
 
 ■9367 
 
 ■9377 
 
 ■9387 
 
 •9397 
 
 20 
 
 I.O 
 
 
 70 
 
 ■9397 
 
 .9407 
 
 .9417 
 
 .9426 
 
 ■9436 
 
 .9446 
 
 •9455 
 
 19 
 
 I.O 
 
 
 71 
 
 ■9455 
 
 •9465 
 
 ■9474 
 
 ■9483 
 
 ■9492 
 
 .9502 
 
 •95" 
 
 18 
 
 0.9 
 
 
 72 
 
 ■95" 
 
 .9520 
 
 .9528 
 
 
 .9546 
 
 ■9555 
 
 •9563 
 
 17 
 
 0.9 
 
 
 73 
 
 •9563 
 
 .9572 
 
 
 .9588 
 
 ■9596 
 
 ■9605 
 
 .9613 
 
 16 
 
 0.8 
 
 
 74 
 
 .9613 
 
 
 19628 
 
 ■9636 
 
 .9644 
 
 .9652 
 
 •9659 
 
 15 
 
 0.8 
 
 
 75 
 
 .9659 
 
 .9667 
 
 .9674 
 
 .9681 
 
 .9689 
 
 .9696 
 
 •9703 
 
 14 
 
 0.7 
 
 
 76 
 
 •9703 
 
 .9710 
 
 .9717 
 
 ■9724 
 
 ■9730 
 
 ■9737 
 
 •9744 
 
 13 
 
 0.7 
 
 
 77 
 
 ■9744 
 
 ■9750 
 
 ■9757 
 
 •9763 
 
 .9769 
 
 ■9775 
 
 .9781 
 
 12 
 
 0.6 
 
 
 78 
 
 .9781 
 
 .9787 
 
 ■9793 
 
 ■9799 
 
 .9805 
 
 .9811 
 
 .9816 
 
 II 
 
 0.6 
 
 
 79 
 
 .9816 
 
 .9822 
 
 .9827 
 
 ■9833 
 
 ■9838 
 
 ■9843 
 
 .9848 
 
 10 
 
 0.5 
 
 
 80 
 
 .9848 
 
 ■9853 
 
 ■9858 
 
 .9863 
 
 .9868 
 
 .9872 
 
 •9877 
 
 9 
 
 0.5 
 
 
 81 
 
 .9877 
 
 .9881 
 
 .9886 
 
 .9890 
 
 .9894 
 
 ■9899 
 
 •9903 
 
 8 
 
 0.4 
 
 
 82 
 
 •9903 
 
 .9907 
 
 ■99" 
 
 .9914 
 
 .9918 
 
 .9922 
 
 •9925 
 
 7 
 
 0.4 
 
 
 ^3 
 
 ■9925 
 
 •9929 
 
 •9932 
 
 ■9936 
 
 ■9939 
 
 ■9942 
 
 •9945 
 
 6 
 
 0-3 
 
 
 84 
 
 ■9945 
 
 
 ■9951 
 
 ■9954 
 
 ■9957 
 
 ■9959 
 
 .9962 
 
 5 
 
 0-3 
 
 
 85 
 
 .9962 
 
 .9964 
 
 .9967 
 
 .9969 
 
 .9971 
 
 •9974 
 
 .9976 
 
 4 
 
 0.2 
 
 
 86 
 
 .9976 
 
 ■9978 
 
 .9980 
 
 .9981 
 
 
 •9985 
 
 .9986 
 
 3 
 
 0.2 
 
 
 87 
 
 .9986 
 
 .9988 
 
 .9989 
 
 .9990 
 
 ■9992 
 
 •9993 
 
 •9994 
 
 2 
 
 0.1 
 
 
 88 
 
 ■9994 
 
 ■9995 
 
 .9996 
 
 ■9997 
 
 ■9997 
 
 •9998 
 
 •9998 
 
 I 
 
 0.1 
 
 
 89 
 
 .9998 
 
 ■9999 
 
 ■9999 
 
 1. 0000 
 
 1. 0000 
 
 1. 0000 
 
 I.OOOO 
 
 
 
 0.0 
 
 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Angle. 
 
 
 
 Smithsonian Tables. 
 
 Natural Cosines. 
 29 
 
Table 8. 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 Natural Tangents. 
 
 Aagle. 
 
 0' 
 
 10' 
 
 zor 
 
 ac 
 
 40' 
 
 5^ 
 
 60' 
 
 Angle. 
 
 Prop. Farts 
 lor 1'. 
 
 
 0° 
 
 .0000 
 
 .0029 I 
 
 .0058 2 
 
 .0087 3 
 
 .01164 
 
 •0145 5 
 
 .01746 
 
 89° 
 
 2.9 
 
 
 I 
 
 .01746 
 
 .0203 6 
 
 .0232 8 
 
 .0261 9 
 
 .0291 
 
 .0320 I 
 
 .0349 2 
 
 88 
 
 2.9 
 
 
 2 
 
 .0349 2 
 
 •0378 3 
 
 .0407 s 
 
 .0436 6 
 
 .0465 8 
 
 .0494 9 
 
 ■0524 I 
 
 87 
 
 2.9 
 
 
 3 
 
 .0524 I 
 
 •0553 3 
 .0728 s 
 
 .0582 4 
 
 .061 1 6 
 
 .0640 8 
 
 .0670 
 
 ■0699 3 
 
 86 
 
 2.9 
 
 
 4 
 
 .06993 
 
 .0757 8 
 
 .0787 
 
 .08163 
 
 .0845 6 
 
 .08749 
 
 85 
 
 2.9 
 
 
 5 
 
 .0874 9 
 
 .0904 2 
 
 ■0933 5 
 
 .0962 9 
 
 .0992 3 
 
 .1021 6 
 
 .1051 
 
 84 
 
 2.9 
 
 
 6 
 
 .1051 
 
 .1080 5 
 
 .11099 
 
 ■I 139 4 
 
 .11688 
 
 .11983 
 
 .12278 
 
 83 
 
 2.9 
 
 
 I 
 
 .12278 
 
 .12574 
 
 .12869 
 
 .13165 
 
 .1346 
 
 ■1376 
 
 .1405 
 
 82 
 
 3^0 
 
 
 8 
 
 .1405 
 
 •1435 
 
 .1465 
 
 •1495 
 
 .1524 
 
 ■1554 
 
 ■1584 
 
 81 
 
 3^0 
 
 
 9 
 
 .1584 
 
 .1614 
 
 .1644 
 
 ■1673 
 
 •1703 
 
 ■1733 
 
 •1763 
 
 80 
 
 3^0 
 
 
 10 
 
 ■1763 
 
 •1793 
 
 .1823 
 
 •1853 
 
 .1883 
 
 .1914 
 
 .1944 
 
 79 
 
 3^0 
 
 
 11 
 
 .1944 
 
 .1974 
 
 .2004 
 
 ■2035 
 
 .2065 
 
 .2095 
 .2278 
 
 .2126 
 
 78 
 
 30 
 
 
 12 
 
 .2126 
 
 .2156 
 
 .2186 
 
 .2217 
 
 .2247 
 
 •2309 
 
 77 
 
 3-1 
 
 
 13 
 
 .2309 
 
 ■2339 
 
 .2370 
 
 .2401 
 
 .2432 
 
 .2462 
 
 
 76 
 
 3^1 
 
 
 '4 
 
 ■2493 
 
 ■2524 
 
 •2555 
 
 .2586 
 
 .2617 
 
 .2648 
 
 .2679 
 
 75 
 
 3^1 
 
 
 15 
 
 .2679 
 
 .2711 
 
 .2742 
 
 •2773 
 
 .2805 
 
 .2836 
 
 .2867 
 
 74 
 
 3'i 
 
 
 i6 
 
 .2867 
 
 .2899 
 
 .293 1 
 
 .2962 
 
 ^ 
 
 .3026 
 
 •3057 
 
 73 
 
 3^2 
 
 
 17 
 
 ■3°S7 
 
 .3089 
 
 .3121 
 
 •3153 
 
 •3217 
 
 •3249 
 
 72 
 
 3^2 
 
 
 i8 
 
 •3249 
 
 .3281 
 
 •3314 
 
 ■3346 
 
 •3378 
 
 ■34" 
 
 ■3443 
 
 71 
 
 3-2 
 
 
 19 
 
 ■3443 
 
 •3476 
 
 .3508 
 
 •3541 
 
 •3574 
 
 .3607 
 
 .3640 
 
 70 
 
 3-3 
 
 
 20 
 
 .3640 
 
 •3673 
 
 .3706 
 
 •3739 
 
 ■3772 
 
 ■3805 
 
 •3839 
 
 69 
 
 3^3 
 
 
 21 
 
 •3839 
 
 .3872 
 
 .3906 
 
 ■3939 
 
 ■3973 
 
 .4006 
 
 .4040 
 
 68 
 
 3-4 
 
 
 22 
 
 .4040 
 
 .4074 
 
 .4108 
 
 .4142 
 
 .4176 
 
 .4210 
 
 .4245 
 
 67 
 
 3^4 
 
 
 23 
 
 .4245 
 
 •4279 
 
 •43'4 
 
 ■4348 
 
 •4383 
 
 .4417 
 
 .4452 
 
 66 
 
 3^5 
 
 
 24 
 
 .4452 
 
 •4487 
 
 .4522 
 
 ■4557 
 
 •4592 
 
 .4628 
 
 .4663 
 
 65 
 
 3' 5 
 
 
 25 
 
 .4663 
 
 .4699 
 
 ■4734 
 
 .4770 
 
 .4806 
 
 .4841 
 
 •4877 
 
 64 
 
 3^6 
 
 
 26 
 
 .4877 
 
 ■4913 
 
 •4950 
 
 .4986 
 
 .5022 
 
 .5059 
 .5280 
 
 •5095 
 
 63 
 
 3-6 
 
 
 27 
 
 •5095 
 
 •5132 
 
 .5169 
 
 .5206 
 
 •5243 
 
 •5317 
 
 62 
 
 3-7 
 
 
 28 
 
 •5317 
 
 ■5354 
 .5581 
 
 •5392 
 
 ■5430 
 
 .5467 
 
 ■5505 
 
 •5543 
 
 61 
 
 3^8 
 
 
 29 
 
 •5543 
 
 .5619 
 
 .5658 
 
 .5696 
 
 •5735 
 
 •5774 
 
 60 
 
 3.8 
 
 
 30 
 
 •5774 
 
 .5812 
 
 .5851 
 
 .5890 
 
 ■5930 
 
 ■5969 
 
 .6009 
 
 59 
 
 3'9 
 
 
 31 
 
 .6009 
 
 .6048 
 
 .6088 
 
 .6128 
 
 .6168 
 
 .6208 
 
 .6249 
 
 58 
 
 4.0 
 
 
 32 
 
 .6249 
 
 .6289 
 
 .6330 
 
 ■6371 
 
 .6412 
 
 ■6453 
 
 .6494 
 
 57 
 
 4.1 
 
 
 33 
 
 .6494 
 
 .6536 
 
 .6577 
 
 .6619 
 
 .6661 
 
 .6703 
 
 •6745 
 
 56 
 
 4.2 
 
 
 34 
 
 •6745 
 
 .6787 
 
 .6830 
 
 .6873 
 
 .6916 
 
 .6959 
 
 .7002 
 
 55 
 
 4-3 
 
 
 35 
 
 .7002 
 
 .7046 
 
 .7089 
 
 ■7133 
 
 .7177 
 
 .7221 
 
 .7265 
 •7536 
 
 54 
 
 4.4 
 
 
 36 
 
 .7265 
 
 ■7310 
 
 ■7355 
 
 .7400 
 
 •7445 
 
 ■7490 
 
 53 
 
 4' 5 
 
 
 3Z 
 
 ■7536 
 
 •75§' 
 
 .7627 
 
 •7673 
 
 .7720 
 
 .7766 
 
 •7813 
 .8098 
 
 52 
 
 4.6 
 
 
 38 
 
 .7860 
 
 .7907 
 
 ■7954 
 
 .8002 
 
 .8050 
 
 51 
 
 4.7 
 
 
 39 
 
 .8098 
 
 .8146 
 
 ■8195 
 
 .8243 
 
 .8292 
 
 .8342 
 
 .8391 
 
 50 
 
 4.9 
 
 
 40 
 
 .8391 
 
 .8441 
 
 .8491 
 
 ;8g47 
 
 .8591 
 
 .8642 
 
 .8693 
 
 49 
 
 5.0 
 
 
 41 
 
 ■8693 
 
 .8744 
 
 .8796 
 
 .8§99 
 
 .8952 
 
 .9004 
 
 48 
 
 5^2 
 
 
 42 
 
 .9004 
 
 .90S7 
 
 .9110 
 
 .9163 
 
 .9217 
 
 .9271 
 
 •9325 
 
 47 
 
 5^4 
 
 
 43 
 
 •9325 
 
 .9380 
 
 ■9435 
 
 .9490 
 
 %\ 
 
 .9601 
 
 .9657 
 
 46 
 
 S-S 
 
 
 44 
 
 .9657 
 
 •9713 
 
 .9770 
 
 .9827 
 
 ■9942 
 
 1. 0000 
 
 45 
 
 5'7 
 
 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Angle. 
 
 
 
 Smithson 
 
 IAN TaBL 
 
 ES. 
 
 
 
 
 
 
 
 
 
 Natural Cotangents. 
 30 
 
NATURAL TANGENTS AND COTANGENTS. 
 
 Natural Tangents. 
 
 Table 8. 
 
 Aseis. 
 
 0' 
 
 10' 
 
 20' 
 
 30' 
 
 40' 
 
 50' 
 
 60' 
 
 Angle. 
 
 Prop. Parts 
 lor 1'. 
 
 
 45° 
 
 1. 0000 
 
 1.0058 
 
 1.OH7 
 
 1.0176 
 
 1.0235 
 
 1.0295 
 
 1-0355 
 
 44° 
 
 5-9 
 
 
 46 
 
 '•0355 
 
 1. 041 6 
 
 1.0477 
 
 1-0538 
 
 1.0599 
 
 1.0661 
 
 1.0724 
 
 43 
 
 6.1 
 
 
 47 
 
 1.0724 
 
 1.0786 
 
 1.0850 
 
 1.0913 
 
 1.0977 
 
 1.1041 
 
 1.1106 
 
 42 
 
 6.4 
 
 
 48 
 
 I.I 106 
 
 1.1171 
 
 1.1237 
 
 1-1303 
 
 1. 1369 
 
 1.1436 
 
 1.1504 
 
 41 
 
 6.6 
 
 
 49 
 
 1.1504 
 
 1.1571 
 
 1. 1640 
 
 1. 1708 
 
 1.1778 
 
 1.1847 
 
 1.1918 
 
 40 
 
 6.9 
 
 
 50 
 
 1.1918 
 
 1.1988 
 
 1.2059 
 
 1-2131 
 
 1.2203 
 
 1.2276 
 
 1.2349 
 
 39 
 
 7.2 
 
 
 51 
 
 1.2349 
 
 1.2423 
 
 1.2497 
 
 1.2572 
 
 1.2647 
 
 1.2723 
 
 1-2799 
 
 38 
 
 7-5 
 
 
 52 
 
 1.2799 
 
 1.2876 
 
 1.2954 
 
 1-3032 
 
 1.3111 
 
 1.3190 
 
 1.3270 
 
 3Z 
 
 l-^ 
 
 
 S3 
 
 1.3270 
 
 '•3351 
 
 J -3432 
 
 I-35H 
 
 1-3597 
 
 1:3680 
 
 1-3764 
 
 36 
 
 8.2 
 
 
 54 
 
 1-3764 
 
 1.3848 
 
 1-3934 
 
 1.4019 
 
 1.4106 
 
 1-4193 
 
 1.4281 
 
 35 
 
 8.5 
 
 
 55 
 
 1.428 1 
 
 1-4370 
 
 1.4460 
 
 1-4550 
 
 1.4641 
 
 1-4733 
 
 1.4826 
 
 34 
 
 9.1 
 
 
 S6 
 
 1.4826 
 
 1.4919 
 
 1.5013 
 
 1. 5108 
 
 1.5204 
 
 1-5301 
 
 1-5399 
 
 33 
 
 9.6 
 
 
 57 
 
 1-5399 
 
 I-54&7 
 
 1-5597 
 
 1-5697 
 
 1.5798 
 
 1.5900 
 
 1.6003 
 
 32 
 
 lO.I 
 
 
 58 
 
 1.6003 
 
 1.6107 
 
 1.6212 
 
 1.6319 
 
 1.6426 
 
 1-6534 
 
 1.6643 
 
 31 
 
 10.7 
 
 
 59 
 
 1.6643 
 
 1-6753 
 
 1.6864 
 
 1.6977 
 
 1.7090 
 
 1.7205 
 
 1.7321 
 
 30 
 
 11-3 
 
 
 60 
 
 1.7321 
 
 1-7437 
 
 1.7556 
 
 1-7675 
 1.8418 
 
 1.7796 
 
 1.7917 
 
 1.8040 
 
 29 
 
 12.0 
 
 
 61 
 
 1.8040 
 
 1.8165 
 
 1. 829 1 
 
 1.8546 
 
 1.8676 
 
 1.8807 
 
 28 
 
 12.8 
 
 
 62 
 
 1.8807 
 
 1.8940 
 
 1.9074 
 
 1.9210 
 
 1-9347 
 
 1.9486 
 
 1.9626 
 
 27 
 
 13.6 
 
 
 63 
 
 1.9626 
 
 1.9768 
 
 1.9912 
 
 2.0057 
 
 2.0204 
 
 2-0353 
 2.1283 
 
 2.0503 
 
 26 
 
 14.6 
 
 
 64 
 
 2.0503 
 
 2.0655 
 
 2.0809 
 
 2.0965 
 
 2.1123 
 
 2.1445 
 
 25 
 
 15-7 
 
 
 65 
 
 2.1 445 
 
 2.1609 
 
 2.1775 
 
 2-1943 
 
 2.2113 
 
 2.2286 
 
 2.2460 
 
 24 
 
 16.9 
 
 
 66 
 
 2.2460 
 
 2.2637 
 
 2.2817 
 
 2.2998 
 
 2.3183 
 
 2-3369 
 
 2-3559 
 
 23 
 
 18.3 
 
 
 67 
 
 2-3559 
 
 2.3750 
 
 2-3945 
 
 2.4142 
 
 2.4342 
 
 2-4545 
 
 2.4751 
 
 22 
 
 19.9 
 
 
 68 
 
 2-4751 
 
 2.4960 
 
 2.5172 
 
 2. 5386 
 
 2.5605 
 
 2.5826 
 
 2.605 1 
 
 21 
 
 21.7 
 
 
 69 
 
 2.6051 
 
 2.6279 
 
 2.6511 
 
 2.6746 
 
 2.6985 
 
 2.7228 
 
 2-7475 
 
 20 
 
 23-7 
 
 
 70 
 
 2-7475 
 
 2.7725 
 
 2.7980 
 
 2.8239 
 
 2.8502 
 
 2.8770 
 
 2.9042 
 
 19 
 
 
 
 71 
 
 2.9042 
 
 2.9319 
 
 2.9600 
 
 2.9887 
 
 3.0178 
 
 3-0475 
 
 3.0777 
 
 18 
 
 
 
 72 
 
 3-0777 
 
 3.1084 
 
 3-1397 
 
 3.1716 
 
 3.2041 
 
 3-2371 
 
 3.2709 
 
 ^l 
 
 
 
 73 
 
 3.2709 
 
 3-3052 
 
 3-3402 
 
 3-3759 
 
 3.4124 
 
 3-4495 
 
 3-4874 
 
 16 
 
 
 
 74 
 
 3-4874 
 
 3.5261 
 
 3-5656 
 
 3.6059 
 
 3.6470 
 
 3.6891 
 
 3-7321 
 
 15 
 
 
 
 75 
 
 3-7321 
 
 3.7760 
 
 3.8208 
 
 3.8667 
 
 3-9136 
 
 3.9617 
 
 4.0108 
 
 14 
 
 
 
 76 
 
 4.0108 
 
 4.061 1 
 
 4.1126 
 
 4-1653 
 
 4-2193 
 
 4.2747 
 
 4-33' 5 
 
 13 
 
 
 
 77 
 
 4-3315 
 
 4-3897 
 
 4-4494 
 
 4.5107 
 
 4-5736 
 
 4.6382 
 
 4.7046 
 
 12 
 
 
 
 78 
 
 4.7046 
 
 4.7729 
 
 4.8430 
 
 4.9152 
 
 4.9894 
 
 5.0658 
 
 5.1446 
 
 11 
 
 
 
 79 
 
 5.1446 
 
 5-2257 
 
 5-3093 
 
 5-3955 
 
 5-4845 
 
 5-5764 
 
 5-6713 
 
 10 
 
 
 
 80 
 
 S-6713 
 
 5-7694 
 
 5-8708 
 
 5-9758 
 
 6.0844 
 
 6.1970 
 
 6-3138 
 
 9 
 
 
 
 81 
 
 6.3138 
 
 6.4348 
 
 6.5606 
 
 6.6912 
 
 6.8269 
 
 6.9682 
 
 7-1154 
 
 8 
 
 
 
 82 
 
 7.1 1 54 
 
 7.2687 
 
 7-4287 
 
 7-5958 
 
 7-7704 
 
 7-9530 
 
 8.1443 
 
 7 
 
 
 
 83 
 
 8.1443 
 
 8.3450 
 
 8-5555 
 
 8-7769 
 
 9.0098 
 
 9-2553 
 
 9.5144 
 
 6 
 
 
 
 84 
 
 9-5144 
 
 9.7882 
 
 10.0780 
 
 10.3854 
 
 10.7119 
 
 11.0594 
 
 11.4301 
 
 5 
 
 
 
 85 
 
 1 1. 4301 
 
 11.8262 
 
 12.2505 
 
 12.7062 
 
 13.1969 
 
 13-7267 
 
 14-3007 
 
 4 
 
 
 
 86 
 
 14.3007 
 
 14.9244 
 
 1 5.6048 
 
 16.3499 
 
 17.1693 
 
 18.0750 
 
 19.0811 
 
 3 
 
 
 
 87 
 
 19.081 1 
 
 20.2056 
 
 21.4704 
 
 22.9038 
 
 24.5418 
 
 26.4316 
 
 28.6363 
 
 2 
 
 
 
 88 
 
 28.6363 
 
 31.2416 
 
 34-3678 
 
 38-1885 
 
 42.9641 
 
 49.1039 
 
 57.2900 
 
 I 
 
 
 
 89 
 
 57.2900 
 
 68.7501 
 
 85-9398 
 
 114.5887 
 
 171.8854 
 
 343-7737 
 
 00 
 
 
 
 
 
 
 60' 
 
 5^ 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Angle. 
 
 
 
 Smithsonian Tables. 
 
 Natural Cotangents. 
 31 
 
Table 9. 
 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. 
 
 1 
 
 3 
 
 <i3 
 u 
 
 c: 
 
 (5 
 
 
 
 
 
 1 
 
 
 
 2 
 
 
 
 u 
 
 .a 
 
 
 1 
 
 c 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 1. 00000 
 
 0.00000 
 
 0.99984 
 
 0.01745 
 
 0.99939 
 
 0.03490 
 
 1 
 
 
 
 2 
 
 2.00000 
 
 0.00000 
 
 1.99969 
 
 0.03490 
 
 1.99878 
 
 0.06980 
 
 2 
 
 
 
 3 
 
 4 
 
 3.00000 
 
 0.00000 
 
 2.99954 
 
 0.052^5 
 
 2.99817 
 
 0.10470 
 
 3 
 
 
 
 4.00000 
 
 0.00000 
 
 3-99939 
 
 0.06980 
 
 3-99756 
 
 0.13960 
 
 4 
 
 
 
 
 
 5.00000 
 
 0.00000 
 
 4-99923 
 
 0.08726 
 
 4.99695 
 
 0.17450 
 
 5 
 
 60 
 
 
 5 
 
 6.00000 
 
 0.00000 
 
 5.99908 
 
 0.1047 1 
 
 S-99634 
 
 0.20940 
 
 6 
 
 
 
 7 
 
 7.00000 
 
 0.00000 
 
 7-99878 
 
 0.12216 
 
 6-99573 
 
 0.24430 
 
 7 
 
 
 
 g 
 
 8.00000 
 
 0.00000 
 
 0.13961 
 
 7.99512 
 
 0.27920 
 
 8 
 
 
 
 9 
 
 9.00000 
 
 0.00000 
 
 8.99862 
 
 0.15707 
 
 8.99451 
 
 0.31410 
 
 9 
 
 
 
 1 
 
 0.99999 
 
 0.00436 
 
 0.99976 
 
 0.02181 
 
 0.99922 
 
 0.03925 
 0.07851 
 
 1 
 
 
 
 2 
 
 1.99998 
 
 0.00872 
 
 1.99952 
 
 0.04363 
 
 1.99845 
 2.99768 
 
 2 
 
 
 
 3 
 
 2.99997 
 
 0.01308 
 
 2.99928 
 
 0.06544 
 
 0.1 1777 
 
 3 
 
 
 
 4 
 
 3.99996 
 
 0.01745 
 
 3-99904 
 
 0.08725 
 
 3.99691 
 
 0.15703 
 
 4 
 
 
 15 
 
 
 4.99995 
 
 0.02 181 
 
 
 0.10907 
 
 4.99614 
 
 0.19629 
 
 5 
 
 45 
 
 
 6 
 
 5-99994 
 
 0.02617 
 
 5-99857 
 
 0.13089 
 
 5-99537 
 
 0.235SS 
 
 6 
 
 
 
 7 
 
 6.99993 
 
 0.03054 
 
 6-99833 
 
 0.15270 
 
 6.99460 
 
 0.27481 
 
 7 
 
 
 
 8 
 
 7.99992 
 
 0.03490 
 
 7-99809 
 
 0.17452 
 
 7-99383 
 
 0.31407 
 
 8 
 
 
 
 9 
 
 8.99991 
 
 0.03926 
 
 8.99785 
 
 0.19633 
 
 8.99306 
 
 0-35.133 
 
 9 
 
 
 
 1 
 
 0.99996 
 
 0.00872 
 
 0.99965 
 
 0.02617 
 
 0.99904 
 
 0.04361 
 
 1 
 
 
 
 2 
 
 1.99992 
 
 0.01745 
 
 1.99931 
 
 0.05235 
 
 1.99809 
 
 0.08723 
 
 2 
 
 
 
 3 
 
 2.99988 
 
 0.02617 
 
 2.99897 
 
 0.07853 
 
 2.99714 
 
 0.13085 
 
 3 
 
 
 
 4 
 
 3.99984 
 
 0.03490 
 
 3-99862 
 
 0.10470 
 
 3.99619 
 
 0. 1 7447 
 
 4 
 
 
 30 
 
 
 4.99981 
 
 0.04363 
 
 4.99828 
 
 0.13088 
 
 4.99524 
 
 0.21809 
 
 S 
 
 30 
 
 
 6 
 
 5-99977 
 
 0.05235 
 
 5-99794 
 
 0.15706 
 
 5.99428 
 
 0.261 7 1 
 
 6 
 
 
 
 7 
 
 6-99973 
 
 0.06108 
 
 6.99760 
 
 0.18323 
 
 6.99333 
 
 0.30533 
 
 7 
 
 
 
 8 
 
 7.99969 
 
 0.06981 
 
 7.99725 
 
 0.20941 
 
 7-99238 
 
 0.34895 
 
 8 
 
 
 
 9 
 
 8.99965 
 
 0.07853 
 
 8.99691 
 
 0.23559 
 
 8.99143 
 
 0.39257 
 
 9 
 
 
 
 1 
 
 0.99991 
 
 0.01308 
 
 0-99953 
 
 0.03053 
 
 0.99884 
 
 0.04797 
 
 1 
 
 
 
 2 
 
 1.99982 
 
 0.02617 
 
 1.99906 
 2.99860 
 
 0.06107 
 
 1.99769 
 
 0.09595 
 
 2 
 
 
 
 3 
 
 2.99974 
 
 0.03926 
 
 0.09161 
 
 2.99654 
 
 0.14393 
 
 3 
 
 
 
 4 
 
 3-99965 
 
 0.05235 
 
 3-99813 
 
 0.12215 
 
 3-99539 
 
 0.19:91 
 
 4 
 
 
 45 
 
 
 4-99957 
 
 0.06544 
 
 4.99766 
 
 0.15269 
 
 4.99424 
 
 0.23989 
 
 S 
 
 '5 
 
 
 6 
 
 
 0.07853 
 
 5.99720 
 
 0.18323 
 
 5-99309 
 
 0.28786 
 
 6 
 
 
 
 7 
 
 6.99940 
 
 0.09162 
 
 6.99673 
 
 0.21376 
 
 6-99193 
 
 0.33584 
 
 7 
 
 
 
 8 
 
 7-9993t 
 
 0.10471 
 
 7.99626 
 
 0.24430 
 
 7.99078 
 
 0.38382 
 
 8 
 
 
 
 9 
 
 8.99922 
 
 0.1 1780 
 
 8.99580 
 
 0.27484 
 
 8.98963 
 
 0.43180 
 
 9 
 
 
 8 
 
 O 
 1 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a 
 
 s 
 
 8 
 
 90 
 
 8 
 
 8° 
 
 8 
 
 7° 
 
 Smithsonian Tables. 
 
 32 
 
Table 9. 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 
 
 1 
 
 1 
 
 5 
 
 3 
 
 
 
 4 
 
 
 
 5 
 
 
 
 <u 
 u 
 
 c 
 
 .a 
 P 
 
 1 
 
 3 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.99863 
 
 0.05233 
 
 0.99756 
 
 0.06975 
 
 0.99619 
 
 0.08715 
 
 1 
 
 
 
 2 
 
 1.99726 
 
 0.10467 
 
 1.99512 
 
 0-1395 1 
 
 1.99238 
 
 0.17431 
 
 2 
 
 
 
 3 
 
 2.99589 
 
 0.15700 
 
 2.99269 
 
 0.20926 
 
 2.98858 
 
 0.26146 
 
 3 
 
 
 
 4 
 
 3-99452 
 
 0.20934 
 
 3-99025 
 
 0.27902 
 
 3-98477 
 
 0.34862 
 
 4 
 
 
 o 
 
 5 
 
 4-99315 
 5-99178 
 
 0.26168 
 
 4.98782 
 
 0.34878 
 
 4.98097 
 
 0-43577 
 
 5 
 
 60 
 
 
 6 
 
 0.31401 
 
 5-98538 
 
 0.41853 
 
 5.97716 
 
 0.52293 
 
 6 
 
 
 
 7 
 
 6.99041 
 
 0.36635 
 
 6.98294 
 
 0.48829 
 
 6-97336 
 
 0.61008 
 
 7 
 
 
 
 8 
 
 7.98904 
 
 0.41868 
 
 7.98051 
 
 0.55805 
 
 7-96955 
 
 0.69724 
 
 8 
 
 
 
 9 
 
 8.98767 
 
 0.47102 
 
 8.97807 
 
 0.62780 
 
 8-96575 
 
 0.78440 
 
 9 
 
 
 
 1 
 
 0.99839 
 
 0.05669 
 
 0.99725 
 
 0.07410 
 
 0.99580 
 
 0.09150 
 
 1 
 
 
 
 2 
 
 1.99678 
 
 0.11338 
 
 1.99450 
 
 0.14821 
 
 1.99160 
 
 0.18300 
 
 2 
 
 
 
 3 
 
 2.99517 
 
 0.17007 
 
 2-99175 
 
 0.22232 
 
 2.98741 
 
 0.27450 
 
 3 
 
 
 
 4 
 
 3-99356 
 
 0.22677 
 
 3.98900 
 
 0.29643 
 
 3.98321 
 
 0.36600 
 
 4 
 
 
 IS 
 
 5 
 
 4.99195 
 
 0.28346 
 
 4.98625 
 
 0.37054 
 
 4.97902 
 
 0.45750 
 
 5 
 
 45 
 
 
 6 
 
 5-99035 
 
 0.34015 
 
 5-98350 
 
 0.44465 
 
 5.97482 
 
 0.54900 
 
 6 
 
 
 
 7 
 
 6.98874 
 
 0.39684 
 
 6.98075 
 
 0.51875 
 
 6.97063 
 
 0.64051 
 
 7 
 
 
 
 8 
 
 7-98713 
 
 0-45354 
 
 7.97800 
 
 0.59286 
 
 7.96643 
 
 0.73201 
 
 8 
 
 
 
 9 
 
 8.98552 
 
 0.51023 
 
 8-97525 
 
 0.66697 
 
 8.96224 
 
 0.82351 
 
 9 
 
 
 
 1 
 
 0.99813 
 
 0.06104 
 
 0.99691 
 
 0.07845 
 
 0-99539 
 
 0.09584 
 
 1 
 
 
 
 2 
 
 1.99626 
 
 0.12209 
 
 1-99383 
 
 0.15691 
 
 
 0.19169 
 
 2 
 
 
 
 3 
 
 2.99440 
 
 0.18314 
 
 
 0-23537 
 
 2;986i8 
 
 0.28753 
 
 3 
 
 
 
 4 
 
 3-99253 
 
 0.24419 
 
 3.98766 
 
 0.31383 
 
 3.98158 
 
 0.38338 
 
 4 
 
 
 30 
 
 5 
 
 4.99067 
 
 0.30524 
 
 4.98458 
 
 0.39229 
 
 4-97698 
 
 0.47922 
 
 5 
 
 30 
 
 
 6 
 
 5.98880 
 
 0.36629 
 
 5.98150 
 
 0.47075 
 
 5-97237 
 
 0-57507 
 
 6 
 
 
 
 7 
 
 6.98694 
 
 0.42733 
 
 6.97842 
 
 0.54921 
 
 6.96777 
 
 0.67092 
 
 7 
 
 
 
 8 
 
 7.98507 
 
 0.48838 
 
 7-97533 
 
 0.62767 
 
 7.96316 
 
 0.76676 
 
 8 
 
 
 
 9 
 
 8.98321 
 
 0-54943 
 
 8.97225 
 
 0.70613 
 
 8.95856 
 
 0.86261 
 
 9 
 
 
 
 1 
 
 0.99785 
 
 0-06540 
 
 0.99656 
 
 0.08280 
 
 0.99496 
 
 0.10018 
 
 1 
 
 
 
 2 
 
 1.9957 1 
 
 0.13080 
 
 I -993 1 3 
 
 0.16561 
 
 1.98993 
 
 0.20037 
 
 2 
 
 
 
 3 
 
 2-99357 
 
 0.19620 
 
 2.98969 
 
 0.24842 
 
 2.98490 
 
 0.30056 
 
 3 
 
 
 
 4 
 
 3-99H3 
 
 0.26161 
 
 3.98626 
 
 0.33123 
 
 3-97987 
 
 0.40075 
 
 4 
 
 
 45 
 
 5 
 
 4.98929 
 
 0.32701 
 
 4.98282 
 
 0.41404 
 
 4.97484 
 
 0.50094 
 
 5 
 
 15 
 
 
 6 
 
 5-98715 
 
 0.39241 
 
 5-97939 
 
 0.49684 
 
 5.96981 
 
 0.601 12 
 
 6 
 
 
 
 7 
 
 6-98501 
 
 0.45782 
 
 6-97595 
 
 0.57965 
 
 6-96477 
 
 0.70131 
 
 7 
 
 
 
 8 
 
 7.98287 
 
 0.52322 
 
 7.97252 
 
 0.66246 
 
 7-95974 
 
 0.80150 
 
 8 
 
 
 
 9 
 
 8.98073 
 
 0.58862 
 
 8.96908 
 
 0.74527 
 
 8.95471 
 
 0.90169 
 
 9 
 
 
 s 
 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 g 
 
 3' 
 
 B 
 
 fD 
 fn 
 
 0)* 
 
 1 
 
 
 
 
 
 
 
 
 !3 
 
 c: 
 
 a 
 01 
 
 8 
 
 6° 
 
 8 
 
 5° 
 
 8 
 
 40 
 
 Smithsonian Tables. 
 
 33 
 
Table 9 
 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 
 
 cn 
 
 3 
 
 c 
 
 6 
 
 
 
 7 
 
 3 
 
 8 
 
 
 
 8 
 
 3 
 
 a 
 
 
 
 
 
 
 
 
 
 1 
 
 5 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 P 
 
 i 
 
 
 
 1 
 
 0.99452 
 
 0.10452 
 
 0.99254 
 
 0.12186 
 
 0.99026 
 
 0.13917 
 
 1 
 
 
 
 
 2 
 
 1.98904 
 
 0.20905 
 
 1.98509 
 
 0.24373 
 
 1.98053 
 
 0.27834 
 
 2 
 
 
 
 
 3 
 
 2.98356 
 
 0.31358 
 
 2.97763 
 
 0.36560 
 
 2.97080 
 
 0.41751 
 
 3 
 
 
 
 
 4 
 
 3.97808 
 
 0.41811 
 
 3.97018 
 
 0.48747 
 
 3.96107 
 
 0.55669 
 
 4 
 
 
 
 O 
 
 
 4.97261 
 
 0.52264 
 
 4-96273 
 
 0.60934 
 
 4-95134 
 
 0.69586 
 
 5 
 
 60 
 
 
 
 6 
 
 5-96713 
 
 0.62717 
 
 5-95519 
 
 0.731 21 
 
 5.94160 
 
 0.83503 
 
 6 
 
 
 
 
 7 
 
 6.96165 
 
 0.73169 
 
 6.94782 
 
 0.85308 
 
 6.93187 
 
 0.97421 
 
 7 
 
 
 
 
 8 
 
 7.95617 
 
 0.83622 
 
 7.94038 
 
 0.97495 
 
 7.92214 
 
 1-11338 
 
 8 
 
 
 
 
 9 
 
 8.95069 
 
 0.94075 
 
 8.93291 
 
 1.09682 
 
 8.91241 
 
 1.25255 
 
 9 
 
 
 
 
 1 
 
 0.99405 
 
 0.10886 
 
 0.99200 
 
 0.12619 
 
 0.98965 
 
 0.14349 
 
 1 
 
 
 
 
 2 
 
 i.^Sii 
 
 0.21773 
 
 1.98400 
 
 0.25239 
 
 1.97930 
 
 0.28698 
 
 2 
 
 
 
 
 3 
 
 2.98216 
 
 0.32660 
 
 2.97601 
 
 0.37859 
 
 2.96895 
 
 0.43047 
 
 3 
 
 
 
 
 4 
 
 3.97622 
 
 0.4354)5 
 
 3.96801 
 
 0.50479 
 
 3.95860 
 
 0.57397 
 
 4 
 
 
 
 IS 
 
 
 4.97028 
 
 0.54433 
 
 4.96002 
 
 0.63099 
 
 4.94825 
 
 0.71746 
 
 5 
 
 45 
 
 
 
 6 
 
 5-96433 
 
 0.65320 
 
 5.95202 
 
 0.75719 
 
 5-93790 
 
 0.86095 
 
 6 
 
 
 
 
 7 
 
 6.95839 
 
 0.76206 
 
 6.94403 
 
 0.88339 
 
 6.92755 
 
 1.00444 
 
 7 
 
 
 
 
 8 
 
 7-95245 
 
 0.87093 
 
 7-93603 
 
 1.00959 
 
 7.91721 
 
 1.14794 
 
 8 
 
 
 
 
 9 
 
 8.94650 
 
 0.97980 
 
 8.92804 
 
 I-13579 
 
 8.90686 
 
 1.29143 
 
 9 
 
 
 
 
 1 
 
 0.99357 
 
 0.1 1320 
 
 0.99144 
 
 0.13052 
 
 0.98901 
 
 0.14780 
 
 1 
 
 
 
 
 2 
 
 1.98714 
 
 0.22640 
 
 1.98288 
 
 0.26105 
 
 1.97803 
 
 0.29561 
 
 2 
 
 
 
 
 3 
 
 2.98071 
 
 0.33960 
 
 2.97433 
 
 0.39157 
 
 2.96704 
 
 0.44342 
 
 3 
 
 
 
 
 4 
 
 3.97428 
 
 0.45281 
 
 3-96577 
 
 0.52210 
 
 3.95606 
 
 0.59123 
 
 4 
 
 
 
 30 
 
 
 4.96786 
 
 0.56601 
 
 4.95722 
 
 0.65263 
 
 4.94508 
 
 0.73904 
 
 5 
 
 30 
 
 
 
 6 
 
 5-96143 
 
 0.67921 
 
 5.94866 
 
 0.78315 
 0.91368 
 
 5-93409 
 
 0.88685 
 
 6 
 
 
 
 
 7 
 
 6.95500 
 
 0.79242 
 
 6.9401 1 
 
 6.923:1 
 
 1.03466 
 
 7 
 
 
 
 
 8 
 
 7.94857 
 
 0.90562 
 1.01882 
 
 7.93155 
 
 1.04420 
 
 7.91212 
 
 i.i8'247 
 
 8 
 
 
 
 
 9 
 
 8.94214 
 
 8.92300 
 
 I-17473 
 
 8.901 14 
 
 1.33028 
 
 9 
 
 
 
 
 1 
 
 0.99306 
 
 O.II7S3 
 
 0.99086 
 
 0.13485 
 
 0.98836 
 
 0.15212 
 
 1 
 
 
 
 
 2 
 
 1.98613 
 
 0.23507 
 
 1-98173 
 
 0.26970 
 
 1.97672 
 
 0.30424 
 
 2 
 
 
 
 
 3 
 
 2.97920 
 
 0.35261 
 
 2-97259 
 
 0.40455 
 
 2.96508 
 
 0.45637 
 
 3 
 
 
 
 
 4 
 
 3.97227 
 
 0.47014 
 
 3-96346 
 
 0.53940 
 
 3-95344 
 
 0.60849 
 
 4 
 
 
 
 45 
 
 
 4-96534 
 5.95841 
 
 0.58768 
 
 4-95432 
 
 0.67425 
 
 4.94180 
 
 0.76061 
 
 5 
 
 15 
 
 
 
 6 
 
 0.70522 
 
 5.94519 
 
 0.80910 
 
 5.93016 
 
 0.91274 
 
 6 
 
 
 
 
 7 
 
 6.95147 
 
 0.82276 
 
 6.93606 
 
 0.94395 
 
 6.91853 
 
 1.06486 
 
 7 
 
 
 
 
 8 
 
 7-94454 
 
 0.94029 
 
 7.92692 
 
 1.07880 
 
 1.21698 
 
 8 
 
 
 
 
 9 
 
 8.93761 
 
 1-05783 
 
 8.91779 
 
 1-21365 
 
 8:8952s 
 
 1-36911 
 
 9 
 
 
 
 e 
 
 CD 
 0] 
 
 
 
 3 
 O 
 CD 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 d 
 
 
 a 
 
 s 
 8 
 
 
 8 
 
 3° 
 
 8 
 
 2° 
 
 8 
 
 1° 
 
 
 Smithsonian Tables. 
 
 34 
 
Table 9. 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 i 
 
 3 
 
 a 
 
 t 
 
 to 
 
 9 
 
 3 
 
 10° 
 
 U 
 
 
 
 1 
 
 to 
 
 
 
 
 
 
 
 
 iS 
 
 Q 
 
 Lat. 
 
 Bep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Q 
 
 i 
 
 
 1 
 
 0.98768 
 
 0.15643 
 
 0.98480 
 
 0.17364 
 
 0.98162 
 
 0.19081 
 
 1 
 
 
 
 2 
 
 1-97537 
 
 0.31286 
 
 1. 96961 
 
 0.34729 
 
 1.96325 
 
 0.38162 
 
 2 
 
 
 
 3 
 
 2.96306 
 
 0.46930 
 
 2.95442 
 
 0.52094 
 
 2.94488 
 
 0-57243 
 
 3 
 
 
 
 4 
 
 3-95075 
 
 0.62573 
 
 3-93923 
 
 0.69459 
 
 3.92650 
 
 0.76324 
 
 4 
 
 
 o 
 
 5 
 
 4.93844 
 
 0.78217 
 
 4.92403 
 
 0.86824 
 
 4.90813 
 
 0.9540s 
 
 5 
 
 60 
 
 
 6 
 
 5.9261 2 
 
 0.93860 
 
 5.90884 
 
 1.04188 
 
 5.88976 
 
 1.14486 
 
 6 
 
 
 
 7 
 
 6.91381 
 
 1.09504 
 
 6-89365 
 
 I-2I553 
 
 6.87139 
 
 1.33566 
 
 7 
 
 
 
 8 
 
 7.90150 
 
 1.25147 
 
 7.87846 
 
 1.38918 
 
 7.85301 
 
 1.52648 
 
 8 
 
 
 
 9 
 
 8.88919 
 
 1.40791 
 
 8.86327 
 
 1.56283 
 
 8.83464 
 
 1.71729 
 
 9 
 
 
 
 1 
 
 0.98699 
 
 0.16074 
 
 0.98404 
 
 0.17794 
 
 0.98078 
 
 0.19509 
 
 1 
 
 
 
 2 
 
 1-97399 
 
 0.32148 
 
 1.96808 
 
 0.35588 
 
 1.96157 
 
 0.39018 
 
 2 
 
 
 
 3 
 
 
 0.48222 
 
 2.95212 
 
 0-53383 
 
 2.9423s 
 
 0.58527 
 
 3 
 
 
 
 4 
 
 3-94798 
 
 0.64297 
 
 3.93616 
 
 0.7 1 1 77 
 
 3-92314 
 
 0.78036 
 
 4 
 
 
 15 
 
 5 
 
 4.93498 
 
 0.80371 
 
 4.92020 
 
 0.88971 
 
 4.90392 
 
 0-97545 
 
 5 
 
 45 
 
 
 6 
 
 5.92197 
 
 0.96445 
 
 5.90424 
 
 1.06766 
 
 5.88471 
 
 1.17054 
 
 6 
 
 
 
 7 
 
 6.90897 
 
 1.12519 
 
 6.88828 
 
 1.24560 
 
 6.86549 
 
 1-36563 
 
 7 
 
 
 
 8 
 
 7.89597 
 
 1.28594 
 
 7.87232 
 
 1.42354 
 
 7.84628 
 
 1.56072 
 
 8 
 
 
 
 9 
 
 8.88296 
 
 1.44668 
 
 8.85636 
 
 1.60149 
 
 8.82706 
 
 1.75581 
 
 9 
 
 
 
 1 
 
 0.98628 
 
 0.16504 
 
 0.9832s 
 
 0.18223 
 
 0.97992 
 
 0.19936 
 
 1 
 
 
 
 2 
 
 1.97257 
 2.958S5 
 
 0.33009 
 
 1.96650 
 
 0.36447 
 
 1-95984 
 
 0.39873 
 
 2 
 
 
 
 3 
 
 0.49514 
 
 2.94976 
 
 0.54670 
 
 2-93977 
 
 0.59810 
 
 3 
 
 
 
 4 
 
 3-945H 
 
 0.66019 
 
 3-933°i 
 
 0.72894 
 
 3.91969 
 
 0.79747 
 
 4 
 
 
 30 
 
 
 4.93142 
 
 0.82523 
 
 4.91627 
 
 0.91117 
 
 4.89962 
 
 0.99683 
 
 5 
 
 30 
 
 
 6 
 
 5.91771 
 
 0.99028 
 
 5.89952 
 
 1. 09341 
 
 5.87954 
 
 1.19620 
 
 6 
 
 
 
 7 
 
 6.90399 
 
 I-ISS33 
 
 6.88278 
 
 1.27564 
 
 6-85947 
 
 1-39557 
 
 7 
 
 
 
 8 
 
 7.89028 
 
 1.32038 
 
 7.86603 
 
 1.45788 
 
 7-83939 
 
 1.59494 
 
 8 
 
 
 
 9 
 
 8.87657 
 
 1.48542 
 
 8.84929 
 
 1.64011 
 
 8.81932 
 
 1-79431 
 
 9 
 
 
 
 1 
 
 0.9855s 
 
 0.1693s 
 
 0.9824s 
 
 0.18652 
 
 0.97904 
 
 0.20364 
 
 1 
 
 
 
 2 
 
 1.97111 
 
 0.33870 
 
 1.96490 
 
 0.37304 
 
 1.95809 
 
 0.40728 
 
 2 
 
 
 
 3 
 
 2.95666 
 
 0.50805 
 
 2-94735 
 
 0-55957 
 
 2.93713 
 
 0.61092 
 
 3 
 
 
 
 4 
 
 3-94222 
 
 0.67740 
 
 3.92980 
 
 0.74609 
 
 3.91618 
 
 0.81456 
 
 4 
 
 
 45 
 
 
 4.92778 
 
 0.84675 
 
 4.91225 
 
 0.93262 
 
 4.89522 
 
 1.01820 
 
 5 
 
 15 
 
 
 6 
 
 
 1.01610 
 
 5.89470 
 
 1.11914 
 
 5-87427 
 
 1.22185 
 
 6 
 
 
 
 7 
 
 6.89889 
 
 1-18545 
 
 6.87715 
 
 1.30566 
 
 6-85331 
 
 1.42549 
 
 7 
 
 
 
 8 
 
 7-88444 
 
 1.35480 
 
 7.85960 
 
 1.49219 
 
 7.83236 
 
 1.62913 
 
 8 
 
 
 
 9 
 
 8.87000 
 
 1-52415 
 
 8.84205 
 
 1.67871 
 
 8.81140 
 
 1.83277 
 
 9 
 
 
 a' 
 S 
 
 xn 
 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 5' 
 
 a 
 
 n 
 
 CO 
 
 8 
 
 T 
 
 7 
 
 go 
 
 7 
 
 B° 
 
 Smithsonian Tables. 
 
 35 
 
''"*^'-^®" TRAVERSE TABLE. 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. 
 
 -Continued. 
 
 s 
 
 6 
 
 a 
 
 .a 
 o 
 
 12° 
 
 13° 
 
 14° 
 
 8 
 
 1 
 
 
 
 1 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 
 1 
 
 0.97814 
 
 0.20791 
 
 0-97437 
 
 0.22495 
 
 0.97029 
 
 0.24192 
 
 1 
 
 
 
 
 2 
 
 1.95629 
 
 0.41582 
 
 1.94874 
 
 0.44990 
 
 1-94059 
 
 0.48384 
 
 2 
 
 
 
 
 3 
 
 2.93444 
 
 0.62373 
 
 2.9231 1 
 
 0.67485 
 
 ^■21 
 
 °-'^Pl^ 
 
 3 
 
 
 
 
 4 
 
 3.91259 
 
 0.83164 
 
 3.89748 
 
 0.89980 
 
 3.881 18 
 
 0.96768 
 
 4 
 
 
 
 o 
 
 
 4.89073 
 
 1-03955 
 
 4.87185 
 
 1-12475 
 
 4.85147 
 
 1. 20961 
 
 5 
 
 60 
 
 
 
 6 
 
 S.8688S 
 
 1.24747 
 
 5.84622 
 
 1-34970 
 
 5.82177 
 
 1-45153 
 
 6 
 
 
 
 
 7 
 
 6.84703 
 
 1-45538. 
 
 6.82059 
 
 1.57465 
 
 6.79206 
 
 1.6934s 
 
 7 
 
 
 
 
 8 
 
 7.82518 
 
 1.66329 
 
 7.79496 
 
 1.79960 
 
 7.76236 
 
 1-93537 
 
 8 
 
 
 
 
 9 
 
 8.80332 
 
 1. 87 1 20 
 
 8.76933 
 
 2.02455 
 
 8.73266 
 
 2.17729 
 
 9 
 
 
 
 
 1 
 
 0.97723 
 
 0.21217 
 
 0.97337 
 
 0.22920 
 
 0.96923 
 
 0.24615 
 
 1 
 
 
 
 
 2 
 
 1.95446 
 
 0.42435 
 
 1.9467s 
 
 0.45840 
 
 1.93846 
 
 0.49230 
 
 2 
 
 
 
 
 3 
 
 2.93169 
 
 0.63653 
 
 2.92013 
 
 0.68760 
 
 2.90769 
 
 0.7384s 
 
 3 
 
 
 
 
 4 
 
 
 0.84871 
 
 mi 
 
 0.91680 
 
 3-87692 
 
 0.98461 
 
 4 
 
 
 
 IS 
 
 
 4.8861 5 
 
 i.o6o88 
 
 1. 14600 
 
 4.84615 
 
 1.23076 
 
 5 
 
 45 
 
 
 
 6 
 
 5-86338 
 
 1.27306 
 
 5-84027 
 
 1.37520 
 
 5-81538 
 
 1-47691 
 
 6 
 
 
 
 
 7 
 
 
 1.48524 
 
 6.81365 
 
 1.60440 
 
 6.78461 
 
 1.72307 
 
 7 
 
 
 
 
 g 
 
 7.81784 
 
 1.69742 
 
 7.78703 
 
 1.83360 
 
 7-75384 
 
 1.96922 
 
 8 
 
 
 
 
 9 
 
 8.79507 
 
 1.90959 
 
 8.76041 
 
 2.06280 
 
 8.72307 
 
 2-21537 
 
 9 
 
 
 
 
 1 
 
 0.97629 
 
 0.21644 
 
 0.97237 
 
 0.23344 
 
 0.96814 
 
 0.25038 
 
 1 
 
 
 
 
 2 
 
 2.92888 
 
 0.43288 
 
 1-94474 
 
 0.46689 
 
 1.93629 
 
 0.50076 
 
 2 
 
 
 
 
 3 
 
 0.64932 
 
 2.91711 
 
 0.70033 
 
 2.90444 
 
 0.75114 
 
 3 
 
 
 
 
 4 
 
 3.90518 
 
 0.86576 
 
 3.88948 
 
 0.93378 
 
 3-87259 
 
 1.00152 
 
 4 
 
 
 
 3° 
 
 
 4.88148 
 
 1.08220 
 
 4.8618s 
 
 1. 16722 
 
 4.84073 
 5.80888 
 
 1.25190 
 
 s 
 
 30 
 
 
 
 6 
 
 5-85777 
 
 1.29864 
 
 5.83422 
 
 1.40067 
 
 1.50228 
 
 6 
 
 
 
 
 7 
 
 6.83407 
 
 1.51508 
 
 6.80659 
 
 1.63411 
 
 6.77703 
 
 1.75266 
 
 7 
 
 
 
 
 g 
 
 7.81036 
 
 1.73152 
 
 7-77896 
 
 1.86756 
 
 7.74518 
 
 2.00304 
 
 8 
 
 
 
 
 9 
 
 8.78666 
 
 1.94796 
 
 8.75133 
 
 2.10100 
 
 8-71332 
 
 2.25342 
 
 9 
 
 
 
 
 1 
 
 0-97534 
 
 0.22069 
 
 0.97134 
 
 0.23768 
 
 0.96704 
 
 0.25460 
 
 1 
 
 
 
 
 2 
 
 1.95068 
 
 0.44139 
 
 1.94268 
 
 0-47537 
 
 1.93409 
 
 0.50920 
 
 2 
 
 
 
 
 3 
 
 2.92602 
 
 0.66209 
 
 2.91402 
 
 0.71305 
 
 2.901 13 
 
 0.76380 
 
 3 
 
 
 
 
 4 
 
 3.90136 
 
 0.88278 
 
 3-88536 
 
 0.95074 
 
 3.86818 
 
 1.01840 
 
 4 
 
 
 
 45 
 
 5 
 
 4.87671 
 
 1. 10348 
 
 4.85671 
 
 1.18843 
 
 4-83523 
 
 1.27301 
 
 5 
 
 15 
 
 
 
 6 
 
 5.85205 
 
 1.32418 
 
 5.82805 
 
 1.42611 
 
 5.80227 
 6.76932 
 
 1.52761 
 
 6 
 
 
 
 
 7 
 
 6.82739 
 
 1.54488 
 
 6-79939 
 
 1.66380 
 
 1.78221 
 
 7 
 
 
 
 
 8 
 
 7.80273 
 
 1-76557 
 
 7-77073 
 
 1.90148 
 
 7-73636 
 
 2.03681 
 
 8 
 
 
 
 
 9 
 
 8.77808 
 
 1.98627 
 
 8.74207 
 
 2.13917 
 
 8.70341 
 
 2.29141 
 
 9 
 
 
 
 
 1 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 1 
 
 s 
 s 
 
 3' 
 
 a 
 
 03 
 
 
 77° 
 
 76° 
 
 75° 
 
 
 Smithsonian Tables. 
 
 36 
 
Table 9. 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 
 
 i 
 
 a 
 e 
 
 8 
 
 1 
 .a 
 
 Q 
 
 15° 
 
 16° 
 
 17° 
 
 
 i 
 
 ■3 
 a 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.96592 
 
 0.25881 
 
 0.96126 
 
 0.27563 
 
 0.95630 
 
 0.29237 
 
 1 
 
 
 
 2 
 
 i-93i8s 
 
 0.51763 
 
 1.92252 
 
 0.55127 
 
 1.91260 
 
 0.58474 
 
 2 
 
 
 
 3 
 
 2.89777 
 
 0.77645 
 
 2.88378 
 
 0.82691 
 
 2.86891 
 
 0.87711 
 
 3 
 
 
 
 4 
 
 3.86370 
 
 1.03527 
 
 3.84504 
 
 1. 10254 
 
 3.82521 
 
 1.16948 
 
 4 
 
 
 
 
 
 4.82962 
 
 1.29409 
 
 4.80630 
 
 1.37818 
 
 4.78152 
 
 1.46185 
 
 5 
 
 60 
 
 
 6 
 
 579555 
 6.76148 
 
 1.55291 
 
 5-76757 
 
 1.65382 
 
 5-73782 
 
 1-75423 
 
 6 
 
 
 
 7 
 
 1-81173 
 
 6.72883 
 
 1.92946 
 
 6.69413 
 
 2.04660 
 
 7 
 
 
 
 8 
 
 7.72740 
 
 2.07055 
 
 7.69009 
 
 2.20509 
 
 7-65043 
 
 2-33897 
 
 8 
 
 
 
 9 
 
 8-69333 
 
 2-32937 
 
 8.6513s 
 
 2.48073 
 
 8.60674 
 
 2.63134 
 
 9 
 
 
 
 1 
 
 0.96478 
 
 0.26303 
 
 0.96005 
 
 0.27982 
 
 0.95502 
 
 0.29654 
 
 1 
 
 
 
 2 
 
 1.92957 
 
 0.52606 
 
 1.92010 
 
 0.55965 
 
 1. 9 1 004 
 
 0.59308 
 
 2 
 
 
 
 3 
 
 2.89436 
 
 0.78909 
 
 2.88015 
 
 0.83948 
 
 2.86506 
 
 0.S8962 
 
 3 
 
 
 
 4 
 
 3.85914 
 
 1.05212 
 
 3.84020 
 
 1-11931 
 
 3.82008 
 
 i.i86i6 
 
 4 
 
 
 «S 
 
 5 
 
 4-82393 
 
 1-31515 
 1.57818 
 
 4.80025 
 
 1.39914 
 
 4.77510 
 
 1.48270 
 
 5 
 
 45 
 
 
 
 
 5.78872 
 
 5.76030 
 
 1.67897 
 
 5.73012 
 
 1.77924 
 
 6 
 
 
 
 7 
 
 6-7535 1 
 
 1.84121 
 
 6.72035 
 
 1.95880 
 
 6.68514 
 
 2.07579 
 
 7 
 
 
 
 8 
 
 7.71829 
 
 2.10424 
 
 7.68040 
 
 2.23863 
 
 7.64016 
 
 2-37233 
 
 8 
 
 
 
 9 
 
 8.68308 
 
 2.36728 
 
 8.6404s 
 
 2.51846 
 
 8.59518 
 
 2.66887 
 
 9 
 
 
 
 1 
 
 0.96363 
 
 0.26723 
 
 0.95882 
 
 0.28401 
 
 0-95371 
 
 0.30070 
 
 1 
 
 
 
 2 
 
 1.92726 
 
 0.53447 
 
 1.91764 
 
 0.56803 
 0.85204 
 
 1-90743 
 
 0.60 1 41 
 
 2 
 
 
 
 3 
 
 2.89089 
 
 0.801 7 1 
 
 2.87646 
 
 2.861 1 5 
 
 0.902 1 1 
 
 3 
 
 
 
 4 
 
 3.85452 
 
 1.06895 
 
 3-83528 
 
 1. 1 3606 
 
 3.81486 
 
 1.20282 
 
 4 
 
 
 30 
 
 5 
 
 4.81815 
 
 1.33619 
 
 4.79410 
 
 1.42007 
 
 4.76858 
 
 1.50352 
 
 5 
 
 30 
 
 
 6 
 
 5.78178 
 
 1.60343 
 
 5.75292 
 
 1.70409 
 
 5.72230 
 
 1.80423 
 
 6 
 
 
 
 7 
 
 6.74541 
 
 1.87066 
 
 6.71174 
 
 1.98810 
 
 6.67601 
 
 2.10494 
 
 7 
 
 
 
 8 
 
 7.70904 
 
 2.13790 
 
 7.67056 
 
 2.27212 
 
 7.62973 
 
 2.40564 
 
 8 
 
 
 
 9 
 
 8.67267 
 
 2.40514 
 
 8.62938 
 
 2-55613 
 
 8.58345 
 
 2.7063s 
 
 9 
 
 
 
 1 
 
 0.96245 
 
 0.27144 
 
 0-95757 
 
 0.28819 
 
 0.95239 
 
 0.30486 
 
 1 
 
 
 
 2 
 
 1.92491 
 
 0.54288 
 
 1.91514 
 
 0.57639 
 
 1.90479 
 
 0.60972 
 
 2 
 
 
 
 3 
 
 2.88736 
 
 0.81432 
 
 2.87271 
 
 0.86458 
 
 2.85718 
 
 0.91459 
 
 3 
 
 
 
 4 
 
 3.84982 
 
 1.08576 
 
 3.83028 
 
 1-15278 
 
 3.80958 
 
 1.21945 
 
 4 
 
 
 45 
 
 
 4.81227 
 
 1.35720 
 
 4.78785 
 
 1.44098 
 
 4-76197 
 
 1.52432 
 
 s 
 
 15 
 
 
 6 
 
 5-77473 
 6.73718 
 
 1.62864 
 
 5.74542 
 
 1.72917 
 
 ^•^5537 
 
 1.82918 
 
 6 
 
 
 
 7 
 
 1.90008 
 
 6.70299 
 
 2.01737 
 
 6.66677 
 
 2.13405 
 
 7 
 
 
 
 8 
 
 7.69964 
 
 2.17152 
 
 7.66057 
 
 2.30557 
 
 7.61916 
 
 2.43891 
 
 8 
 
 
 
 9 
 
 8.66209 
 
 2,44296 
 
 8.61814 
 
 2.59376 
 
 8.57156 
 
 2.74377 
 
 9 
 
 
 
 a 
 
 5' 
 
 1 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 re 
 en 
 
 7 
 
 1° 
 
 7 
 
 i° 
 
 7J 
 
 i° 
 
 Smithsonian Tables. 
 
 37 
 
^"■■^ ^' TRAVERSE TABLE. 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 
 i 
 
 .2 
 P 
 
 18° 
 
 IS 
 
 ° 
 
 20 
 
 ° 
 
 1 
 
 1 ■ 
 
 3 - 
 
 
 3 
 
 .s 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 (A 
 
 5 
 
 g 
 
 k 
 
 
 
 1 
 
 2 
 
 0.9510S 
 1. 902 1 1 
 2.85316 
 3.80422 
 4.75528 
 
 0.30901 
 0.61803 
 
 0.94551 
 1.89103 
 
 0-32556 
 0.651 13 
 
 0.93969 
 1.87938 
 
 0.34202 
 0.68404 
 1.02606 
 1.36808 
 
 1 
 2 
 
 
 
 
 3 
 4 
 
 0.92705 
 1.23606 
 
 2.83655 
 3.78207 
 
 0.97670 
 1.30227 
 
 2.81907 
 
 3 
 4 
 
 60 
 
 
 o 
 
 
 1.54508 
 
 4-72759 
 
 1.62784 
 
 4.69846 
 
 1.71010 
 
 6 
 
 
 
 g 
 
 5.70633 
 
 1.85410 
 
 5-673" 
 
 1.95340 
 
 5-63815 
 
 2.05212 
 
 
 
 
 7 
 
 6.6';7'!g 
 
 2.16311 
 
 6.61863 
 
 2.27897 
 
 6.57784 
 
 2.39414 
 2.73616 
 3.07818 
 
 7 
 
 
 
 
 7.6084s 
 
 2.47213 
 
 7.56414 
 
 2.60454 
 
 7-51754 
 
 8 
 
 
 
 
 9 
 
 8.55950 
 
 2.781 1 5 
 
 8.50966 
 
 2.9301 1 
 
 8.45723 
 
 9 
 
 
 
 
 1 
 
 0.94969 
 
 0.31316 
 
 0.94408 
 
 0.32969 
 
 0.93819 
 
 0.34611 
 
 1 
 
 
 
 
 2 
 
 1.89939 
 
 0.62632 
 
 1.88817 
 
 0.65938 
 
 1.87638 
 
 0.69223 
 
 2 
 
 
 
 
 3 
 
 2.84909 
 
 0-93949 
 
 2.83226 
 
 0.98907 
 
 2.81457 
 
 1.0383s 
 1.38446 
 
 3 
 
 
 
 
 4 
 
 379879 
 
 1.25265 
 
 3-77635 
 
 1.31876 
 
 3-75276 
 
 4 
 
 
 
 IS 
 
 
 4.74849 
 
 1.56581 
 1.87898 
 
 4.72044 
 
 1.64845 
 
 4.69095 
 
 1-73058 
 
 5 
 
 45 
 
 
 6 
 
 5.69819 
 
 5.66453 
 
 1.97814 
 
 5.62914 
 
 2.07670 
 
 6 
 
 
 
 
 7 
 
 6.64789 
 
 2.19214 
 
 6.60862 
 
 2.30783 
 
 6-56733 
 
 2.44281 
 
 7 
 
 
 
 
 8 
 
 7 -597 59 
 
 2.50531 
 2.81847 
 
 7.55271 
 
 2.63752 
 
 7-50553 
 
 2.76893 
 
 8 
 
 
 
 
 9 
 
 8.54729 
 
 8.49680 
 
 2.96721 
 
 8.44372 
 
 3-I150S 
 
 9 
 
 
 
 
 1 
 
 0.94832 
 
 0.31730 
 
 0.94264 
 
 0.33380 
 
 0.93667 
 
 0.35020 
 
 1 
 
 
 
 
 2 
 
 1.89664 
 
 0.63460 
 
 1.88528 
 
 0.66761 
 
 1-87334 
 
 0.70041 
 
 2 
 
 
 
 
 3 
 
 2.84497 
 
 0.95191 
 
 2.82792 
 
 1.00142 
 
 2.81001 
 
 1.05062 
 
 3 
 
 
 
 
 4 
 
 3-79329 
 
 1.26921 
 
 3.77056 
 
 1-33522 
 
 3.74668 
 
 1.40082 
 
 4 
 
 
 
 3° 
 
 
 4.74161 
 
 1.58652 
 
 4.71320 
 
 1.66903 
 
 4-68336 
 
 I-75103 
 
 5 
 
 30 
 
 
 
 6 
 
 S-68994 
 
 1.90382 
 
 5.65584 
 6.59849 
 
 2.00284 
 
 5.62003 
 
 2.10124 
 
 6 
 
 
 
 
 y 
 
 6.63826 
 
 2.221 1 3 
 
 2.33664 
 
 6.55670 
 
 2.45145 
 
 7 
 
 
 
 
 8 
 
 7.58658 
 
 2.53843 
 
 7-54113 
 
 2.67045 
 
 7-49337 
 
 2.80165 
 
 8 
 
 
 
 
 9 
 
 8-S349I 
 
 2.85574 
 
 8.48377 
 
 3.00426 
 
 8.43004 
 
 3.15186 
 
 9 
 
 
 
 
 1 
 
 0.94693 
 
 0.32143 
 
 0.94117 
 
 0-3379" 
 
 0.93513 
 
 0.35429 
 
 1 
 
 
 
 
 2 
 
 1.89386 
 
 0.64287 
 
 1.88235 
 
 0.67583 
 
 1.87027 
 
 0.70858 
 
 2 
 
 
 
 
 3 
 
 2.84079 
 
 0.96431 
 
 2.82352 
 
 1.01375 
 
 2.80540 
 
 1.06287 
 
 3 
 
 
 
 
 4 
 
 3-78772 
 
 1.28575 
 
 ■3.76470 
 
 1.35166 
 1.68958 
 
 3.74054 
 
 1.41716 
 
 4 
 
 
 
 45 
 
 5 
 
 4-73465 
 5.68158 
 
 1.60719 
 
 4.70588 
 
 4.67567 
 
 1-77145 
 
 5 
 
 IS 
 
 
 
 6 
 
 1.92863 
 
 5.64705 
 
 2.02750 
 
 5.61081 
 
 2.12574 
 
 6 
 
 
 
 
 7 
 
 6.62851 
 
 2.25007 
 
 6.58823 
 
 2.36541 
 
 6.54594 
 
 2.48003 
 
 7 
 
 
 
 
 8 
 
 7-57544 
 
 2.571 51 
 2.89295 
 
 7.52940 
 
 2.70333 
 
 7.48108 
 
 2.83432 
 
 8 
 
 
 
 
 9 
 
 8.52237 
 
 8.47058 
 
 3.04125 
 
 8.41621 
 
 3.18861 
 
 9 
 
 
 
 S 
 
 d 
 
 Crt' 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a 
 
 
 1 
 
 
 71° 
 
 70° 
 
 69° 
 
 
 Smithsonian Tables. 
 
 38 
 
Table 9. 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 en 
 
 •1 
 
 V 
 
 u 
 
 1 
 
 "tn 
 
 
 
 21° 
 
 22° 
 
 23° 
 
 
 
 1 
 
 5 
 
 to 
 
 C 
 
 i 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.93358 
 
 0.35836 
 
 0.92718 
 
 0.37460 
 
 0.92050 
 
 0-39073 
 
 1 
 
 
 
 2 
 
 1. 867 1 6 
 
 0.71673 
 
 1.85436 
 
 0.74921 
 
 1.84100 
 
 0.78146 
 
 2 
 
 
 
 3 
 
 2.80074 
 
 1.07510 
 
 2.78155 
 
 1.12381 
 1.49842 
 
 2.761 51 
 
 1.17219 
 
 3 
 
 
 
 4 
 
 3-73432 
 
 1-43347 
 
 3-70873 
 
 3.68201 
 
 1.56292 
 
 4 
 
 
 o 
 
 5 
 
 4.66790 
 
 1.79183 
 
 4.63591 
 
 1.87303 
 
 4.60252 
 
 1-95365 
 
 5 
 
 60 
 
 
 6 
 
 5.60148 
 
 2.15020 
 
 5.56310 
 
 2.24763 
 
 5-52302 
 
 2.34438 
 
 6 
 
 
 
 7 
 
 0.53506 
 7.46864 
 
 2.50857 
 
 6.49028 
 
 2.62224 
 
 6.44353 
 
 2.735" 
 
 7 
 
 
 
 8 
 
 
 7.41747 
 
 2.99685 
 
 7-36403 
 
 3.12584 
 
 8 
 
 
 
 9 
 
 8.40222 
 
 3-22531 
 
 8.34465 
 
 3-37145 
 
 8.28454 
 
 3-51657 
 
 9 
 
 
 
 1 
 
 0.93200 
 
 0.36243 
 
 0.92554 
 
 0.37864 
 
 0.91879 
 
 0.39474 
 
 1 
 
 
 
 2 
 
 1. 86401 
 
 0.72487 
 
 1.85108 
 
 0.75729 
 
 1.83758 
 
 0.78948 
 
 2 
 
 
 
 3 
 
 2.79602 
 
 1. 08731 
 
 2.77662 
 
 I-I3S94 
 
 2.75637 
 
 1. 18423 
 
 3 
 
 
 
 4 
 
 3.72803 
 
 1.44975 
 
 3.70216 
 
 1-51459 
 1.89324 
 
 3.67516 
 
 1-57897 
 
 4 
 
 
 IS 
 
 5 
 
 4.66004 
 
 1.81219 
 
 4.62770 
 
 4.59395 
 
 1-97372 
 2.36846 
 
 5 
 
 45 
 
 
 6 
 
 5.59204 
 
 2.17462 
 
 5-55324 
 
 2.27189 
 
 5-51274 
 
 6 
 
 
 
 7 
 
 6.52405 
 
 2.53706 
 
 6.47878 
 
 2.65054 
 
 6-43153 
 
 2.76320 
 
 7 
 
 
 
 8 
 
 7.45606 
 
 2.89950 
 
 7.40432 
 8.32986 
 
 3.02918 
 
 7-35032 
 
 3-15795 
 
 8 
 
 
 
 9 
 
 8.38807 
 
 3.26194 
 
 3-40783 
 
 8.26912 
 
 3.55269 
 
 9 
 
 
 
 1 
 
 o-93^i 
 
 0.36650 
 
 0.92388 
 
 0.38268 
 
 0.91706 
 
 0.39874 
 
 1 
 
 
 
 2 
 
 
 0.73300 
 
 1.84776 
 
 0.76536 
 
 1. 8341 2 
 
 0.79749 
 
 2 
 
 
 
 3 
 
 2.79125 
 
 1.09950 
 
 2.77164 
 
 1. 14805 
 
 2.751 18 
 
 1.19624 
 
 3 
 
 
 
 4 
 
 3.72167 
 
 1.46600 
 
 3.69552 
 
 1-53073 
 
 3.66824 
 
 1-59499 
 
 4 
 
 
 30 
 
 5 
 
 4.65208 
 5.58250 
 
 1.83250 
 
 4.61940 
 
 1.91341 
 
 4.58530 
 
 1-99374 
 
 5 
 
 30 
 
 
 6 
 
 2.19900 
 
 S-54328 
 
 2.29610 
 
 5-50236 
 
 2.39249 
 
 6 
 
 
 
 7 
 
 6.51292 
 
 2.56550 
 
 6.46716 
 
 2.67878 
 
 6.41942 
 
 2.79124 
 
 7 
 
 
 
 8 
 
 7-44334 
 
 2.93200 
 
 7-39104 
 
 3.06146 
 
 7.33648 
 
 3.18999 
 
 8 
 
 
 
 9 
 
 8-37375 
 
 3-29851 
 
 8.31492 
 
 3-+441S 
 
 8.25354 
 
 3.58874 
 
 9 
 
 
 
 1 
 
 0.92881 
 
 0-37055 
 
 0.92220 
 
 0.38671 
 
 0.91 531 
 
 0.40274 
 
 1 
 
 
 
 2 
 
 1.85762 
 2.78643 
 
 0.74111 
 
 1.84440 
 
 0.77342 
 
 1.83062 
 
 0.80549 
 
 2 
 
 
 
 3 
 
 1.11167 
 
 2.76660 
 
 1.16013 
 
 2.74593 
 
 1.20824 
 
 3 
 
 
 
 4 
 
 3-71524 
 
 1.48222 
 
 3.68880 
 
 1.54684 
 
 3.66124 
 
 1.61098 
 
 4 
 
 
 45 
 
 5 
 
 4.64405 
 
 1.85278 
 
 4.61100 
 
 I-9335S 
 
 4-57655 
 
 2.01373 
 
 5 
 
 IS 
 
 
 6 
 
 5.57286 
 
 2.22334 
 
 5-53320 
 
 2.32026 
 
 5.49186 
 
 2.41648 
 
 6 
 
 
 
 7 
 
 6.50167 
 
 2.59390 
 
 6.45540 
 
 2.70697 
 
 6.40718 
 
 2.81922 
 
 7 
 
 
 
 8 
 
 7.43048 
 
 2.96445 
 
 7.37760 
 
 3.09368 
 
 7.32249 
 
 3.22197 
 
 8 
 
 
 
 9 
 
 8-35929 
 
 3-33501 
 
 
 
 8.29980 
 
 3-48039 
 
 8.23780 
 
 3-62472 
 
 9 
 
 
 g 
 
 i 
 
 a 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 P 
 
 a 
 c 
 n 
 
 CO 
 
 68° 
 
 6 
 
 7° 
 
 6 
 
 6° 
 
 Smithsonian Tables. 
 
 39 
 
Table 9. 
 
 
 
 
 
 
 
 
 
 
 
 
 TRAVERSE TABLE. 
 
 
 
 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 1 
 .5 
 
 <l3 
 O 
 
 .1 
 
 24° 
 
 25° 
 
 26° 
 
 i 
 
 0) 
 
 s 
 c 
 
 
 
 
 
 
 
 in 
 
 S 
 
 fl 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 i 
 
 
 1 
 
 0.91354 
 
 0.40673 
 
 0.90630 
 
 0.42261 
 
 0.89879 
 
 0.43837 
 
 1 
 
 
 
 2 
 
 1.82709 
 
 0.81347 
 
 1.81261 
 
 0.84523 
 
 1.79758 
 
 0.87674 
 
 2 
 
 
 
 3 
 
 2.74063 
 
 1.22020 
 
 2.71892 
 
 1.26785 
 
 2.69638 
 
 1.31511 
 
 3 
 
 
 
 4 
 
 3.65418 
 
 1.62694 
 
 3.62523 
 
 1.69047 
 
 3-59517 
 
 1-75348 
 
 4 
 
 
 
 
 5 
 
 4.56772 
 
 2.03368 
 
 4-53153 
 
 2.11309 
 
 4-49397 
 
 2.19185 
 
 S 
 
 60 
 
 
 o 
 
 5.48127 
 
 2.44041 
 
 5-43784 
 
 2-53570 
 2.95832 
 
 3.80356 
 
 5.39276 
 
 2.63022 
 
 6 
 
 
 
 7 
 
 6.39481 
 
 2.84715 
 
 6.34415 
 
 6.29155 
 
 3.06859 
 
 7 
 
 
 
 8 
 
 7.30836 
 
 3-25389 
 
 7.25046 
 
 7-19035 
 
 3.50696 
 
 8 
 
 
 
 9 
 
 8.22190 
 
 3.66062 
 
 8.15677 
 
 8.08914 
 
 3-94533 
 
 9 
 
 
 
 1 
 
 0.91 176 
 
 0.41071 
 
 0.90445 
 
 0.42656 
 
 0.89687 
 
 0.44228 
 
 1 
 
 
 
 2 
 
 1.82352 
 
 0.82143 
 
 1.80891 
 
 0-85313 
 
 1-79374 
 
 0.88457 
 
 2 
 
 
 
 3 
 
 2.73528 
 
 1-23215 
 
 2.71336 
 
 1.27970 
 
 2.69061 
 
 1.32686 
 
 3 
 
 
 
 4 
 
 3.64704 
 
 1.64287 
 
 3.61782 
 
 1.70627 
 
 3-58749 
 
 1.76915 
 
 4 
 
 
 'S 
 
 5 
 
 4.55881 
 
 2.05359 
 
 4.52227 
 
 2.13284 
 
 4.48436 
 
 2.21144 
 
 
 45 
 
 
 6 
 
 5-47057 
 
 2.46431 
 
 5-42673 
 
 2.55941 
 2.98598 
 
 5-38123 
 5.27810 
 
 2.65373 
 
 g 
 
 
 
 7 
 
 6.38233 
 
 2.87503 
 
 6.331 18 
 
 3.09602 
 
 7 
 
 
 
 8 
 
 7.29409 
 
 3-28575 
 
 7-23564 
 
 3-41254 
 
 7.17498 
 
 3-53830 
 3.98059 
 
 8 
 
 
 
 9 
 
 8.20585 
 
 3.69647 
 
 8.14009 
 
 3-839" 
 
 8.07185 
 
 9 
 
 
 
 1 
 
 0.90996 
 
 0.41469 
 
 0.90258 
 
 0.43051 
 
 0.89493 
 
 0.44619 
 
 1 
 
 
 
 2 
 
 1.81992 
 
 0.82938 
 
 1.80517 
 
 0.86102 
 
 1.78986 
 
 0.89239 
 
 2 
 
 
 
 3 
 
 2.72988 
 
 1.24407 
 
 2.70775 
 
 1.29153 
 
 2.68480 
 
 1-33859 
 
 3 
 
 
 
 4 
 
 3-63984 
 
 1.65877 
 
 3.61034 
 
 1.72204 
 
 3-57973 
 
 1.78479 
 
 4 
 
 
 30 
 
 5 
 
 4.54980 
 
 2.07346 
 
 4.51292 
 
 2.15255 
 
 4.47467 
 
 2.23098 
 
 
 30 
 
 
 6 
 
 5.45976 
 
 2.48815 
 
 5-41551 
 
 2.58306 
 
 5.36960 
 
 2.67718 
 
 6 
 
 
 
 7 
 
 6.36972 
 
 2.90285 
 
 6.31809 
 
 3-01357 
 
 6.26454 
 
 3-12338 
 
 7 
 
 
 
 8 
 
 7.27969 
 
 3-31754 
 
 7.22068 
 
 3-44408 
 
 7-15947 
 
 3-56958 
 
 8 
 
 
 
 9 
 
 8.18965 
 
 3-73223 
 
 8.12326 
 
 3-87459 
 
 8.05440 
 
 4.01578 
 
 9 
 
 
 
 1 
 
 0.90814 
 
 0.41866 
 
 0.90069 
 
 0-43444 
 
 0.89297 
 
 0.45009 
 
 1 
 
 
 
 2 
 
 1.81628 
 
 0.83732 
 
 1. 801 39 
 
 0.86889 
 
 1.78595 
 2.67893 
 
 0.90019 
 
 2 
 
 
 
 3 
 
 2.72442 
 
 1.25598 
 
 2.70209 
 
 1-30333 
 
 1.35029 
 
 -J 
 
 
 
 4 
 
 3-63257 
 
 1.67464 
 
 3-60279 
 
 1-73778 
 
 3-57191 
 
 1.80039 
 
 4 
 
 
 45 
 
 5 
 
 4.54071 
 
 2.09330 
 
 4-50349 
 5.40418 
 
 2.17222 
 
 4-46489 
 
 2.25049 
 
 
 IS 
 
 
 6 
 
 5-44885 
 
 2.51196 
 
 2.60667 
 
 5-35787 
 
 2.70059 
 
 g 
 
 
 7 
 
 6.35700 
 
 2.93062 
 
 6.30488 
 
 3.04111 
 
 6.25085 
 
 3.15068 
 
 7 
 
 
 
 8 
 
 7.26514 
 
 3-34928 
 
 7.20558 
 8.10628 
 
 3-47556 
 
 7-14383 
 
 3.60078 
 
 8 
 
 
 
 9 
 
 8.17328 
 
 3-76794 
 
 3.91000 
 
 8.03681 
 
 4.05088 
 
 9 
 
 
 5' 
 
 O 
 
 ..... n 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a 
 
 en 
 
 S 
 
 
 fD 
 
 c 
 
 0) 
 
 6£ 
 
 r 
 
 >° 
 
 64° 
 
 63 
 
 ° 
 
 40 
 
TRAVERSE TABLE. ^'^'■^ ®' 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 
 
 1 
 
 a3 
 1 
 
 CO 
 
 27° 
 
 28° 
 
 29° 
 
 
 s 
 
 1 
 
 
 
 
 
 
 
 ■i 
 
 P 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 
 i 
 
 
 1 
 
 0.89100 
 
 0-45399 
 0.90798 
 
 0.88294 
 
 0.46947 
 
 0.87462 
 
 0.48481 
 
 1 
 
 
 
 2 
 
 1.78201 
 
 1.76589 
 
 0.93894 
 
 1.74924 
 
 0.96962 
 
 2 
 
 
 
 3 
 
 2.67301 
 
 1-36197 
 
 2.64884 
 
 1.40841 
 
 2.62386 
 
 1-45443 
 
 3 
 
 
 
 4 
 
 3.56402 
 
 1.81596 
 
 3-53179 
 
 1.87788 
 
 1.93924 
 
 4 
 
 
 
 
 S 
 
 4-45503 
 
 2.26995 
 
 4-41473 
 
 2-34735 
 
 4-37310 
 
 2.42405 
 
 
 60 
 
 
 6 
 
 5-34603 
 
 2.72394 
 
 S-29768 
 
 2.81682 
 
 5.24772 
 
 2.90886 
 
 6 
 
 
 
 7 
 
 6.23704 
 
 3-17793 
 
 6.18063 
 
 3.28630 
 
 6.12234 
 
 3-39367 
 
 7 
 
 
 
 8 
 
 7.12805 
 
 3-63193 
 4.08591 
 
 7-06358 
 
 3-75577 
 
 6.99696 
 
 3.87848 
 
 8 
 
 
 
 9 
 
 8.01905 
 
 7.94652 
 
 4.22524 
 
 7.87156 
 
 4-36329 
 
 9 
 
 
 
 1 
 
 0.88901 
 
 0.45787 
 
 0.88089 
 
 0-47332 
 
 0.87249 
 
 0.48862 
 
 1 
 
 
 
 2 
 
 1.77803 
 
 0.91574 
 
 1.76178 
 
 0.94664 
 
 1.74499 
 
 0-97724 
 
 2 
 
 
 
 3 
 
 2.66705 
 
 1-37362 
 
 2.64267 
 
 1.41996 
 
 2.61748 
 
 1.46566 
 
 3 
 
 
 
 4 
 
 3-55606 
 
 1.83149 
 
 3-52356 
 
 1.89328 
 
 3.48998 
 
 1.95448 
 
 4 
 
 
 IS 
 
 5 
 
 4.44508 
 
 2.28937 
 
 4.40445 
 
 2.36660 
 
 4.36248 
 
 2.44310 
 
 
 45 
 
 
 6 
 
 5-33410 
 
 2.74724 
 
 5-28534 
 
 2.83992 
 
 5-23497 
 
 2.93172 
 
 6 
 
 
 
 7 
 
 6.2231 1 
 
 3.205U 
 
 6.16623 
 
 3-31324 
 
 6.10747 
 
 3-42034 
 
 7 
 
 
 
 8 
 
 7.11213 
 
 3.66299 
 
 7.04712 
 
 3.78656 
 
 6-97996 
 
 3.90896 
 
 8 
 
 
 
 9 
 
 8.00H5 
 
 4.12086 
 
 7.92801 
 
 4.25988 
 
 7-85246 
 
 4-39759 
 
 9 
 
 
 
 1 
 
 0.88701 
 
 0.46174 
 
 0.87881 
 
 0.47715 
 
 0.87035 
 
 0.49242 
 
 1 
 
 
 
 2 
 
 1.77402 
 
 0.92349 
 
 1-75763 
 
 0-95431 
 
 1.74071. 
 
 0.98484 
 
 2 
 
 
 
 3 
 
 2.66103 
 
 1.38524, 
 
 2.63645 
 
 1-43147 
 
 2.61106 
 
 1.47727 
 
 3 
 
 
 
 4 
 
 3.54804 
 
 1.84699 
 
 3-S1526 
 
 1.90863 
 
 3.48142 
 
 1.96969 
 
 4 
 
 
 30 
 
 5 
 
 4-43505 
 
 2.30874 
 
 4.39408 
 
 2.38579 
 
 4-35177 
 
 2.4621 1 
 
 
 30 
 
 
 6 
 
 5.32206 
 
 2.77049 
 
 5.27290 
 
 2.86295 
 
 5.22213 
 
 2-95454 
 
 6 
 
 
 
 7 
 
 6.20907 
 
 3.23224 
 
 6.1517: 
 
 3-340II 
 
 6.09248 
 
 3.44696 
 
 7 
 
 
 
 8 
 
 7.09608 
 
 3-69398 
 
 7-03053 
 
 3.81727 
 
 6.96284 
 
 3-93938 
 
 8 
 
 
 
 9 
 
 7.98309 
 
 4-15573 
 
 7-90935 
 
 4.29442 
 
 7.83320 
 
 4.43181 
 
 9 
 
 
 
 1 
 
 0.88498 
 
 0.46561 
 
 0.87672 
 
 0.48098 
 
 0.86819 
 
 0.49621 
 
 1 
 
 
 
 2 
 
 1.76997 
 
 0.93122 
 
 1-75345 
 
 0.96197 
 
 1-73639 
 
 0-99243 
 
 2 
 
 
 
 3 
 
 2.65496 
 
 1.39684 
 
 2.63018 
 
 1.44296 
 
 2.60459 
 
 1.48864 
 
 3 
 
 
 
 4 
 
 3-53995 
 
 1.86245 
 
 3.50690 
 
 1.92395 
 
 3-47279 
 
 1.98486 
 
 4 
 
 
 45 
 
 5 
 
 4.42493 
 
 2.32807 
 
 4-38363 
 
 2.40494 
 
 4.34099 
 
 2.48108 
 
 5 
 
 IS 
 
 
 6 
 
 5.30992 
 
 2.79368 
 
 5.26036 
 
 2-88593 
 
 5.20919 
 
 2.97729 
 
 6 
 
 
 
 7 
 
 6.19491 
 
 3-25930 
 
 6.13708 
 
 3.36692 
 
 6.07739 
 
 3-47351 
 
 7 
 
 
 
 8 
 
 7.07990 
 
 3.72491 
 
 7.01381 
 
 3.84791 
 
 6.94559 
 
 3-96973 
 
 8 
 
 
 
 9 
 
 7-96488 
 
 4.19053 
 
 7.89054 
 
 4.32889 
 
 7.81378 
 
 4.46594 
 
 9 
 
 
 a 
 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 s 
 
 3" 
 
 B 
 a 
 
 CO 
 
 S 
 
 n 
 re 
 
 
 
 
 
 
 
 i 
 
 5 
 
 p 
 
 n 
 en 
 
 6 
 
 2° 
 
 6 
 
 L° 
 
 6 
 
 0° 
 
 Smithsonian Tables. 
 
 41 
 
"^•^"-^ ®' TRAVERSE TABLE. 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 1 
 
 c 
 
 i 
 
 1 
 
 to 
 
 5 
 
 30° 
 
 31° 
 
 32° 
 
 8 
 
 § 
 .2 
 P 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.86602 
 
 0.50000 
 
 0.85716 
 
 0.51503 
 
 0.84804 
 
 0.52991 
 
 1 
 
 
 
 2 
 
 1.7320S 
 
 1. 00000 
 
 1-71433 
 
 1.03007 
 
 1.69609 
 
 I-OW83 
 
 2 
 
 
 
 3 
 
 2.59807 
 
 1.50000 
 
 2.57150 
 
 1.54511 
 
 2.54414 
 
 1-58975 
 
 3 
 
 
 
 4 
 
 3.46410 
 
 2.00000 
 
 3.42866 
 
 2.06015 
 
 3-39219 
 
 2.1 1967 
 
 4 
 
 
 o 
 
 
 4.33012 
 
 2.50000 
 
 4.28583 
 
 2-57519 
 
 4.24024 
 
 2.64959 
 
 1 
 
 60 
 
 
 6 
 
 5.19615 
 
 3.00000 
 
 5.14300 
 
 3.09022 
 
 5.08828 
 
 3-17951 
 
 6 
 
 
 
 7 
 
 6.06217 
 
 3.50000 
 
 6.00017 
 
 3.60526 
 
 S-93633 
 6.78438 
 
 3-70943 
 
 7 
 
 
 
 8 
 
 6.92820 
 
 4.00000 
 
 6-85733 
 
 4.12030 
 
 4-23935 
 
 8 
 
 
 
 9 
 
 7.79422 
 
 4.50000 
 
 7.71450 
 
 4-63534 
 
 7-63243 
 
 4-76927 
 
 9 
 
 
 
 1 
 
 0.86383 
 
 0.50377 
 
 0.85491 
 
 0.51877 
 
 0.84572 
 
 0-53361 
 
 1 
 
 
 
 2 
 
 1.72767 
 
 1.00754 
 
 1.70982 
 
 1-03754 
 
 1.6914s 
 
 1.06722 
 
 2 
 
 
 
 3 
 
 2.59150 
 
 1.51132 
 
 2.56473 
 
 I-55631 
 
 2.53718 
 
 1.60084 
 
 3 
 
 
 
 4 
 
 3-45534 
 
 2.01509 
 
 3.41964 
 
 2.07509 
 
 3.38291 
 
 2.I344S 
 
 4 
 
 
 IS 
 
 
 4-31917 
 
 2.51887 
 
 4.27456 
 
 2.59386 
 
 4.22863 
 
 2.66S07 
 
 1 
 
 45 
 
 
 6 
 
 5.18301 
 
 3.02264 
 
 5.12947 
 
 3.1 1263 
 
 5.07436 
 
 3.20168 
 
 
 
 7 
 
 6.04684 
 
 3.52641 
 
 5.98438 
 
 3-63141 
 
 5.92009 
 
 4.26891 
 
 7 
 
 
 
 8 
 
 6.91068 
 
 4.03019 
 
 6.83929 
 
 4.15018 
 
 6.76582 
 
 8 
 
 
 
 9 
 
 7-774SI 
 
 4-53396 
 
 7.69420 
 
 4.6689s 
 
 7.61155 
 
 4.80253 
 
 9 
 
 
 
 1 
 
 0.86162 
 
 0.50753 
 
 0.85264 
 
 0.52249 
 
 0.84339 
 
 0.53730 
 
 1 
 
 
 
 2 
 
 '•72325 
 
 1. 01 507 
 
 1.70528 
 
 1.04499 
 
 1.68678 
 
 1.07460 
 
 2 
 
 
 
 3 
 
 2.58488 
 
 1.52261 
 
 2-55792 
 
 1.56749 
 
 2-53017 
 
 1.61190 
 
 3 
 
 
 
 4 
 
 3.44651 
 
 2.03015 
 
 3.41056 
 
 2.08999 
 
 3-37356 
 
 2.14920 
 
 4 
 
 
 30 
 
 
 4.30814 
 
 2.53769 
 
 4.26320 
 
 2.61249 
 
 4.21695 
 
 2.68650 
 
 1 
 
 30 
 
 
 6 
 
 5.16977 
 
 3-04523 
 
 S-11584 
 
 3-13499 
 
 5.06034 
 
 3.22380 
 
 
 
 7 
 
 6.03140 
 
 3-55276 
 
 6.82 II 2 
 
 3-65749 
 
 5-90373 
 
 3-761 10 
 
 7 
 
 
 
 8 
 
 6.89303 
 
 4.06030 
 4.56784 
 
 4-17998 
 
 6-74713 
 
 4.29840 
 
 8 
 
 
 
 9 
 
 7.75466 
 
 7.67376 
 
 4.70248 
 
 7.59052 
 
 4.83570 
 
 9 
 
 
 
 1 
 
 0.85940 
 
 0.51 129 
 
 0.85035 
 
 0.52621 
 
 0.84103 
 
 0.54097 
 
 1 
 
 
 
 2 
 
 I.7I88I 
 
 1.02258 
 r--53387 
 
 1.70070 
 
 1.05242 
 
 1.68207 
 
 1.08194 
 
 2 
 
 
 
 3 
 
 2.57821 
 
 2-55105 
 
 1.57864 
 
 2.5231 1 
 
 1.62292 
 2.16389 
 
 3 
 
 
 
 4 
 
 3-43762 
 
 2.04517 
 
 3.40140 
 
 2.10485 
 
 3-36415 
 
 4 
 
 
 4S 
 
 5 
 
 4.29703 
 
 2.55646 
 
 4.25176 
 
 2.63107 
 
 4.20519 
 
 2.70487 
 
 5 
 
 'S 
 
 
 6 
 
 5-15643 
 
 3-06775 
 
 5.102H 
 
 3.15728 
 3-68349 
 
 5.04623 
 
 3.24584 
 
 6 
 
 
 
 7 
 
 6.01584 
 
 3-57905 
 
 5.95246 
 
 5.88827 
 
 3.78682 
 
 7 
 
 
 
 8 
 
 6.87525 
 
 4.09034 
 
 6.80281 
 
 4.20971 
 
 6.72831 
 
 4.32779 
 
 8 
 
 
 
 9 
 
 7-73465 
 
 4.60163 
 
 7.65316 
 
 4-73592 
 
 7-56935 
 
 4.86877 
 
 9 
 
 
 § 
 
 a 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a 
 
 s 
 
 5 
 
 c 
 
 
 
 
 
 
 
 
 K 
 
 1 
 
 
 
 
 
 
 
 01 
 
 s 
 
 5 
 
 3° 
 
 5 
 
 3° 
 
 5 
 
 70 
 
 P 
 
 p 
 
 Smithsonian Tables. 
 
 42 
 
TRAVERSE TABLE. ^*°'-^ ®' 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 i 
 
 u 
 
 § 
 
 CO 
 
 33° 
 
 34° 
 
 35° 
 
 (D 
 (J 
 C 
 
 CO 
 
 CO 
 
 B 
 
 a 
 a 
 
 
 
 
 
 
 
 i 
 
 5 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Q 
 
 § 
 
 
 1 
 
 0.83867 
 
 0.54463 
 
 0.82903 
 
 0.55919 
 
 0.81915 
 
 0.57357 
 
 1 
 
 
 
 2 
 
 1.67734 
 
 1.08927 
 
 1.65807 
 
 1.11838 
 
 1.63830 
 
 1.14715 
 
 2 
 
 
 
 3 
 
 2.51601 
 
 1.63391 
 
 2.48711 
 
 1.67757 
 
 2.45745 
 
 1.72072 
 
 3 
 
 
 
 4 
 
 3-35468 
 
 2.17855 
 
 3-31615 
 4.14518 
 
 2.23677 
 
 3.27660 
 
 2.29430 
 
 4 
 
 
 
 
 5 
 
 4-19335 
 
 2.72319 
 
 2.79596 
 
 4.09576 
 
 2.86788 
 
 5 
 
 60 
 
 
 6 
 
 5.03202 
 
 3-26783 
 
 4.97422 
 
 3-35515 
 
 4.91491 
 
 3-44145 
 
 6 
 
 
 
 7 
 
 5.87069 
 
 3.81247 
 
 5.80326 
 
 3-91435 
 
 5.73406 
 
 4.01503 
 
 7 
 
 
 
 8 
 
 6.70936 
 
 4.35711 
 
 6.63230 
 
 4-47354 
 
 6-55321 
 
 4.58861 
 
 8 
 
 
 
 9 
 
 7.54803 
 
 4.90175 
 
 7-46133 
 
 5-03273 
 
 7-37236 
 
 5.16218 
 
 9 
 
 
 
 1 
 
 0.83628 
 
 0.54829 
 
 0.82659 
 
 0.56280 
 
 0.81664 
 
 0.57714 
 
 1 
 
 
 
 2 
 
 1.67257 
 
 1.09658 
 
 1.65318 
 
 1. 12560 
 
 1.63328 
 
 1.15429 
 
 2 
 
 
 
 3 
 
 2.50885 
 
 1.64487 
 
 2.47977 
 
 1.68841 
 
 2.44992 
 
 1-73143 
 
 3 
 
 
 
 4 
 
 3-34514 
 
 219317 
 
 3-30636 
 
 2.25121 
 
 3.26656 
 
 2.30858 
 
 4 
 
 
 IS 
 
 5 
 
 4-18143 
 
 2.74146 
 
 4.13295 
 
 2.81402 
 
 4.08320 
 
 2.88572 
 
 
 45 
 
 
 6 
 
 5.01771 
 
 3.28975 
 
 4.95954 
 5.78613 
 
 3-37682 
 
 4.89984 
 
 3.46287 
 
 6 
 
 
 
 7 
 
 5-85400 
 
 3.83805 
 
 3-93963 
 
 5-71649 
 
 4.04001 
 
 7 
 
 
 
 8 
 
 6.69028 
 
 4.38634 
 
 6.61272 
 
 4.50243 
 
 6-53313 
 
 4.61 7 16 
 
 8 
 
 
 
 9 
 
 7.52657 
 
 4.93463 
 
 7-43931 
 
 5.06524 
 
 7-34977 
 
 5.19430 
 
 9 
 
 
 
 1 
 
 0.83388 
 
 0-55193 
 
 0.82412 
 
 0.56640 
 
 0.81411 
 
 0.58070 
 
 1 
 
 
 
 2 
 
 1.66777 
 
 1. 1 0387 
 
 1.64825 
 
 1.13281 
 
 1.62823 
 
 1.16140 
 
 2 
 
 
 
 3 
 
 2.50165 
 
 1.65581 
 
 2.47237 
 
 I. 69921 
 
 2.44234 
 
 1.74210 
 
 3 
 
 
 
 4 
 
 3-33554 
 
 2.20774 
 
 3.29650 
 
 2.26562 
 
 3.25646 
 
 2.32281 
 
 4 
 
 
 30 
 
 5 
 
 4.16942 
 
 2.75968 
 
 4.12063 
 
 2.83203 
 
 4.07057 
 
 2.90351 
 
 5 
 
 30 
 
 
 6 
 
 5-00331 
 
 3.31 162 
 
 4.94475 
 
 3-39843 
 
 4.88469 
 
 3.48421 
 
 6 
 
 
 
 7 
 
 5.83720 
 
 3-86355 
 
 5.76888 
 
 3.96484 
 
 5.69880 
 
 4.06492 
 
 7 
 
 
 
 8 
 
 6.67108 
 
 4.41549 
 
 6.59300 
 
 4.53124 
 
 6.51292 
 
 4.64562 
 
 8 
 
 
 
 9 
 
 7.50497 
 
 4.96743 
 
 7.41713 
 
 5-09765 
 
 7.32703 
 
 5.22632 
 
 9 
 
 
 
 1 
 
 0.83147 
 
 0-55557 
 
 0.82164 
 
 5-56999 
 
 0.81 1 57 
 
 0.58425 
 
 1 
 
 
 
 2 
 
 1.66294 
 
 1.11114 
 
 1.64329 
 
 1-13999 
 
 1.62314 
 
 1. 16850 
 
 2 
 
 
 
 3 
 
 2.49441 
 
 1. 6667 1 
 
 2.46494 
 
 
 2.43472 
 
 1.75275 
 
 3 
 
 
 
 4 
 
 3.32588 
 
 2.22228 
 
 3.28658 
 
 2.27998 
 
 3.24629 
 
 2.33700 
 
 4 
 
 
 45 
 
 5 
 
 4-' 5735 
 4.98882 
 
 2.77785 
 
 4.10823 
 
 2.84998 
 
 4.05787 
 
 2.92125 
 
 
 15 
 
 
 6 
 
 
 4.92988 
 
 3-41998 
 
 4.86944 
 
 3-50550 
 
 6 
 
 
 
 7 
 
 5.82029 
 
 3.88899 
 
 5-75152 
 
 3-98997 
 
 5.68101 
 
 4.08975 
 
 7 
 
 
 
 8 
 
 6.65176 
 
 4.44456 
 
 6.57317 
 
 4-55997 
 
 6.49260 
 
 4.67400 
 
 8 
 
 
 
 9 
 
 7.48323 
 
 5.00013 
 
 7.39482 
 
 5.12997 
 
 7.30416 
 
 5.25825 
 
 9 
 
 
 t 
 
 
 
 S-' 
 p 
 
 3 
 
 n 
 
 CD 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a 
 
 c 
 n 
 
 CO 
 
 5 
 
 5° 
 
 5 
 
 5° 
 
 5 
 
 1° 
 
 Smithsonian Tables. 
 
 43 
 
Table 9. 
 
 TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 s. 
 
 3 
 
 .3 
 
 Q 
 
 36° 
 
 37° 
 
 38° 
 
 CO 
 
 S 
 
 3 
 
 .a 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.80901 
 
 0.58778 
 
 0.79863 
 
 0.60181 
 
 0.78801 
 
 0.61 566 
 
 1 
 
 
 
 2 
 
 1.61803 
 
 I-I75S7 
 
 1.59727 
 
 1.20363 
 
 1.57602 
 
 1. 23132 
 
 2 
 
 
 
 3 
 
 2.42705 
 
 176335 
 
 2.39590 
 
 1.80544 
 
 2.36403 
 
 1.84698 
 
 3 
 
 
 
 4 
 
 3.23606 
 
 2.35114 
 
 3-19454 
 
 2.40726 
 
 3.15204 
 
 2.46264 
 
 4 
 
 
 
 
 
 4.04508 
 
 2.93892 
 
 3-99317 
 
 3.00907 
 
 3-9400S 
 
 3.07830 
 
 s 
 
 60 
 
 
 6 
 
 4.85410 
 
 3.52671 
 
 4.79181 
 
 3.61089 
 
 4.72806 
 
 3-69396 
 
 6 
 
 
 
 7 
 
 5.6631 1 
 
 4.1 1449 
 
 5-59044 
 
 4.21270 
 
 5.51607 
 
 4-30963 
 
 7 
 
 
 
 8 
 
 6.47213 
 
 4.70228 
 
 6.38908 
 
 4.81452 
 
 6.30408 
 
 4.92529 
 
 8 
 
 
 
 9 
 
 7.28115 
 
 5.29006 
 
 7.18771 
 
 S-41633 
 
 7.09209 
 
 s- 54095 
 
 9 
 
 
 
 1 
 
 0.80644 
 
 0.59130 
 
 0.79600 
 
 0.60529 
 
 1-78531 
 
 0.61909 
 
 1 
 
 
 
 2 
 
 1. 61 288 
 
 1.18261 
 
 1.59200 
 
 1.21058 
 1.81588 
 
 1.57063 
 
 1.23818 
 
 2 
 
 
 
 3 
 
 2.41933 
 
 1-77392 
 
 2.38800 
 
 2-35595 
 
 1.85728 
 
 3 
 
 
 
 4 
 
 3.22577 
 
 2.36523 
 
 3.18400 
 
 2.421 17 
 
 3.14126 
 
 2.47637 
 
 4 
 
 
 IS 
 
 5 
 
 4.03222 
 
 2.95654 
 3-54785 
 
 3.98001 
 
 3.02647 
 
 3.92658 
 
 3-09547 
 
 1 
 
 45 
 
 
 6 
 
 4.83866 
 
 4.77601 
 
 3-63176 
 
 4.71 190 
 
 3-71456 
 
 6 
 
 
 
 7 
 
 5.6451 1 
 
 4.13916 
 
 5.57201 
 
 4-23705 
 
 S.49721 
 
 4-33365 
 
 7 
 
 
 
 8 
 
 6.4515s 
 
 4-73047 
 
 6.36801 
 
 4.84235 
 
 6.28253 
 7.06785 
 
 4-95275 
 
 8 
 
 
 
 9 
 
 7.25800 
 
 5.32178 
 
 7.16401 
 
 5.44764 
 
 5-57184 
 
 9 
 
 
 
 1 
 
 0.80385 
 
 0.59482 
 
 0-79335 
 
 0.60876 
 
 0.78260 
 
 0.62251 
 
 1 
 
 
 
 2 
 
 1.6077 1 
 
 1. 1 8964 
 
 1.58670 
 
 1.21752 
 
 1.56521 
 
 1.24502 
 
 2 
 
 
 
 3 
 
 2.41 1 57 
 
 1.78446 
 
 2.38005 
 
 1.82628 
 
 2.34782 
 
 1.86754 
 
 3 
 
 
 
 4 
 
 3.21542 
 
 2.37929 
 
 3-17341 
 
 2.43504 
 
 3-13043 
 
 2.49005 
 
 4 
 
 
 3° 
 
 5 
 
 4.01928 
 
 2.9741 1 
 
 3.96676 
 
 3.04380 
 
 3-91304 
 
 3.1 1 257 
 
 5 
 
 30 
 
 
 6 
 
 4.82314 
 
 3-56893 
 
 4.7601 1 
 
 3.65256 
 
 4-69564 
 
 3-73508 
 
 6 
 
 
 
 7 
 
 5.62699 
 
 4-16375 
 
 5-S5347 
 
 4.26132 
 
 5-47825 
 
 4-35760 
 4.9801 1 
 
 7 
 
 
 
 8 
 
 6.43085 
 
 4.75858 
 
 6.34682 
 
 4.87009 
 
 6.26086 
 
 8 
 
 
 
 9 
 
 7-23471 
 
 5-35340 
 
 7-14017 
 
 S-47885 
 
 7-04347 
 
 5.60263 
 
 9 
 
 
 
 1 
 
 0.80125 
 
 0.59832 
 
 0.79068 
 
 0.61 221 
 
 0.77988 
 
 0.62592 
 
 r 
 
 
 
 2 
 
 1.60250 
 
 1.19664 
 
 1-58137 
 
 1.22443 
 
 1-55946 
 
 1. 25184 
 
 2 
 
 
 
 3 
 
 2.40376 
 
 1.79497 
 
 2.37206 
 
 1.83665 
 
 2-33965 
 
 1-87777 
 
 3 
 
 
 
 4 
 
 3.20501 
 
 2.39329 
 
 3.16275 
 
 2.44886 
 
 3-1 1953 
 
 2.50369 
 
 4 
 
 
 45 
 
 
 4.00626 
 
 2.99162 
 
 3-95344 
 
 3.06108 
 
 3.89942 
 
 3.1 2961 
 
 5 
 
 IS 
 
 
 6 
 
 4.80752 
 
 3.58994 
 
 4-74413 
 
 3-67330 
 
 4.67930 
 
 3-75554 
 
 6 
 
 
 
 7 
 
 5.60877 
 
 4.18827 
 
 S-53482 
 
 4.28552 
 
 5.45919 
 
 4.38146 
 
 7 
 
 
 
 8 
 
 6.41003 
 
 4.78659 
 
 6-32551 
 
 4-89773 
 
 6.23907 
 
 5-00738 
 
 8 
 
 
 
 9 
 
 7.21128 
 
 5.38492 
 
 7.1 1620 
 
 5-50995 
 
 7.01896 
 
 5-63331 
 
 9 
 
 
 1' 
 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 1 
 
 1 
 
 1 
 
 5 
 
 3° 
 
 5 
 
 2° 
 
 S 
 
 1° 
 
 g 
 S 
 
 1 
 
 Smithsonian Tables. 
 
 44 
 
TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. 
 
 Table 9. 
 
 -Continued. 
 
 .a 
 
 en 
 
 S 
 
 39° 
 
 40° 
 
 41° 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.77714 
 
 0.62932 
 
 0.76604 
 
 0.64278 
 
 0.75470 
 
 0.65605 
 
 1 
 
 
 
 2 
 
 1-55429 
 
 11^796 
 
 1.53208 
 
 1.28557 
 
 1.50941 
 
 1.31211 
 
 2 
 
 
 
 3 
 
 2.33143 
 
 2.29813 
 
 1.92836 
 
 2.26412 
 
 1.96S17 
 
 3 
 
 
 
 4 
 
 3.10858 
 
 2.51728 
 
 3.06417 
 
 2-57115 
 
 3.01883 
 
 2.62423 
 
 4 
 
 
 o 
 
 S 
 
 3-88573 
 
 3.14660 
 
 3.83022 
 
 3-21393 
 
 3-77354 
 
 3.28029 
 
 5 
 
 60 
 
 
 6 
 
 4.66287 
 
 3-77592 
 
 4.59626 
 
 3.85672 
 
 4.52825 
 
 3-93635 
 
 
 
 
 
 7 
 
 5.44002 
 
 4.40524 
 
 S-36231 
 6.12835 
 
 4.49951 
 
 5.28296 
 
 4.59241 
 
 7 
 
 
 
 8 
 
 6.21716 
 
 ml 
 
 5.14230 
 
 6.03767 
 
 5-24847 
 
 8 
 
 
 
 9 
 
 6.99431 
 
 6-89439 
 
 5.78508 
 
 6.79238 
 
 5-90453 
 
 9 
 
 
 
 1 
 
 0-77439 
 
 0.63270 
 
 0.76323 
 
 0.64612 
 
 0.75184 
 
 0.65934 
 
 1 
 
 
 
 2 
 
 1.54878 
 
 1.26 541 
 1.89811 
 
 1.52646 
 
 1.29224 
 
 1.50368 
 
 1.31869 
 
 2 
 
 
 
 3 
 
 2-32317 
 
 2.28969 
 
 1-93837 
 
 2.25552 
 
 1.97803 
 
 3 
 
 
 
 4 
 
 309757 
 
 2.53082 
 
 3-05293 
 
 2.58449 
 
 3.00736 
 
 2.63738 
 
 4 
 
 
 IS 
 
 
 3.87196 
 
 3.16352 
 
 3.81616 
 
 3.23062 
 
 3.75920 
 
 3.29672 
 
 5 
 
 45 
 
 
 6 
 
 4.64635 
 
 3-79623 
 
 4-57939 
 
 3.87674 
 
 4.51104 
 
 3.95607 
 
 6 
 
 
 
 7 
 
 5.42074 
 
 4.42893 
 
 5.34262 
 
 4.52286 
 
 5.26288 
 
 4.61542 
 
 7 
 
 
 
 8 
 
 6.19514 
 
 5.06164 
 
 6.10586 
 
 5.16899 
 
 6.01472 
 
 5-27476 
 
 8 
 
 
 
 9 
 
 6-96953 
 
 s-69434 
 
 6.86909 
 
 5-815" 
 
 6.76656 
 
 5-934" 
 
 9 
 
 
 
 1 
 
 0.77162 
 
 0.63607 
 
 0.76040 
 
 0.64944 
 
 0.7489s 
 
 0.66262 
 
 1 
 
 
 
 2 
 
 1-54324 
 
 1.27215 
 
 1.52081 
 
 1.29889 
 
 1. 49791 
 
 1.32524 
 
 2 
 
 
 
 3 
 
 2.31487 
 
 1.90823 
 
 2.28121 
 
 1.94834 
 
 2.24686 
 
 1.98786 
 
 3 
 
 
 
 4 
 
 3.08649 
 
 2.54431 
 
 3.04162 
 
 2-59779 
 
 2.99582 
 
 2.65048 
 
 4 
 
 
 3° 
 
 5 
 
 3.85812 
 
 3.18039 
 
 3.80203 
 
 3.24724 
 
 3-74477 
 
 3-31310 
 
 5 
 
 30 
 
 
 6 
 
 4.62974 
 
 3.81646 
 
 4.56243 
 
 3.89668 
 
 4-49373 
 
 3-97572 
 
 6 
 
 
 
 7 
 
 5-40137 
 
 4-45254 
 
 5-32284 
 
 4.54613 
 5.19558 
 
 5.24268 
 
 4-63834 
 
 7 
 
 
 
 8 
 
 6.17299 
 
 5.08862 
 
 6.08324 
 
 5.99164 
 
 5.30096 
 
 8 
 
 
 
 9 
 
 6.94462 
 
 5.72470 
 
 6.84365 
 
 5-84503 
 
 6.74060 
 
 5.96358 
 
 9 
 
 
 
 1 
 
 0.76884 
 
 0.63943 
 
 0.75756 
 
 0.65276 
 
 0.74605 
 
 0.66588 
 
 1 
 
 
 
 2 
 
 1.53768 
 
 1.27887 
 
 1-S1513 
 
 1.30552 
 1.95828 
 
 1. 492 1 1 
 
 1.33176 
 
 2 
 
 
 
 3 
 
 2.30652 
 
 1.91831 
 
 2.27269 
 
 2.23817 
 2.98422 
 
 1.99764 
 
 3 
 
 
 
 4 
 
 3-07536 
 
 2-55775 
 
 3.03026 
 
 2.61 104 
 
 2.66352 
 
 4 
 
 
 45 
 
 S 
 
 3.84420 
 
 3-19719 
 
 3-78782 
 
 2.26380 
 
 3-73028 
 
 3-32940 
 
 5 
 
 15 
 
 
 6 
 
 4.61305 
 
 3-83663 
 
 4-54539 
 
 3.91656 
 
 4-47634 
 
 3-99529 
 
 6 
 
 
 
 7 
 
 5.38189 
 
 4.47607 
 
 5-30295 
 
 4.56932 
 
 5.22240 
 
 4.661 17 
 
 7 
 
 
 
 8 
 
 6.15073 
 
 5.1 1 551 
 
 6.06052 
 
 5.22208 
 
 5.98845 
 
 5-32705 
 
 8 
 
 
 
 9 
 
 6.91957 
 
 S-75495 
 
 6.81808 
 
 5.87484 
 
 6.71451 
 
 5-99293 
 
 9 
 
 
 5' 
 
 B 
 
 a 
 
 
 
 1 
 
 o 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 s 
 
 c 
 
 5 
 
 0° 
 
 4 
 
 9° 
 
 4 
 
 8° 
 
 Smithsonian Tables. 
 
 45 
 
''"*^'-^ ®' TRAVERSE TABLE. 
 
 DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 
 
 CO 
 
 1 
 
 .a 
 
 42° 
 
 43° 
 
 44° 
 
 i 
 
 .a 
 
 p 
 
 1 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 1 
 
 0.74314 
 
 0.66913 
 
 0-73135 
 
 0.68199 
 
 0.71933 
 
 0.69465 
 
 1 
 
 
 
 2 
 
 1.48628 
 
 1.33826 
 
 1.46270 
 
 1-36399 
 
 1.43867 
 
 1-38931 
 
 2 
 
 
 
 3 
 
 2.22943 
 
 2.00739 
 
 2.19406 
 
 2.04599 
 
 2.15801 
 
 2.08397 
 
 3 
 
 
 
 4 
 
 2.97257 
 
 2.67652 
 
 2.92541 
 
 2.72799 
 
 2-87735 
 
 2.77863 
 
 4 
 
 
 
 
 5 
 
 3.71572 
 4.45886 
 
 3-34565 
 4.01478 
 
 3.65676 
 
 3.40999 
 
 3-59669 
 
 3-47329 
 
 5 
 
 60 
 
 
 6 
 
 4.38812 
 
 4.09199 
 
 4.31603 
 
 4-16795 
 
 6 
 
 
 
 7 
 
 5.20201 
 
 4.68391 
 
 5- "947 
 
 4-77398 
 
 5-03537 
 
 4.86260 
 
 7 
 
 ' 
 
 
 8 
 
 5-94515 
 
 5-35304 
 
 5.8W82 
 
 5-45598 
 
 5-75471 
 
 5-55726 
 
 8 
 
 
 
 9 
 
 6.68830 
 
 6.02217 
 
 6.58218 
 
 6.13798 
 
 6.47405 
 
 6.25192 
 
 9 
 
 
 
 1 
 
 0.74021 
 
 0.67236 
 
 0.72837 
 
 0.68518 
 
 0.71630 
 
 0.69779 
 1-39558 
 
 1 
 
 
 
 2 
 
 1.48043 
 
 1-34473 
 
 1-45674 
 
 1-37036 
 
 1.43260 
 
 2 
 
 
 
 3 
 
 2.22065 
 
 2.017 10 
 
 2.18511 
 
 2-05554 
 
 2.14890 
 
 2-09337 
 
 3 
 
 
 
 4 
 
 2.96087 
 
 2.68946 
 
 2.91348 
 
 2.74073 
 
 2.86520 
 
 2.79116 
 
 4 
 
 
 15 
 
 5 
 
 3.70109 
 
 3-36183 
 
 3.64185 
 
 3.42591 
 
 3-58151 
 4.29781 
 
 3.48895 
 
 5 
 
 45 
 
 
 6 
 
 4.44130 
 
 4.03420 
 
 4.37022 
 
 4.H109 
 
 4.18674 
 
 6 
 
 
 
 7 
 
 5.18152 
 
 4.70656 
 
 5.09859 
 
 4.79628 
 
 5-01411 
 
 4-88453 
 
 7 
 
 
 
 8 
 
 5.92174 
 
 5-37893 
 
 5.82696 
 
 5.48146 
 
 5-73041 
 
 5.58232 
 
 8 
 
 
 
 9 
 
 6.66196 
 
 6.05130 
 
 6-55533 
 
 6.16664 
 
 6.44671 
 
 6.28011 
 
 9 
 
 
 
 1 
 
 0.73727 
 
 0.67559 
 
 0.72537 
 
 0.68835 
 
 0.71325 
 
 0.70090 
 
 1 
 
 
 
 2 
 
 1-47455 
 
 i-35"8 
 
 1.45074 
 
 1.37670 
 
 1.42650 
 
 1.40181 
 
 2 
 
 
 
 3 
 
 2.21183 
 
 2.02677 
 
 2.17612 
 
 2.06506 
 
 2-13975 
 
 2.10272 
 
 3 
 
 
 
 4 
 
 2.94910 
 
 2.70236 
 
 2.90149 
 
 2-75341 
 
 2.85300 
 
 2.80363 
 
 4 
 
 
 3° 
 
 5 
 
 3.68638 
 
 3-37795 
 
 3.62687 
 
 3-44177 
 
 3.56625 
 
 3-50454 
 
 5 
 
 30 
 
 
 6 
 
 4.42366 
 
 4-05354 
 
 4.35224 
 
 '•■l^'i^l 
 
 4.27950 
 
 4.2054s 
 
 6 
 
 
 
 7 
 
 5.16094 
 
 4.72913 
 
 5-07762 
 
 4.81848 
 
 4.99275 
 
 4.90636 
 
 7 
 
 
 
 8 
 
 5.89821 
 
 5.40472 
 
 5.80299 
 6.52836 
 
 5-50683 
 
 5.70600 
 
 5.60727 
 
 8 
 
 
 
 9 
 
 6-63549 
 
 6.08031 
 
 6.19519 
 
 6.41925 
 
 6.30818 
 
 9 
 
 
 
 1 
 
 0-73432 
 
 0.67880 
 
 0.72236 
 
 0.69151 
 
 0.71018 
 
 0.70401 
 
 1 
 
 
 
 2 
 
 1.46864 
 
 1.35760 
 
 1.44472 
 
 1.38302 
 
 1.42037 
 
 1.40802 
 
 2 
 
 
 
 3 
 
 2.20296 
 
 2.03640 
 
 2.16709 
 
 2.07453 
 
 2.13055 
 
 2.11204 
 
 3 
 
 
 
 4 
 
 2.93729 
 
 2.71520 
 
 2.88945 
 
 2.76605 
 3-45756 
 
 2.84074 
 
 2.81605 
 
 4 
 
 
 45 
 
 5 
 
 3.67 1 61 
 
 3-39400 
 
 3.61182 
 
 3-55092 
 
 3.52007 
 
 I 
 
 IS 
 
 
 6 
 
 4-40593 
 
 4.07280 
 
 4-33418 
 
 4.14907 
 
 4.26111 
 
 4.22408 
 
 
 
 7 
 
 5.14025 
 
 4.75160 
 
 5-05654 
 
 4.84059 
 
 4.97129 
 
 4.92810 
 
 7 
 
 
 
 8 
 
 S.87458 
 
 5-43040 
 
 5.77891 
 
 5.53210 
 
 5.68148 
 
 5.63211 
 
 8 
 
 
 
 9 
 
 6.60890 
 
 6.10920 
 
 6.50127 
 
 6.22361 
 
 6.39166 
 
 6.33613 
 
 9 
 
 
 g 
 
 a 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 g 
 
 a 
 a 
 
 1 
 
 
 
 
 
 
 
 5' 
 % 
 
 5' 
 
 c 
 
 4 
 
 70 
 
 4 
 
 6° 
 
 4 
 
 5° 
 
 Smithsonian Tables. 
 
 46 
 
TRAVERSE TABLE. 
 DIFFERENCES OF LATITUDE AND DEPARTURE. 
 
 Table 9. 
 
 -Continued. 
 
 
 oi 
 
 u 
 
 i 
 
 .2 
 Q 
 
 45° 
 
 
 
 i 
 
 CO 
 
 5 
 
 
 Lat. 
 
 Dep. 
 
 1 
 
 0.70710 
 
 0.70710 
 
 1 
 
 
 2 
 
 1.41421 
 
 1.41421 
 
 2 
 
 
 
 3 
 
 2.12132 
 
 2.12132 
 
 3 
 
 
 
 4 
 
 2.82842 
 
 2.82842 
 
 4 
 
 
 
 5 
 
 3-53553 
 
 3-53553 
 
 5 
 
 
 
 6 
 
 4.24264 
 
 4.24264 
 
 6 
 
 
 
 7 
 
 4.94974 
 
 4.94974 
 
 7 
 
 
 
 8 
 
 5.65685 
 
 5-65685 
 
 8 
 
 
 
 9 
 
 6.36396 
 
 6.36396 
 
 9 
 
 
 
 P 
 
 Dep. 
 
 Lat. 
 
 
 
 4. 
 
 5° 
 
 Smithsonian Tables. 
 
 47 
 
Table: 10. 
 
 LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ IN ENGLISH 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 10° 
 
 P. P. 
 
 
 0' 
 
 I 
 2 
 
 3 
 4 
 
 1 
 1 
 
 9 
 10 
 II 
 
 12 
 
 13 
 
 H 
 ■5 
 i6 
 
 il 
 19 
 
 20 
 
 21 
 22 
 
 23 
 24 
 
 2I 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 
 33 
 
 34 
 
 35 
 36 
 
 11 
 39 
 
 40 
 
 4" 
 42 
 43 
 
 44 
 
 11 
 
 47 
 48 
 49 
 
 50 
 
 S' 
 52 
 S3 
 
 54 
 55 
 56 
 
 57 
 S8 
 59 
 
 60 
 
 7.317 
 
 7379 
 
 7.317 
 
 7392 
 
 7.317 
 
 7433 
 
 7.317 
 
 7300 
 
 7.317 
 
 7593 
 
 7.317 
 
 77M 
 
 7.317 
 
 7861 
 
 7.317 
 
 8034 
 
 7.317 
 
 8233 
 
 7.317 
 
 8458 
 
 7.317 
 
 8709 
 
 1 
 
 
 7379 
 7379 
 7379 
 
 7379 
 7379 
 7379 
 
 7379 
 7379 
 7379 
 
 7392 
 7393 
 7394 
 
 7394 
 7395 
 7395 
 
 7396 
 7396 
 7397 
 
 7434 
 743S 
 7436 
 
 7437 
 7438 
 7438 
 
 7439 
 7440 
 7441 
 
 7501 
 7503 
 7504 
 7506 
 7507 
 7508 
 
 7510 
 7SII 
 75'3 
 
 7595 
 7597 
 7599 
 7600 
 7602 
 7604 
 
 7606 
 7608 
 7610 
 
 7716 
 7719 
 7721 
 
 7723 
 7726 
 7728 
 
 7730 
 7732 
 7735 
 
 7864 
 7866 
 7869 
 7872 
 7875 
 7877 
 7S80 
 7883 
 7885 
 
 8037 
 8040 
 8043 
 8046 
 S050 
 8053 
 8056 
 
 £1? 
 
 8237 
 8240 
 
 8244 
 8247 
 8251 
 8255 
 8258 
 8262 
 8265 
 
 8462 
 S466 
 8470 
 
 8474 
 8478 
 8482 
 
 8486 
 8490 
 8494 
 
 87r8 
 8722 
 
 8727 
 873- 
 873s 
 8740 
 8744 
 8749 
 
 
 10 
 20 
 50 
 40 
 50 
 60 
 
 .2 
 •3 
 •5 
 
 .7 
 .8 
 
 I.O 
 
 
 7379 
 
 7397 
 
 7442 
 
 7514 
 
 7612 
 
 7737 
 
 78S8 
 
 8065 
 
 8269 
 
 8498 
 
 8753 
 
 
 7379 
 7379 
 7379 
 
 7379 
 7380 
 7380 
 
 7380 
 7380 
 7380 
 
 7398 
 7398 
 7399 
 
 7399 
 7400 
 7401 
 
 7401 
 7402 
 7402 
 
 7443 
 7444 
 7445 
 7446 
 7447 
 7448 
 
 7449 
 7450 
 7451 
 
 7515 
 75'7 
 75-8 
 
 7S20 
 7521 
 7522 
 
 7524 
 7525 
 7527 
 
 7614 
 7616 
 7618 
 
 7619 
 7621 
 7623 
 
 7625 
 7627 
 7629 
 
 7739 
 7742 
 7744 
 7746 
 7749 
 7751 
 
 7753 
 7755 
 7757 
 
 •7891 
 7894 
 78,6 
 
 7899 
 7902 
 7905 
 7908 
 7910 
 7913 
 
 8068 
 8071 
 807s 
 8078 
 8081 
 8084 
 
 8087 
 8091 
 8094 
 
 8273 
 8276 
 8280 
 
 8283 
 8287 
 8291 
 
 8294 
 8298 
 8301 
 
 i'502 
 S506 
 8510 
 
 8514 
 8518 
 8523 
 
 8527 
 8531 
 853s 
 
 8758 
 8762 
 8767 
 
 8771 
 8776 
 8780 
 
 8785 
 8789 
 8794 
 
 2 
 
 
 10 
 20 
 30 
 40 
 50 
 6a 
 
 •3 
 ■ 7 
 1.0 
 ■•3 
 1-7 
 2.0 
 
 
 7380 
 
 7403 
 
 7452 
 
 7528 
 
 7631 
 
 7760 
 
 7916 
 
 8097 
 
 8305 
 
 8539 
 
 8798 
 
 
 7380 
 7380 
 7381 
 
 7381 
 7381 
 7381 
 
 7381 
 7382 
 7382 
 
 7404 
 7404 
 7405 
 
 7405 
 7406 
 7407 
 
 7407 
 7408 
 7408 
 
 7453 
 7454 
 7455 
 7456 
 7458 
 7459 
 7460 
 7461 
 7462 
 
 7530 
 7531 
 7533 
 
 7534 
 7535 
 7537 
 7538 
 7540 
 7541 
 
 7633 
 7635 
 7637 
 
 7638 
 7640 
 7642 
 
 7644 
 7646 
 7648 
 
 7762 
 
 7765 
 7767 
 
 7770 
 7772 
 7774 
 
 7777 
 7779 
 7782 
 
 7919 
 7922 
 7924 
 7927 
 7930 
 7933 
 
 7936 
 7938 
 7941 
 
 8100 
 8104 
 8107 
 
 Siio 
 
 81 14 
 8117 
 
 8120 
 8123 
 8127 
 
 8309 
 8312 
 8316 
 8320 
 8324 
 8327 
 
 8331 
 
 8543 
 8547 
 85S» 
 
 8555 
 8559 
 8554 
 
 8568 
 8572 
 8576 
 
 8803 
 8807 
 8812 
 
 8816 
 8821 
 8826 
 
 883a 
 8839 
 
 3 
 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 •5 
 1.0 
 
 ■•5 
 2.0 
 
 2-5 
 
 30 
 
 
 7382 
 
 7409 
 
 7463 
 
 7543 
 
 7650 
 
 7784 
 
 7944 
 
 8130 
 
 8342 
 
 8580 
 
 8844 
 
 
 7382 
 7383 
 7383 
 
 7383 
 7384 
 7384 
 
 7384 
 7384 
 7385 
 
 7410 
 7410 
 741 1 
 
 7412 
 7413 
 7413 
 
 7414 
 7415 
 7415 
 
 7464 
 7465 
 7466 
 
 7467 
 7469 
 7470 
 
 7471 
 7472 
 7473 
 
 7545 
 7546 
 7548 
 
 7549 
 7551 
 7553 
 
 7554 
 7556 
 7SS7 
 
 7652 
 7654 
 7656 
 
 7658 
 7661 
 7663 
 
 7665 
 7667 
 7669 
 
 7786 
 7789 
 7791 
 
 7794 
 7796 
 7799 
 7801 
 7804 
 7806 
 
 7947 
 79S<> 
 7953 
 7956 
 7959 
 7961 
 7964 
 7967 
 7970 
 
 ■8.33 
 8137 
 8140 
 
 8144 
 8147 
 8150 
 
 8154 
 8157 
 8161 
 
 8346 
 8350 
 8353 
 
 8357 
 8361 
 836s 
 8369 
 8372 
 8376 
 
 8584 
 8588 
 8593 
 
 8597 
 8601 
 8605 
 
 8609 
 8614 
 8618 
 
 8849 
 8853 
 8858 
 
 8862 
 8867 
 8872 
 
 S876 
 8881 
 8885 
 
 4 
 
 L 
 
 
 10 
 20 
 30 
 40 
 
 60 
 
 •7 
 1-3 
 2.0 
 2-7 
 3-3 
 4.0 
 
 
 738s 
 
 7416 
 
 7474 
 
 7559 
 
 7671 
 
 7809 
 
 7973 
 
 8164 
 
 8380 
 
 8622 
 
 8890 
 
 
 7385 
 7386 
 7386 
 
 7386 
 ■7387 
 7387 
 
 7387 
 7388 
 
 74'7 
 7418 
 74-8 
 
 7419 
 7420 
 7421 
 
 7422 
 7422 
 7423 
 
 7475 
 7476 
 7478 
 
 7482 
 
 7483 
 7484 
 7486 
 
 7561 
 7562 
 7564 
 7566 
 7567 
 7569 
 
 7S7I 
 7573 
 7574 
 
 7673 
 7675 
 7677 
 
 7679 
 7682 
 7684 
 7686 
 7688 
 7690 
 
 7811 
 7814 
 7816 
 
 7819 
 7821 
 7824 
 
 7826 
 7829 
 7831 , 
 
 7976 
 7979 
 7982 
 
 7985 
 7988 
 7991 
 
 7994 
 7997 
 Sooo 
 
 8003 
 
 8167 
 8171 
 8174 
 8178 
 8181 
 8184 
 
 8188 
 8191 
 8195. 
 
 8384 
 8388 
 8392 
 
 8396 
 8400 
 8403 
 8407 
 8411 
 841S 
 
 8626 
 8631 
 8635 
 
 8639 
 8643 
 8648 
 
 8652 
 8656 
 8661 
 
 8895 
 8899 
 8904 
 
 8909 
 8914 
 8918 
 
 8923 
 8928 
 8932 
 
 6 
 
 
 10 
 20 
 30 
 40 
 50 
 5o 
 
 .8 
 '•7 
 
 2-5 
 
 3-3 
 4.2 
 50 
 
 
 738S 
 
 7424 
 
 7487 
 
 7576 
 
 7692 
 
 7834 
 
 8198 
 
 8419 
 
 8665 
 
 8937 
 
 
 7388 
 7389 
 7389 
 
 7390 
 7390 
 7390 
 
 7391 
 7391 
 7392 
 
 7425 
 7426 
 
 7427 
 
 7428 
 
 7429 
 7429 
 
 7430 
 7431 
 7432 
 
 7488 
 7489 
 7490 
 
 7491 
 7493 
 7494 
 7495 
 7497 
 7498 
 
 7578 
 7579 
 7581 
 
 7583 
 7584 
 7586 
 
 7588 
 7590 
 7591 
 
 7694 
 7696 
 7699 
 
 7701 
 7703 
 7705 
 
 7707 
 7710 
 7712 
 
 7837 
 7839 
 7842 
 
 7845 
 7848 
 7850 
 
 7853 
 7856 
 7858 
 
 8006 
 8009 
 8012 
 
 8015 
 8019 
 S022 
 
 8025 
 8028 
 8031 
 
 8201 
 8205 
 8208 
 
 8212 
 8215 
 8219 
 
 8222 
 8226 
 8229 
 
 8423 
 8427 
 8431 
 
 8435 
 8439 
 8442 
 
 8446 
 8450 
 8454 
 
 8669 
 8674 
 8678 
 
 8683 
 8687 
 8691 
 
 8696 
 8700 
 8705 
 
 8942 
 8947 
 8951 
 
 8956 
 8961 
 8966 
 
 8971 
 8975 
 8980 
 
 
 
 7392 
 
 7433 
 
 7500 
 
 7593 
 
 7714 
 
 7861 
 
 8034 
 
 8233 
 
 8458 
 
 8709 
 
 898s 
 
 
 SniTh 
 
 SONIAN 
 
 Tables 
 
 . 
 
 
 
 
 
 
 
 
 
 — — 
 
 
 
 48/ 
 
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Table 10. 
 IN ENGLISH 
 
 Lat. 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 15° 
 
 16° 
 
 17° 
 
 i8° 
 
 19° 
 
 20° 
 
 P.P. 
 
 
 
 7.317 
 
 7.317 
 
 7.317 
 
 7.317 
 
 7.318 
 
 7.318 
 
 7.318 
 
 7.318 
 
 7.318 
 
 7.318 
 
 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 8985 
 
 9285 
 
 9611 
 
 9960 
 
 0333 
 
 0730 
 
 "49 
 
 1591 
 
 2054 
 
 2539 
 
 
 4 
 
 
 S9C10 
 8995 
 8999 
 
 9290 
 9296 
 9301 
 
 9617 
 9622 
 9628 
 
 9966 
 9972 
 9978 
 
 0340 
 '346 
 0353 
 
 0737 
 0744 
 0750 
 
 1156 
 "63 
 1171 
 
 159=1 
 1606 
 1614 
 
 2062 
 2070 
 2078 
 
 2547 . 
 
 2556 
 
 2564 
 
 
 10 
 
 ■7 
 
 
 4 
 I 
 
 9004 
 9009 
 9014 
 
 9306 
 9312 
 9317 
 
 9633 
 9639 
 9645 
 
 9984 
 9990 
 91.96 
 
 0319 
 0366 
 0372 
 
 0757 
 0764 
 0771 
 
 1178 
 ii«S 
 1 192 
 
 1621 
 
 1629 
 1637 
 
 2086 
 2094 
 2102 
 
 2572 
 2580 
 2589 
 
 20 
 30 
 40 
 50 
 60 
 
 1-3 
 2.0 
 2.7 
 3-3 
 4.0 
 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 
 ■3 
 
 9019 
 9023 
 9028 
 
 9322 
 9327 
 9333 
 
 9650 
 9656 
 9661 
 
 *0002 
 
 *ooo8 
 *ooi4 
 
 0379 
 0385 
 0392 
 
 0778 
 0784 
 0791 
 
 "99 
 1207 
 I2I4 
 
 1644 
 1652 
 1659 
 
 2110 
 2118 
 2126 
 
 2597 
 2605 
 2614 
 
 
 6 
 
 
 9033 
 
 9338 
 
 9667 
 
 *0020 
 
 0398 
 
 0798 
 
 1221 
 
 1667 
 
 2134 
 
 2622 
 
 
 9038 
 9043 
 9048 
 
 9343 
 9349 
 9354 
 
 9673 
 9678 
 9684 
 
 *<XJ26 
 *0O32 
 
 *ao39 
 
 0404 
 041 1 
 0418 
 
 ■0805 
 o8i2 
 
 0819 
 
 122S 
 1236 
 1243 
 
 1675 
 1682 
 1690 
 
 2142 
 2150 
 2.58 
 
 2630 
 2639 
 2647 
 
 
 10 
 
 .8 
 
 
 ■4 
 15 
 i6 
 
 9°53 
 9058 
 9062 
 
 9359 
 9365 
 9370 
 
 9690 
 9696 
 9701 
 
 *oo45 
 *oo5i 
 *oos7 
 
 0424 
 0430 
 0437 
 
 0S26 
 0833 
 0839 
 
 1250 
 
 1258 
 1265 
 
 1697 
 1705 
 1713 
 
 2166 
 2174 
 2182 
 
 Itll 
 2672 
 
 20 
 30 
 40 
 50 
 
 1-7 
 2.5 
 3-3 
 4.2 
 
 
 11 
 '9 
 
 20 
 
 21 
 22 
 
 23 
 
 9067 
 9072 
 9077 
 
 9375 
 9380 
 9386 
 
 9707 
 9713 
 9718 
 
 *oo63 
 *0070 
 •0076 
 
 o<43 
 0450 
 0456 
 
 0846 
 0853 
 0860 
 
 1272 
 1279 
 1287 
 
 1720 
 1728 
 1735 
 
 2190 
 219S 
 2206 
 
 2680 
 2688 
 2697 
 
 6a 
 
 50 
 
 
 6 
 
 
 9082 
 
 9391 
 
 9724 
 
 »oo82 
 
 0463 
 
 0867 
 
 1294 
 
 1743 
 
 2214 
 
 2705 
 
 
 9087 
 9092 
 9097 
 
 9396 
 9402 
 9407 
 
 9730 
 9736 
 9741 
 
 *oo88 
 *tx394 
 
 *OIOI 
 
 0470 
 0476 
 0483 
 
 0874 
 0881 
 
 o888 
 
 1301 
 1309 
 1316 
 
 1751 
 1758 
 1766 
 
 2222 
 2230 
 2238 
 
 2713 
 2722 
 2730 
 
 
 10 
 
 I. a 
 
 
 24 
 2S 
 26 
 
 9102 
 9107 
 5112 
 
 9413 
 9418 
 9423 
 
 9747 
 9753 
 9759 
 
 *oio7 
 *oii3 
 *oii9 
 
 0489 
 0496 
 0503 
 
 0895 
 0902 
 0909 
 
 1323 
 1330 
 1338 
 
 1774 
 1781 
 1789 
 
 2246 
 2254 
 2262 
 
 2739 
 2747 
 2755 
 
 30 
 40 
 50 
 
 3.0 
 4.0 
 5-0 
 
 
 27 
 
 2g 
 
 29 
 30 
 
 31 
 32 
 33 
 
 91 17 
 9122 
 9127 
 
 9429 
 9434 
 944° 
 
 9765 
 9770 
 9776 
 
 *OI25 
 
 *oi32 
 *oi38 
 
 0509 
 0516 
 0522 
 
 0916 
 0923 
 0930 
 
 1345 
 1352 
 1360 
 
 1797 
 1805 
 1812 
 
 2270 
 
 2278 
 22S6 
 
 2764 
 2772 
 2781 
 
 60 
 
 6.a 
 
 
 7 
 
 
 9132 
 
 9445 
 
 9782 
 
 *oi44 
 
 0529 
 
 0937 
 
 .367 
 
 1820 
 
 2294 
 
 2789 
 
 
 9137 
 9142 
 9147 
 
 9450 
 9456 
 9451 
 
 9788 
 9794 
 9800 
 
 *oiso 
 *o.s6 
 *oi63 
 
 0536 
 0542 
 OS49 
 
 0944 
 0951 
 0938 
 
 1374 
 1382 
 1389 
 
 1828 
 1835 
 1843 
 
 2302 
 2310 
 2318 
 
 2797 
 2806 
 2814 
 
 
 10 
 
 1.2 
 
 
 34 
 35 
 36 
 
 9152 
 9' 57 
 9162 
 
 9467 
 9472 
 9477 
 
 9806 
 9812 
 98.7 
 
 *oi69 
 *oi75 
 *oi8i 
 
 OSSS 
 0562 
 0569 
 
 0965 
 
 0972 
 
 \0979 
 
 1397 
 1404 
 141 1 
 
 185 1 
 1858 
 1866 
 
 2326 
 
 2334 
 2343 
 
 2823 
 2831 
 2840 
 
 30 
 40 
 50 
 
 3-5 
 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 9167 
 9172 
 9177 
 
 9483 
 948S 
 9494 
 
 9823 
 9829 
 9835 
 
 *oi87 
 *oi94 
 
 *0200 
 
 "575 
 0582 
 0388 
 
 0986 
 0993 
 1000 
 
 1419 
 1426 
 1434 
 
 1874 
 1S82 
 1889 
 
 2351 
 2359 
 2367 
 
 2848 
 2857 
 2865 
 
 60 
 
 7.0 
 
 
 8 
 
 
 9182 
 
 9499 
 
 9841 
 
 »0206 
 
 059s 
 
 1007 
 
 1441 
 
 1897 
 
 2375 
 
 2874 
 
 
 9187 
 9192 
 9197 
 
 9505 
 95 10 
 
 9516 
 
 9847 
 9853 
 9859 
 
 *0212 
 *02T9 
 *0225 
 
 0602 
 0608 
 0615 
 
 1014 
 
 I02I 
 1028 
 
 1448 
 1456 
 1463 
 
 1905 
 1913 
 1920 
 
 2383 
 2391 
 
 2400 
 
 2882 
 2S91 
 2899 
 
 
 10 
 
 1-3 
 
 
 44 
 45 
 45 
 
 9202 
 9207 
 9213 
 
 9521 
 9527 
 9533 
 
 986s 
 9871 
 9876 
 
 *023I 
 
 *o238 
 *0244 
 
 0622 
 0629 
 0635 
 
 "035 
 1042 
 1050 
 
 147 1 
 1479 
 I486 
 
 1928 
 1936 
 1944 
 
 2408 
 2416 
 
 2424 
 
 2908 
 2916 
 2925 
 
 30 
 40 
 50 
 
 4.0 
 5-3 
 6.7 
 
 
 49 
 60 
 SI 
 
 52 
 
 S3 
 
 9218 
 9223 
 922S 
 
 9538 
 9544 
 9549 
 
 9882 
 9888 
 9894 
 
 *0250 
 
 *c.256 
 *o263 
 
 0642 
 0649 
 0655 
 
 1057 
 1064 
 1071 
 
 1494 
 1501 
 1509 
 
 1952 
 1959 
 1967 
 
 2432 
 2441 
 2449 
 
 2933 
 2942 
 2950 
 
 60 
 
 8.a 
 
 
 9 
 
 
 9233 
 
 9555 
 
 9900 
 
 *0269 
 
 0662 
 
 1078 
 
 1516 
 
 1975 
 
 2457 
 
 2959 
 
 
 9238 
 9243 
 9249 
 
 9561 
 9566 
 9572 
 
 9906 
 9912 
 9918 
 
 *0275 
 *0282 
 
 *o288 
 
 0669 
 0676 
 o582 
 
 1085 
 1092 
 1099 
 
 1524 
 1531 
 1539 
 
 1983 
 1991 
 1999 
 
 2465 
 2473 
 2482 
 
 2968 
 2976 
 
 
 ID 
 
 1-5 
 30 
 4-5 
 6.0 
 7-5 
 
 
 54 
 55 
 56 
 
 9254 
 9259 
 9264 
 
 9577 
 95''3 
 9589 
 
 9924 
 9930 
 9936 
 
 *o29S 
 *o3oi 
 *0307 
 
 0689 
 0696 
 0703 
 
 1 106 
 1113 
 1121 
 
 1546 
 1561 
 
 2007 
 2014 
 2022 
 
 2490 
 2498 
 2506 
 
 2993 
 3002 
 301 1 
 
 30 
 40 
 50 
 
 
 57 
 58 
 59 
 
 60 
 
 9269 
 927s 
 9280 
 
 9594 
 9600 
 960s 
 
 9942 
 9948 
 9954 
 
 *03i4 
 •03 2d 
 »o327 
 
 0710 
 0716 
 0723 
 
 1128 
 
 "35 
 1 142 
 
 1569 
 1576 
 1584 
 
 2030 
 2038 
 2046 
 
 2514 
 2523 
 2531 
 
 3019 
 3028 
 3036 
 
 6a 
 
 90 
 
 
 
 
 9285 
 
 9611 
 
 9960 
 
 *o333 
 
 0730 
 
 1149 
 
 -1591 
 
 2054 
 
 2539 
 
 304s 
 
 
 Smithsonian Tables. 
 
 49 
 
Table lO. 
 
 LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„, 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 IN ENGLISH 
 
 Lat. 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 30° 
 
 p.p. 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 9 
 10 
 
 11 
 
 12 
 
 "3 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 20 
 
 21 
 
 22 
 23 
 24 
 
 =5 
 26 
 
 li 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 44 
 45 
 45 
 
 47 
 48 
 49 
 
 60 
 
 SI 
 
 52 
 
 53 
 
 54 
 55 
 56 
 
 57 
 58 
 59 
 
 60 
 
 7.318 
 
 304s 
 
 7.318 
 
 3570 
 
 7.318 
 
 4115 
 
 7.318 
 
 4678 
 
 7.318 
 
 5259 
 
 7.318 
 
 5858 
 
 7.318 
 
 6474 
 
 7.318 
 
 7105 
 
 7.318 
 
 7751 
 
 7.318 
 
 8412 
 
 8 
 
 
 3053 
 3062 
 3070 
 
 3079 
 3088 
 3096 
 
 3105 
 3113 
 3122 
 
 3579 
 
 3597 
 3606 
 3614 
 3623 
 
 3632 
 3641 
 3650 
 
 4124 
 4>33 
 4142 
 
 4152 
 4161 
 4170 
 
 4179 
 4189 
 4198 
 
 4688 
 4697 
 4707 
 
 4716 
 4726 
 4735 
 
 4745 
 4754 
 4764 
 
 5269 
 5279 
 5289 
 
 5299 
 5309 
 5319 
 5328 
 5338 
 5348 
 
 5868 
 
 % 
 
 5899 
 5909 
 5919 
 
 5929 
 5939 
 5949 
 
 6484 
 6494 
 6505 
 65 IS 
 6526 
 6536 
 6546 
 6557 
 6567 
 
 71 16 
 7126 
 7137 
 7148 
 7158 
 7169 
 
 7180 
 7190 
 7201 
 
 7762 
 7784 
 
 7795 
 7806 
 7817 
 7828 
 7839 
 7850 
 
 8423 
 8434 
 8445 
 
 8457 
 8468 
 8479 
 
 8490 
 8501 
 8512 
 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 1.3 
 2.6 
 4.0 
 5-3 
 6.7 
 S.o 
 
 
 3131 
 
 3659 
 
 4207 
 
 4774 
 
 5358 
 
 5960 
 
 6578 
 
 7212 
 
 7860 
 
 8523 
 
 
 3139 
 3148 
 3157 
 316s 
 3174 
 3183 
 
 3191 
 3200 
 3209 
 
 3668 
 3677 
 3686 
 
 3695 
 ■3704 
 37'3 
 3722 
 3731 
 3740 
 
 4216 
 4226 
 4235 
 
 4244 
 4254 
 4263 
 
 4272 
 4282 
 4291 
 
 4783 
 4793 
 4802 
 
 4812 
 4822 
 4831 
 
 4841 
 4851 
 4860 
 
 5368 
 5378 
 538S 
 
 5398 
 5408 
 5417 
 
 5427 
 5437 
 5447 
 
 5970 
 59S0 
 5990 
 
 6000 
 6011 
 6021 
 
 6031 
 6041 
 6051 
 
 6588 
 6599 
 6609 
 
 6620 
 6630 
 6640 
 
 6651 
 6661 
 6672 
 
 7222 
 7233 
 7244 
 
 7254 
 7263 
 7276 
 
 7287 
 7297 
 7308 
 
 7871 
 7882 
 7893 
 
 7904 
 7915 
 7926 
 
 7937 
 7948 
 7959 
 
 8557 
 8568 
 8579 
 8591 
 
 8602 
 8613 
 8624 
 
 
 9 
 
 
 10 
 20 
 30 
 
 40 
 
 1^ 
 
 1-5 
 3.0 
 4-5 
 6.0 
 7-S 
 9.0 
 
 
 3217 
 
 3749 
 
 4300 
 
 487Q 
 
 5457 
 
 6062 
 
 6682 
 
 7319 
 
 7970 
 
 8635 
 
 
 3226 
 3235 
 3244 
 
 3252 
 3261 
 3270 
 
 3278 
 3287 
 3296 
 
 3758 
 3767 
 3776 
 
 3785 
 
 3813 
 3822 
 3831 
 
 4310 
 4319 
 4328 
 
 4338 
 4347 
 4356 
 4366 
 4375 
 4384 
 
 4899 
 490S 
 4918 
 4928 
 
 4937 
 4947 
 4937 
 
 5467 
 5477 
 5487 
 
 5497 
 5507 
 5517 
 
 5527 
 5537 
 5347 
 
 6072 
 6082 
 6092 
 
 6ro2 
 6113 
 6123 
 
 6133 
 6143 
 6154 
 
 6693 
 6703 
 6714 
 
 6724 
 673s 
 6745 
 6756 
 6766 
 6777 
 
 7329 
 7340 
 7351 
 7362 
 
 7383 
 
 7394 
 7403 
 7416 
 
 79S1 
 7992 
 8003 
 
 8014 
 8025 
 8036 
 
 8047 
 8058 
 8069 
 
 8647 
 8658 
 8669 
 
 8680 
 8691 
 8703 
 
 8714 
 S725 
 8736 
 
 
 10 
 
 
 10 
 20 
 30 
 40 
 5° 
 60 
 
 1-7 
 3-3 
 S-o 
 6.7 
 8-3 
 
 lO.O 
 
 
 33°S 
 
 3840 
 
 4394 
 
 4966 
 
 5557 
 
 6164 
 
 6787 
 
 7426 
 
 8080 
 
 8747 
 
 
 3313 
 3322 
 3331 
 
 3340 
 3349 
 3357 
 3366 
 3375 
 3384 
 
 3849 
 3858 
 3067 
 
 3876 
 3885 
 3894 
 
 3904 
 3913 
 3922 
 
 4403 
 4413 
 4422 
 
 4431 
 
 4441 
 4450 
 
 4460 
 4469 
 4479 
 
 4976 
 4986 
 4996 
 
 5005 
 5015 
 5025 
 
 5034 
 5044 
 5054 
 
 SS67 
 
 5587 
 
 5 '597 
 5607 
 5617 
 
 5627 
 5637 
 5647 
 
 6174 
 5i8s 
 6195 
 6205 
 
 6226 
 
 6236 
 6246 
 6256 
 
 6798 
 6808 
 6819 
 
 6829 
 6840 
 6851 
 6861 
 
 7437 
 7448 
 7459 
 7469 
 7480 
 7491 
 7502 
 7513 
 7523 
 
 8091 
 8102 
 8113 
 
 8124 
 8135 
 8146 
 
 8157 
 8168 
 8179 
 
 8759 
 
 i^i 
 
 8815 
 
 8826 
 8838 
 8849 
 
 
 11 
 
 
 10 
 20 
 30 
 40 
 
 1.8 
 3-7 
 5-5 
 7-3 
 9.2 
 
 II.O 
 
 
 3393 
 
 3931 
 
 4488 
 
 5064 
 
 5657 
 
 6267 
 
 6893 
 
 7534 
 
 8190 
 
 8860 
 
 
 34°i 
 3410 
 
 3419 
 3428 
 3437 
 3446 
 
 3463 
 3472 
 
 3940 
 3949 
 395S 
 3967 
 3977 
 3986 
 
 3995 
 4004 
 
 4<"3 
 
 4498 
 4507 
 4516 
 
 4526 
 4S3S 
 4545 
 
 4554 
 4564 
 4373 
 
 5073 
 5083 
 5093 
 
 5103 
 5112 
 5122 
 
 5132 
 5142 
 5'5i 
 
 5667 
 5677 
 5687 
 
 5697 
 5707 
 5717 
 
 S727 
 5737 
 5747 
 
 6277 
 6287 
 6298 
 
 6308 
 6318 
 6329 
 
 6339 
 
 tis. 
 
 6903 
 6914 
 6924 
 
 693s 
 6946 
 6956 
 6967 
 
 7545 
 7556 
 7567 
 
 7578 
 7588 
 7599 
 7610 
 7621 
 7632 
 
 8201 
 8212 
 8223 
 
 8234 
 8246 
 8257 
 8268 
 
 llll 
 
 8871 
 8883 
 8894 
 
 8905 
 8916 
 8928 
 
 8939 
 8950 
 S962 
 
 
 12 
 
 
 10 
 20 
 30 
 40 
 so 
 6a 
 
 2.0 
 4.0 
 6.0 
 8.0 
 10.0 
 
 12.0 
 
 
 3481 
 
 4022 
 
 4583 
 
 S161 
 
 5757 
 
 6370 
 
 6999 
 
 7643 
 
 8301 
 
 8973 
 
 
 3490 
 3499 
 3508 
 
 3516 
 3525 
 3S34 
 
 3543 
 3552 
 3561 
 
 4032 
 4041 
 4050 
 
 4059 
 4068 
 4078 
 
 4086 
 4096 
 410S 
 
 4592 
 4602 
 4611 
 
 4621 
 4630 
 4640 
 
 4649 
 4659 
 4668 
 
 5171 
 5181 
 5191 
 
 52tXl 
 
 5210 
 
 5220 
 
 5230 
 5240 
 5250 
 
 5767 
 
 5787 
 
 5798 
 5808 
 5818 
 
 5828 
 5838 
 5848 
 
 6380 
 6391 
 
 6401 
 
 64II 
 
 6422 
 
 6432 
 6442 
 6453 
 6463 
 
 7009 
 7020 
 
 7030 
 7041 
 7052 
 
 7062 
 
 7073 
 7084 
 7094 
 
 7653 
 7664 
 7675 
 
 7685 
 7697 
 7708 
 
 7719 
 7729 
 7740 
 
 8312 
 8323 
 8334 
 
 llti 
 8368 
 
 8379 
 8390 
 8401 
 
 89S4 
 8996 
 9007 
 
 9018 
 9030 
 9041 
 
 9052 
 9064 
 907s 
 
 
 
 
 3570 
 
 4115 
 
 4678 
 
 5259 
 
 5858 
 
 6474 
 
 7105 
 
 7751 
 
 8412 
 
 9086 
 
 
 Smithsonian Tables. 
 
 50 
 
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Table 10. 
 IN ENGLISH 
 
 Lat. 
 
 31° 
 
 32= 
 
 33° 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 40° 
 
 P.P. 
 
 
 7.318 
 
 7.318 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 go86 
 
 9773 
 
 0472 
 
 1182 
 
 1902 
 
 2631 
 
 3369 
 
 4114 
 
 4866 
 
 5623 
 
 11 
 
 9098 
 9109 
 9120 
 
 9785 
 9796 
 9807 
 
 0484 
 049s 
 0507 
 
 1194 
 
 I205 
 
 I2I8 
 
 1914 
 1926 
 1938 
 
 2643 
 2656 
 2658 
 
 3381 
 3394 
 3406 
 
 4126 
 4139 
 4151 
 
 4878 
 4891 
 4904 
 
 5636 
 5649 
 5661 
 
 4 
 I 
 
 9132 
 9143 
 9134 
 
 9819 
 9831 
 9843 
 
 0519 
 0531 
 0542 
 
 1230 
 I24I 
 1253 
 
 1950 
 1962 
 ■974 
 
 2680 
 2692 
 2705 
 
 3418 
 3431 
 3443 
 
 4164 
 4176 
 4189 
 
 49i5 
 4929 
 4941 
 
 5674 
 5687 
 5699 
 
 
 10 
 
 1.8 
 
 9 
 10 
 
 II 
 
 12 
 
 13 
 
 9165 
 9177 
 9189 
 
 9854 
 9866 
 9877 
 
 0554 
 0566 
 0577 
 
 1265 
 
 1277 
 1289 
 
 1986 
 
 ■999 
 20H 
 
 2717 
 2729 
 2741 
 
 3455 
 3468 
 3480 
 
 4201 
 4214 
 4226 
 
 4954 
 4966 
 4979 
 
 5712 
 5725 
 5737 
 
 30 
 40 
 50 
 
 5-5 
 7-3 
 9.2 
 
 II.O 
 
 9200 
 
 9889 
 
 0590 
 
 I30I 
 
 2023 
 
 2753 
 
 3492 
 
 4239 
 
 4992 
 
 5750 
 
 9211 
 9223 
 9234 
 
 9900 
 9912 
 9924 
 
 o6qi 
 0613 
 0625 
 
 I3I3 
 1325 
 1337 
 
 2035 
 2047 
 2059 
 
 2766 
 2778 
 2790 
 
 3S°5 
 3517 
 3530 
 
 4251 
 4264 
 4276 
 
 5004 
 5017 
 5029 
 
 5763 
 5775 
 5788 
 
 
 14 
 
 9245 
 9257 
 9268 
 
 9935 
 9947 
 9958 
 
 0637 
 0648 
 0660 
 
 1349 
 
 I36I 
 
 1373 
 
 2071 
 2083 
 209s 
 
 2803 
 2815 
 28»7 
 
 3542 
 3567 
 
 4289 
 4301 
 4314 
 
 5042 
 S°55 
 5067 
 
 5801 
 5813 
 5826 
 
 
 '9 
 20 
 
 21 
 22 
 
 23 
 
 9280 
 9291 
 9302 
 
 9970 
 9982 
 9993 
 
 0672 
 0684 
 0696 
 
 138s 
 
 '397 
 1409 
 
 2108 
 2120 
 2132 
 
 2839 
 2852 
 2S64 
 
 3579 
 3592 
 3604 
 
 4326 
 4339 
 4351 
 
 5080 
 5092 
 5 'OS 
 
 5839 
 5851 
 5864 
 
 12 
 
 9314 
 
 *ooo5 
 
 0707 
 
 1421 
 
 2144 
 
 2876 
 
 3616 
 
 4364 
 
 5118 
 
 5877 
 
 9325 
 9337 
 9348 
 
 *ooi6 
 •0028 
 *0040 
 
 0719 
 0731 
 0743 
 
 1433 
 1445 
 1457 
 
 2156 
 2i68 
 218a 
 
 2888 
 2901 
 2913 
 
 3629 
 3641 
 3654 
 
 4376 
 4389 
 4401 
 
 5130 
 5M3 
 5156 
 
 5890 
 5902 
 59'5 
 
 24 
 
 25 
 26 
 
 9360 
 9371 
 9382 
 
 *oo5i 
 ♦0063 
 *0075 
 
 0755 
 0766 
 0778 
 
 1469 
 1481 
 1493 
 
 2192 
 2205 
 2217 
 
 292s 
 2938 
 2950 
 
 3666 
 3678 
 3691 
 
 4414 
 4426 
 4439 
 
 S'68 
 5181 
 S193 
 
 592S 
 5940 
 5953 
 
 
 10 
 
 2.0 
 
 11 
 
 29 
 30 
 
 31 
 32 
 
 33 
 
 9393 
 9405 
 9417 
 
 *oo86 
 »<»98 
 *oiio 
 
 0790 
 0802 
 0814 
 
 1505 
 1517 
 1529 
 
 2229 
 2241 
 2253 
 
 2962 
 2974 
 2987 
 
 3703 
 3716 
 3728 
 
 445' 
 4464 
 4477 
 
 5206 
 5219 
 5231 
 
 5966 
 5978 
 5991 
 
 30 
 40 
 50 
 60 
 
 4.0 
 6.0 
 8.0 
 
 lO.O 
 IZ.O 
 
 9428 
 
 *OI2I 
 
 0826 
 
 1541 
 
 2265 
 
 2999 
 
 3741 
 
 4489 
 
 5244 
 
 6004 
 
 9440 
 945' 
 9463 
 
 *oi33 
 *oi44 
 •0156 
 
 0837 
 0849 
 086. 
 
 1565 
 IS77 
 
 2278 
 2290 
 2302 
 
 301 1 
 3024 
 3036 
 
 3753 
 3765 
 3778 
 
 4502 
 4514 
 4527 
 
 5256 
 5269 
 5282 
 
 6017 
 6029 
 6042 
 
 
 34 
 
 9474 
 9485 
 9497 
 
 *oi68 
 *oi79 
 *oi9i 
 
 0873 
 0885 
 0897 
 
 1589 
 1601 
 1613 
 
 2314 
 2326 
 2338 
 
 3048 
 3060 
 3073 
 
 3790 
 3803 
 3815 
 
 4539 
 4552 
 4564 
 
 5294 
 5307 
 5320 
 
 6055 
 6067 
 6080 
 
 
 39 
 40 
 
 4" 
 42 
 43 
 
 9508 
 9520 
 9531 
 
 *0203 
 
 •0214 
 
 *0226 
 
 0908 
 0920 
 0932 
 
 1625 
 1637 
 1649 
 
 2351 
 2363 
 2375 
 
 3085 
 3097 
 3110 
 
 3828 
 3840 
 3852 
 
 4S77 
 4589 
 4602 
 
 5332 
 5345 
 5358 
 
 6093 
 6106 
 6118 
 
 13 
 
 9543 
 
 *0238 
 
 0944 
 
 1661 
 
 2387 
 
 3122 
 
 3865 
 
 4614 
 
 5370 
 
 6131 
 
 9554 
 9566 
 9577 
 
 9589 
 9600 
 9612 
 
 *0249 
 
 •0261 
 
 *o273 
 
 *028s 
 ♦0296 
 *0308 
 
 0956 
 0968 
 0980 
 
 1673 
 1685 
 1697 
 
 2399 
 2411 
 2424 
 
 2436 
 2448 
 2460 
 
 3134 
 3147 
 3159 
 
 3877 
 3890 
 3902 
 
 391S 
 3927 
 3939 
 
 4627 
 4640 
 4652 
 
 4665 
 4677 
 4690 
 
 5383 
 539S 
 
 5421 
 5433 
 5446 
 
 6144 
 6156 
 6169 
 
 6/82 
 6195 
 6207 
 
 44 
 11 
 
 1003 
 1015 
 
 1721 
 >733 
 
 3>84 
 3196 
 
 10 
 20 
 
 2.2 
 4.3 
 
 47 
 48 
 49 
 
 50 
 
 51 
 52 
 53 
 
 9623 
 9635 
 9646 
 
 *0320 
 
 *033i 
 *0343 
 
 1027 
 1039 
 1051 
 
 1745 
 .>757 
 1769 
 
 2472 
 2485 
 2497 
 
 3208 
 3221 
 3233 
 
 3952 
 3964 
 3977 
 
 4702 
 4715 
 4727 
 
 5459 
 5471 
 5484 
 
 6220 
 6233 
 6245 
 
 30 
 40 
 so 
 60 
 
 |-S 
 
 10.8 
 13.0 
 
 9658 
 
 *035S 
 
 1063 
 
 1781 
 
 2509 
 
 324s 
 
 3989 
 
 4740 
 
 5497 
 
 6258 
 
 9669 
 9'i8i 
 9692 
 
 *0366 
 *o378 
 *0390 
 
 i°7S 
 1087 
 1098 
 
 J 793 
 1805 
 1S17 
 
 2521 
 2533 
 2546 
 
 3258 
 3270 
 3282 
 
 4002 
 4014 
 4027 
 
 4753 
 4765 
 4778 
 
 5509 
 5522 
 5535 
 
 6271 
 6284 
 6296 
 
 
 54 
 11 
 
 9704 
 97"S 
 9727 
 
 *O402 
 
 *o4i3 
 *o425 
 
 mo 
 1122 
 1134 
 
 1829 
 1841 
 1854 
 
 2558 
 2570 
 2582 
 
 329s 
 3307 
 3319 
 
 4039 
 4052 
 4064 
 
 4790 
 4803 
 4815 
 
 5347 
 5560 
 5573 
 
 6309 
 6322 
 6335 
 
 
 % 
 59 
 
 60 
 
 9739 
 975° 
 9762 
 
 *0437 
 *0449 
 *046o 
 
 1146 
 1158 
 1170 
 
 1S66 
 1878 
 1890 
 
 2594 
 2607 
 26x9 
 
 3332 
 3344 
 3356 
 
 4077 
 4089 
 4101 
 
 4828 
 4841 
 4853 
 
 5585 
 5598 
 561 1 
 
 6347 
 6360 
 
 6373 
 
 
 9773 
 
 *o472 
 
 1 182 
 
 igo2 
 
 2631 
 
 3369 
 
 4114 
 
 4866 
 
 5623 
 
 6385 
 
 
 1 
 
 Smithsonian Tables. 
 
 51 
 
Table 10. 
 
 LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ IN ENGLISH 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 4S° 
 
 46° 
 
 .47° 
 
 48° 
 
 49° 
 
 50° 
 
 P.P. 
 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.319 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 
 0' 
 
 1 
 
 6385 
 
 7152 
 
 7921 
 
 8692 
 
 9464 
 
 0236 
 
 1007 
 
 1776 
 
 2543 
 
 3306 
 
 
 6398 
 
 7164 
 
 7933 
 
 8704 
 
 9476 
 
 0248 
 
 1020 
 
 1789 
 
 2556 
 
 33 '9 
 
 2 
 
 641 1 
 
 7177 
 
 7946 
 
 8717 
 
 9489 
 
 0261 
 
 1033 
 
 1802 
 
 2569 
 
 3331 
 
 
 3 
 
 5424 
 
 7190 
 
 7959 
 
 8730 
 
 9502 
 
 0274 
 
 104s 
 
 1815 
 
 2581 
 
 3344 
 
 
 4 
 
 6436 
 
 7203 
 
 7972 
 
 8743 
 
 95^5 
 
 0287 
 
 1058 
 
 1827 
 
 2594 
 
 3357 
 
 
 1 
 
 6449 
 
 7216 
 
 7985 
 
 8756 
 
 9528 
 
 0300 
 
 1071 
 
 1840 
 
 2607 
 
 3369 
 
 
 6 
 
 6462 
 
 7228 
 
 7998 
 
 8769 
 
 9541 
 
 0313 
 
 1084 
 
 1853 
 
 2619 
 
 3382 
 
 
 7 
 
 6475 
 
 7241 
 
 8010 
 
 8782 
 
 9554 
 
 0326 
 
 1097 
 
 1866 
 
 2632 
 
 3395 
 
 
 8 
 
 6487 
 
 7254 
 
 8023 
 
 8794 
 
 9566 
 
 0338 
 
 mo 
 
 1879 
 
 2645 
 
 3407 
 
 
 9 
 10 
 
 II 
 
 6500 
 
 7267 
 
 8036 
 
 8807 
 
 9579 
 
 0351 
 
 1122 
 
 1892 
 
 265S 
 
 3420 
 
 12 
 
 6513 
 
 7280 
 
 8049 
 
 8820 
 
 9592 
 
 0364 
 
 "35 
 
 1904 
 
 2670 
 
 3433 
 
 6526 
 
 7292 
 
 8062 
 
 8833 
 
 9605 
 
 0377 
 
 1 148 
 
 1917 
 
 2683 
 
 3445 
 
 
 
 12 
 
 6538 
 
 7305 
 
 807s 
 
 8846 
 
 9618 
 
 0390 
 
 1 161 
 
 1930 
 
 2696 
 
 3458 
 
 
 
 13 
 
 6551 
 
 7318 
 
 8087 
 
 8859 
 
 9631 
 
 0403 
 
 1174 
 
 1943 
 
 2709 
 
 3471 
 
 30 
 40 
 
 4.0 
 
 14 
 
 6564 
 
 7331 
 
 8100 
 
 8872 
 
 9644 
 
 0416 
 
 1 187 
 
 1955 
 
 2721 
 
 34S3 
 
 8.0 
 
 "5 
 
 6577 
 
 7344 
 
 8113 
 
 8884 
 
 9657 
 
 0429 
 
 1199 
 
 1968 
 
 2734 
 
 3496 
 
 50 
 
 
 i6 
 
 6589 
 
 7356 
 
 8126 
 
 S897 
 
 9669 
 
 0441 
 
 1212 
 
 1981 
 
 2747 
 
 3509 
 
 60 
 
 12.0 
 
 \l 
 
 6602 
 6615 
 
 7369 
 7382 
 
 8139 
 8152 
 
 8910 
 8923 
 
 9682 
 9695 
 
 0454 
 0467 
 
 1225 
 123S 
 
 1994 
 2007 
 
 2760 
 2772 
 
 3521 
 3534 
 
 
 
 
 19 
 20 
 
 21 
 
 6628 
 
 7395 
 
 8165 
 
 8936 
 
 9708 
 
 0480 
 
 1251 
 
 2019 
 
 2785 
 
 3547 
 
 
 6640 
 
 7408 
 
 8177 
 
 8949 
 
 9721 
 
 0493 
 
 1264 
 
 2032 
 
 2798 
 
 3559 
 
 itll 
 
 7420 
 
 8igo 
 
 8962 
 
 9734 
 
 0506 
 
 1276 
 
 2°45 
 
 2811 
 
 3572 
 
 22 
 
 6666 
 
 7433 
 
 8203 
 
 8975 
 
 9747 
 
 0519 
 
 1289 
 
 2058 
 
 2823 
 
 3585 
 
 
 n 
 
 6679 
 
 7446 
 
 8216 
 
 8987 
 
 9760 
 
 0531 
 
 1302 
 
 2071 
 
 2836 
 
 3597 
 
 
 24 
 
 6692 
 
 7459 
 
 8229 
 
 9000 
 
 9772 
 
 0544 
 
 1315 
 
 2083 
 
 2849 
 
 3610 
 
 
 ^1 
 
 6704 
 
 7472 
 
 8242 
 
 9013 
 
 9785 
 
 0557 
 
 1328 
 
 2096 
 
 2861 
 
 3623 
 
 
 26 
 
 6717 
 
 7485 
 
 8254 
 
 9026 
 
 9798 
 
 0570 
 
 1341 
 
 2109 
 
 2874 
 
 3635 
 
 
 27 
 
 6730 
 
 7497 
 
 8267 
 
 9039 
 
 9811 
 
 0583 
 
 1353 
 
 2122 
 
 28S7 
 
 3648 
 
 
 28 
 
 6743 
 
 7510 
 
 8280 
 
 9052 
 
 9824 
 
 0596 
 
 1366 
 
 2134 
 
 2900 
 
 3661 
 
 
 29 
 
 30 
 
 31 
 
 675s 
 
 7523 
 
 8293 
 
 9065 
 
 9837 
 
 0609 
 
 1379 
 
 2147 
 
 2912 
 
 3673 
 
 13 
 
 6768 
 
 7536 
 
 8306 
 
 9077 
 
 9850 
 
 0621 
 
 1392 
 
 2160 
 
 2925 
 
 3686 
 
 6781 
 
 7549 
 
 8319 
 
 9090 
 
 9862 
 
 0634 
 
 1405 
 
 2173 
 
 2938 
 
 3699 
 
 32 
 33 
 
 6794 
 6806 
 
 7561 
 7574 
 
 S332 
 8344 
 
 9103 
 91 16 
 
 ^^^i 
 
 0647 
 0660 
 
 1418 
 1430 
 
 2186 
 2198 
 
 2950 
 2963 
 
 37" 
 3724 
 
 
 
 
 34 
 
 6819 
 
 7587 
 
 8357 
 
 9129 
 
 9901 
 
 0673 
 
 1443 
 
 221 1 
 
 2976 
 
 3736 
 
 10 
 
 2.2 
 
 35 
 
 6832 
 
 7600 
 
 8370 
 
 9142 
 
 9914 
 
 0686 
 
 1456 
 
 2224 
 
 2989 
 
 3749 
 
 
 4-3 
 
 35 
 
 6844 
 
 7613 
 
 8383 
 
 9155 
 
 9927 
 
 0699 
 
 1469 
 
 2237 
 
 3001 
 
 3762 
 
 30 
 40 
 50 
 
 6.5 
 8.7 
 
 37 
 
 6858 
 
 7626 
 
 8396 
 
 9168 
 
 9940 
 
 0711 
 
 1482 
 
 2249 
 
 3014 
 
 3774 
 
 38 
 
 6870 
 
 7638 
 
 8409 
 
 9180 
 
 9953 
 
 0724 
 
 1494 
 
 2262 
 
 3027 
 
 3787 
 
 60 
 
 13.0 
 
 39 
 40 
 
 41 
 
 6883 
 
 7651 
 
 8422 
 
 9193 
 
 9965 
 
 0737 
 
 1507 
 
 2275 
 
 3039 
 
 3800 
 
 
 
 6896 
 
 7664 
 
 8434 
 
 9206 
 
 997S 
 
 0750 
 
 1520 
 
 2288 
 
 3052 
 
 3812 
 
 
 6909 
 
 7677 
 
 8447 
 
 9219 
 
 9991 
 
 0763 
 
 1533 
 
 2301 
 
 3065 
 
 3825 
 
 42 
 
 6921 
 
 7690 
 
 8460 
 
 9232 
 
 *ooo4 
 
 0776 
 
 1546 
 
 2313 
 
 3078 
 
 3838 
 
 
 43 
 
 6934 
 
 7702 
 
 8473 
 
 9245 
 
 *ooi7 
 
 0788 
 
 1559 
 
 2326 
 
 3090 
 
 3850 
 
 
 44 
 
 6947 
 
 7715 
 
 8486 
 
 9258 
 
 *oo3o 
 
 0801 
 
 1571 
 
 2339 
 
 3103 
 
 3863 
 
 
 45 
 
 6960 
 
 7728 
 
 8499 
 
 9270 
 
 *oo43 
 
 0814 
 
 1584 
 
 2352 
 
 
 3875 
 
 
 46 
 
 6973 
 
 7741 
 
 8512 
 
 9283 
 
 *oo55 
 
 0827 
 
 1597 
 
 2364 
 
 3128 
 
 3888 
 
 
 *l 
 
 698s 
 
 7754 
 
 S524 
 
 9296 
 
 *oo68 
 
 0840 
 
 i6ia 
 
 
 
 
 
 48 
 
 6998 
 
 7767 
 
 8537 
 
 9309 
 
 *oo8i 
 
 0853 
 
 1623 
 
 2390 
 
 3^54 
 
 39^3 
 
 
 49 
 60 
 5> 
 
 701 1 
 
 7779 
 
 8550 
 
 9322 
 
 *oo94 
 
 0866 
 
 1635 
 
 2403 
 
 3166 
 
 3926 
 
 
 7024 
 
 7792 
 
 8563 
 
 9335 
 
 *oi07 
 
 0878 
 
 1648 
 
 241S 
 
 3179 
 
 3938 
 
 7036 
 
 780s 
 
 IH^ 
 
 9348 
 
 *0I20 
 
 0891 
 
 1661 
 
 2428 
 
 
 
 52 
 
 7049 
 
 7818 
 
 8589 
 
 9361 
 
 *oi33 
 
 0904 
 
 1674 
 
 2441 
 
 3205 
 
 3964 
 
 
 53 
 
 
 7831 
 
 
 9373 
 
 •0146 
 
 0917 
 
 1687 
 
 2454 
 
 3217 
 
 3976 
 
 
 54 
 
 7'c^i 
 7100 
 
 7844 
 lit. 
 
 8614 
 8627 
 8640 
 
 9356 
 
 9399 
 9412 
 
 *oi58 
 *oi7i 
 •0184 
 
 0930 
 0943 
 09SS 
 
 1699 
 1712 
 1725 
 
 2466 
 2479 
 2492 
 
 3230 
 3243 
 3255 
 
 3989 
 4002 
 4014 
 
 
 57 
 58 
 
 7113 
 7126 
 
 7882 
 7895 
 
 8653 
 8666 
 
 9425 
 9438 
 
 *oi97 
 
 *0210 
 
 0968 
 0981 
 
 1738 
 1751 
 
 2503 
 2517 
 
 3268 
 3281 
 
 4027 
 4039 
 4052 
 
 
 59 
 60 
 
 7139 
 
 7908 
 
 8679 
 
 9451 
 
 *0223 
 
 C994 
 
 1763 
 
 2530 
 
 3293 
 
 
 7132 
 
 7921 
 
 8692 
 
 9464 
 
 *0236 
 
 1007 
 
 1776 
 
 2543 
 
 3306 
 
 4065 
 
 Smiths 
 
 DNIAN T 
 
 ABLES. 
 
 
 
 
 
 
 
 
 
 
 
 52 
 
Table 10. 
 LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 Si° 
 
 52° 
 
 S3° 
 
 54° 
 
 ss° 
 
 S6° 
 
 57° 
 
 58° 
 
 59° 
 
 60° 
 
 P.P. 
 
 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.321 
 
 
 
 C 
 
 I 
 
 2 
 
 3 
 
 4065 
 
 4817 
 
 5564 
 
 6303 
 
 7034 
 
 7756 
 
 8467 
 
 9168 
 
 9857 
 
 0534 
 
 
 
 4077 
 4090 
 4102 
 
 4829 
 4842 
 4854 
 
 5576 
 5589 
 5601 
 
 6315 
 6327 
 6340 
 
 7046 
 7058 
 7070 
 
 7768 
 7780 
 7792 
 
 8479 
 8491 
 8502 
 
 9180 
 9191 
 9203 
 
 9868 
 9891 
 
 0545 
 0556 
 0567 
 
 
 4 
 5 
 
 6 
 
 4115 
 4127 
 4140 
 
 4867 
 4879 
 4892 
 
 5613 
 ^63! 
 
 6352 
 
 7082 
 7094 
 7107 
 
 7804 
 7815 
 7827 
 
 lilt 
 8538 
 
 9214 
 9226 
 9238 
 
 9903 
 9914 
 9925 
 
 0578 
 
 
 
 
 
 
 9 
 10 
 II 
 
 12 
 
 13 
 
 4152 
 4165 
 
 4177 
 
 4904 
 4917 
 4929 
 
 5650 
 5662 
 5675 
 
 6388 
 6401 
 6413 
 
 7119 
 7131 
 7143 
 
 7839 
 7851 
 7863 
 
 8550 
 8561 
 8573 
 
 9249 
 9261 
 9272 
 
 9937 
 9948 
 9960 
 
 0612 
 0623 
 0634 
 
 20 
 30 
 
 40 
 
 5° 
 60 
 
 4-3 
 |-5 
 
 10.8 
 13.0 
 
 
 4190 
 
 4942 
 
 5687 
 
 642s 
 
 7155 
 
 787s 
 
 8585 
 
 9284 
 
 9971 
 
 0645 
 
 
 4203 
 4215 
 4228 
 
 4954 
 4967 
 4979 
 
 5699 
 5712 
 5724 
 
 6437 
 6449 
 6462 
 
 7167 
 7179 
 7191 
 
 7887 
 7899 
 791 1 
 
 8597 
 8608 
 8620 
 
 9295 
 9307 
 9318 
 
 9982 
 ^9994 
 •0005 
 
 0656 
 0667 
 0678 
 
 
 
 
 H 
 IS 
 i6 
 
 4240 
 
 4992 
 5004 
 5017 
 
 5737 
 5749 
 5761 
 
 6474 
 6486 
 6498 
 
 7203 
 7215 
 7227 
 
 7923 
 7934 
 7946 
 
 8632 
 8643 
 8655 
 
 9330 
 9341 
 9353 
 
 •0016 
 
 *0027 
 
 *oo39 
 
 0689 
 0701 
 0712 
 
 
 
 19 
 
 20 
 
 21 
 22 
 23 
 
 4278 
 4291 
 4303 
 
 5029 
 5042 
 5054 
 
 5774 
 5786 
 5799 
 
 6510 
 6523 
 6535 
 
 7239 
 7251 
 7263 
 
 7958 
 7970 
 7982 
 
 8667 
 8679 
 8690 
 
 9364 
 9376 
 9387 
 
 •0050 
 *oo6i 
 •cx>73 
 
 0723 
 0734 
 0745 
 
 
 
 4316 
 
 5067 
 
 581. 
 
 6547 
 
 7275 
 
 7994 
 
 8702 
 
 9399 
 
 •0084 
 
 0756 
 
 
 4328 
 4341 
 4353 
 
 5079 
 S092 
 
 5104 
 
 5823 
 5848 
 
 6559 
 till 
 
 7287 
 7299 
 7311 
 
 8006 
 8018 
 8030 
 
 8714 
 8725 
 8737 
 
 9410 
 9422 
 9433 
 
 *°°95 
 *oi07 
 *oii8 
 
 0767 
 0778 
 0789 
 
 
 24 
 25 
 26 
 
 4366 
 4378 
 4391 
 
 5117 
 5129 
 5141 
 
 5860 
 5872 
 5885 
 
 6596 
 6608 
 6620 
 
 7323 
 7335 
 7348 
 
 S042 
 8053 
 8065 
 
 8749 
 8760 
 8772 
 
 9445 
 9456 
 9468 
 
 *OI29 
 
 *oi40 
 
 *OIS2 
 
 0800 
 0812 
 0823 
 
 12 
 
 
 
 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 4403 
 4416 
 4428 
 
 5154 
 5166 
 5179 
 
 5897 
 5909 
 5922 
 
 6632 
 6645 
 6657 
 
 7360 
 7384 
 
 8077 
 8101 
 
 8784 
 8796 
 8807 
 
 9479 
 9491 
 9502 
 
 *oi63 
 
 0834 
 0845 
 0856 
 
 20 
 30 
 40 
 50 
 60 
 
 4.0 
 6.0 
 8.0 
 
 lO.O 
 
 12.0 
 
 
 4441 
 
 5191 
 
 5934 
 
 6669 
 
 7396 
 
 81 13 
 
 8819 
 
 9514 
 
 •0197 
 
 0867 
 
 
 4454 
 4466 
 
 4479 
 
 5203 
 5216 
 5228 
 
 5946 
 5959 
 5971 
 
 6681 
 6693 
 6706 
 
 7408 
 7420 
 7432 
 
 8125 
 8137 
 8148 
 
 8831 
 8842 
 88S4 
 
 9525 
 9537 
 9548 
 
 *0208 
 *0219 
 
 *023I 
 
 0878 
 0889 
 0900 
 
 
 
 
 34 
 35 
 36 
 
 4491 
 4504 
 
 4517 
 
 5241 
 
 5983 
 5995 
 6008 
 
 6718 
 6730 
 6742 
 
 7456 
 7468 
 
 8160 
 8172 
 8184 
 
 8866 
 8877 
 8889 
 
 9560 
 9571 
 9583 
 
 *0242 
 •0253 
 ♦0264 
 
 0911 
 0922 
 0933 
 
 
 
 11 
 39 
 
 40 
 
 41 
 
 42 
 43 
 
 4529 
 4542 
 4554 
 
 5278 
 5291 
 5303 
 
 6020 
 6032 
 6045 
 
 6754 
 6767 
 6779 
 
 7480 
 7492 
 7504 
 
 8196 
 8207 
 8219 
 
 8901 
 8913 
 8924 
 
 9617 
 
 *027S 
 
 •0287 
 •0298 
 
 0944 
 09SS 
 0966 
 
 11 
 
 
 4567 
 
 5316 
 
 6057 
 
 6791 
 
 7516 
 
 8231 
 
 8936 
 
 9629 
 
 *0309 
 
 0977 
 
 
 4579 
 4592 
 4604 
 
 5328 
 5341 
 5353 
 
 6069 
 6082 
 6094 
 
 6803 
 6815 
 6828 
 
 7528 
 7540 
 7552 
 
 8243 
 8255 
 8266 
 
 8948 
 8959 
 8971 
 
 9640 
 9652 
 9663 
 
 *0320 
 
 *°332 
 *o343 
 
 0988 
 0999 
 
 lOIO 
 
 
 44 
 45 
 46 
 
 4617 
 4629 
 4642 
 
 5366 
 5378 
 5390 
 
 6106 
 61 18 
 6131 
 
 684a 
 6852 
 6864 
 
 7564 
 7576 
 7588 
 
 8278 
 8290 
 8302 
 
 8982 
 8994 
 9006 
 
 9675 
 9686 
 9697 
 
 *0354 
 •0365 
 *o377 
 
 102 1 
 1032 
 
 1043 
 
 
 
 10 
 
 1.8 
 
 
 47 
 48 
 49 
 
 50 
 
 5' 
 52 
 53 
 
 4654 
 4667 
 4679 
 
 ■5403 
 5428 
 
 6143 
 6155 
 6168 
 
 6876 
 6889 
 6goi 
 
 7600 
 7612 
 7624 
 
 8314 
 8325 
 8337 
 
 9017 
 9029 
 9040 
 
 9709 
 9720 
 9732 
 
 *0388 
 *0399 
 ♦0411 
 
 1054 
 1065 
 1076 
 
 20 
 30 
 40 
 
 5° 
 60 
 
 3-7 
 5-5 
 7-3 
 9-2 
 
 II.O 
 
 
 4692 
 
 5440 
 
 6180 
 
 6913 
 
 7636 
 
 8349 
 
 9052 
 
 9743 
 
 *0422 
 
 1087 
 
 
 4704 
 4717 
 4729 
 
 5452 
 5465 
 5477 
 
 6192 
 6205 
 6217 
 
 6925 
 
 6937 
 6949 
 
 7648 
 7660 
 7672 
 
 8361 
 8373 
 8384 
 
 9064 
 9075 
 9087 
 
 9754 
 9766 
 9777 
 
 *°433 
 *o444 
 *0456 
 
 1098 
 
 1109 
 
 II20 
 
 
 
 54 
 1^ 
 
 4742 
 4754 
 4767 
 
 5490 
 5502 
 
 55'4 
 
 6229 
 6241 
 6254 
 
 6961 
 6973 
 6986 
 
 7684 
 7696 
 7708 
 
 8396 
 8408 
 8420 
 
 9098 
 9110 
 9122 
 
 9789 
 9800 
 981 1 
 
 •0467 
 
 II31 
 1 142 
 
 "53 
 
 
 
 11 
 59 
 
 60 
 
 4779 
 4792 
 4804 
 
 5527 
 5539 
 5552 
 
 6266 
 6278 
 6291 
 
 6998 
 7010 
 7022 
 
 7720 
 7732 
 7744 
 
 8432 
 8443 
 845s 
 
 9133 
 9145 
 9156 
 
 9823 
 9834 
 9846 
 
 *o5oo 
 
 *OSI2 
 
 *o523 
 
 1 164 
 
 1175 
 1 186 
 
 
 
 4817 
 
 5564 
 
 6303 
 
 7034 
 
 7756 
 
 8467 
 
 916S 
 
 9857 
 
 *o534 
 
 1 197 
 
 
 
 
 Smithsonian Tables. 
 
 S3 
 
Table 10. 
 
 LOGARITHMS OF MERIDIAN RADIUS OP CURVATURE p„ 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 IN ENGLISH 
 
 Lat. 
 
 61° 
 
 62° 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 68° 
 
 69° 
 
 70° 
 
 P.P. 
 
 0' 
 
 1 
 
 2 
 
 3 
 4 
 
 7 
 8 
 
 9 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 15 
 16 
 
 •7 
 18 
 
 19 
 
 20 
 
 21 
 22 
 23 
 
 24 
 25 
 26 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 11 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 45 
 46 
 
 :^ 
 49 
 
 60 
 
 51 
 52 
 53 
 
 54 
 55 
 56 
 
 11 
 59 
 
 60 
 
 7.321 
 
 1 197 
 
 7.321 
 
 1845 
 
 7.321 
 
 2479 
 
 7.321 
 
 3097 
 
 7.321 
 
 3698 
 
 7.321 
 
 4282 
 
 7.321 
 
 4848 
 
 7.321 
 
 5396 
 
 7.321 
 
 * 5924 
 
 7.321 
 
 6432 
 
 11 
 
 1208 
 1219 
 1230 
 
 1241 
 1262 
 
 1284 
 129s 
 
 1856 
 1866 
 1877 
 
 i838 
 1898 
 1909 
 
 1920 
 193 1 
 1941 
 
 2489 
 2500 
 2510 
 
 2521 
 2531 
 2541 
 
 2552 
 2562 
 2573 
 
 3107 
 3117 
 3127 
 
 3137 
 3147 
 3158 
 
 3168 
 3178 
 3188 
 
 3708 
 3718 
 3728 
 
 3738 
 3747 
 3757 
 
 3767 
 3777 
 3787 
 
 4292 
 4301 
 43" 
 4320 
 4330 
 4340 
 
 4349 
 
 'Ml 
 4876 
 
 4885 
 4894 
 4904 
 
 4913 
 4922 
 4932 
 
 5405 
 5414 
 5423 
 
 5432 
 5440 
 5449 
 
 5458 
 5467 
 5476 
 
 5933 
 5941 
 5950 
 
 5958 
 5967 
 5975 
 
 5984 
 5993 
 6001 
 
 6440 
 6448 
 6457 
 6465 
 
 t%\ 
 
 6480 
 649S 
 6506 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 1.8 
 3-7 
 5.5 
 7-3 
 9.2 
 ll.o 
 
 1306 
 
 1952 
 
 2583 
 
 3198 
 
 3797 
 
 4378 
 
 494' 
 
 5485 
 
 6010 
 
 6514 
 
 1317 
 1328 
 1338 
 
 1349 
 1360 
 
 1371 
 1382 
 1392 
 1403 
 
 1963 
 1973 
 1984 
 
 1994 
 2005 
 2016 
 
 2026 
 2037 
 2047 
 
 2593 
 2604 
 2614 
 
 2625 
 2635 
 2645 
 2656 
 2666 
 2677 
 
 3208 
 3218 
 3228 
 
 3238 
 3248 
 3259 
 3269 
 3279 
 3289 
 
 3807 
 3817 
 3826 
 
 3836 
 3846 
 3856 
 3866 
 
 3885 
 
 4387 
 4397 
 4406 
 
 4416 
 4425 
 4435 
 
 4444 
 4454 
 4463 
 
 4950 
 4959 
 4969 
 
 4978 
 4987 
 ,4996 
 5005 
 S015 
 5024 
 
 5494 
 5503 
 5512 
 
 5521 
 5529 
 5538 
 
 5547 
 5556 
 5565 
 
 6018 
 6027 
 6035 
 6044 
 6052 
 6061 
 
 6069 
 6078 
 6086 
 
 6522 
 6530 
 6539 
 6547 
 
 till 
 6571 
 6580 
 6588 
 
 10 
 
 1414 
 
 2058 
 
 2687 
 
 3299 
 
 3895 
 
 4473 
 
 S033 
 
 5574 
 
 6095 
 
 6596 
 
 10 
 
 20 
 30 
 40 
 50 
 60 
 
 '•7 
 3-3 
 5.0 
 
 8.3 
 10,0 
 
 1425 
 1436 
 1447 
 
 1% 
 1479 
 1490 
 1501 
 1512 
 
 2069 
 2079 
 2090 
 
 2100 
 2111 
 2122 
 
 2132 
 2143 
 2153 
 
 2697 
 2708 
 2718 
 
 2728 
 2738 
 2749 
 
 2759 
 2769 
 2780 
 
 3309 
 3319 
 3329 
 
 3339 
 3349 
 3360 
 
 3370 
 33S0 
 339° 
 
 3905 
 3915 
 3924 
 
 3934 
 3944 
 3954 
 
 3964 
 3973 
 3983 
 
 4482 
 4492 
 4501 
 
 45" 
 4520 
 4530 
 
 4539 
 4549 
 4558 
 
 5042 
 5051 
 5060 
 
 5069 
 5078 
 5088 
 
 5097 
 5106 
 5"5 
 
 5583 
 5592 
 5600 
 
 5609 
 5618 
 5627 
 
 5636 
 5644 
 5653 
 
 6103 
 6112 
 6120 
 
 6129 
 
 6171 
 
 6604 
 6612 
 6621 
 
 6629 
 6637 
 664s 
 
 6653 
 6662 
 6670 
 
 9 
 
 1523 
 
 2164 
 
 2790 
 
 3400 
 
 3993 
 
 4568 
 
 5124 
 
 5662 
 
 6180 
 
 6678 
 
 1534 
 1545 
 1555 
 1566 
 1577 
 1588 
 
 1599 
 1009 
 1620 
 
 2175 
 2185 
 2196 
 
 2206 
 2217 
 2228 
 
 2238 
 
 2249 
 2259 
 
 2S00 
 2811 
 2821 
 
 2831 
 2841 
 2852 
 
 2862 
 2872 
 2883 
 
 3410 
 3420 
 3430 
 
 3440 
 3450 
 3460 
 
 3470 
 3480 
 3490 
 
 4003 
 4012 
 4022 
 
 4032 
 4041 
 4051 
 
 4061 
 4071 
 4080 
 
 4577 
 4587 
 4596 
 
 4606 
 4615 
 4624 
 
 4634 
 4643 
 4653 
 
 5'33 
 5 '42 
 5151 
 5160 
 5169 
 5179 
 5188 
 5'97 
 5206 
 
 5671 
 5680 
 5688 
 
 5697 
 5706 
 5715 
 
 5724 
 5732 
 5741 
 
 6188 
 6197 
 6205 
 
 6214 
 6222 
 6230 
 
 6239 
 6247 
 6256 
 
 6686 
 6694 
 6702 
 
 6710 
 6718 
 6727 
 
 6735 
 6743 
 6751 
 
 10 
 20 
 30 
 40 
 
 ■•5 
 3.0 
 
 ti 
 
 7-5 
 9.0 
 
 1631 
 
 2270 
 
 2893 
 
 3500 
 
 4090 
 
 4662 
 
 5215 
 
 575° 
 
 6264 
 
 6759 
 
 
 8 
 
 1642 
 1652 
 1663 
 
 1674 
 1684 
 1695 
 1706 
 1717 
 1727 
 
 2280 
 2291 
 2301 
 2312 
 2322 
 2333 
 
 2343 
 2354 
 2364 
 
 2903 
 2913 
 2924 
 
 2934 
 2944 
 2954 
 2964 
 
 2975 
 2985 
 
 3510 
 3520 
 3530 
 
 3540 
 3549 
 3559 
 
 3569 
 3579 
 3589 
 
 4100 
 4109 
 4119 
 
 4128 
 4138 
 4148 
 
 4157 
 4167 
 4176 
 
 fell 
 4690 
 
 4S99 
 4708 
 4718 
 
 ■4727 
 4736 
 4745 
 
 5224 
 5233 
 5242 
 
 5251 
 5260 
 5270 
 
 5297 
 
 5759 
 5767 
 5776 
 
 5785 
 5793 
 5802 
 
 5811 
 5820 
 5828 
 
 6272 
 6281 
 6289 
 
 6298 
 6306 
 6314 
 
 6323 
 6331 
 6340 
 
 6767 
 till 
 
 6791 
 6799 
 6807 
 
 6815 
 6823 
 6831 
 
 10 
 20 
 30 
 40 
 50 
 6a 
 
 '•3 
 
 2.6 
 4.0 
 5-3 
 6.7 
 8.0 
 
 1738 
 
 2375 
 
 2995 
 
 3599 
 
 4185 
 
 4755 
 
 5306 
 
 5837 
 
 6348 
 
 6839 
 
 1749 
 1759 
 1770 
 
 1781 
 1791 
 1802 
 
 1813 
 1824 
 1834 
 
 2385 
 2396 
 
 2406 
 
 2417 
 2427 
 2437 
 2448 
 
 2458 
 2469 
 
 3005 
 3015 
 3026 
 
 3036 
 3046 
 3056 
 3066 
 3077 
 3087 
 
 3609 
 3619 
 3629 
 
 3639 
 364i 
 3658 
 3668 
 
 4196 
 4205 
 4215 
 4224 
 4234 
 4244 
 
 4253 
 4263 
 4272 
 
 4764 
 4774 
 4783 
 
 4792 
 4801 
 481 1 
 
 4820 
 4829 
 4839 
 
 53'5 
 5324 
 5333 
 5342 
 535' 
 5360 
 
 5369 
 5378 
 5387 
 
 5846 
 5854 
 5863 
 
 §z 
 
 5889 
 5898 
 
 5907 
 5915 
 
 6356 
 6365 
 6373 
 6382 
 6390 
 6398. 
 
 6407 
 641S 
 6424 
 
 6847 
 6855 
 6863 
 
 6871 
 6879 
 6887 
 
 6895 
 6903 
 691 1 
 
 
 1845 
 
 2479 
 
 3097 
 
 3698 
 
 4282 
 
 4848 
 
 5396 
 
 5924 
 
 6432 
 
 6919 
 
 Smiths 
 
 ONIAN T 
 
 ABLES. 
 
 
 
 
 
 
 
 
 
 
 
 S4 
 
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ 
 
 FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Table 10. 
 IN ENGLISH 
 
 Lat. 
 
 71° 
 
 72° 
 
 73° 
 
 74° 
 
 75° 
 
 76° 
 
 77° 
 
 78° 
 
 79° 
 
 80° 
 
 P.P. 
 
 0' 
 I 
 
 2 
 
 3 
 
 4 
 5 
 6 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 
 "3 
 14 
 \l 
 
 '7 
 iS 
 '9 
 
 20 
 
 21 
 22 
 23 
 
 24 
 
 11 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 
 % 
 49 
 
 50 
 
 51 
 52 
 53 
 
 54 
 % 
 
 I 
 59 
 
 60 
 
 7.321 
 
 6919 
 
 7.321 
 
 7385 
 
 7.321 
 
 7829 
 
 7.321 
 
 8251 
 
 7.321 
 
 8650 
 
 7.321 
 
 9025 
 
 7.321 
 
 9377 
 
 7.321 
 
 9704 
 
 7.322 
 
 0007 
 
 7.322 
 
 0284 
 
 
 6927 
 6935 
 6943 
 
 6951 
 6958 
 6966 
 
 6974 
 6982 
 6990 
 
 7392 
 7400 
 7407 
 
 7415 
 7422 
 7430 
 
 7437 
 7445 
 7452 
 
 7836 
 7843 
 7851 
 
 7858 
 786s 
 7872 
 
 7879 
 7887 
 7894 
 
 8258 
 8265 
 8271 
 
 8278 
 8285 
 8292 
 
 8299 
 830s 
 8312 
 
 8656 
 8663 
 8669 
 
 8676 
 8682 
 8688 
 
 8695 
 8701 
 8708 
 
 9°3i 
 9037 
 9043 
 
 9049 
 9°55 
 9061 
 
 9067 
 9073 
 9079 
 
 9383 
 9388 
 9394 
 
 9399 
 9405 
 9411 
 
 9416 
 9422 
 9427 
 
 9709 
 
 9714 
 9720 
 
 9725 
 9730 
 9735 
 9740 
 9746 
 9751 
 
 0012 
 0017 
 
 CX)2I 
 
 0026 
 0031 
 0036 
 
 0041 
 0045 
 0050 
 
 0288 
 0293 
 0297 
 
 0302 
 0306 
 0310 
 
 0315 
 0319 
 0324 
 
 10 
 20 
 30 
 40 
 
 lo 
 
 1.3 
 
 2.6 
 
 4.0 
 
 5-3 
 
 %■'' 
 8.0 
 
 6998 
 
 7460 
 
 7901 
 
 8319 
 
 8714 
 
 9085 
 
 9433 
 
 9756 
 
 0055 
 
 0328 
 
 7 
 
 7006 
 7014 
 7021 
 
 7029 
 7037 
 7045 
 
 7053 
 7060 
 7068 
 
 7467 
 
 7475 
 7482 
 
 7490 
 7497 
 7505 
 
 7512 
 7520 
 7527 
 
 7908 
 7915 
 7922 
 
 7929 
 7936 
 7944 
 
 7951 
 795S 
 7965 
 
 8326 
 8332 
 8339 
 8346 
 8353 
 8359 
 8366 
 8373 
 8379 
 
 8720 
 8727 
 8733 
 
 8739 
 8745 
 8752 
 
 8758 
 8764 
 8771 
 
 9091 
 9097 
 9103 
 
 9109 
 911S 
 9121 
 
 9127 
 9133 
 9139 
 
 9438 
 9444 
 9449 
 
 9455 
 9460 
 9466 
 
 9471 
 9477 
 9482 
 
 9761 
 9766 
 9771 
 9776 
 9781 
 9787 
 
 9792 
 9797 
 9802 
 
 0060 
 0064 
 0069 
 
 0074 
 0078 
 0083 
 
 0088 
 0093 
 0097 
 
 0332 
 0337 
 0341 
 
 0345 
 0349 
 0354 
 
 0358 
 0362 
 0367 
 
 10 
 20 
 
 30 
 40 
 
 1.2 
 2.3 
 3-5 
 
 li 
 7.0 
 
 7076 
 
 7535 
 
 7972 
 
 8385 
 
 8777 
 
 9145 
 
 9488 
 
 9807 
 
 0102 
 
 0371 
 
 7084 
 7092 
 7099 
 7107 
 711S 
 7123 
 
 7131 
 7138 
 7146 
 
 7542 
 7550 
 7557 
 
 7565 
 
 7580 
 
 7587 
 7595 
 7602 
 
 7993 
 8000 
 8007 
 8014 
 
 802I 
 
 8028 
 8035 
 
 8393 
 8399 
 8406 
 
 8413 
 8419 
 S426 
 
 8433 
 8440 
 8446 
 
 8783 
 8790 
 8796 
 8802 
 8808 
 881S 
 
 8821 
 8827 
 8834 
 
 9151 
 9157 
 9163 
 
 9169 
 9174 
 9180 
 
 9186 
 9192 
 9198 
 
 9493 
 9499 
 9504 
 
 9510 
 9515 
 9521 
 
 9526 
 9532 
 9537 
 
 9812 
 9817 
 9822 
 
 9827 
 9832 
 9838 
 
 9843 
 9848 
 9853 
 
 0107 
 GUI 
 OI16 
 
 0120 
 0125 
 0130 
 
 0134 
 0139 
 0143 
 
 0375 
 0379 
 0384 
 0388 
 0392 
 0396 
 
 0400 
 0405 
 0409 
 
 6 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 I.O 
 
 2.0 
 
 30 
 4.0 
 5.0 
 6.0 
 
 7154 
 
 7610 
 
 8042 
 
 8453 
 
 8840 
 
 9204 
 
 9543 
 
 9858 
 
 0148 
 
 0413 
 
 7162 
 7170 
 7177 
 
 718s 
 7193 
 7201 
 
 7209 
 7216 
 7224 
 
 7617 
 7625 
 7632 
 
 7639 
 7646 
 7654 
 7661 
 7668 
 7676 
 
 8056 
 
 8070 
 8077 
 8084 
 
 8091 
 8098 
 8105 
 
 8460 
 8466 
 8473 
 
 8493 
 
 8499 
 8506 
 8512 
 
 8846 
 8852 
 8859 
 886s 
 8871 
 8877 
 8883 
 8890 
 8896 
 
 9210 
 9216 
 9221 
 
 9227 
 9233 
 9239 
 
 9245 
 9250 
 9256 
 
 9548 
 9554 
 9559 
 
 9565 
 9570 
 9575 
 
 9586 
 9592 
 
 9863 
 9868 
 9873 
 
 9883 
 988S 
 
 9893 
 9898 
 9903 
 
 0153 
 0157 
 0162 
 
 0166 
 OI7I 
 0176 
 
 0180 
 0185 
 0189 
 
 0417 
 0421 
 0426 
 
 0430 
 0434 
 0438 
 
 0442 
 0447 
 0451 
 
 6 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 .8 
 1-7 
 2-5 
 3-3 
 4.2 
 5-0 
 
 7232 
 
 7683 
 
 8112 
 
 8519 
 
 S902 
 
 9262 
 
 9597 
 
 9908 
 
 0194 
 
 0455 
 
 7240 
 7247 
 7255 
 
 7263 
 7270 
 7278 
 
 7286 
 7294 
 7301 
 
 7690 
 7798 
 7705 
 7712 
 7719 
 7727 
 
 7734 
 7741 
 7749 
 
 8119 
 8126 
 8133 
 8140 
 8147 
 8154 
 8161 
 8168 
 8175 
 
 8526 
 8532 
 S539 
 
 8545 
 8552 
 8SS9 
 
 8565 
 8572 
 8578 
 
 8908 
 8914 
 8921 
 
 8927 
 8933 
 8939 
 
 8945 
 8952 
 8958 
 
 9268 
 9274 
 9279 
 9285 
 9291 
 9297 
 
 9303 
 9308 
 9314 
 
 9602 
 9608 
 9613 
 9619 
 9624 
 9629 
 
 963 s 
 9640 
 9646 
 
 9913 
 9918 
 9923 
 9928 
 9933 
 9938 
 
 9943 
 9948 
 9953 
 
 0199 
 0203 
 0208 
 
 0212 
 0217 
 0222 
 
 0226 
 0231 
 023s 
 
 0459 
 0463 
 0467 
 
 0471 
 
 0475 
 0480 
 
 0484 
 0488 
 0492 
 
 4 
 
 10 
 20 
 30 
 40 
 
 lo 
 
 •7 
 1-3 
 2.0 
 2.7 
 3-3 
 4.0 
 
 7309 
 
 7756 
 
 8182 
 
 8585 
 
 8964 
 
 9320 
 
 9651 
 
 9953 
 
 0240 
 
 0496 
 
 7317 
 7324 
 7332 
 
 7339 
 7347 
 7355 
 7362 
 7370 
 7377 
 
 7763 
 7771 
 7778 
 
 7785 
 7792 
 7800 
 
 7807 
 7814 
 7822 
 
 8189 
 8196 
 8203 
 
 8210 
 8216 
 8223 
 
 8230 
 8237 
 8244 
 
 8598 
 8604 
 
 86ri 
 8617 
 8624 
 8630 
 8637 
 8643 
 
 8970 
 
 8988 
 8994 
 9001 
 
 9007 
 9°i3 
 9019 
 
 9326 
 9331 
 9337 
 
 9343 
 9348 
 9354 
 9360 
 9366 
 9371 
 
 9656 
 9662 
 9667 
 
 9672 
 9677 
 9683 
 
 9688 
 9693 
 9699 
 
 9963 
 9968 
 9973 
 
 9987 
 
 9992 
 
 9997 
 
 *boo2 
 
 0244 
 0249 
 0253 
 
 0258 
 0262 
 0266 
 
 0271 
 0275 
 0280 
 
 0500 
 0504 
 0508 
 
 0512 
 0516 
 0520 
 
 0524 
 0528 
 0532 
 
 
 7385 
 
 7829 
 
 8251 
 
 8650 
 
 9025 
 
 9377 
 
 9704 
 
 *ooo7 
 
 0284 
 
 0536 
 
 Smithsonian Tables. 
 
 55 
 
Table 10. 
 LOGARITHMS OF MERrOIAN RADIUS OF CURVATURE p^ IN ENGLISH 
 
 FEET. 
 
 [Derivation of table explained on p. xlv,] 
 
 Lat. 
 
 8l° 
 
 82° 
 
 83° 
 
 84° 
 
 Ss" 
 
 86° 
 
 87° 
 
 88° 
 
 89° 
 
 P. 
 
 P. 
 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 
 
 0' 
 
 I 
 3 
 
 0536 
 
 0763 
 
 0963 
 
 1138 
 
 1285 
 
 1407 
 
 1501 
 
 1569 
 
 1609 
 
 4 
 
 0540 
 0544 
 0548 
 
 0766 
 0770 
 0773 
 
 0966 
 0969 
 0972 
 
 1 141 
 1 143 
 1146 
 
 1287 
 1289 
 1292 
 
 1409 
 1410 
 1412 
 
 1502 
 1504 
 1505 
 
 1570 
 ■571 
 1571 
 
 i6og 
 1610 
 1610 
 
 
 
 4 
 1 
 
 0552 
 0555 
 0560 
 
 0777 
 0780 
 0784 
 
 0975 
 0978 
 0982 
 
 1148 
 1151 
 "54 
 
 1294 
 1296 
 1298 
 
 1414 
 
 1415 
 1417 
 
 1506 
 1507 
 1509 
 
 ■572 
 ■573 
 ■574 
 
 1611 
 1611 
 1611 
 
 10 
 20 
 30 
 40 
 
 50 
 60 
 
 •7 
 ■•3 
 2.0 
 2-7 
 3.3 
 4.0 
 
 7 
 8 
 9 
 
 10 
 
 II 
 
 12 
 
 '3 
 
 0564 
 0568 
 0572 
 
 0787 
 0791 
 0794 
 
 ^11 
 0991 
 
 1156 
 IIS9 
 1 161 
 
 1300 
 1303 
 I3°5 
 
 1419 
 1421 
 1422 
 
 1510 
 1511 
 1513 
 
 ■575 
 ■576 
 
 1612 
 1612 
 1613 
 
 0576 
 
 0798 
 
 0994 
 
 1 164 
 
 1307 
 
 1424 
 
 1514 
 
 ■577 
 
 1613 
 
 
 
 0580 
 
 I'M 
 
 0801 
 0805 
 0808 
 
 0997 
 1000 
 1003 
 
 1167 
 1 169 
 1 172 
 
 1309 
 1311 
 13 14 
 
 1426 
 1427 
 1429 
 
 I5'5 
 1517 
 1518 
 
 1578 
 ■579 
 ■579 
 
 1613 
 1614 
 1614 
 
 14 
 
 0592 
 0595 
 0599 
 
 0812 
 0815 
 0819 
 
 1006 
 1009 
 1012 
 
 1 174 
 1 180 
 
 1316 
 1318 
 1320 
 
 1431 
 1432 
 1434 
 
 1519 
 1520 
 1522 
 
 1580 
 1581 
 1582 
 
 1615 
 1615 
 1615 
 
 
 
 17 
 l8 
 19 
 
 20 
 
 ZI 
 22 
 
 23 
 
 0603 
 0607 
 0611 
 
 0822 
 0826 
 0829 
 
 1015 
 1018 
 1021 
 
 1182 
 1185 
 1 187 
 
 1322 
 1325 
 1327 
 
 1435 
 1438 
 1439 
 
 1523 
 1524 
 1526 
 
 ■583 
 ■583 
 1584 
 
 1616 
 1616 
 
 1617 
 
 fl 
 
 
 0615 
 
 0833 
 
 1024 
 
 1190 
 
 1329 
 
 1441 
 
 • 1527 
 
 ■585 
 
 1617 
 
 10 
 20 
 30 
 40 
 
 5° 
 60 
 
 .5 
 
 I.O 
 
 ■■5 
 2.0 
 
 0619 
 0623 
 0626 
 
 0836 
 0840 
 0843 
 
 1027 
 1030 
 1033 
 
 1 192 
 1 195 
 1 197 
 
 1331 
 1333 
 133s 
 
 1443 
 1444 
 1446 
 
 1528 
 1529 
 1530 
 
 1585 
 1586 
 ■587 
 
 1617 
 1617 
 1618 
 
 24 
 
 0630 
 0634 
 0638 
 
 0846 
 0849 
 0853 
 
 1036 
 1039 
 1042 
 
 1200 
 1202 
 1205 
 
 1337 
 1339 
 1341 
 
 1447 
 1449 
 1451 
 
 1531 
 1532 
 ■534 
 
 1588 
 ■S88 
 ■589 
 
 1618 
 1618 
 1618 
 
 
 
 29 
 30 
 
 31 
 32 
 33 
 
 0642 
 0645 
 0649 
 
 0856 
 0859 
 0863 
 
 1045 
 1048 
 105 1 
 
 1207 
 1210 
 1212 
 
 1343 
 1345 
 1347 
 
 1452 
 1454 
 1455 
 
 ■535 
 ■536 
 ■537 
 
 ■59° 
 1591 
 ■591 
 
 1618 
 1619 
 1619 
 
 3 
 
 
 0653 
 
 0866 
 
 1054 
 
 1215 
 
 1349 
 
 '457 
 
 ■538 
 
 1592 
 
 1619 
 
 0660 
 0664 
 
 0S69 
 0873 
 0876 
 
 1057 
 1060 
 1062 
 
 1217 
 1220 
 1222 
 
 1351 
 1353 
 1355 
 
 1459 
 1460 
 1462 
 
 1539 
 ■540 
 1541 
 
 ■593 
 ■593 
 ■594 
 
 1619 
 1619 
 1620 
 
 34 
 11 
 
 o658 
 0671 
 0675 
 
 0879 
 0882 
 o885 
 
 1065 
 1068 
 1071 
 
 1225 
 1227 
 1229 
 
 1357 
 1359 
 1361 
 
 1463 
 1465 
 1467 
 
 1542 
 ■543 
 ■545 
 
 ■595 
 '595 
 1595 
 
 1620 
 1620 
 1620 
 
 
 
 10 
 
 ■3 
 
 11 
 39 
 
 40 
 
 41 
 42 
 43 
 
 0679 
 0683 
 0686 
 
 0889 
 0892 
 0895 
 
 1074 
 1076 
 1079 
 
 1232 
 1234 
 1237 
 
 1363 
 1365 
 1367 
 
 1468 
 1470 
 1471 
 
 1546 
 1548 
 
 ■597 
 1598 
 ■598 
 
 1620 
 1621 
 
 1621 
 
 20 
 30 
 40 
 50 
 60 
 
 ■7 
 1.0 
 ■•3 
 ■■7 
 2.0 
 
 0690 
 
 0899 
 
 1082 
 
 1239 
 
 1369 
 
 1473 
 
 ■549 
 
 ■599 
 
 162 1 
 
 0694 
 0697 
 0701 
 
 0902 
 ogo6 
 0909 
 
 108s 
 1088 
 1090 
 
 1241 
 1244 
 1246 
 
 1371 
 1373 
 >375 
 
 1474 
 1476 
 1477 
 
 ■550 
 ■55^ 
 ■552 
 
 ■599 
 1600 
 1600 
 
 1621 
 1621 
 162 z 
 
 
 
 44 
 45 
 46 
 
 0705 
 0708 
 0712 
 
 0912 
 0915 
 0919 
 
 1093 
 1096 
 1099 
 
 1249 
 1251 
 "S3 
 
 ■377 
 1378 
 1380 
 
 >479 
 1480 
 148. 
 
 ■553 
 ■554 
 ■555 
 
 1601 
 1601 
 1602 
 
 1621 
 1621 
 
 1622 
 
 
 
 49 
 60 
 
 51 
 52 
 53 
 
 0716 
 0720 
 0723 
 
 0922 
 0925 
 0929 
 
 1 102 
 1 104 
 1 107 
 
 1256 
 1258 
 1261 
 
 1382 
 ■384 
 1386 
 
 1483 
 \1M 
 
 ■556 
 ■557 
 1558 
 
 1602 
 1603 
 1603 
 
 1622 
 1622 
 1622 
 
 1 
 
 
 0727 
 
 0932 
 
 mo 
 
 1263 
 
 1388 
 
 1487 
 
 ■559 
 
 1604 
 
 1622 
 
 10 
 20 
 30 
 40 
 
 .2 
 •3 
 '5 
 ■7 
 
 0731 
 0734 
 0738, 
 
 0935 
 0938 
 0941 
 
 1113 
 1116 
 1118 
 
 1265 
 1267 
 1270 
 
 1390 
 1392 
 1394 
 
 1488 
 1490 
 1491 
 
 1560 
 1561 
 1562 
 
 1604 
 1605 
 1605 
 
 1622 
 1622 
 1622 
 
 54 
 11 
 
 0741 
 
 0745 
 0749 
 
 0944 
 0947 
 0951 
 
 1121 
 1124 
 1 127 
 
 1272 
 1274 
 1275 
 
 1396 
 1397 
 1399 
 
 1493 
 1494 
 
 1495 
 
 ■563 
 1564 
 1565 
 
 1606 
 i6o5 
 1607 
 
 1622 
 1622 
 1623 
 
 so 
 60 
 
 1.0 
 
 
 
 11 
 59 
 
 60 
 
 0752 
 0756 
 0759 
 
 0954 
 0957 
 0960 
 
 1 130 
 1 132 
 1135 
 
 1278 
 1281 
 1283 
 
 1401 
 1403 
 1405 
 
 1497 
 1498 
 1500 
 
 156S 
 1567 
 1568 
 
 1607 
 l6o8 
 1608 
 
 1623 
 1623 
 
 1623 
 
 
 
 0763 
 
 0963 
 
 1138 
 
 1285 
 
 1407 
 
 1 501 
 
 1569 
 
 1609 
 
 1623 
 
 Smithsoni 
 
 KM Tab 
 
 LES. 
 
 
 
 
 
 
 
 
 
 
 S6 
 
Table 1 1 , 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ (N ENGLISH FEET. 
 
 [Derivatiou of table explained on p. xlv.] 
 
 Lat. 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 10° 
 
 P.P. 
 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 
 I 
 
 2 
 
 3 
 
 6875 
 
 6S80 
 
 6893 
 
 6916 
 
 6947 
 
 6987 
 
 7036 
 
 7094 
 
 7160 
 
 7235 
 
 7319 
 
 
 687s 
 6875 
 6875 
 
 6880 
 6880 
 6880 
 
 6893 
 6894 
 6894 
 
 6916 
 6917 
 6917 
 
 6948 
 6949 
 6949 
 
 6988 
 6989 
 6989 
 
 7037 
 7038 
 7039 
 
 7095 
 7096 
 7097 
 
 7161 
 7162 
 7164 
 
 7236 
 7238 
 7239 
 
 7320 
 7322 
 7323 
 
 4 
 5 
 6 
 
 687s 
 6875 
 6875 
 
 6880 
 6881 
 6881 
 
 6894 
 6894 
 6895 
 
 6918 
 6918 
 6918 
 
 6950 
 6950 
 6951 
 
 6990 
 6991 
 6992 
 
 7040 
 7041 
 7041 
 
 7098 
 7099 
 7100 
 
 7165 
 7166 
 7167 
 
 7240 
 7241 
 7243 
 
 7325 
 7326 
 7327 
 
 
 I 
 9 
 
 10 
 
 ir 
 12 
 13 
 
 6875 
 6875 
 
 6881 
 6881 
 68S1 
 
 6895 
 6895 
 6896 
 
 6919 
 6919 
 6920 
 
 6951 
 6952 
 6953 
 
 6993 
 6993 
 6994 
 
 7042 
 7043 
 7044 
 
 7101 
 7102 
 7103 
 
 7168 
 7170 
 7171 
 
 7244 
 7245 
 7247 
 
 7329 
 7330 
 7332 
 
 1 
 
 6875 
 
 68S1 
 
 6896 
 
 6920 
 
 6953 
 
 699s 
 
 7045 
 
 7104 
 
 7172 
 
 7248 
 
 7333 
 
 687s 
 6875 
 6875 
 
 6881 
 6881 
 6882 
 
 6896 
 6897 
 6897 
 
 6920 
 6921 
 6921 
 
 6954 
 6955 
 6955 
 
 6996 
 6996 
 6997 
 
 7046 
 7047 
 7048 
 
 7105 
 7106 
 7107 
 
 7173 
 7174 
 7176 
 
 7249 
 7251 
 7252 
 
 7334 
 7336 
 7338 
 
 
 
 14 
 l6 
 
 6875 
 6876 
 6875 
 
 6882 
 6882 
 6882 
 
 6898 
 689S 
 6898 
 
 6922 
 6922 
 6923 
 
 6956 
 6957 
 
 6998 
 6999 
 6999 
 
 7049 
 70S0 
 7050 
 
 7108 
 7109 
 7111 
 
 7177 
 7178 
 7'79 
 
 7254 
 7255 
 7256 
 
 7339 
 7341 
 7342 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 .2 
 ■3 
 .5 
 
 -.1 
 
 I.O 
 
 20 
 
 21 
 22 
 23 
 
 6876 
 6876 
 6876 
 
 6882 
 6883 
 6883 
 
 6899 
 6899 
 6goo 
 
 6923 
 6924 
 6924 
 
 6957 
 6958 
 6959 
 
 7000 
 7001 
 7cx)i 
 
 7051 
 7052 
 7053 
 
 7112 
 7113 
 7114 
 
 7180 
 7182 
 7-83 
 
 7258 
 7259 
 7261 
 
 7343 
 7345 
 7346 
 
 6876 
 
 6883 
 
 6goo 
 
 692s 
 
 69S9 
 
 7002 
 
 7054 
 
 711S 
 
 7184 
 
 7262 
 
 7348 
 
 
 6876 
 6876 
 6876 
 
 6883 
 6883 
 6884 
 
 6900 
 6901 
 6901 
 
 6925 
 6926 
 6926 
 
 6960 
 6960 
 6961 
 
 7003 
 7004 
 7004 
 
 7055 
 7056 
 
 7057 
 
 7116 
 7117 
 7118 
 
 718s 
 7186 
 7188 
 
 7263 
 7265 
 7266 
 
 7350 
 7351 
 7353 
 
 24 
 
 25 
 26 
 
 6876 
 6876 
 6876 
 
 6884 
 6884 
 6884 
 
 6901 
 6902 
 6902 
 
 6927 
 6927 
 6928 
 
 6962 
 6962 
 6963 
 
 7005 
 7006 
 7007 
 
 7058 
 
 7059 
 7060 
 
 71 19 
 7120 
 7122 
 
 71S9 
 7190 
 7191 
 
 7268 
 7269 
 7270 
 
 7354 
 7356 
 7358 
 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 32 
 33 
 
 6876 
 6876 
 6876 
 
 6884 
 688s 
 6885 
 
 6962 
 6902 
 6903 
 
 6928 
 6929 
 6929 
 
 696s 
 6965 
 
 7008 
 7Cxj8 
 7009 
 
 7061 
 7062 
 7063 
 
 7123 
 7124 
 7125 
 
 7192 
 
 7 "94 
 7195 
 
 7272 
 7273 
 7275 
 
 7359 
 7361 
 7362 
 
 
 6876 
 
 6885 
 
 6903 
 
 6930 
 
 6966 
 
 7010 
 
 7064 
 
 7126 
 
 7196 
 
 7276 
 
 7364 
 
 6877 
 6877 
 6877 
 
 688s 
 6886 
 6886 
 
 6903 
 6904 
 6904 
 
 6930 
 6931 
 6931 
 
 6967 
 6967 
 6968 
 
 701 1 
 7012 
 7013 
 
 7065 
 7066 
 7067 
 
 7127 
 7128 
 7129 
 
 7197 
 7199 
 7200 
 
 7277 
 
 7366 
 7367 
 7368 
 
 34 
 35 
 36 
 
 6877 
 6877 
 6877 
 
 6886 
 6887 
 6887 
 
 690s 
 6905 
 6905 
 
 6932 
 6932 
 6933 
 
 6969 
 6969 
 6970 
 
 7014 
 701S 
 701S 
 
 7068 
 7069 
 7070 
 
 7130 
 7131 
 7133 
 
 7201 
 7202 
 7204 
 
 7282 
 7283 
 7284 
 
 7370 
 737' 
 7373 
 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 6877 
 6877 
 6877 
 
 6887 
 6887 
 6888 
 
 6906 
 6906 
 6907 
 
 6933 
 6934 
 6935 
 
 6971. 
 6972 
 6972 
 
 7016 
 7017 
 7018 
 
 7070 
 7071 
 7072 
 
 7134 
 7135 
 7136 
 
 7206 
 7208 
 
 7286 
 7287 
 7289 
 
 7374 
 7376 
 7377 
 
 2 
 
 6877 
 
 6888 
 
 6907 
 
 6935 
 
 6973 
 
 7019 
 
 7073 
 
 7137 
 
 7209 
 
 7290 
 
 7379 
 
 10 
 
 20 
 30 
 
 40 
 
 .3 
 •7 
 1.0 
 1-3 
 
 6877 
 6877 
 6877 
 
 6888 
 6888 
 6889 
 
 6909 
 
 6936 
 6936 
 6937 
 
 6974 
 6974 
 697s 
 
 7020 
 7021 
 7021 
 
 7074 
 
 7075 
 7076 
 
 7138 
 7139 
 7140 
 
 7210 
 7212 
 7213 
 
 7291 
 7293 
 7294 
 
 7381 
 7382 
 7384 
 
 44 
 
 % 
 49 
 
 50 
 
 5' 
 52 
 53 
 
 6877 
 6878 
 6878 
 
 6878 
 6878 
 ■6878 
 
 6889 
 6889 
 6889 
 
 6889 
 68qo 
 6890 
 
 6909 
 6910 
 6910 
 
 6910 
 69II 
 691 1 
 
 6937 
 6938 
 6938 
 
 6939 
 6939 
 6940 
 
 6976 
 6976 
 6977 
 
 6978 
 6979 
 6979 
 
 7022 
 7023 
 7024 
 
 702s 
 7025 
 7026 
 
 7077 
 7078 
 7079 
 
 7080 
 ■7081 
 7082 
 
 7141 
 7142 
 7144 
 
 7"4| 
 7146 
 7147 
 
 7214 
 7216 
 7217 
 
 72>8 
 7219 
 7221 
 
 7296 
 7297 
 7298 
 
 7300 
 7301 
 7303 
 
 7385 
 7387 
 7389 
 7390 
 7392 
 7393 
 
 so 
 60 
 
 1-7 
 2.0 
 
 
 6878 
 
 6890 
 
 6gri 
 
 6941 
 
 6980 
 
 7027 
 
 70S3 
 
 7148 
 
 7222 
 
 7304 
 
 7395 
 
 6878 
 6878 
 6879 
 
 6890 
 6891 
 6891 
 
 6gi2 
 691Z 
 6913 
 
 6942 
 6942 
 6943 
 
 6981 
 6981 
 6982 
 
 702S 
 7029 
 7030 
 
 7084 
 7085 
 7086 
 
 7149 
 7150 
 7152 
 
 7223 
 7225 
 7226 
 
 7305 
 7307 
 7308 
 
 7397 
 7398 
 7400 
 
 54 
 11 
 
 6879 
 6879 
 6879 
 
 6891 
 6892 
 6892 
 
 6913 
 6914 
 6914 
 
 6943 
 6944 
 6944 
 
 6983 
 6983 
 6984 
 
 7031 
 7032 
 7032 
 
 7087 
 7088 
 7090 
 
 7153 
 7154 
 715s 
 
 7227 
 7228 
 7230 
 
 7310 
 7311 
 7313 
 
 7401 
 7403 
 7405 
 
 
 11 
 59 
 
 60 
 
 6879 
 6880 
 6880 
 
 6892 
 6892 
 6893 
 
 6915 
 6915 
 6916 
 
 6945 
 694s 
 6946 
 
 6985 
 6986 
 6986 
 
 7033 
 7034 
 7035 
 
 7091 
 7092 
 7°93 
 
 7156 
 7158 
 7159 
 
 7231 
 7232 
 7234 
 
 7314 
 7316 
 7317 
 
 7406 
 7408 
 7409 
 
 
 688a 
 
 6893 
 
 6916 
 
 6947 
 
 6987 
 
 7036 
 
 7094 
 
 7160 
 
 7235 
 
 7319 
 
 741 1 
 
 
 -1 
 
 Smithsonian Tables. 
 
 57 
 
Table 1 1 . 
 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 iS° 
 
 16° 
 
 17° 
 
 i8° 
 
 19° 
 
 20° 
 
 p.p. 
 
 0' 
 
 X 
 2 
 
 3 
 4 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 ■3 
 
 ■4 
 IS 
 l6 
 
 I's' 
 '9 
 
 20 
 
 21 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 'I 
 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 
 35 
 36 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 45 
 46 
 
 47 
 48 
 49 
 
 60 
 
 51 
 52 
 S3 
 
 54 
 55 
 S6 
 
 57 
 58 
 59 
 
 60 
 
 7.320 
 
 741 1 
 
 7.320 
 
 75" 
 
 7.320 
 
 7619 
 
 7.320 
 
 7736 
 
 7.320 
 
 7860 
 
 7.320 
 
 7992 
 
 7.320 
 
 8132 
 
 7.320 
 
 8279 
 
 7.320 
 
 8434 
 
 7.320 
 
 8595 
 
 1 
 
 7413 
 7414 
 7416 
 
 7417 
 7419 
 7421 
 
 7422 
 7424 
 7425 
 
 7513 
 75-4 
 7516 
 7518 
 75 '9 
 7521 
 
 7523 
 7525 
 7526 
 
 7621 
 7623 
 7625 
 7627 
 7628 
 7630 
 
 7632 
 7634 
 7636 
 
 7738 
 7740 
 7742 
 
 7744 
 7746 
 7748 
 
 7750 
 7752 
 7754 
 
 7862 
 7864 
 7867 
 
 7869 
 7871 
 7873 
 
 7875 
 7878 
 7880 
 
 7994 
 7997 
 7999 
 8001 
 8003 
 8006 
 
 8008 
 80 lO 
 8013 
 
 8134 
 8137 
 8139 
 8142 
 8.44 
 8146 
 
 8149 
 8151 
 8.54 
 
 8282 
 8284 
 8287 
 
 8289 
 8292 
 8295 
 8297 
 8300 
 8302 
 
 8437 
 8439 
 8442 
 
 8444 
 8447 
 8450 
 
 8452 
 8455 
 8457 
 
 8603 
 
 R606 
 8609 
 8612 
 
 861S 
 8617 
 8620 
 
 JO 
 
 20 
 30 
 40 
 
 lo 
 
 .2 
 
 ■ 3 
 
 •5 
 
 :l 
 
 I.O 
 
 7427 
 
 7528 
 
 7638 
 
 7756 
 
 7882 
 
 801S 
 
 8156 
 
 8305 
 
 8460 
 
 8623 
 
 7429 
 7430 
 7432 
 
 7433 
 7435 
 7437 
 7438 
 7440 
 7441 
 
 7530 
 7532 
 7533 
 
 7535 
 7537 
 7539 
 
 7541 
 7542 
 7544 
 
 7640 
 7642 
 7644 
 
 7646 
 7647 
 7649 
 
 7651 
 7653 
 7655 
 
 7758 
 7760 
 7762 
 
 7768 
 
 7770 
 7772 
 7774 
 
 7884 
 7886 
 7888 
 
 7890 
 7892 
 7895 
 
 7897 
 7899 
 7901 
 
 8017 
 802a 
 8022 
 
 8024 
 8026 
 8029 
 
 8031 
 8033 
 8036 
 
 8158 
 8161 
 8163 
 
 8166 
 8168 
 8170 
 
 8173 
 817s 
 8178 
 
 8307 
 831a 
 8312 
 
 8315 
 8317 
 8320 
 
 8322 
 832s 
 8327 
 
 8463 
 
 HI 
 
 8471 
 8473 
 8476 
 8479 
 8482 
 
 8484 
 
 8626 
 8629 
 8631 
 
 8634 
 8637 
 8640 
 
 8643 
 8645 
 8648 
 
 a 
 
 7443 
 
 7546 
 
 7657 
 
 7776 
 
 7903 
 
 8038 
 
 8180 
 
 8330 
 
 8487 
 
 8651 
 
 7445 
 7445 
 7448 
 74SO 
 7451 
 7453 
 
 7455 
 7457 
 7458 
 
 7548 
 7550 
 7551 
 
 7553 
 7555 
 7557 
 
 7559 
 7560 
 7562 
 
 7659 
 7661 
 7663 
 
 ^666 
 766S 
 
 7670 
 7672 
 7674 
 
 7778 
 7780 
 7782 
 
 7784 
 7786 
 7789 
 
 7791 
 7793 
 7795 
 
 7905 
 7907 
 7910 
 
 7912 
 79"4 
 7916 
 
 7918 
 7921 
 7923 
 
 8040 
 8043 
 8045 
 
 8047 
 S049 
 8052 
 
 tit 
 8059 
 
 8182 
 8185 
 8187 
 
 8 [90 
 8192 
 8195 
 8197 
 8200 
 8202 
 
 8333 
 
 till 
 8340 
 8343 
 8346 
 
 8348 
 8351 
 8353 
 
 8490 
 8492 
 8495 
 
 8498 
 8500 
 8503 
 8506 
 8509 
 8511 
 
 8654 
 8657 
 8659 
 8662 
 
 8668 
 
 S671 
 8673 
 8676 
 
 10 
 20 
 30 
 
 40 
 
 lo 
 
 .3 
 •7 
 
 x.o 
 
 1.3 
 
 1-7 
 2.0 
 
 7460 
 
 7564 
 
 7676 
 
 7797 
 
 7925 
 
 8061 
 
 8205 
 
 8356 
 
 8514 
 
 8679 
 
 7462 
 7463 
 7465 
 7466 
 7468 
 7470 
 
 7471 
 7473 
 7474 
 
 7566 
 7568 
 7569 
 
 7571 
 7573 
 7575 
 
 7577 
 7578 
 7580 
 
 7678 
 7680 
 7682 
 
 7684 
 7686 
 7688 
 
 7690 
 7692 
 7694 
 
 7799 
 7801 
 7803 
 
 7805 
 7807 
 7810 
 7812 
 7814 
 7816 
 
 7927 
 7929 
 7932 
 
 7934 
 7936 
 7938 
 7940 
 7943 
 7945 
 
 8063 
 8066 
 8068 
 8071 
 8073 
 8075 
 8078 
 8080 
 8083 
 
 8207 
 8210 
 8212 
 
 8215 
 8217 
 8219 
 
 8222 
 8224 
 8227 
 
 8358 
 8361 
 8363 
 8366 
 8368 
 8371 
 8373 
 
 8378 
 
 8517 
 8519 
 8522 
 
 8525 
 8527 
 8530 
 
 8536 
 8538 
 
 8682 
 868s 
 8687 
 
 869a 
 8693 
 8696 
 
 8699 
 8701 
 8704 
 
 3 
 
 7476 
 
 7582 
 
 7696 
 
 7818 
 
 7947 
 
 8085 
 
 8229 
 
 8381 
 
 8541 
 
 8707 
 
 7478 
 7479 
 7481 
 
 7483 
 7484 
 7486 
 
 7488 
 7490 
 7491 
 
 7584 
 7586 
 7588 
 
 7590 
 7591 
 7593 
 
 7595 
 7597 
 7599 
 
 769S 
 7700 
 7702 
 
 7704 
 7706 
 7708 
 
 7710 
 7712 
 7714 
 
 7820 
 7822 
 7824 
 
 7826 
 7828 
 7831 
 
 7833 
 7835 
 7837 
 
 7949 
 7952 
 7954 
 7956 
 7958 
 7961 
 
 7963 
 7965 
 7968 
 
 8087 
 8090 
 8092 
 
 ^,t 
 8099 
 
 8101 
 8103 
 3io6 
 
 823. 
 8234 
 8236 
 
 8239 
 8241 
 8244 
 
 8246 
 8249 
 8251 
 
 8384 
 8386 
 8389 
 
 8391 
 8394 
 8397 
 
 8399 
 8402 
 8404 
 
 8544 
 8546 
 8549 
 
 8552 
 8554 
 8557 
 8560 
 8563 
 856s 
 
 8710 
 87-3 
 87.S 
 
 8718 
 8721 
 8724 
 
 8727 
 8729 
 8732 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 •5 
 1.0 
 '■5 
 
 2.0 
 2.5 
 3.0 
 
 7493 
 
 7601 
 
 7716 
 
 7839 
 
 7970 
 
 8108 
 
 8254 
 
 8407 
 
 8568 
 
 8735 
 
 7495 
 7497 
 7498 
 
 7500 
 7502 
 7504 
 7506 
 7507 
 7509 
 
 7603 
 7605 
 7606 
 
 7608 
 7610 
 7612 
 
 7614 
 76'5 
 7617 
 
 77 '8 
 7720 
 7722 
 
 7724 
 7726 
 7728 
 
 7730 
 7732 
 7734 
 
 7841 
 7843 
 7845 
 
 7847 
 7849 
 7852 
 
 I! 
 7858 
 
 7972 
 7974 
 7977 
 
 7979 
 7981 
 7983 
 
 7985 
 7988 
 7990 
 
 81 10 
 81 13 
 81 15 
 
 8118 
 81 20 
 
 8l22 
 
 8125 
 
 8.27 
 8130 
 
 8256 
 8259 
 8261 
 
 8264 
 8266 
 8269 
 
 8271 
 8274 
 8276 
 
 8410 
 84.2 
 84-5 
 8418 
 8420 
 8423 
 8425 
 8429 
 8431 
 
 8571 
 8573 
 8576 
 
 8579 
 8581 
 8584 
 
 8587 
 8590 
 8592 
 
 8738 
 8741 
 8743 
 8746 
 8749 
 8752 
 
 8755 
 
 
 75" 
 
 7619 
 
 7736 
 
 7860 
 
 7992 
 
 8132 
 
 8279 
 
 8434 
 
 8595 
 
 8763 
 
 Smiths 
 
 ONIAN 1 
 
 'ables. 
 
 
 
 
 
 
 
 
 
 
 
 S8 
 
LOGARITHMS OF 
 
 Table 1 1 . 
 RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 210 
 
 22? 
 
 23° 
 
 24° 
 
 2S° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 30° 
 
 P. P. 
 
 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.320 
 
 7.321 
 
 7.321 
 
 7.321 
 
 
 
 0' 
 
 X 
 2 
 
 3 
 
 8763 
 
 8939 
 
 9120 
 
 9308 
 
 9502 
 
 9701 
 
 9907 
 
 0117 
 
 0332 
 
 0553 
 
 
 
 8766 
 8769 
 8772 
 
 8942 
 894s 
 8948 
 
 9"3 
 9126 
 9129 
 
 9311 
 9314 
 9318 
 
 9505 
 9508 
 9512 
 
 9705 
 9708 
 9712 
 
 9910 
 99'3 
 9917 
 
 0121 
 0124 
 0128 
 
 0336 
 0340 
 0343 
 
 0556 
 0560 
 0564 
 
 
 4 
 5 
 6 
 
 877s 
 8778 
 8780 
 
 8951 
 8953 
 8956 
 
 9132 
 9136 
 9139 
 
 9321 
 9324 
 9327 
 
 95'S 
 95-8 
 9521 
 
 9715 
 9718 
 9722 
 
 9920 
 9924 
 9927 
 
 0131 
 0135 
 0138 
 
 0347 
 0351 
 0354 
 
 0567 
 0571 
 0575 
 
 
 
 7 
 8 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 8789 
 
 8959 
 8962 
 8965 
 
 9142 
 
 9'45 
 9148 
 
 9330 
 9333 
 9337 
 
 952s 
 9528 
 9531 
 
 9725 
 9728 
 9732 
 
 993" 
 9934 
 9938 
 
 0142 
 0145 
 0149 
 
 0358 
 0361 
 0365 
 
 0579 
 0582 
 0586 
 
 2 
 
 
 S792 
 
 8968 
 
 9151 
 
 9340 
 
 9535 
 
 9735 
 
 9941 
 
 0153 
 
 0369 
 
 0590 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 •3 
 •7 
 l.o 
 1-3 
 
 
 8795 
 8798 
 8800 
 
 8971 
 8974 
 8977 
 
 9'54 
 9157 
 9160 
 
 9343 
 9346 
 9349 
 
 9538 
 9541 
 9545 
 
 9739 
 9742 
 9745 
 
 9945 
 9948 
 9952 
 
 0156 
 0159 
 0163 
 
 0372 
 0376 
 0380 
 
 0594 
 0597 
 0601 
 
 
 14 
 
 ■ 5 
 i6 
 
 8804 
 8807 
 
 88ia 
 
 8980 
 8983 
 8986 
 
 9163 
 9167 
 
 9170 
 
 9353 
 9356 
 9359 
 
 9548 
 9551 
 9554. 
 
 9749 
 9752 
 9756 
 
 9955 
 9959 
 9962 
 
 0167 
 0170 
 0174 
 
 0383 
 0387 
 0391 
 
 0605 
 0608 
 0612 
 
 2.0 
 
 
 
 
 17 
 
 i8 
 '9 
 
 20 
 
 21 
 22 
 23 
 
 88i2 
 88.5 
 8818 
 
 8989 
 8992 
 899s 
 
 9173 
 9176 
 9179 
 
 9362 
 936s 
 9368 
 
 9SS8 
 9561 
 9564 
 
 9759 
 9762 
 9766 
 
 9966 
 9969 
 9973 
 
 0177 
 0181 
 0185 
 
 0394 
 0398 
 0402 
 
 0616 
 0620 
 0623 
 
 
 
 8821 
 
 S998 
 
 9182 
 
 9372 
 
 9568 
 
 9769 
 
 9976 
 
 018S 
 
 0405 
 
 0627 
 
 
 8824 
 8827 
 8830 
 
 9001 
 9004 
 9007 
 
 9191 
 
 9375 
 9378 
 9381 
 
 9571 
 9574 
 9578 
 
 9773 
 9776 
 9779 
 
 9980 
 9983 
 9987 
 
 0192 
 0195 
 0199 
 
 0409 
 0413 
 0416 
 
 0631 
 063s 
 0638 
 
 
 24 
 25 
 26 
 
 11 
 29 
 
 30 
 
 3' 
 32 
 33 
 
 8833 
 8836 
 8839 
 
 8841 
 8844 
 8847 
 
 9010 
 9013 
 9016 
 
 9020 
 9023 
 9026 
 
 9195 
 919S 
 9201 
 
 9204 
 9207 
 9210 
 
 9384 
 9388 
 9391 
 
 9394 
 9398 
 9401 
 
 9581 
 9584 
 9588 
 
 9591 
 9594 
 9598 
 
 9783 
 9786 
 9790 
 
 9793 
 9796 
 9800 
 
 9990 
 9994 
 9997 
 
 *QOOI 
 
 *ooo4 
 *ooo8 
 
 0203 
 0206 
 0210 
 
 0213 
 0217 
 0220 
 
 0420 
 0424 
 0427 
 
 0431 
 0435 
 0438 
 
 0642 
 0646 
 0649 
 
 0653 
 0657 
 o66i 
 
 3 
 
 
 10 
 20 
 30 
 40 
 
 ■5 
 
 I.O 
 
 1-5 
 2.0 
 2-5 
 3.0 
 
 
 8850 
 
 9029 
 
 9213 
 
 9404 
 
 9601 
 
 9803 
 
 *OOTI 
 
 0224 
 
 0442 
 
 0664 
 
 
 8853 
 8856 
 8859 
 
 9032 
 9035 
 9038 
 
 9216 
 9220 
 9223 
 
 9407 
 9411 
 94>4 
 
 9604 
 9608 
 9611 
 
 9807 
 9810 
 9814 
 
 *aoi5 
 *ooi8 
 
 *0022 
 
 0228 
 0231 
 023s 
 
 0446 
 0449 
 04S3 
 
 0668 
 0672 
 0676 
 
 
 
 
 34 
 35 
 36 
 
 8862 
 8865 
 8868 
 
 9041 
 9044 
 9047 
 
 9226 
 9229 
 9232 
 
 9417 
 9420 
 9424 
 
 9614 
 9618 
 9621 
 
 9817 
 9820 
 9824 
 
 *0025 
 *0029 
 *0032 
 
 0238 
 0242 
 0246 
 
 0457 
 0460 
 0464 
 
 0679 
 0683 
 0687 
 
 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 11 
 
 8871 
 8874 
 8877 
 
 9050 
 9053 
 9056 
 
 923s 
 9238 
 9242 
 
 9427 
 9430 
 9433 
 
 9624 
 9628 
 9631 
 
 9827 
 9831 
 9834 
 
 •0036 
 
 *oo39 
 *oo43 
 
 0249 
 0253 
 0256 
 
 0468 
 0471 
 0475 
 
 0691 
 0694 
 0698 
 
 4 
 
 
 8879 
 
 9059 
 
 924s 
 
 9437 
 
 9634 
 
 9838 
 
 *oo46 
 
 0260 
 
 0479 
 
 0702 
 
 
 8882 
 8885 
 8888 
 S891 
 
 ?!'♦ 
 8S97 
 
 9062 
 906s 
 9068 
 
 9071 
 9074 
 9077 
 
 9248 
 9251 
 9254 
 
 9257 
 9260 
 9264 
 
 9440 
 9443 
 9446 
 
 945° 
 9453 
 9456 
 
 963S 
 9641 
 9644 
 
 9648 
 9651 
 9654 
 
 9841 
 9844 
 9848 
 
 9851 
 9855 
 9858 
 
 *oo5o 
 •0053 
 *oo57 
 
 *oo6o 
 *oo64 
 *oo67 
 
 0264 
 0267 
 0271 
 
 0274 
 0278 
 0282 
 
 0482 
 0486 
 0490 
 
 0493 
 0497 
 0501 
 
 0706 
 0710 
 0713 
 
 0717 
 0721 
 0725 
 
 
 10 
 20 
 
 •7 
 1-3 
 
 
 47 
 48 
 49 
 
 60 
 
 51 
 52 
 53 
 
 8900 
 8903 
 8906 
 
 908a 
 9083 
 9086 
 
 9267 
 9270 
 9273 
 
 9459 
 9466 
 
 9658 
 9661 
 9664 
 
 9862 
 9865 
 9869 
 
 *oo7l 
 *oo74 
 ♦0078 
 
 0285 
 0289 
 0293 
 
 0505 
 050S 
 0512 
 
 0728 
 0732 
 0736 
 
 40 
 5° 
 60 
 
 2-7 
 3-3 
 4.0 
 
 
 8909 
 
 9089 
 
 9276 
 
 9469 
 
 9668 
 
 9872 
 
 *co82 
 
 0296 
 
 0516 
 
 0740 
 
 
 
 
 8912 
 S915 
 8918 
 
 9093 
 9096 
 9099 
 
 9279 
 9283 
 9286 
 
 9472 
 9476 
 9479 
 
 9671 
 9674 
 9678 
 
 9875 
 9879 
 9882 
 
 *oo85 
 *oo39 
 ♦0092 
 
 0300 
 0303 
 0307 
 
 05.9 
 0523 
 0527 
 
 0743 
 0747 
 0751 
 
 
 54 
 55 
 56 
 
 8921 
 8924 
 8927 
 
 9102 
 910S 
 9108 
 
 9289 
 9292 
 9295 
 
 9482 
 9485 
 9489 
 
 9681 
 9685 
 9688 
 
 9886 
 9889 
 9893 
 
 •0096 
 *oo99 
 *oi03 
 
 0311 
 0314 
 0318 
 
 0530 
 0534 
 0538 
 
 0755 
 0759 
 0762 
 
 
 
 57 
 58 
 59 
 
 60 
 
 8930 
 8933 
 8936 
 
 9111 
 91 14 
 9117 
 
 9298 
 9302 
 9305 
 
 9492 
 9495 
 9498 
 
 9691 
 9695 
 969S 
 
 9896 
 9900 
 9903 
 
 *oro6 
 
 *OIIO 
 
 *oii3 
 
 0322 
 0325 
 0329 
 
 0542 
 054s 
 0549 
 
 0766 
 0770 
 0774 
 
 
 
 8939 
 
 9120 
 
 9308 
 
 9502 
 
 9701 
 
 9907 
 
 *oii7 
 
 0332 
 
 0553 
 
 0777 
 
 
 
 
 Smithsonian Tables. 
 
 59 
 
Table 1 1 . 
 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 31° 
 
 32° 
 
 33° 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 40° 
 
 P.P. 
 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 0777 
 
 1006 
 
 1239 
 
 1476 
 
 1716 
 
 1959 
 
 2205 
 
 2453 
 
 2704 
 
 2956 
 
 3 
 
 0781 
 0785 
 0789 
 
 1010 
 1014 
 I0i8 
 
 1243 
 1247 
 1251 
 
 1480 
 1484 
 1488 
 
 1720 
 1724 
 1728 
 
 1963 
 1967 
 1971 
 
 2209 
 2213 
 2217 
 
 2457 
 2462 
 2466 
 
 2708 
 2712 
 2716 
 
 2961 
 2965 
 2969 
 
 4 
 1 
 
 0793 
 0796 
 0800 
 
 1022 
 1026 
 1029 
 
 1255 
 I2S9 
 1263 
 
 1492 
 1496 
 1500 
 
 1732 
 1736 
 1740 
 
 1975 
 1979 
 1983 
 
 2221 
 2226 
 2230 
 
 2470 
 2474 
 2478 
 
 2721 
 2725 
 2729 
 
 2973 
 2978 
 2982 
 
 
 10 
 
 •5 
 
 I 
 9 
 
 10 
 
 iz 
 
 12 
 
 13 
 
 0804 
 0808 
 081 1 
 
 1033 
 1037 
 IO41 
 
 1267 
 1271 
 1275 
 
 1504 
 1S08 
 1512 
 
 1744 
 1748 
 1752 
 
 .988 
 1992 
 1996 
 
 2234 
 223S 
 2242 
 
 2482 
 2487 
 2491 
 
 2733 
 2737 
 2742 
 
 2986 
 2990 
 2994 
 
 20 
 30 
 40 
 so 
 60 
 
 1.0 
 i-S 
 2.0 
 25 
 3.0 
 
 081S 
 
 I04S 
 
 1279 
 
 1516 
 
 1756 
 
 2000 
 
 2246 
 
 2495 
 
 2746 
 
 2999 
 
 0S19 
 0823 
 0827 
 
 1049 
 IOS3 
 IOS7 
 
 1282 
 1286 
 1290 
 
 1520 
 
 1760 
 1764 
 1768 
 
 2004 
 2008 
 2012 
 
 2250 
 2254 
 
 2259 
 
 2499 
 2503 
 2507 
 
 2750 
 2754 
 2758 
 
 3003 
 3007 
 3011 
 
 
 >4 
 IS 
 l6 
 
 0830 
 
 fslt 
 
 1060 
 1064 
 1068 
 
 1294 
 1298 
 1302 
 
 1532 
 1536 
 iS-P 
 
 1772 
 1776 
 1780 
 
 2016 
 2020 
 2024 
 
 2263 
 2267 
 
 2271 
 
 2512 
 2516 
 2520 
 
 2763 
 2767 
 2771 
 
 3016 
 3020 
 3024 
 
 
 11 
 '9 
 
 20 
 
 21 
 22 
 23 
 
 0842 
 0846 
 0849 
 
 1072 
 1076 
 1080 
 
 .306 
 1310 
 1314 
 
 1 544 
 1548 
 1552 
 
 1784 
 1789 
 1793 
 
 2028 
 2033 
 2037 
 
 2275 
 2279 
 2283 
 
 2524 
 2528 
 2532 
 
 2775 
 2779 
 2784 
 
 3028 
 3032 
 3037 
 
 4 
 
 o8s3 
 
 1084 
 
 13 iS 
 
 1556 
 
 1797 
 
 2041 
 
 2287 
 
 2537 
 
 2788 
 
 3041 
 
 0857 
 0861 
 0865 
 
 1087 
 1091 
 1095 
 
 1322 
 1326 
 1330 
 
 1560 
 1564 
 1368 
 
 1801 
 1805 
 1809 
 
 2045 
 2049 
 
 2053 
 
 2292 
 2296 
 2300 
 
 2541 
 2545 
 2549 
 
 2792 
 2796 
 2800 
 
 3045 
 3049 
 3054 
 
 24 
 
 0869 
 0872 
 0876 
 
 1099 
 U03 
 1 107 
 
 1334 
 1337 
 1341 
 
 1572 
 1576 
 1580 
 
 1813 
 1817 
 1821 
 
 2057 
 2061 
 2065 
 
 2304 
 2308 
 2312 
 
 2553 
 2557 
 2562 
 
 2805 
 2809 
 28.3 
 
 3058 
 3062 
 3066 
 
 
 10 
 
 •7 
 
 27 
 
 28 
 
 39 
 
 30 
 
 31 
 32 
 33 
 
 0880 
 0884 
 0888 
 
 nil 
 1115 
 iiiS 
 
 1345 
 1349 
 1353 
 
 1584 
 1588 
 1592 
 
 1825 
 1829 
 1833 
 
 2069 
 2073 
 2077 
 
 2316 
 2321 
 2325 
 
 2566 
 2570 
 2574 
 
 2817 
 2822 
 2826 
 
 3071 
 3075 
 3079 
 
 20 
 30 
 40 
 50 
 60 
 
 '•3 
 2.0 
 2-7 
 3-3 
 4.0 
 
 0891 
 
 1 122 
 
 1357 
 
 ■596 
 
 1837 
 
 2082 
 
 2329 
 
 2578 
 
 2830 
 
 30S3 
 
 089s 
 0899 
 0903 
 
 X126 
 1130 
 1134 
 
 1361 
 1365 
 1369 
 
 1600 
 1604 
 1608 
 
 1841 
 1845 
 1849 
 
 2086 
 2090 
 2094 
 
 2333 
 2337 
 2341 
 
 2583 
 2587 
 2591 
 
 2834 
 2838 
 2843 
 
 3087 
 3092 
 3096 
 
 
 34 
 35 
 
 36 
 
 0907 
 0910 
 0914 
 
 1 138 
 1142 
 1146 
 
 1373 
 1377 
 1381 
 
 1612 
 1616 
 1620 
 
 1853 
 1857 
 1861 
 
 209S 
 2102 
 2106 
 
 234s 
 2350 
 
 2354 
 
 2595 
 2599 
 2603 
 
 2847 
 2851 
 
 2855 
 
 3100 
 3104 
 3109 
 
 
 39 
 40 
 
 4« 
 
 42 
 43 
 
 0918 
 0922 
 0926 
 
 1 150 
 "53 
 "57 
 
 1385 
 1389 
 1393 
 
 1624 
 1628 
 1632 
 
 1865 
 1870 
 1874 
 
 2110 
 2114 
 2119 
 
 2358 
 2362 
 2366 
 
 2608 
 2612 
 2616 
 
 2859 
 2864 
 2868 
 
 3113 
 3117 
 3121 
 
 6 
 
 0930 
 
 1 161 
 
 1397 
 
 1636 
 
 1878 
 
 2123 
 
 2370 
 
 2620 
 
 2872 
 
 3126 
 
 °933 
 0937 
 0941 
 
 1 165 
 1169 
 1 173 
 
 1401 
 1405 
 1409 
 
 1640 
 
 1882 
 1886 
 1890 
 
 2127 
 213 1 
 2135 
 
 2374 
 2379 
 2383 
 
 2624 
 2629 
 2633 
 
 2876 
 2880 
 2885 
 
 3130 
 3138 
 
 44 
 4S 
 46 
 
 % 
 49 
 
 60 
 
 s> 
 
 52 
 S3 
 
 54 
 
 5S 
 S6 
 
 0945 
 0949 
 0953 
 
 0956 
 0960 
 0964 
 
 M77 
 1181 
 1185 
 
 1189 
 119Z 
 1196 
 
 1412 
 1416 
 X420 
 
 1428 
 1432 
 
 1652 
 1656 
 1660 
 
 1664 
 1668 
 1672 
 
 1894 
 1898 
 1902 
 
 1906 
 1910 
 1914 
 
 2139 
 2143 
 2147 
 
 2151 
 2156 
 2160 
 
 2387 
 2391 
 2395 
 
 2399 
 2403 
 2408 
 
 2637 
 2641 
 264s 
 2649 
 
 lilt 
 
 2889 
 2893 
 2897 
 
 2902 
 2906 
 2910 
 
 3143 
 3147 
 3151 
 
 3 '55 
 3160 
 3164 
 
 
 10 
 20 
 
 30 
 40 
 
 .8 
 '•7 
 2-5 
 3<3 
 4.2 
 5.0 
 
 0968 
 
 1200 
 
 1436 
 
 1676 
 
 1918 
 
 2164 
 
 2412 
 
 2662 
 
 2914 
 
 3 '68 
 
 0972 
 0976 
 0979 
 
 0983 
 0987 
 0991 
 
 1204 
 1208 
 1212 
 
 12 16 
 1220 
 1224 
 
 1440 
 1444 
 1448 
 
 1452 
 1436 
 1460 
 
 1680 
 1684 
 1688 
 
 1692 
 1696 
 1700 
 
 1922 
 1926 
 1931 
 
 1935 
 1939 
 1943 
 
 2168 
 2172 
 2176 
 
 218a 
 2184 
 2i88 
 
 2416 
 2420 
 2424 
 
 2428 
 2433 
 2437 
 
 2666 
 2670 
 267s 
 
 2687 
 
 2918 
 2923 
 2927 
 
 2931 
 2935 
 2940 
 
 3 '72 
 3181 
 
 3185 
 3189 
 3 '93 
 
 
 57 
 58 
 59 
 
 60 
 
 0995 
 0999 
 1003 
 
 1228 
 1231 
 I23S 
 
 1464 
 1468 
 '472 
 
 1704 
 1708 
 1712 
 
 1947 
 1951 
 1955 
 
 2193 
 2197 
 2201 
 
 2441 
 2445 
 2449 
 
 2691 
 2696 
 2700 
 
 2944 
 2948 
 2952 
 
 3198 
 3202 
 3206 
 
 
 iao6 
 
 1239 
 
 1476 
 
 1716 
 
 1959 
 
 2205 
 
 2453 
 
 2704 
 
 2956 
 
 3210 
 
 Smiths 
 
 ONIAN 7 
 
 ABLES. 
 
 
 
 
 
 
 
 
 
 
 
 60 
 
Table 1 1 . 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 45° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 So° 
 
 P.P. 
 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 3210 
 
 3466 
 
 3722 
 
 3979 
 
 4236 
 
 4494 
 
 4751 
 
 5007 
 
 5263 
 
 5517 
 
 
 3215 
 3219 
 3223 
 
 3470 
 3474 
 3479 
 
 3726 
 3731 
 3735 
 
 3988 
 3992 
 
 4241 
 4245 
 4249 
 
 4498 
 4502 
 4507 
 
 4760 
 4764 
 
 5012 
 5016 
 5020 
 
 5267 
 5271 
 5276 
 
 5522 
 
 5526 
 5530 
 
 4 
 I 
 
 3227 
 3232 
 3236 
 
 3483 
 3487 
 3491 
 
 3739 
 3744 
 3748 
 
 3996 
 4001 
 4005 
 
 4254 
 4258 
 4262 
 
 45" 
 4515 
 4520 
 
 4768 
 4772 
 4777 
 
 5024 
 5029 
 5033 
 
 5280 
 5284 
 5288 
 
 5534 
 5538 
 5543 
 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 
 "3 
 
 3240 
 3244 
 3249 
 
 3496 
 3500 
 3504 
 
 3752 
 3756 
 3761 
 
 4Q09 
 4013 
 4018 
 
 4267 
 4271 
 4275 
 
 4524 
 4528 
 4532 
 
 4781 
 4785 
 4789 
 
 5037 
 5042 
 5046 
 
 5293 
 5297 
 5301 
 
 5547 
 5551 
 5555 
 
 4 
 
 3253 
 
 3508 
 
 3765 
 
 4022 
 
 4279 
 
 4537 
 
 4794 
 
 5050 
 
 5305 
 
 5560 
 
 3257 
 3261 
 3266 
 
 3513 
 3517 
 3521 
 
 3769 
 3774 
 3778 
 
 4026 
 4031 
 4035 
 
 4284 
 4288 
 4292 
 
 4541 
 4545 
 4550 
 
 4798 
 4802 
 4807 
 
 5054 
 5059 
 5063 
 
 5310 
 5314 
 53'8 
 
 5568 
 5572 
 
 14 
 
 3270 
 3274 
 3278 
 
 3526 
 3530 
 3534 
 
 3782 
 3786 
 3791 
 
 4039 
 4043 
 4048 
 
 4297 
 4301 
 4305 
 
 4554 
 4558 
 4562 
 
 4S11 
 481S 
 4819 
 
 5067 
 S071 
 5076 
 
 5322 
 5327 
 5331 
 
 5576 
 5581 
 5585 
 
 
 10 
 
 •7 
 
 19 
 20 
 
 21 
 22 
 23 
 
 32S3 
 3287 
 3291 
 
 3538 
 3543 
 3547 
 
 3795 
 3799 
 3803 
 
 4052 
 4056 
 4061 
 
 4309 
 4314 
 4318 
 
 4567 
 4571 
 4575 
 
 4824 
 4828 
 4832 
 
 5080 
 S084 
 5088 
 
 5335 
 5339 
 5344 
 
 5589 
 5593 
 5598 
 
 30 
 40 
 
 50 
 60 
 
 2.0 
 2.7 
 3-3 
 4.0 
 
 3295 
 
 3S5I 
 
 3808 
 
 4065 
 
 4322 
 
 4580 
 
 4837 
 
 5093 
 
 5348 
 
 5602 
 
 3300 
 3304 
 3308 
 
 3555 
 3560 
 3564 
 
 3812 
 3816 
 3821 
 
 4069 
 4°73 
 4078 
 
 4327 
 4331 
 4335 
 
 4584 
 4588 
 4592 
 
 4841 
 4845 
 4S49 
 
 5097 
 5101 
 
 S«o5 
 
 5352 
 5356 
 5361 
 
 5606 
 5610 
 5614 
 
 
 24 
 
 11 
 
 3312 
 3317 
 3321 
 
 3568 
 3573 
 3577 
 
 3825 
 3829 
 3833 
 
 4082 
 4086 
 4091 
 
 4339 
 4344 
 434S 
 
 4597 
 4601 
 4605 
 
 4854 
 
 5 no 
 
 5II4 
 SII8 
 
 5365 
 5369 
 5373 
 
 5619 
 5623 
 5627 
 
 
 11 
 
 29 
 
 30 
 
 31 
 32 
 33 
 
 3325 
 3329 
 3334 
 
 3581 
 3585 
 3590 
 
 3S38 
 3842 
 3846 
 
 409s 
 4099 
 4104 
 
 4352 
 4357 
 4361 
 
 4610 
 4614 
 4618 
 
 4866 
 4871 
 4875 
 
 5123 
 5127 
 5I3I 
 
 5378 
 5382 
 5386 
 
 5631 
 5636 
 5640 
 
 
 3338 
 
 3594 
 
 3851 
 
 4108 
 
 4365 
 
 4622 
 
 4879 
 
 5135 
 
 5390 
 
 5644 
 
 3342 
 3347 
 335' 
 
 3607 
 
 3855 
 3859 
 3863 
 
 4112 
 41 16 
 4121 
 
 4369 
 4374 
 4378 
 
 4627 
 4631 
 4635 
 
 4884 
 4888 
 4892 
 
 SI40 
 
 5-44 
 5148 
 
 5395 
 5399 
 5403 
 
 5648 
 5652 
 5657 
 
 34 
 
 - 33 5S 
 33 S9 
 3364 
 
 361 1 
 3615 
 3620 
 
 3868 
 3876 
 
 412s 
 4129 
 4134 
 
 4382 
 4387 
 4391 
 
 4640 
 4644 
 -4648 
 
 4896 
 4901 
 4905 
 
 5152 
 5157 
 5 161 
 
 5407 
 5412 
 5416 
 
 5661 
 5665 
 5669 
 
 
 39 
 40 
 
 4' 
 42 
 43 
 
 336S 
 3372 
 3376 
 
 3624 
 3628 
 3632 
 
 3881 
 3SS5 
 3889 
 
 4138 
 4142 
 4146 
 
 4395 
 4399 
 4404 
 
 4652 
 4661 
 
 4909 
 4913 
 491S 
 
 5165 
 5 169 
 5174 
 
 5420 
 5424 
 5428 
 
 5673 
 5678 
 5682 
 
 6 
 
 10 
 20 
 30 
 40 
 
 1^ 
 
 .8 
 1-7 
 2-5 
 3-3 
 4.2 
 
 3381 
 
 3637 
 
 3893 
 
 4151 
 
 4408 
 
 4665 
 
 4922 
 
 5178 
 
 5433 
 
 5686 
 
 3385 
 3389 
 3393 
 
 3641 
 3645 
 3649 
 
 3898 
 
 ■ 3902 
 
 3906 
 
 4155 
 4159 
 4164 
 
 4412 
 4417 
 442 1 
 
 4670 
 4674 
 467S 
 
 4926 
 4931 
 4935 
 
 5182 
 5186 
 5191 
 
 5437 
 5441 
 5445 
 
 5690 
 5694 
 5699 
 
 44 
 45 
 46 
 
 3398 
 3402 
 3406 
 
 3658 
 3662 
 
 39" 
 391S 
 3919 
 
 4168 
 4172 
 4176 
 
 4425 
 4430 
 4434 
 
 4682 
 4687 
 4691 
 
 4939 
 4943 
 4948 
 
 5195 
 5 '99 
 5203 
 
 5450 
 5454 
 5458 
 
 5703 
 5707 
 57" 
 
 
 
 
 47 
 
 48 
 49 
 
 60 
 
 51 
 52 
 53 
 
 3410 
 3415 
 3419 
 
 3667 
 3671 
 3675 
 
 3923 
 3928 
 3932 
 
 4181 
 4>85 
 4189 
 
 4438 
 4442 
 4447 
 
 4695 
 4700 
 4704 
 
 4952 
 4956 
 4960 
 
 5208 
 52,2 
 5216 
 
 5462 
 5467 
 5471 
 
 5716 
 5720 
 5724 
 
 
 3423 
 
 3679 
 
 3936 
 
 4194 
 
 4451 
 
 4708 
 
 4965 
 
 5220 
 
 5475 
 
 5728 
 
 3427 
 3432 
 3436 
 
 3684 
 368S 
 3692 
 
 3941 
 3945 
 3949 
 
 419S 
 4202 
 4206 
 
 4455 
 4460 
 4464 
 
 4712 
 4717 
 4721 
 
 4969 
 4973 
 4978 
 
 5225 
 5229 
 5233 
 
 5479 
 5488 
 
 5732 
 5737 
 5741 
 
 54 
 11 
 
 3440 
 3445 
 3449 
 
 3697 
 3701 
 
 37°S 
 
 3953 
 3958 
 3962 
 
 42II 
 
 421S 
 4219 
 
 4468 
 4472 
 4477 
 
 472s 
 4730 
 4734 
 
 4982 
 4986 
 4990 
 
 5237 
 5242 
 5246 
 
 5492 
 5496 
 5500 
 
 5745 
 5749 
 5753 
 
 
 S7 
 58 
 59 
 
 60 
 
 3453 
 3457 
 3462 
 
 3709 
 3714 
 3718 
 
 3966 
 3971 
 3975 
 
 4224 
 422S 
 4232 
 
 4481 
 448s 
 449° 
 
 4738 
 4742 
 4747 
 
 4995 
 4999 
 
 5o°3 
 
 5250 
 5254 
 5259 
 
 5505 
 5509 
 5513 
 
 5758 
 5762 
 5766 
 
 
 3466 
 
 3722 
 
 3979 
 
 4236 
 
 4494 
 
 4751 
 
 5007 
 
 5263 
 
 SS17 
 
 5770 
 
 
 
 Smithsonian Tables. 
 
 61 
 
Table 1 1 . 
 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 
 Pn 
 
 IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 Si° 
 
 52° 
 
 53° 
 
 54° 
 
 55° 
 
 56° 
 
 57" 
 
 58° 
 
 59° 
 
 60° 
 
 P.P. 
 
 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 7.321 
 
 
 
 0' 
 
 X 
 2 
 
 3 
 
 S770 
 
 6021 
 
 6270 
 
 6517 
 
 6760 
 
 7001 
 
 7238 
 
 7472 
 
 7701 
 
 7927 
 
 
 
 5774 
 577S 
 5783 
 
 6025 
 6029 
 6034 
 
 6274 
 6278 
 6282 
 
 6521 
 6525 
 6529 
 
 6764 
 6768 
 6772 
 
 7005 
 7009 
 7013 
 
 7242 
 7246 
 7250 
 
 7476 
 7480 
 7483 
 
 770s 
 7709 
 7712 
 
 7931 
 7934 
 7938 
 
 
 4 
 I 
 
 5787 
 5791 
 5795 
 
 6038 
 6042 
 6046 
 
 6286 
 6290 
 6295 
 
 6533 
 6537 
 6541 
 
 6775 
 6780 
 6785 
 
 7017 
 7021 
 7025 
 
 7254 
 7261 
 
 7487 
 7491 
 7495 
 
 7716 
 7720 
 7724 
 
 7942 
 7945 
 7949 
 
 
 
 
 .8 
 1-7 
 
 2-5 
 
 3-3 
 4.2 
 
 5-0 
 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 
 '3 
 
 5799 
 5804 
 5808 
 
 6050 
 6055 
 6059 
 
 6299 
 6303 
 6307 
 
 654s 
 6549 
 6553 
 
 6789 
 6793 
 6797 
 
 7029 
 7033 
 7037 
 
 726s 
 7269 
 7273 
 
 7499 
 7502 
 7506 
 
 7728 
 7735 
 
 7953 
 7957 
 7960 
 
 20 
 30 
 40 
 
 IS 
 
 
 5812 
 
 6063 
 
 6311 
 
 6557 
 
 6801 
 
 7041 
 
 7277 
 
 75 'o 
 
 7739 
 
 7964 
 
 
 5816 
 5820 
 5825 
 
 6067 
 6071 
 6075 
 
 6315 
 6319 
 6324 
 
 6561 
 6569 
 
 6805 
 6809 
 6813 
 
 7045 
 7049 
 7053 
 
 7281 
 728s 
 7289 
 
 75U 
 7518 
 7522 
 
 7743 
 7747 
 77SO 
 
 7968 
 7971 
 7975 
 
 
 
 
 '4 
 
 16 
 
 5829 
 5833 
 5837 
 
 6079 
 6083 
 6088 
 
 6328 
 6332 
 6336 
 
 6573 
 6577 
 6582 
 
 6817 
 6821 
 6825 
 
 7057 
 7060 
 7064 
 
 7293 
 7296 
 7300 
 
 7526 
 7529 
 7533 
 
 7754 
 7758 
 7762 
 
 7979 
 7982 
 7986 
 
 
 
 "7 
 18 
 •9 
 
 20 
 
 21 
 
 22 
 23 
 
 5841 
 5846 
 5850 
 
 6092 
 6096 
 6100 
 
 6340 
 6345 
 6349 
 
 6586 
 6590 
 6594 
 
 6829 
 6837 
 
 7068 
 7072 
 7076 
 
 73°4 
 73°8 
 7312 
 
 7537 
 7S4I 
 7545 
 
 7766 
 7769 
 7773 
 
 7990 
 7994 
 7997 
 
 
 
 5854 
 
 6104 
 
 6353 
 
 6598 
 
 6841 
 
 7080 
 
 7316 
 
 7549 
 
 7777 
 
 8001 
 
 
 5858 
 5862 
 5867 
 
 6108 
 61 12 
 6117 
 
 6357 
 6361 
 6365 
 
 6602 
 6606 
 6610 
 
 6845 
 
 7092 
 
 7320 
 7324 
 7328 
 
 7552 
 7557 
 7560 
 
 7781 
 
 8005 
 8008 
 8012 
 
 
 24 
 
 11 
 
 5871 
 5875 
 5879 
 
 6121 
 6125 
 6129 
 
 6369 
 6373 
 6378 
 
 6614 
 6618 
 6623 
 
 SI? 
 686s 
 
 7096 
 
 7100 
 
 7104 
 
 7332 
 7335 
 7339 
 
 7564 
 7568 
 7572 
 
 7792 
 7796 
 7800 
 
 8016 
 8019 
 8023 
 
 4 
 
 
 
 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 5883 
 5888 
 5892 
 
 6133 
 6138 
 6142 
 
 6382 
 6386 
 6390 
 
 6627 
 6631 
 6635 
 
 6869 
 
 6873 
 6877 
 
 7108 
 7II2 
 7II6 
 
 7343 
 7347 
 7351 
 
 7576 
 7579 
 7583 
 
 7804 
 7807 
 781 1 
 
 8027 
 8031 
 8034 
 
 10 
 20 
 30 
 40 
 
 lo 
 
 •7 
 1-3 
 2.0 
 2-7 
 3-3 
 4.0 
 
 
 5896 
 
 6146 
 
 6394 
 
 6639 
 
 6881 
 
 7120 
 
 7355 
 
 7587 
 
 7815 
 
 8038 
 
 
 S900 
 5904 
 5909 
 
 6150 
 6154 
 6158 
 
 6398 
 6402 
 6406 
 
 6647 
 6651 
 
 6885 
 6889 
 6893 
 
 7124 
 7128 
 7132 
 
 7359 
 7363 
 7367 
 
 7591 
 7595 
 7598 
 
 7819 
 7822 
 7826 
 
 8042 
 8045 
 8049 
 
 
 
 
 34 
 35 
 36 
 
 5913 
 5917 
 5921 
 
 6162 
 6i66 
 6171 
 
 6410 
 6414 
 6419 
 
 6655 
 6659 
 6663 
 
 6897 
 6901 
 6905 
 
 7136 
 7139 
 7143 
 
 7371 
 7374 
 7378 
 
 7602 
 7605 
 7610 
 
 7830 
 7833 
 7837 
 
 8053 
 8056 
 8060 
 
 
 
 11 
 39 
 
 40 
 
 41 
 42 
 43 
 
 5925 
 5930 
 5934 
 
 6175 
 6179 
 6183 
 
 6423 
 6427 
 6431 
 
 6667 
 6671 
 6675 
 
 6909 
 
 6913 
 6917 
 
 7147 
 7I5I 
 
 7155 
 
 7386 
 7390 
 
 7614 
 7617 
 7621 
 
 7841 
 7845 
 7848 
 
 8064 
 8068 
 8071 
 
 3 
 
 
 5938 
 
 6187 
 
 6435 
 
 6679 
 
 6921 
 
 7159 
 
 7394 
 
 7625 
 
 7852 
 
 8075 
 
 
 5942 
 5946 
 
 5951 
 
 6191 
 6195 
 6200 
 
 6439 
 6443 
 6447 
 
 6683 
 6687 
 6691 
 
 6925 
 6929 
 6933 
 
 7163 
 7167 
 7I7I 
 
 7398 
 7402 
 7406 
 
 7629 
 7633 
 7635 
 
 7856 
 7860 
 7863 
 
 8079 
 8082 
 8086 
 
 
 44 
 
 5955 
 5959 
 5963 
 
 6204 
 6208 
 6212 
 
 6451 
 645s 
 6460 
 
 669s 
 6699 
 6704 
 
 6937 
 6941 
 6945 
 
 7175 
 7179 
 7183 
 
 7410 
 7413 
 7417 
 
 7640 
 
 7867 
 7871 
 7875 
 
 8089 
 8093 
 8097 
 
 
 
 .5 
 
 I.O 
 
 '•5 
 2.0 
 
 2-5 
 
 3.0 
 
 
 % 
 49 
 
 60 
 
 51 
 52 
 53 
 
 5967 
 5972 
 5976 
 
 6216 
 6221 
 6225 
 
 6464 
 6468 
 6472 
 
 6708 
 6712 
 6716 
 
 6949 
 6953 
 6957 
 
 7187 
 7I9I 
 
 7'9S 
 
 7421 
 7425 
 7429 
 
 7652 
 7655 
 7659 
 
 7882 
 7886 
 
 8100 
 8104 
 8107 
 
 20 
 40 
 
 
 5980 
 
 6229 
 
 6476 
 
 6720 
 
 6961 
 
 7199 
 
 7433 
 
 7663 
 
 7890 
 
 8111 
 
 
 5992 
 
 6233 
 6237 
 6241 
 
 6480 
 6484 
 6488 
 
 6724 
 6728 
 6732 
 
 6965 
 6969 
 6973 
 
 7203 
 7207 
 7211 
 
 7437 
 744' 
 7445 
 
 7667 
 7671 
 7674 
 
 7894 
 7897 
 7901 
 
 8115 
 8118 
 8122 
 
 
 
 
 54 
 
 5996 
 6000 
 6005 
 
 6245 
 6249 
 6254 
 
 6492 
 6496 
 6501 
 
 6736 
 6740 
 6744 
 
 6977 
 6981 
 6985 
 
 7215 
 7218 
 7222 
 
 7449 
 7452 
 7456 
 
 7678 
 7682 
 7686 
 
 7905 
 7908 
 7912 
 
 8126 
 8129 
 8133 
 
 
 
 59 
 60 
 
 6009 
 6013 
 6017 
 
 6258 
 6262 
 6266 
 
 6505 
 6509 
 65 '3 
 
 6748 
 6752 
 6756 
 
 6989 
 6993 
 6997 
 
 7226 
 7230 
 7234 
 
 7460 
 7464 
 7468 
 
 7690 
 7693 
 
 7916 
 7920 
 7923 
 
 8.37 
 8141 
 8.44 
 
 * 
 
 
 6021 
 
 6270 
 
 6517 
 
 6760 
 
 7001 
 
 7238 
 
 7472 
 
 7701 
 
 7927 
 
 8148 
 
 
 Smiths 
 
 ONIAN 1 
 
 'ables. 
 
 
 
 
 
 
 
 
 
 
 
 
 62 
 
Table 1 1 . 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 61° 
 
 62° 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 68° 
 
 69° 
 
 70° 
 
 P.P. 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 4 
 5 
 6 
 
 7 
 S 
 
 9 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 
 20 
 
 21 
 22 
 23 
 
 24 
 
 :i 
 
 27 
 28 
 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 
 11 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 44 
 45 
 46 
 
 47 
 48 
 49 
 
 50 
 
 SI 
 52 
 53 
 54 
 
 11 
 
 % 
 59 
 
 60 
 
 7.321 
 
 8148 
 
 7.321 
 
 8364 
 
 7.321 
 
 8575 
 
 7.321 
 
 8781 
 
 7.321 
 
 8982 
 
 7.321 
 
 9175 
 
 7.321 
 
 9365 
 
 7.321 
 
 9548 
 
 7.321 
 
 9724 
 
 7.321 
 
 9893 
 
 4 
 
 
 8x52 
 8155 
 8159 
 
 8162 
 8166 
 8170 
 
 8173 
 8177 
 8180 
 
 8368 
 8371 
 837s 
 
 8378 
 8382 
 8386 
 
 8389 
 8393 
 839S 
 
 It 
 8585 
 
 8589 
 8606 
 
 8784 
 8788 
 8791 
 
 879s 
 8798 
 8801 
 
 880s 
 8808 
 8812 
 
 S'f5 
 8989 
 8992 
 
 8995 
 8998 
 9002 
 
 9008 
 9012 
 
 9179 
 9182 
 9186 
 
 9189 
 9192 
 9195 
 9198 
 9202 
 9205 
 
 9368 
 9371 
 9374 
 
 9377 
 9380 
 9384 
 
 9387 
 9390 
 9393 
 
 9551 
 9554 
 9557 
 9560 
 9562 
 9565 
 9568 
 9571 
 9574 
 
 9727 
 9730 
 9732 
 
 9735 
 9738 
 9741 
 
 9744 
 9746 
 9749 
 
 9896 
 9898 
 9901 
 
 9904 
 9906 
 9909 
 
 9912 
 9915 
 9917 
 
 
 8184 
 
 8400 
 
 8610 
 
 8815 
 
 9015 
 
 9208 
 
 9396 
 
 9577 
 
 9752 
 
 9920 
 
 10 
 20 
 30 
 40 
 
 io 
 
 •7 
 1.3 
 
 2.0 
 2.7 
 3.3 
 4.0 
 
 
 8188 
 8191 
 819s 
 8198 
 8202 
 8206 
 
 8209 
 8213 
 8215 
 
 8403 
 8407 
 8410 
 
 84.4 
 8417 
 8421 
 
 8428 
 8431 
 
 8613 
 8617 
 8620 
 
 8624 
 8627 
 8631 
 
 8534 
 8638 
 8641 
 
 8818 
 8822 
 882s 
 
 8829 
 8832 
 8835 
 
 8839 
 8842 
 8845 
 
 9018 
 go2i 
 9025 
 
 9028 
 9°3i 
 9034 
 
 9°37 
 9041 
 9044 
 
 9211 
 9214 
 9218 
 
 9221 
 9224 
 9227 
 
 9230 
 9234 
 9237 
 
 9399 
 9402 
 9405 
 
 9408 
 941 1 
 9415 
 9418 
 9421 
 9424 
 
 9580 
 9583 
 9586 
 
 9589 
 9592 
 9595 
 
 9598 
 9601 
 9604 
 
 9755 
 9758 
 9761 
 
 9764 
 9766 
 9769 
 9772 
 9775 
 9778 
 
 9923 
 9926 
 9928 
 
 9931 
 9934 
 9937 
 9940 
 9942 
 9945 
 
 
 3 
 
 
 S220 
 
 8435 
 
 8645 
 
 8849 
 
 9047 
 
 9240 
 
 9427 
 
 9607 
 
 9781 
 
 9948 
 
 
 8224 
 8227 
 8231 
 
 8238 
 8242 
 
 8245 
 8250 
 8253 
 
 8438 
 8442 
 S445 
 8449 
 8452 
 8456 
 
 8459 
 8463 
 8465 
 
 8648 
 8652 
 8655 
 
 8659 
 8662 
 8665 
 
 8669 
 8672 
 8676 
 
 8852 
 8856 
 8859 
 8862 
 8865 
 8869 
 
 8872 
 
 8879 
 
 9050 
 9054 
 9057 
 
 9060 
 9063 
 9067 
 
 9070 
 9073 
 9077 
 
 9243 
 9246 
 9250 
 
 9253 
 9256 
 
 9259 
 9262 
 9266 
 9269 
 
 9430 
 9433 
 9436 
 
 9439 
 9442 
 9445 
 9448 
 9451 
 9454 
 
 9610 
 9613 
 9616 
 
 9619 
 9621 
 9624 
 
 9627 
 9630 
 9633 
 
 9784 
 9787 
 9789 
 
 9792 
 9795 
 9798 
 
 9801 
 
 9951 
 9953 
 9956 
 
 9959 
 9961 
 9964 
 
 9967 
 9970 
 9972 
 
 
 10 
 20 
 3° 
 40 
 
 50 
 60 
 
 •5 
 r.o 
 
 1-5 
 2.0 
 
 2-5 
 
 3.0 
 
 
 8257 
 
 847° 
 
 8679 
 
 8882 
 
 9080 
 
 9272 
 
 9457 
 
 9636 
 
 9809 
 
 9975 
 
 
 8261 
 8264 
 8268 
 
 8271 
 8275 
 8279 
 8282 
 8285 
 8289 
 
 8473 
 8477 
 848a 
 
 8484 
 8487 
 8491 
 
 8494 
 8498 
 
 8501 
 
 8682 
 8686 
 8689 
 
 8693 
 8696 
 8699 
 
 8703 
 8706 
 8710 
 
 88Ss 
 8889 
 8892 
 
 8896 
 8899 
 8902 
 
 8906 
 8909 
 8913 
 
 9083 
 9086 
 9090 
 
 9°93 
 9096 
 9099 
 
 9102 
 9106 
 9109 
 
 9275 
 9278 
 9281 
 
 9284 
 9287 
 9291 
 
 9294 
 9297 
 9300 
 
 9460 
 9463 
 9466 
 
 9469 
 9472 
 9475 
 9478 
 9481 
 9484 
 
 9639 
 9642 
 9645 
 
 9648 
 9651 
 9654 
 
 9660 
 9663 
 
 9812 
 981S 
 9817 
 
 9820 
 9823 
 9826 
 
 9829 
 983' 
 9834 
 
 9978 
 9980 
 9983 
 
 9986 
 998S 
 9991 
 
 9994 
 9997 
 9999 
 
 
 2 
 
 
 8293 
 
 8505 
 
 8713 
 
 8916 
 
 9112 
 
 9303 
 
 9487 
 
 9666 
 
 9837 
 
 #0002 
 
 
 S296 
 8300 
 8303 
 
 8307 
 8310 
 8314 
 
 8317 
 8321 
 8324 
 
 8508 
 8512 
 8515 
 8519 
 8522 
 8526 
 
 8529 
 8533 
 8536 
 
 8716 
 8720 
 8723 
 
 8727 
 8730 
 8733 
 
 8737 
 8740 
 8744 
 
 8919 
 
 8926 
 
 8929 
 8932 
 8936 
 
 8939 
 8942 
 8946 
 
 9115 
 9118 
 9122 
 
 912s 
 9128 
 9131 
 
 9134 
 9138 
 9141 
 
 9306 
 9309 
 9312 
 
 9315 
 93 iS 
 9322 
 
 9325 
 9328 
 9331 
 
 9490 
 9493 
 9496 
 
 9499 
 9502 
 9506 
 
 9509 
 9512 
 9515 
 
 9669 
 9672 
 967s 
 
 9678 
 9680 
 9683 
 
 9686 
 9689 
 9692 
 
 9840 
 9843 
 9845 
 9848 
 9851 
 9854 
 
 9857 
 
 *ooos 
 *ooo7 
 *ooio 
 
 *ooi3 
 *coiS 
 *ooi8 
 
 *002I 
 *0024 
 *0026 
 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 ■3 
 
 ■7 
 
 I.O 
 
 1-3 
 1-7 
 2.0 
 
 
 8328 
 
 8540 
 
 8747 
 
 8949 
 
 9144 
 
 9334 
 
 9S<8 
 
 9695 
 
 986s 
 
 *0029 
 
 
 
 
 8332 
 8335 
 8339 
 8342 
 8346 
 8350 
 
 8353 
 8357 
 8360 
 
 8543 
 8547 
 8550 
 
 8554 
 8561 
 
 8564 
 8568 
 8571 
 
 8750 
 8754 
 8757 
 8761 
 8764 
 S767 
 
 8771 
 8778 
 
 8952 
 8956 
 8959 
 
 8962 
 8965 
 8969 
 8972 
 8975 
 8979 
 
 9147 
 9150 
 9154 
 
 9157 
 9160 
 9163 
 
 9166 
 9170 
 9173 
 
 9337 
 9340 
 9343 
 9346 
 9349 
 9353 
 
 9356 
 9359 
 9362 
 
 9521 
 9524 
 9527 
 
 9530 
 9533 
 9536 
 
 9539 
 9542 
 9545 
 
 9698 
 9701 
 9704 
 
 9707 
 9709 
 9712 
 
 9715 
 9718 
 9721 
 
 9868 
 9S71 
 9873 
 9876 
 9879 
 9882 
 
 9885 
 9887 
 9890 
 
 *C032 
 
 *oo34 
 *oo37 
 
 *oo39 
 
 *0042 
 
 *oo45 
 *oo47 
 *oo5o 
 #0052 
 
 
 8364 
 
 8575 
 
 8781 
 
 8982 
 
 9176 
 
 9365 
 
 9548 
 
 9724 
 
 9893 
 
 *ooS5 
 
 
 
 
 Smithsonian Tables. 
 
 63 
 
Table 1 1 . 
 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 p„ IN ENGLISH FEET. 
 
 [Derivatton of table explained on p. xlv.] 
 
 Lat. 
 
 71" 
 
 72° 
 
 73° 
 
 74° 
 
 75° 
 
 76° 
 
 77° 
 
 78° 
 
 79° 
 
 80° 
 
 P.P. 
 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 * 
 
 0' 
 
 X 
 2 
 
 3 
 
 005S 
 
 0210 
 
 03 59 
 
 0499 
 
 0632 
 
 0757 
 
 0875 
 
 0984 
 
 1085 
 
 1177 
 
 
 0058 
 0060 
 0063 
 
 0213 
 0215 
 0218 
 
 0361 
 0364 
 0366 
 
 0501 
 0504 
 0506 
 
 0634 
 0636 
 0639 
 
 0759 
 0761 
 0763 
 
 0877 
 0879 
 0880 
 
 0986 
 0987 
 0989 
 
 1087 
 1088 
 1090 
 
 1178 
 ii8a 
 1181 
 
 4 
 
 i 
 I 
 
 9 
 10 
 
 II 
 
 12 
 
 13 
 
 0066 
 006S 
 0071 
 
 0074 
 0077 
 0079 
 
 0220 
 0223 
 '0226 
 
 0228 
 0231 
 0233 
 
 0369 
 0371 
 0373 
 0376 
 0378 
 0381 
 
 0508 
 0510 
 0513 
 0515 
 0517 
 0520 
 
 0641 
 0643 
 064s 
 0647 
 0650 
 0652 
 
 0765 
 0767 
 0769 
 
 0771 
 0773 
 0775 
 
 0882 
 0884 
 0886 
 
 0888 
 0889 
 0891 
 
 0991 
 0992 
 0994 
 
 0995 
 0998 
 0999 
 
 1091 
 1093 
 109s 
 
 1Q96 
 1098 
 1099 
 
 1 183 
 1184 
 1186 
 
 1 187 
 1 189 
 1190 
 
 3 
 
 JO 
 
 20 
 30 
 40 
 
 ■5 
 
 I.O 
 
 ■•5 
 2.0 
 
 2-5 
 
 3.0 
 
 0082 
 
 0236 
 
 0383 
 
 0522 
 
 0654 
 
 0777 
 
 0893 
 
 IGOI 
 
 IIOI 
 
 1192 
 
 0085 
 0087 
 0090 
 
 0238 
 0241 
 0243 
 
 0385 
 0388 
 0390 
 
 0524 
 0526 
 0529 
 
 0656 
 0658 
 0660 
 
 0779 
 0781 
 0783 
 
 0895 
 0897 
 0899 
 
 1003 
 1004 
 1006 
 
 II02 
 
 1 104 
 
 1 105 
 
 ■■93 
 1 19s 
 1 196 
 
 
 14 
 IS 
 l6 
 
 0092 
 009s 
 0098 
 
 0246 
 0248 
 0251 
 
 0392 
 0394 
 0397 
 
 0531 
 0533 
 0535 
 
 0662 
 0664 
 0667 
 
 0785 
 0787 
 0789 
 
 ogoi 
 0902 
 0904 
 
 1008 
 1009 
 lOIl 
 
 1 107 
 
 1 108 
 
 mo 
 
 1 198 
 
 ■■99 
 1200 
 
 
 20 
 
 21 
 22 
 
 23 
 
 0100 
 
 OIG3 
 
 0105 
 
 0253 
 0256 
 0258 
 
 0399 
 0401 
 0404 
 
 0537 
 0540 
 0542 
 
 0669 
 0671 
 0673 
 
 0791 
 0793 
 0795 
 
 0906 
 0908 
 0910 
 
 IOI3 
 
 lois 
 1016 
 
 nil 
 1113 
 1114 
 
 1202 
 1203 
 ■205 
 
 
 oioS 
 
 0261 
 
 0406 
 
 0544 
 
 067s 
 
 0797 
 
 0912 
 
 1018 
 
 1 1 16 
 
 1206 
 
 GUI 
 OII3 
 OI16 
 
 0263 
 0266 
 0268 
 
 □408 
 041 1 
 0413 
 
 0546 
 0549 
 0551 
 
 0677 
 0679 
 0681 
 
 0799 
 0801 
 0803 
 
 0914 
 0916 
 0917 
 
 1020 
 1021 
 1023 
 
 1118 
 1119 
 1121 
 
 1207 
 1209 
 1210 
 
 24 
 
 11 
 11 
 
 29 
 
 30 
 
 31 
 32 
 33 
 
 aii8 
 0121 
 0124 
 
 0126 
 0129 
 0131 
 
 0271 
 0273 
 0276 
 
 0278 
 0281 
 0283 
 
 0416 
 0418 
 0420 
 
 0423 
 0425 
 0428 
 
 0553 
 
 0555 
 0558 
 
 0560 
 0562 
 0565 
 
 0683 
 0685 
 068S 
 
 o6go 
 0692 
 0694 
 
 0805 
 0807 
 0809 
 
 081 1 
 0813 
 081S 
 
 0919 
 0921 
 0923 
 
 0925 
 0926 
 0928 
 
 1025 
 1026 
 1028 
 
 1030 
 1032 
 1033 
 
 1122 
 
 1126 
 
 1 127 
 1 129 
 1130 
 
 1212 
 1213 
 1214 
 
 1216 
 1217 
 1219 
 
 a 
 
 10 
 20 
 30 
 40 
 
 1: 
 
 .3 
 •7 
 1.0 
 ■■3 
 ».? 
 
 I.O 
 
 0134 
 
 0286 
 
 0430 
 
 0567 
 
 0696 
 
 0817 
 
 0930 
 
 1035 
 
 1 132 
 
 1220 
 
 1221 
 1223 
 1224 
 
 0137 
 0139 
 0142 
 
 0288 
 0291 
 0293 
 
 0432 
 0435 
 0437 
 
 0569 
 0571 
 0574 
 
 0698 
 0700 
 0702 
 
 0819 
 0821 
 0823 
 
 0932 
 0934 
 0935 
 
 1037 
 1038 
 1040 
 
 ■■33 
 ■ ■35 
 1136 
 
 
 34 
 11 
 
 0144 
 0147 
 0150 
 
 0296 
 0298 
 0300 
 
 0439 
 0441 
 0444 
 
 0575 
 0578 
 0580 
 
 0704 
 0706 
 0708 
 
 0825 
 0826 
 0828 
 
 0937 
 0939 
 
 0941 
 
 1042 
 1043 
 1045 
 
 1138 
 
 ■■39 
 1141 
 
 1226 
 1227 
 1228 
 
 
 11 
 39 
 
 40 
 
 41 
 42 
 43 
 
 0152 
 0155 
 0157 
 
 0303 
 0305 
 0308 
 
 0446 
 0448 
 0451 
 
 0582 
 0585 
 0587 
 
 0710 
 0712 
 0714 
 
 0830 
 
 0832 
 0834 
 
 0943 
 0944 
 0945 
 
 1047 
 1049 
 1050 
 
 1 142 
 ■■44 
 ■■45 
 
 1230 
 123 1 
 ■233 
 
 
 0160 
 
 0310 
 
 0453 
 
 0589 
 
 0716 
 
 0835 
 
 0948 
 
 1052 
 
 ■■47 
 
 ■234 
 
 0162 
 0165 
 0167 
 
 0312 
 0315 
 0317 
 
 0455 
 0458 
 0460 
 
 0591 
 0593 
 0596 
 
 0718 
 0720 
 0722 
 
 0838 
 0840 
 0842 
 
 0950 
 0952 
 0953 
 
 1054 
 1055 
 1057 
 
 1 148 
 1150 
 ■■5^ 
 
 ■235 
 ■237 
 .238 
 
 44 
 
 49 
 60 
 
 5' 
 52 
 53 
 
 0170 
 0172 
 0175 
 
 0177 
 oi8a 
 0182 
 
 0185 
 
 0320 
 0322 
 0324 
 
 0327 
 0329 
 0332 
 
 0462 
 0464 
 0467 
 
 0469 
 0471 
 0474 
 
 0598 
 c6oo 
 0602 
 
 0604 
 0607 
 0609 
 
 0724 
 0726 
 0729 
 
 0731 
 0733 
 0735 
 
 0844 
 0846 
 0848 
 
 0850 
 0852 
 0854 
 
 0955 
 0957 
 0959 
 
 0961 
 0962 
 09S4 
 
 1058 
 1060 
 1062 
 
 1063 
 1065 
 1066 
 
 '■53 
 1 154 
 1 156 
 
 ■■57 
 ■■59 
 1160 
 
 1240 
 1241 
 
 1242 
 
 ■244 
 ■245 
 ■247 
 
 1 
 
 10 
 20 
 30 
 40 
 50 
 60 
 
 .2 
 ■3 
 ■5 
 
 i 
 
 I.O 
 
 0334 
 
 0476 
 
 061 1 
 
 0737 
 
 0856 
 
 0966 
 
 1068 
 
 1 162 
 
 1248 
 
 0187 
 0190 
 0192 
 
 0336 
 0339 
 0341 
 
 0478 
 0481 
 0483 
 
 0613 
 0615 
 0617 
 
 0739 
 0741 
 0743 
 
 0858 
 086a 
 0862 
 
 0968 
 0970 
 0971 
 
 1070 
 107 1 
 1073 
 
 ■ ■63 
 1 165 
 1166 
 
 ■249 
 125 1 
 
 I2S2 
 
 
 54 
 11 
 
 0195 
 0197 
 0200 
 
 0344 
 0346 
 0349 
 
 0485 
 0487 
 0490 
 
 0619 
 0621 
 0624 
 
 0745 
 0747 
 0749 
 
 □864 
 0865 
 0867 
 
 0973 
 0975 
 0977 
 
 •075 
 1076 
 1078 
 
 1 168 
 1169 
 ■ 171 
 
 ■253 
 ■254 
 1256 
 
 
 11 
 59 
 
 60 
 
 0202 
 
 0205 
 0207 
 
 0351 
 0354 
 0356 
 
 0492 
 0494 
 0497 
 
 0626 
 0628 
 0630 
 
 0751 
 0753 
 0755 
 
 0869 
 0871 
 0873 
 
 0979 
 0980 
 0982 
 
 1080 
 1082 
 1083 
 
 1 172 
 
 ■■74 
 ■■75 
 
 ■257 
 ■ 258 
 1260 
 
 
 0210 
 
 0359 
 
 0499 
 
 0632 
 
 0757 
 
 0875 
 
 0984 
 
 1085 
 
 ■■77 
 
 I261 
 
 Smiths 
 
 ONIAN 1 
 
 'ables. 
 
 
 
 
 
 
 
 
 
 
 
 64 
 
Table 11. 
 LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION 
 
 Pn 
 
 IN ENGLISH FEET. 
 
 [Derivation of table explained on p. xlv.] 
 
 Lat. 
 
 81° 
 
 82° 
 
 83° 
 
 84° 
 
 850 
 
 86° 
 
 87° 
 
 88° 
 
 89° 
 
 p.p. 
 
 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 7.322 
 
 
 
 0' 
 
 I 
 
 2 
 
 3 
 
 1261 
 
 1337 
 
 1403 
 
 1461 
 
 1511 
 
 '55' 
 
 1583 
 
 1605 
 
 1619 
 
 
 
 1262 
 1264 
 1265 
 
 1338 
 1339 
 1340 
 
 1404 
 1405 
 1406 
 
 1462 
 1463 
 1464 
 
 1512 
 1512 
 I5'3 
 
 1552 
 1552 
 1553 
 
 '583 
 '584 
 '584 
 
 1605 
 1606 
 i6o6 
 
 1619 
 1619 
 1619 
 
 
 4 
 I 
 
 1266 
 1267 
 1269 
 
 1341 
 1342 
 1344 
 
 1407 
 1408 
 1410 
 
 1465 
 1465 
 1465 
 
 1514 
 ■514 
 1515 
 
 ■553 
 ■554 
 1555 
 
 '585 
 1585 
 
 ■585 
 
 1606 
 1606 
 1607 
 
 1619 
 1619 
 1620 
 
 
 
 I 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 1270 
 1271 
 1273 
 
 1345 
 1346 
 1347 
 
 1411 
 1412 
 1413 
 
 1467 
 1468 
 1469 
 
 1516 
 1517 
 1517 
 
 ■555 
 ■555 
 1556 
 
 ■586 
 1586 
 ■587 
 
 1607 
 1607 
 160S 
 
 1620 
 1620 
 1620 
 
 2 
 
 
 1274 
 
 1348 
 
 1414 
 
 1470 
 
 1518 
 
 ■557 
 
 ■587 
 
 1608 
 
 1620 
 
 
 1275 
 1277 
 1278 
 
 1349 
 1350 
 
 1352 
 
 1415 
 1416 
 1417 
 
 1471 
 1472 
 1473 
 
 1519 
 1519 
 1520 
 
 ■558 
 1558 
 '559 
 
 '587 
 1588 
 1588 
 
 1608 
 1609 
 1609 
 
 1620 
 1620 
 1620 
 
 
 H 
 
 1279 
 1280 
 1282 
 
 1353 
 
 1354 
 1355 
 
 1418 
 
 1419 
 1420 
 
 1474 
 1474 
 1475 
 
 1521 
 1521 
 1522 
 
 ■559 
 1560 
 1561 
 
 ■589 
 ■589 
 15S9 
 
 1609 
 i6og 
 1610 
 
 1620 
 1620 
 1621 
 
 
 
 10 
 
 •3 
 
 
 17 
 i8 
 19 
 
 20 
 
 21 
 22 
 
 23 
 
 1283 
 1284 
 1286 
 
 1356 
 1358 
 >359 
 
 142 1 
 
 1422 
 1423 
 
 1476 
 1477 
 1478 
 
 1523 
 1524 
 1524 
 
 1561 
 ■562 
 1562 
 
 1590 
 '59° 
 1591 
 
 1610 
 1610 
 161 1 
 
 162 1 
 162 1 
 1621 
 
 20 
 30 
 40 
 so 
 60 
 
 ■7 
 1.0 
 ■■3 
 ■■7 
 2.0 
 
 
 1287 
 
 1360 
 
 1424 
 
 1479 
 
 1525 
 
 '563 
 
 ■591 
 
 1611 
 
 1621 
 
 
 1288 
 
 I2go 
 1291 
 
 1361 
 1362 
 1363 
 
 1425 
 1426 
 1427 
 
 1480 
 1481 
 1481 
 
 1526 
 1526 
 .1527 
 
 1563 
 1564 
 1564 
 
 I59^ 
 ■592 
 ■592 
 
 1611 
 1611 
 1612 
 
 162 z 
 1621 
 1621 
 
 
 
 24 
 
 26 
 
 1292 
 1293 
 1295 
 
 1364 
 1365 
 1367 
 
 142S 
 1429 
 1430 
 
 1482 
 1483 
 1484 
 
 152S 
 1528 
 1529 
 
 1565 
 
 ■593 
 '593 
 1593 
 
 1612 
 1612 
 i6ia 
 
 1621 
 I62I 
 1622 
 
 
 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 1296 
 1297 
 1299 
 
 1368 
 1369 
 1370 
 
 1431 
 1432 
 1433 
 
 1485 
 1485 
 i486 
 
 1530 
 1531 
 1531 
 
 1566 
 1567 
 1567 
 
 ■594 
 ■594 
 '595 
 
 1612 
 1613 
 1613 
 
 1622 
 1622 
 1622 
 
 
 
 1300 
 
 1371 
 
 ■434 
 
 1487 
 
 1532 
 
 1568 
 
 ■595 
 
 1613 
 
 1622 
 
 
 1301 
 1302 
 1304 
 
 1372 
 '373 
 1374 
 
 1435 
 1436 
 1437 
 
 1488 
 1489 
 1489 
 
 1533 
 1533 
 '534 
 
 1568 
 1569 
 '569 
 
 '595 
 ■596 
 ■596 
 
 1613 
 1613 
 
 1614 
 
 1622 
 1622 
 1622 
 
 
 34 
 35 
 36 
 
 1305 
 1306 
 1307 
 
 1376 
 1378 
 
 1438 
 1438 
 1439 
 
 1490 
 1491 
 1492 
 
 1535 
 1535 
 1536 
 
 '57° 
 1570 
 ■571 
 
 1597 
 ■597 
 ■ 597 
 
 1614 
 1614 
 1614 
 
 1622 
 1622 
 1623 
 
 
 
 37 
 38 
 39 
 
 40 
 
 41 
 42 
 43 
 
 1308 
 13 10 
 1311 
 
 "379 
 1380 
 1381 
 
 1440 
 1441 
 1442 
 
 1493 
 1493 
 1494 
 
 1537 
 '538 
 1538 
 
 1571 
 1572 
 1572 
 
 ■598 
 ■598 
 '599 
 
 1614 
 1615 
 
 1615 
 
 1623 
 1623 
 
 1623 
 
 1 
 
 
 ID 
 20 
 30 
 40 
 
 fo 
 
 .2 
 •3 
 ■5 
 .7 
 .8 
 1.0 
 
 
 13 12 
 
 1382 
 
 1443 
 
 I49S 
 
 1539 
 
 '573 
 
 ■599 
 
 1615 
 
 1623 
 
 
 1313 
 1315 
 13 15 
 
 1383 
 1384 
 138s 
 1386 
 1387 
 1389 
 
 1444 
 1445 
 1446 
 
 1496 
 1497 
 1497 
 1498 
 1499 
 1500 
 
 1540 
 '54° 
 154' 
 
 1573 
 ■ 574 
 ■574 
 
 '599 
 1600 
 1600 
 
 1615 
 
 1615 
 1616 
 
 1616 
 1616 
 1616 
 
 1623 
 1623 
 1623 
 
 1623 
 1623 
 1623 
 
 
 tl 
 
 13 18 
 1320 
 
 1447 
 1448 
 
 1542 
 1543 
 
 ■575 
 ■576 
 
 1600 
 1601 
 
 
 
 tl 
 49 
 
 60 
 
 51 
 52 
 53 
 
 1321 
 1322 
 1324 
 
 1390 
 1391 
 1392 
 
 1449 
 1450 
 1451 
 
 1501 
 1 501 
 1502 
 
 1543 
 1544 
 1544 
 
 1576 
 ■577 
 ■577 
 
 1601 
 1601 
 1602 
 
 1616 
 1617 
 1617 
 
 1623 
 1623 
 1623 
 
 
 
 1325 
 
 1393 
 
 1452 
 
 1503 
 
 1545 
 
 ■578 
 
 1602 
 
 1617 
 
 1623 
 
 
 1326 
 1327 
 1329 
 
 1394 
 1395 
 1396 
 
 1453 
 1454 
 1455 
 
 1504 
 1505 
 1505 
 
 1546 
 1546 
 '547 
 
 '578 
 1579 
 '579 
 
 1602 
 1603 
 1603 
 
 1617 
 
 1617 
 1618 
 
 1623 
 1623 
 1623 
 
 
 54 
 55 
 S6 
 
 >330 
 >33i 
 1332 
 
 1397 
 1398 
 1399 
 
 1456 
 1456 
 1457 
 
 1506 
 1507 
 1508 
 
 1547 
 1548 
 '549 
 
 ■ 580 
 1580 
 1581 
 
 1603 
 ■603 
 1604 
 
 1618 
 1618 
 1618 
 
 1623 
 1623 
 1623 
 
 
 
 57 
 58 
 59 
 
 60 
 
 1333 
 1335 
 1336 
 
 1400 
 1401 
 1402 
 
 1458 
 1459 
 1460 
 
 1509 
 1509 
 1510 
 
 1549 
 '55° 
 1550 
 
 1581 
 '582 
 1582 
 
 1604 
 1604 
 1605 
 
 16.8 
 1619 
 
 i6ig 
 
 1623 
 1623 
 1623 
 
 
 
 1337 
 
 1403 
 
 1461 
 
 1511 
 
 '55' 
 
 ■583 
 
 1605 
 
 i6ig 
 
 1623 
 
 
 
 
 Smithsonian Tables. 
 
 fie 
 
Table 12. 
 
 LOGARITHMS OF RADIUS OF CURVATURE Pa (IN METRES) OF SECTION 
 OF EARTH'S SURFACE INCLINED TO MERIDIAN AT AZIMUTH a. 
 
 [Formula for pa given on p. xlv,] 
 
 
 
 
 
 
 LATITUDE 
 
 
 
 
 
 
 Azimuth. 
 
 
 
 
 
 
 
 
 
 
 
 
 22° 
 
 23° 
 
 24° 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 30° 
 
 31° 
 
 
 0° 
 
 6.80237 
 
 6.80242 
 
 6.80248 
 
 6.80254 
 
 6.80260 
 
 6.80266 
 
 6.80272 
 
 6.80279 
 
 6.80285 
 
 6.80292 
 
 
 5 
 
 10 
 
 IS 
 
 239 
 244 
 254 
 
 244 
 250 
 259 
 
 250 
 264 
 
 256 
 261 
 270 
 
 262 
 267 
 276 
 
 268 
 
 273 
 282 
 
 274 
 275 
 
 280 
 
 285 
 294 
 
 287 
 292 
 300 
 
 $ 
 
 
 20 
 
 3° 
 
 266 
 282 
 300 
 
 271 
 287 
 305 
 
 277 
 292 
 
 309 
 
 282 
 297 
 314 
 
 288 
 302 
 319 
 
 293 
 308 
 
 324 
 
 299 
 313 
 330 
 
 305 
 319 
 335 
 
 3" 
 
 325 
 340 
 
 317 
 331 
 346 
 
 
 35 
 40 
 
 45 
 
 320 
 364 
 
 324 
 
 329 
 350 
 371 
 
 333 
 
 354 
 375 
 
 338 
 358 
 379 
 
 343 
 362 
 
 383 
 
 387 
 
 353 
 372 
 391 
 
 358 
 396 
 
 363 
 382 
 
 400 
 
 
 50 
 
 386 
 407 
 427 
 
 389 
 410 
 
 430 
 
 392 
 
 413 
 432 
 
 396 
 416 
 
 435 
 
 399 
 420 
 
 438 
 
 403 
 423 
 442 
 
 407 
 426 
 445 
 
 411 
 
 430 
 448 
 
 415 
 434 
 451 
 
 419 
 437 
 455 
 
 
 65 
 70 
 
 75 
 
 445 
 461 
 
 473 
 
 476 
 
 450 
 465 
 478 
 
 480 
 
 455 
 470 
 482 
 
 458 
 473 
 484 
 
 461 
 
 475 
 487 
 
 464 
 478 
 489 
 
 467 
 481 
 492 
 
 470 
 484 
 494 
 
 
 80 
 
 85 
 90 
 
 483 
 489 
 490 
 
 485 
 490 
 492 
 
 487 
 492 
 494 
 
 489 
 
 494 
 496 
 
 491 
 496 
 498 
 
 4p 
 500 
 
 495 
 5°' 
 502 
 
 498 
 
 503 
 504 
 
 500 
 
 5°5 
 507 
 
 502 
 
 507 
 509 
 
 
 Azimuth. 
 
 LATITUDE. 
 
 
 32° 
 
 33° 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 40° 
 
 41° 
 
 
 0° 
 
 6.80299 
 
 6.80306 
 
 6.80313 
 
 6.80320 
 
 6.80327 
 
 6.8033s 
 
 6.80342 
 
 6.80350 
 
 6.80357 
 
 6.80365 
 
 
 5 
 10 
 
 '5 
 
 300 
 305 
 313 
 
 307 
 
 312 
 
 320 
 
 314 
 319 
 326 
 
 322 
 326 
 333 
 
 329 
 333 
 340 
 
 336 
 
 340 
 348 
 
 344 
 
 348 
 355 
 
 351 
 III 
 
 f 
 
 369 
 
 366 
 370 
 376 
 
 
 20 
 
 25 
 30 
 
 324 
 337 
 352 
 
 330 
 
 337 
 349 
 364 
 
 343 
 355 
 370 
 
 350 
 362 
 
 376 
 
 382 
 
 364 
 11^ 
 
 III 
 394 
 
 37| 
 388 
 
 401 
 
 385 
 395 
 407 
 
 
 35 
 40 
 
 45 
 
 369 
 386 
 405 
 
 374 
 392 
 410 
 
 380 
 
 397 
 414 
 
 385 
 402 
 419 
 
 391 
 407 
 424 
 
 397 
 412 
 429 
 
 402 
 418 
 434 
 
 408 
 423 
 439 
 
 414 
 429 
 444 
 
 420 
 434 
 449 
 
 
 50 
 II 
 
 423 
 441 
 458 
 
 428 
 
 445 
 462 
 
 432 
 449 
 465 
 
 436 
 469 
 
 441 
 
 457 
 472 
 
 461 
 476 
 
 450 
 465 
 480 
 
 454 
 469 
 484 
 
 459 
 474 
 487 
 
 464 
 478 
 491 
 
 
 65 
 
 70 
 
 75 
 
 486 
 497 
 
 476 
 489 
 Soo 
 
 480 
 492 
 
 502 
 
 483 
 495 
 505 
 
 486 
 498 
 S08 
 
 489 
 501 
 510 
 
 493 
 504 
 513 
 
 496 
 
 507 
 S16 
 
 500 
 510 
 519 
 
 503 
 514 
 522 
 
 
 80 
 
 85 
 90 
 
 505 
 510 
 5" 
 
 507 
 512 
 
 514 
 
 510 
 5H 
 516 
 
 512 
 517 
 518 
 
 515 
 519 
 521 
 
 517 
 522 
 
 523 
 
 520 
 524 
 526 
 
 523 
 527 
 528 
 
 525 
 529 
 53' 
 
 528 
 532 
 533 
 
 
 Smithsonia 
 
 N Table 
 
 s. 
 
 
 
 
 
 
 
 
 
 
 66. 
 
Table 12. 
 LOGARITHMS OF RADIUS OF CURVATURE Pa. (IN METRES) OF SECTION 
 OF EARTH'S SURFACE INCLINED TO MERIDIAN AT AZIMUTH a. 
 
 [Formula for pa given on p. xlv.] 
 
 
 
 
 LATITUDE. 
 
 
 
 
 Azimuth. 
 
 
 
 
 
 
 
 42° 
 
 43° 
 
 44° 
 
 45° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 50° 
 
 51° 
 
 o° 
 
 6.80373 
 
 6.80380 
 
 6.80388 
 
 6.80396 
 
 6.80404 
 
 6.8041 1 
 
 6.8041 
 
 9 6.80426 
 
 6.80434 
 
 6.80442 
 
 S 
 
 10 
 
 IS 
 
 374 
 384 
 
 382 
 385 
 391 
 
 389 
 393 
 399 
 
 
 397 
 400 
 406 
 
 404 
 408 
 413 
 
 412 
 
 415 
 
 420 
 
 42 
 
 42 
 42 
 
 428 
 3 430 
 8 435 
 
 438 
 442 
 
 443 
 445 
 450 
 
 20 
 
 3° 
 
 392 
 
 402 
 
 413 
 
 408 
 420 
 
 406 
 426 
 
 
 413 
 422 
 
 433 
 
 420 
 429 
 439 
 
 427 
 
 436 
 446 
 
 4: 
 44 
 4. 
 
 4 441 
 2 449 
 2 458 
 
 448 
 456 
 465 
 
 463 
 471 
 
 35 
 40 
 
 45 
 
 426 
 440 
 454 
 
 432 
 446 
 
 459 
 
 438 
 464 
 
 
 444 
 457 
 470 
 
 450 
 462 
 
 475 
 
 456 
 468 
 480 
 
 At 
 
 >2 468 
 
 4 479 
 
 5 490 
 
 474 
 485 
 495 
 
 480 
 490 
 500 
 
 SO 
 
 468 
 482 
 495 
 
 %l 
 499 
 
 478 
 490 
 S02 
 
 
 482 
 
 487 
 
 499 
 510 
 
 492 
 503 
 SU 
 
 4c 
 5c 
 5' 
 
 6 501 
 8 512 
 8 522 
 
 506 
 
 526 
 
 SIO 
 520 
 
 530 
 
 65 
 70 
 
 75 
 
 S07 
 S17 
 525 
 
 510 
 520 
 528 
 
 514 
 523 
 530 
 
 
 534 
 
 520 
 529 
 536 
 
 524 
 532 
 539 
 
 52 
 
 s: 
 
 54 
 
 8 531 
 
 6 539 
 
 t2 545 
 
 S34 
 542 
 548 
 
 538 
 545 
 551 
 
 8o 
 
 85 
 90 
 
 531 
 534 
 536 
 
 534 
 537 
 538 
 
 536 
 540 
 541 
 
 
 539 
 542 
 544 
 
 542 
 
 545 
 546 
 
 544 
 548 
 549 
 
 54 
 5. 
 5. 
 
 ^7 55° 
 
 553 
 
 1 554 
 
 553 
 
 555 
 556 
 
 '3 
 
 559 
 
 
 
 
 LATITUDE. 
 
 
 
 
 Azimuth. 
 
 
 
 
 
 
 
 52° 
 
 53° 
 
 54° 
 
 55° 
 
 S6° 
 
 57° 
 
 58° 
 
 59° 
 
 60° 
 
 0° 
 
 6.80449 
 
 6.80457 
 
 6.80464 
 
 6.80471 
 
 6.80479 
 
 6.80486 
 
 6.80493 
 
 6.80500 
 
 6.80506 
 
 5 
 10 
 
 '5 
 
 450 
 453 
 457 
 
 464 
 
 465 
 467 
 471 
 
 472 
 474 
 478 
 
 479 
 481 
 
 485 
 
 486 
 488 
 492 
 
 493 
 495 
 498 
 
 500 
 502 
 
 505 
 
 507 
 509 
 
 5" 
 
 20 
 25 
 30 
 
 462 
 469 
 477 
 
 469 
 476 
 484 
 
 476 
 482 
 
 49° 
 
 483 
 489 
 496 
 
 489 
 495 
 502 
 
 496 
 
 501 
 S08 
 
 502 
 508 
 514 
 
 509 
 514 
 519 
 
 515 
 520 
 
 525 
 
 35 
 40 
 
 45 
 
 486 
 496 
 505 
 
 492 
 SOI 
 Sio 
 
 498 
 506 
 
 515 
 
 S°3 
 512 
 520 
 
 509 
 517 
 525 
 
 51S 
 522 
 
 530 
 
 520 
 
 527 
 534 
 
 525 
 532 
 539 
 
 S3I 
 
 537 
 543 
 
 SO 
 1^ 
 
 515 
 524 
 533 
 
 520 
 S28 
 537 
 
 524 
 533 
 541 
 
 528 
 537 
 544 
 
 533 
 548 
 
 537 
 545 
 552 
 
 542 
 548 
 SSS 
 
 546 
 552 
 558 
 
 556 
 562 
 
 6s 
 
 70 
 75 
 
 548 
 554 
 
 545 
 SSI 
 557 
 
 548 
 SS4 
 559 
 
 551 
 SS7 
 562 
 
 1 
 
 565 
 
 5|^ 
 gi 
 
 561 
 566 
 570 
 
 564 
 569 
 
 573 
 
 567 
 
 572 
 
 575 
 
 80 
 
 85 
 90 
 
 S6? 
 
 563 
 564 
 
 566 
 
 S66 
 568 
 569 
 
 568 
 
 570 
 571 
 
 571 
 573 
 574 
 
 573 
 575 
 576 
 
 576 
 
 S78 
 578 
 
 578 
 580 
 580 
 
 Smithsonian Tables. 
 
 67 
 
Table 13. 
 
 LOGARITHMS 
 
 OF FACTORS t^stFOR COMPUTING SPHEROIDAL 
 EXCESS OF TRIANGLES. 
 
 UNIT = THE ENGLISH FOOT. 
 
 [Derivation and use of table explained on p. Iviii.] 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 <!> 
 
 change per 
 minute. 
 
 <* 
 
 change per 
 minute. 
 
 1> 
 
 change per 
 minute. 
 
 
 
 change per 
 minute. 
 
 
 0° 
 
 0.37498 
 — 0.00 
 
 20° 
 
 0.37429 
 
 — O.I2 
 
 40° 
 
 0'372S5 
 — 0.18 
 
 60° 
 
 0.37056 
 — 0.15 
 
 
 I 
 
 498 
 — 0.02 
 
 21 
 
 422 
 — 0.12 
 
 41 
 
 244 
 — 0.17 
 
 61 
 
 047 
 — 0.15 
 
 
 2 
 
 497 
 — 0.02 
 
 22 
 
 41 S 
 
 — 0.12 
 
 42 
 
 234 
 — 0.17 
 
 62 
 
 038 
 — 0.13 
 
 
 3 
 
 496 
 — 0.02 
 
 23 
 
 40S 
 — 0.12 
 
 43 
 
 224 
 — 0.17 
 
 63 
 
 030 
 — 0.13 
 
 
 4 
 
 49S 
 — 0.03 
 
 24 
 
 401 
 — 0.13 
 
 44 
 
 214 „ 
 
 — 0.18 
 
 64 
 
 022 
 — 0.13 
 
 
 5 
 
 493 
 — 0.03 
 
 25 
 
 393 
 — 0.13 
 
 45 
 
 203 
 — 0.17 
 
 65 
 
 014 
 — 0.13 
 
 
 6 
 
 491 
 — 0.03 
 
 26 
 
 385 
 — 0.13 
 
 46 
 
 193 
 
 — 0.17 
 
 66 
 
 006 
 — 0.13 
 
 
 7 
 
 489 
 — 0.03 
 
 27 
 
 377 
 — 0.15 
 
 47 
 
 183 
 
 — 0.17 
 
 67 
 
 0.36998 
 — 0.12 
 
 
 8 
 
 487 
 — 0.05 
 
 28 
 
 368 
 — 0.13 
 
 48 
 
 '73 „ 
 — 0.18 
 
 68 
 
 991 
 — 0.12 
 
 
 9 
 
 484 
 — 0.07 
 
 29 
 
 360 
 — 0.15 
 
 49 
 
 162 
 — 0.17 
 
 69 
 
 984 
 — 0.12 
 
 
 10 
 
 480 
 — 0.07 
 
 30 
 
 3SI 
 — 0.15 
 
 50 
 
 152 
 — 0.17 
 
 70 
 
 977 
 — 0.10 
 
 
 II 
 
 476 
 — 0.07 
 
 31 
 
 342 
 — 0.1S 
 
 SI 
 
 142 
 — 0.17 
 
 71 
 
 971 
 — 0.12 
 
 
 12 
 
 472 
 
 32 
 
 333 
 — 0.17 
 
 52 
 
 132 
 — 0.17 
 
 72 
 
 964 
 — 0.08 
 
 
 '3 
 
 468 
 —0.08 
 
 33 
 
 323 
 — 0.15 
 
 53 
 
 122 
 — 0.17 
 
 73 
 
 959 
 — 0.10 
 
 
 14 
 
 463 
 — 0.07 
 
 34 
 
 314 
 — 0.17 
 
 54 
 
 112 
 — 0.15 
 
 74 
 
 953 „ 
 — 0.08 
 
 
 15 
 
 4S9 
 
 — O.IO 
 
 35 
 
 304 
 — 0.15 
 
 55 
 
 103 
 — 0.17 
 
 75 
 
 948 
 — 0.08 
 
 
 i6 
 
 453 „ 
 — 0.08 
 
 36 
 
 295 
 — 0.17 
 
 56 
 
 093 
 — 0.17 
 
 76 
 
 943 „ 
 — 0.08 
 
 
 17 
 
 448 
 
 — O.IO 
 
 37 
 
 285 
 — 0.17 
 
 57 
 
 083 
 — 0.1S 
 
 77 
 
 938 
 — 0.07 
 
 
 i8 
 
 442 
 
 — O.IO 
 
 38 
 
 275 
 — 0.17 
 
 58 
 
 074 
 — 0.1S 
 
 78 
 
 934 
 — 0.07 
 
 
 19 
 
 436 
 
 — 0.12 
 
 39 
 
 26s 
 — 0.17 
 
 59 
 
 065 
 — 0.1s 
 
 79 
 
 930 
 — 0.07 
 
 
 20 
 
 429 
 
 — 0.12 
 
 40 
 
 — 0.18 
 
 60 
 
 056 
 — 0.IS 
 
 80 
 
 926 
 
 
 Smithsonian Tables. 
 
 68 
 
_e«_ 
 
 Table 14. 
 LOGARITHMS OF FACTORS-j^^ FOR COMPUTING SPHEROIDAL 
 EXCESS OF TRIANGLES. 
 
 UNIT==THE METRE. 
 
 [Derivation and use of table explained on p, Iviii.] 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 log. factor and 
 
 
 i- 
 
 change per 
 minute. 
 
 * 
 
 change per 
 minute. 
 
 <!> 
 
 change per 
 minute. 
 
 
 
 change per 
 minute. 
 
 
 0° 
 
 1.40695 
 — 0.00 
 
 20° 
 
 1.40626 
 — 0.12 
 
 40° 
 
 1.40452 
 — 0.18 
 
 60° 
 
 1.40253 
 — 0.15 
 
 
 I 
 
 695 
 — 0.02 
 
 21 
 
 619 
 — 0.12 
 
 41 
 
 441 
 — 0.17 
 
 61 
 
 244 
 — 0.15 
 
 
 2 
 
 694 
 — 0.02 
 
 22 
 
 612 
 — 0.12 
 
 42 
 
 431 
 — 0.17 
 
 62 
 
 23s 
 — 0.13 
 
 
 3 
 
 693 
 — 0.02 
 
 23 
 
 605 
 — 0.13 
 
 43 
 
 421 
 — 0.17 
 
 63 
 
 227 
 — 0.13 
 
 
 4 
 
 692 
 — 0.03 
 
 .24 
 
 597 
 — 0.12 
 
 44 
 
 4" „ 
 — 0.18 
 
 64 
 
 219 
 — CIS 
 
 
 5 
 
 690 
 — 0.03 
 
 25 
 
 590 
 — 0.13 
 
 45 
 
 400 
 — 0.17 
 
 65 
 
 210 
 — 0.12 
 
 
 6 
 
 688 
 — 0.03 
 
 26 
 
 582 
 — 0.1S 
 
 46 
 
 390 
 — 0.17 
 
 66 
 
 203 
 — 0.13 
 
 
 7 
 
 686 
 — 0.05 
 
 27 
 
 573 
 — C.13 
 
 47 
 
 380 
 — 0.18 
 
 67 
 
 195 
 — 0.12 
 
 
 8 
 
 683 
 — 0.05 
 
 28 
 
 S6S 
 — 0.1S 
 
 48 
 
 369 
 — 0.17 
 
 68 
 
 188 
 — 0.12 
 
 
 9 
 
 680 
 — 0.05 
 
 29 
 
 SS6 
 
 —0.13 
 
 49 
 
 359 
 — 0.17 
 
 69 
 
 181 
 — 0.12 
 
 
 10 
 
 677 
 — 0.07 
 
 30 
 
 548 
 —0.15 
 
 50 
 
 349 
 — 0.17 
 
 70 
 
 174 
 — 0.10 
 
 
 II 
 
 673 
 — 0.07 
 
 31 
 
 539 
 — 0.15 
 
 SI 
 
 339 
 — 0.17 
 
 71 
 
 168 
 — 0.12 
 
 
 12 
 
 669 
 
 — 0.07 
 
 32 
 
 530 
 — 0.17 
 
 52 
 
 329 
 — 0.17 
 
 72 
 
 161 
 
 — O.IO 
 
 
 13 
 
 665 ^ 
 — 0.08 
 
 33 
 
 520 
 — 0.15 
 
 S3 
 
 319 
 — 0.17 
 
 73 
 
 '55 „ 
 — 0.08 
 
 
 14 
 
 660 
 — 0.08 
 
 34 
 
 — 0.17 
 
 S4 
 
 309 
 — 0.17 
 
 74 
 
 150 
 
 — O.IO 
 
 
 15 
 
 ^55 „ 
 — 0.08 
 
 35 
 
 SOI 
 — 0.17 
 
 55 
 
 299 
 -0.15 
 
 75 
 
 144 
 — 0.08 
 
 
 i6 
 
 650 
 
 — O.IO 
 
 36 
 
 491 
 — 0.15 
 
 S6 
 
 290 
 — 0.17 
 
 76 
 
 139 
 
 — 0.07 
 
 
 17 
 
 644 
 — 0.08 
 
 37 
 
 482 
 — 0.17 
 
 S7 
 
 280 
 -0.15 
 
 77 
 
 '35 „ 
 — 0.08 
 
 
 i8 
 
 639 
 — 0.12 
 
 38 
 
 472 
 — 0.17 
 
 S8 
 
 271 
 — 0.1S 
 
 78 
 
 130 
 — 0.07 
 
 
 19 
 
 632 
 
 — O.IO 
 
 39 
 
 462 
 — 0.17 
 
 59 
 
 262 
 — 0.15 
 
 79 
 
 126 
 — 0.05 
 
 
 20 
 
 626 
 
 — 0.12 
 
 40 
 
 452 
 — 0.18 
 
 60 
 
 253 
 -0.15 
 
 80 
 
 123 
 
 
 Smithsonian Tables. 
 
 69 
 
Table 15. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANCULATION. 
 
 UNIT=THE ENGLISH FOOT. 
 
 [Derivation and use of table explained on p. Ix.] 
 
 
 
 ai 
 
 ^i=a 
 
 «2 
 
 *2 
 
 ^2 
 
 
 
 01 
 
 h = CX 
 
 oa 
 
 *2 
 
 f2 
 
 o°oo' 
 
 7.99669 
 
 7-99374 
 
 00 
 
 00 
 
 0.372 
 
 io°oo' 
 
 7-99655 
 
 7-99369 
 
 9.621 
 
 9.926 
 
 0.398 
 
 10 
 
 669 
 
 374 
 
 7-839 
 
 8.137 
 
 0.372 
 
 10 
 
 6SS 
 
 369 
 
 9.628 
 
 9-933 
 
 0.399 
 
 20 
 
 669 
 
 374 
 
 8.140 
 
 8.438 
 
 0.372 
 
 20 
 
 654 
 
 369 
 
 9.636 
 
 9.941 
 
 0.400 
 
 3° 
 
 669 
 
 374 
 
 8.316 
 
 8.614 
 
 0.372 
 
 30 
 
 654 
 
 369 
 
 9-643 
 
 9-948 
 
 0.401 
 
 40 
 
 669 
 
 374 
 
 8.441 
 
 Im 
 
 0.372 
 
 40 
 
 654 
 
 369 
 
 9-650 
 
 9-955 
 
 0.402 
 
 SO 
 
 669 
 
 374 
 
 8.538 
 
 8.836 
 
 0.372 
 
 SO 
 
 653 
 
 369 
 
 9-657 
 
 9-963 
 
 0.403 
 
 I 00 
 
 669 
 
 374 
 
 8.617 
 
 8.915 
 
 0-372 
 
 II 00 
 
 653 
 
 3i 
 
 9-663 
 
 9.970 
 
 0.404 
 
 10 
 
 
 374 
 
 8.684 
 
 8.982 
 
 0.372 
 
 10 
 
 652 
 
 ^fo 
 
 9.670 
 
 9-977 
 
 0.404 
 
 20 
 
 668 
 
 374 
 
 8.742 
 
 9.040 
 
 0-372 
 
 20 
 
 652 
 
 368 
 
 9.677 
 
 9-983 
 
 0.405 
 
 30 
 
 668 
 
 374 
 
 ^•^93 
 
 9.091 
 
 0-373 
 
 30 
 
 651 
 
 ^fo 
 
 9-683 
 
 9.990 
 
 0.406 
 
 40 
 
 668 
 
 374 
 
 8.839 
 8.880 
 
 9-137 
 
 0-373 
 
 40 
 
 651 
 
 368 
 
 9.690 
 
 9-997 
 
 0.407 
 
 50 
 
 668 
 
 374 
 
 9.179 
 
 0-373 
 
 so 
 
 650 
 
 368 
 
 9.696 
 
 0.003 
 
 0.408 
 
 2 00 
 
 668 
 
 374 
 
 8.918 
 
 9.216 
 
 0-373 
 
 12 00 
 
 650 
 
 367 
 
 9.702 
 
 0.010 
 
 0.409 
 
 10 
 
 668 
 
 373 
 
 1%^ 
 
 9.251 
 
 0-373 
 
 10 
 
 649 
 
 367 
 
 9.708 
 
 0.016 
 
 0.410 
 
 20 
 
 668 
 
 373 
 
 8.985 
 
 9.283 
 
 0-373 
 
 20 
 
 649 
 
 367 
 
 9-714 
 
 0.023 
 
 0.412 
 
 30 
 
 668 
 
 373 
 
 9.015 
 
 9-314 
 
 0.374 
 
 30 
 
 648 
 
 367 
 
 9.720 
 
 0.029 
 
 0.413 
 
 40 
 
 668 
 
 373 
 
 9-043 
 
 9-342 
 
 0-374 
 
 40 
 
 648 
 
 367 
 
 9.726 
 
 0-035 
 
 0.414 
 
 SO 
 
 668 
 
 373 
 
 9.069 
 
 9-368 
 
 0.374 
 
 SO 
 
 647 
 
 367 
 
 9-732 
 
 0.041 
 
 0.415 
 
 300 
 
 668 
 
 373 
 
 9.094 
 
 9-393 
 
 0.374 
 
 1300 
 
 646 
 
 366 
 
 9-738 
 
 0.048 
 
 0.416 
 
 10 
 
 667 
 
 373 
 
 9.1 18 
 
 9.417 
 
 0-37 5 
 
 10 
 
 646 
 
 366 
 
 9-744 
 
 0.054 
 
 0.417 
 
 20 
 
 667 
 
 373 
 
 9.140 
 
 9-439 
 
 0-37S 
 
 20 
 
 645 
 
 366 
 
 9-749 
 
 0.060 
 
 0.418 
 
 30 
 
 667 
 
 373 
 
 9.161 
 
 9.460 
 
 0-37 5 
 
 30 
 
 645 
 
 366 
 
 9-7S5 
 
 0.065 
 
 0.419 
 
 40 
 
 667 
 
 373 
 
 9.182 
 
 9.481 
 
 0.376 
 
 40 
 
 644 
 
 366 
 
 9-761 
 
 0.071 
 
 0.420 
 
 SO 
 
 667 
 
 373 
 
 9.201 
 
 9.500 
 
 0-376 
 
 SO 
 
 644 
 
 36s 
 
 9.766 
 
 0.077 
 
 0.422 
 
 400 
 
 667 
 
 373 
 
 9.220 
 
 9.519 
 
 0.376 
 
 1400 
 
 643 
 
 365 
 
 9.771 
 
 0.083 
 
 0.423 
 
 lO 
 
 666 
 
 373 
 
 9-237 
 
 9-537 
 
 0-377 
 
 10 
 
 642 
 
 36s 
 
 9-777 
 
 0.088 
 
 0.424 
 
 20 
 
 666 
 
 373 
 
 9-254 
 
 9-554 
 
 0-377 
 
 20 
 
 642 
 
 365 
 
 9.782 
 
 0.094 
 
 0.425 
 
 30 
 
 666 
 
 373 
 
 9.271 
 
 9- 570 
 
 0.377 
 
 30 
 
 641 
 
 36s 
 
 9.787 
 
 0.100 
 
 0.426 
 
 40 
 
 666 
 
 373 
 
 9.287 
 
 
 0.378 
 
 40 
 
 640 
 
 364 
 
 9.792 
 
 0.105 
 
 0.428 
 
 SO 
 
 666 
 
 373 
 
 9.302 
 
 9.602 
 
 0.378 
 
 50 
 
 640 
 
 364 
 
 9.798 
 
 0.1 11 
 
 0.429 
 
 SOD 
 
 ffs 
 
 373 
 
 9-317 
 
 9.617 
 
 0.379 
 
 15 00 
 
 639 
 
 364 
 
 t^ 
 
 0.116 
 
 0.430 
 
 10 
 
 66s 
 
 373 
 
 9-331 
 
 9.631 
 
 0.379 
 
 10 
 
 639 
 
 364 
 
 0.121 
 
 0.431 
 
 20 
 
 665 
 
 372 
 
 9-34S 
 
 9-645 
 
 0.379 
 
 20 
 
 638 
 
 363 
 
 9.813 
 
 0.127 
 
 0.433 
 
 30 
 
 f.^ 
 
 372 
 
 9-358 
 
 9.659 
 
 0.380 
 
 30 
 
 637 
 
 363 
 
 9.818 
 
 0.132 
 
 0.434 
 
 40 
 
 664 
 
 372 
 
 9-372 
 
 9.672 
 
 0.380 
 
 40 
 
 637 
 
 363 
 
 9.822 
 
 0.137 
 
 0.435 
 
 so 
 
 664 
 
 372 
 
 9-384 
 
 9.685 
 
 0.381 
 
 50 
 
 636 
 
 363 
 
 9.827 
 
 0.142 
 
 0.437 
 
 600 
 
 664 
 
 372 
 
 9-397 
 
 9.697 
 
 0.381 
 
 16 00 
 
 63s 
 
 363 
 
 9.832 
 
 0.147 
 
 0.438 
 
 10 
 
 664 
 
 372 
 
 9-409 
 
 9.709 
 
 0.382 
 
 10 
 
 63s 
 
 362 
 
 9-837 
 
 0-153 
 
 0-439 
 
 20 
 
 663 
 
 372 
 
 9.420 
 
 9.721 
 
 0.383 
 
 20 
 
 634 
 
 362 
 
 9.841 
 
 0.158 
 
 0.441 
 
 30 
 
 f? 
 
 372 
 
 9-432 
 
 9-732 
 
 0.383 
 
 30 
 
 633 
 
 362 
 
 9.846 
 
 0.163 
 0.168 
 
 0.442 
 
 40 
 
 663 
 
 372 
 
 9-443 
 
 9-744 
 
 0.384 
 
 40 
 
 632 
 
 362 
 
 9.851 
 
 0.443 
 
 so 
 
 662 
 
 372 
 
 9-4S3 
 
 9-755 
 
 0.384 
 
 SO 
 
 632 
 
 ■361 
 
 9.855 
 
 0.173 
 
 0-44S 
 
 7 00 
 
 662 
 
 372 
 
 9.464 
 
 9-765 
 
 0.385 
 
 17 00 
 
 631 
 
 361 
 
 9.860 
 
 0.178 
 
 0.446 
 
 10 
 
 662 
 
 371 
 
 9-474 
 
 9.776 
 
 0.386 
 
 10 
 
 630 
 
 36r 
 
 9.864 
 
 0.182 
 
 0.448 
 
 20 
 
 662 
 
 371 
 
 9.484 
 
 9.786 
 
 0.386 
 
 20 
 
 630 
 
 361 
 
 9.869 
 
 0.187 
 
 0.449 
 
 30 
 
 661 
 
 371 
 
 9.494 
 
 9.796 
 
 0.387 
 
 30 
 
 629 
 
 360 
 
 9-873 
 
 0.192 
 
 0.450 
 
 40 
 
 661 
 
 371 
 
 9-504 
 
 9.806 
 
 °-^ll 
 
 40 
 
 628 
 
 360 
 
 9.878 
 
 0.197 
 
 0.452 
 
 50 
 
 661 
 
 371 
 
 9-513 
 
 9.816 
 
 0.388 
 
 so 
 
 627 
 
 360 
 
 9.882 
 
 0.202 
 
 0-4S3 
 
 800 
 
 660 
 
 371 
 
 9-523 
 
 9.825 
 
 0.389 
 
 1800 
 
 627 
 
 360 
 
 9.886 
 
 0.206 
 
 0.455 
 
 10 
 
 660 
 
 371 
 
 9-532 
 
 9-834 
 
 0.389 
 
 10 
 
 626 
 
 359 
 
 9.890 
 
 0.21 1 
 
 0.456 
 
 20 
 
 659 
 
 371 
 
 9.541 
 
 9-843 
 
 0.390 
 
 20 
 
 625 
 
 359 
 
 9.895 
 
 0.216 
 
 0.458 
 
 30 
 
 659 
 
 371 
 
 9-549 
 
 9.852 
 
 0.391 
 
 30 
 
 624 
 
 359 
 
 9-899 
 
 0.220 
 
 0.459 
 
 40 
 
 ^l 
 
 370 
 
 9-5|| 
 
 9.861 
 
 0.392 
 
 40 
 
 624 
 
 359 
 
 9-903 
 
 0.225 
 
 0.461 
 
 SO 
 
 658 
 
 370 
 
 9.566 
 
 9.870 
 
 0.392 
 
 so 
 
 623 
 
 3S8 
 
 9-907 
 
 0.229 
 
 0.463 
 
 900 
 
 658 
 
 370 
 
 9-575 
 
 9-f2? 
 
 0.393 
 
 1900 
 
 622 
 
 358 
 
 9.911 
 
 0.234 
 
 0.464 
 
 10 
 
 i^'' 
 
 370 
 
 9-583 
 
 9.886 
 
 0-394 
 
 10 
 
 621 
 
 358 
 
 9-915 
 
 0.239 
 
 0.466 
 
 20 
 
 657 
 
 370 
 
 9.591 
 
 9-895 
 
 0-395 
 
 20 
 
 620 
 
 3S8 
 
 9.919 
 
 0.243 
 
 0.467 
 
 30 
 
 \\ 
 
 370 
 
 ^•M 
 
 9-903 
 
 0.396 
 
 30 
 
 620 
 
 357 
 
 9.923 
 
 0.248 
 
 0.469 
 
 40 
 
 656 
 
 ^I^ 
 
 9.606 
 
 9.910 
 
 0.396 
 
 40 
 
 619 
 
 357 
 
 9.927 
 
 0.252 
 
 0.470 
 
 SO 
 
 656 
 
 369 
 
 9.614 
 
 9.918 
 
 0.397 
 
 50 
 
 618 
 
 3S7 
 
 9-931 
 
 0.256 
 
 0.472 
 
 1000 
 
 6SS 
 
 369 
 
 9.621 
 
 9.926 
 
 0.398 
 
 20.00 
 
 617 
 
 3S7 
 
 9-935 
 
 0.261 
 
 0.474 
 
 Smithso 
 
 NIAN Tab 
 
 LES. 
 
 
 
 
 
 
 
 
 
 
 70 
 
Table 15. 
 LOGARITHMS' OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE. LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION. 
 
 UNIT = THE ENGLISH FOOT. 
 
 [Derivation and use of table explained on p. Ix.] 
 
 20°00' 
 
 01 
 
 h\==cx 
 
 02 
 
 *2 
 
 ^2 
 
 
 
 ax 
 
 *2 = f2 
 
 02 
 
 *2 
 
 f2 
 
 
 7.99617 
 
 7-99357 
 
 9-935 
 
 0.261 
 
 0.474 
 
 3o°oo' 
 
 7-99558 
 
 7-99337 
 
 0.13s 
 0.138 
 
 0.496 
 
 0.593 
 
 
 10 
 
 616 
 
 356 
 
 9-939 
 
 0.265 
 
 0.47 s 
 
 10 
 
 ^\ 
 
 337 
 
 0.500 
 
 0.595 
 
 
 20 
 
 61 S 
 
 356 
 
 9-943 
 
 0.270 
 
 0.477 
 
 20 
 
 556 
 
 336 
 
 0.141 
 
 0.503 
 
 0.598 
 
 
 3° 
 
 615 
 
 356 
 
 9-947 
 
 0.274 
 
 0.479 
 
 30 
 
 555 
 
 336 
 
 0.144 
 
 0.507 
 
 0.600 
 
 
 40 
 
 614 
 
 355 
 
 9.951 
 
 0.278 
 
 0.480, 
 
 40 
 
 554 
 
 335 
 
 0.146 
 
 O.5H 
 
 0.603 
 
 
 5° 
 
 613 
 
 355 
 
 9-955 
 
 0.282 
 
 0.482 
 
 50 
 
 553 
 
 335 
 
 0.149 
 
 0.514 
 
 0.605 
 
 
 21 00 
 
 612 
 
 355 
 
 9-958 
 
 0.287 
 
 0.484 
 
 31 00 
 
 552 
 
 335 
 
 0.152 
 
 0.518 
 
 0.607 
 
 
 10 
 
 611 
 
 355 
 
 9.962 
 
 0.291 
 
 0.486 
 
 10 
 
 550 
 
 334 
 
 0.1 55 
 
 0.522 
 
 0.610 
 
 
 20 
 
 610 
 
 354 
 
 9.966 
 
 0.295 
 
 0.487 
 
 20 
 
 549 
 
 334 
 
 0.158 
 
 0.525 
 
 0.612 
 
 
 3° 
 
 609 
 
 354 
 
 9.970 
 
 0.299 
 
 0.489 
 
 30 
 
 548 
 
 333 
 
 0.161 
 
 0.529 
 
 0.615 
 
 
 40 
 
 608 
 
 354 
 
 9-973 
 
 0.304 
 
 0.491 
 
 40 
 
 547 
 
 333 
 
 0.164 
 
 0.532 
 
 0.617 
 
 
 SO 
 
 608 
 
 353 
 
 9-977 
 
 0.308 
 
 0.493 
 
 50 
 
 546 
 
 333 
 
 0.166 
 
 0.536 
 
 0.619 
 
 
 22 00 
 
 607 
 
 353 
 
 9.981 
 
 0.312 
 
 0.494 
 
 32 00 
 
 545 
 
 332 
 
 0.169 
 
 0.540 
 
 0.622 
 
 
 10 
 
 606 
 
 353 
 
 9.984 
 
 0.316 
 
 0.496 
 
 10 
 
 544 
 
 332 
 
 0.172 
 
 0.543 
 
 0.624 
 
 
 20 
 
 60s 
 
 353 
 
 9.988 
 
 0.320 
 
 0.498 
 
 20 
 
 542 
 
 332 
 
 0-I7S 
 
 0.547 
 
 0.627 
 
 
 3° 
 
 604 
 
 352 
 
 9-991 
 
 0.324 
 
 0.500 
 
 30 
 
 541 
 
 331 
 
 0.177 
 
 0.550 
 
 0.629 
 
 
 4° 
 
 603 
 
 352 
 
 9-995 
 9.998 
 
 0.328 
 
 0.502 
 
 40 
 
 540 
 
 331 
 
 O.IOO 
 
 0.554 
 
 0.632 
 
 
 5° 
 
 602 
 
 352 
 
 0.332 
 
 0.503 
 
 50 
 
 539 
 
 330 
 
 0.183 
 
 0.558 
 
 0.634 
 
 
 2300 
 
 601 
 
 351 
 
 0.002 
 
 0.336 
 
 0.505 
 
 3300 
 
 538 
 
 330 
 
 0.186 
 
 0.561 
 
 0.637 
 
 
 10 
 
 600 
 
 351 
 
 0.005 
 
 0.340 
 
 0.507 
 
 10 
 
 537 
 
 330 
 
 0.188 
 
 It^ 
 
 0.639 
 
 
 20 
 
 600 
 
 351 
 
 0.009 
 
 0.344 
 
 0.509 
 
 20 
 
 535 
 
 329 
 
 0.I9I 
 
 0.642 
 
 
 3° 
 
 599 
 
 350 
 
 0.012 
 
 0.348 
 
 0.51 1 
 
 30 
 
 534 
 
 329 
 
 0.194 
 
 0.572 
 
 0.644 
 
 
 40 
 
 cg8 
 
 35° 
 
 0.016 
 
 0.352 
 
 0.513 
 
 40 
 
 533 
 
 328 
 
 0.197 
 
 0.S7S 
 
 0.647 
 
 
 5° 
 
 597 
 
 350 
 
 0.019 
 
 0-356 
 
 0.51 5 
 
 50 
 
 532 
 
 328 
 
 0.199 
 
 0.579 
 
 0.650 
 
 
 2400 
 
 596 
 
 349 
 
 0.023 
 
 0.360 
 
 0.517 
 
 3400 
 
 531 
 
 328 
 
 0.202 
 
 0.583 
 
 0.652 
 
 
 10 
 
 59S 
 
 349 
 
 0.026 
 
 0.364 
 
 0.518 
 
 10 
 
 529 
 
 327 
 
 0.205 
 0.208 
 
 0.586 
 
 0.65s 
 
 
 20 
 
 594 
 
 349 
 
 0.029 
 
 0.368 
 
 0.520 
 
 20 
 
 528 
 
 327 
 
 0.590 
 
 0.657 
 
 
 30 
 
 593 
 
 348 
 
 0.033 
 
 0.372 
 
 0.522 
 
 30 
 
 527 
 
 326 
 
 0.210 
 
 0.593 
 
 0.660 
 
 
 40 
 
 592 
 
 348 
 
 0.036 
 
 0.376 
 
 0.524 
 
 40 
 
 526 
 
 326 
 
 0.213 
 
 0.597 
 
 °-f^ 
 
 
 5° 
 
 591 
 
 348 
 
 0.039 
 
 0.380 
 
 0.526 
 
 SO 
 
 525 
 
 326 
 
 0.216 
 
 0.600 
 
 0.665 
 
 
 2500 
 
 
 347 
 
 0.043 
 
 0.384 
 
 0.528 
 
 35 00 
 
 523 
 
 325 
 
 0.218 
 
 0.604 
 
 0.668 
 
 
 10 
 
 589 
 
 347 
 
 0.046 
 
 0.388 
 
 0.530 
 
 10 
 
 522 
 
 325 
 
 0.221 
 
 0.608 
 
 0.671 
 0.673 
 
 
 20 
 
 588 
 
 347 
 
 0.049 
 
 0.392 
 
 0.532 
 
 20 
 
 521 
 
 324 
 
 0.224 
 
 0.61 1 
 
 
 30 
 
 587 
 
 346 
 
 0.052 
 
 0.396 
 
 0.534 
 
 30 
 
 520 
 
 324 
 
 0.226 
 
 0.615 
 
 0.676 
 0.679 
 
 o.68i 
 
 
 40 
 
 586 
 
 346 
 
 0.056 
 
 0.399 
 
 0.536 
 
 40 
 
 519 
 
 324 
 
 0.229 
 
 0.618 
 
 
 5° 
 
 58s 
 
 346 
 
 0.059 
 
 0.403 
 
 0.538 
 
 SO 
 
 517 
 
 323 
 
 0.232 
 
 0.622 
 
 
 2600 
 
 584 
 
 345 
 
 0.062 
 
 0.407 
 
 0.540 
 
 3600 
 
 516 
 
 323 
 
 0.234 
 
 0.625 
 
 0.684 
 
 
 10 
 
 583 
 
 345 
 
 0.065 
 0.068 
 
 0.41 1 
 
 0-S43 
 
 10 
 
 515 
 
 322 
 
 0.237 
 
 0.629 
 
 0.687 
 ^0 
 
 
 20 
 
 S82 
 
 345 
 
 0.415 
 
 0.54s 
 
 20 
 
 514 
 
 322 
 
 0.239 
 
 0.632 
 
 0.689 
 
 
 3° 
 
 581 
 
 344 
 
 0.072 
 
 0.418 
 
 0.547 
 
 30 
 
 512 
 
 322 
 
 0.242 
 
 0.636 
 
 0.692 
 
 
 40 
 
 580 
 
 344 
 
 0.075 
 
 0.422 
 
 0.549 
 
 40 
 
 5" 
 
 321 
 
 0.245 
 
 0.640 
 
 
 5° 
 
 579 
 
 344 
 
 0.078 
 
 0.426 
 
 0.551 
 
 50 
 
 510 
 
 321 
 
 0.247 
 
 0.643 
 
 
 27 00 
 
 578 
 
 343 
 
 0.081 
 
 0.430 
 
 0-5S3 
 
 3700 
 
 509 
 
 320 
 
 0.250 
 
 0.647 
 
 0.700 
 
 
 10 
 
 577 
 
 343 
 
 0.084 
 
 0.433 
 
 0.555 
 
 10 
 
 S°2 
 
 320 
 
 0.253 
 
 0.650 
 
 0.703 
 0.706 
 
 
 20 
 
 576 
 
 343 
 
 0.087 
 
 0.437 
 
 0.557 
 
 20 
 
 506 
 
 320 
 
 0.25s 
 
 0.654 
 
 
 30 
 
 575 
 
 342 
 
 0.090 
 
 0.441 
 
 0.559 
 
 30 
 
 505 
 
 319 
 
 0.258 
 
 0.657 
 0.661 
 
 0.709 
 
 
 40 
 
 574 
 
 342 
 
 0.093 
 
 0.445 
 
 0.562 
 
 40 
 
 504 
 
 319 
 
 0.260 
 
 0.712 
 
 
 5° 
 
 573 
 
 342 
 
 0.096 
 
 0.448 
 
 0.564 
 
 SO 
 
 503 
 
 318 
 
 0.263 
 
 0.665 
 
 0.715 
 
 
 2800 
 
 571 
 
 341 
 
 0.099 
 
 0.452 
 
 0.566 
 
 3800 
 
 501 
 
 318 
 
 0.266 
 
 0.668 
 
 0.717 
 
 
 10 
 
 570 
 
 341 
 
 0.102 
 
 0.456 
 
 0.568 
 
 10 
 
 500 
 
 317 
 
 0.268 
 
 0.672 
 
 0.720 
 
 
 20 
 
 569 
 
 341 
 
 0.105 
 
 0.460 
 
 0.570 
 
 20 
 
 499 
 
 317 
 
 0.271 
 
 0.675 
 
 0.723 
 
 
 30 
 
 568 
 
 340 
 
 0.108 
 
 0.463 
 
 0.573 
 
 30 
 
 498 
 
 317 
 
 0.273 
 
 0.679 
 
 0.726 
 
 
 40 
 
 567 
 
 340 
 
 0.1 1 1 
 
 0.467 
 
 0-575 
 
 40 
 
 496 
 
 316 
 
 0.276 
 
 °-^«^ 
 
 0.729 
 
 
 50 
 
 566 
 
 340 
 
 0.1 14 
 
 0.471 
 
 0.577 
 
 50 
 
 495 
 
 316 
 
 0.278 
 
 0.686 
 
 0.732 
 
 
 2900 
 
 56s 
 
 339 
 
 0.117 
 
 0.474 
 
 0.579 
 
 3900 
 
 494 
 
 315 
 
 0.281 
 
 0.690 
 0.693 
 
 V.?k 
 
 
 10 
 
 y 
 
 564 
 
 339 
 
 0.120 
 
 0.478 
 
 0.582 
 
 10 
 
 492 
 
 315 
 
 0.284 
 
 0.738 
 
 
 20 
 
 563 
 
 338 
 
 0.123 
 
 0.482 
 
 0.584 
 
 20 
 
 491 
 
 315 
 
 0.286 
 
 0.697 
 
 0.741 
 
 
 30 
 
 562 
 
 338 
 
 0.126 
 
 0.485 
 
 0.586 
 
 30 
 
 490 
 
 314 
 
 0.289 
 
 0.701 
 
 0.744 
 
 
 40 
 
 S6i 
 
 338 
 
 0.129 
 
 0.489 
 
 0.588 
 
 40 
 
 489 
 
 314 
 
 0.291 
 
 0.704 
 
 0.747 
 
 
 50 
 
 560 
 
 337 
 
 0.132 
 
 0.493 
 
 0.591 
 
 50 
 
 487 
 
 313 
 
 0.294 
 
 0.708 
 
 0.750 
 
 
 3000 
 
 558 
 
 337 
 
 0.135 
 
 0.496 
 
 0.593 
 
 40 00 
 
 486 
 
 313 
 
 0.296 
 
 0.71 1 
 
 0-7S3 
 
 
 Smithsonian Tables. 
 
 71 
 
Table 15. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION. 
 
 UNIT = THE ENGLISH FOOT. 
 
 [Derivation and use of table explained on p. be.] 
 
 <p 
 
 31 
 
 *i=a 
 
 02 
 
 h 
 
 <^z 
 
 ^ 
 
 31 
 
 h=<^i 
 
 02 
 
 h 
 
 <^2 
 
 
 40°oo' 
 
 7.99486 
 
 7-99313 
 
 0.296 
 
 0.7 1 1 
 
 0.752 
 
 5o°oo' 
 
 7.99409 
 
 7-99287 
 
 0.448 
 
 0-939 
 
 0-955 
 
 
 10 
 
 485 
 
 312 
 
 0.299 
 
 0-715 
 
 0-755 
 
 10 
 
 408 
 
 287 
 
 0.450 
 
 0.944 
 
 0.958 
 
 
 20 
 
 484 
 
 312 
 
 0.301 
 
 0.719 
 
 0-759 
 
 20 
 
 407 
 
 287 
 
 0-4S3 
 
 0.948 
 
 0.962 
 
 
 30 
 
 482 
 
 312 
 
 0.304 
 
 0.722 
 
 0.762 
 
 30 
 
 406 
 
 286 
 
 0.45s 
 
 0.952 
 
 0.966 
 
 
 40 
 
 481 
 
 3" 
 
 0.307 
 
 0.726 
 
 0.765 
 0.768 
 
 40 
 
 404 
 
 286 
 
 0-458 
 
 0.956 
 
 0.970 
 
 
 5° 
 
 480 
 
 3" 
 
 0.309 
 
 0.730 
 
 50 
 
 403 
 
 28s 
 
 0.960 
 
 0.974 
 
 
 41 CX3 
 
 479 
 
 310 
 
 0.312 
 
 0-733 
 
 0.771 
 
 51 00 
 
 402 
 
 28s 
 
 0.463 
 
 0.964 
 
 0.978 
 
 
 10 
 
 477 
 
 310 
 
 0.314 
 
 0.737 
 
 0.774 
 
 10 
 
 401 
 
 284 
 
 0.466 
 
 0.968 
 
 0.982 
 
 
 20 
 
 476 
 
 309 
 
 0.317 
 
 0.740 
 
 0-777 
 
 20 
 
 399 
 
 284 
 
 0.468 
 
 0.972 
 
 0.985 
 
 
 30 
 
 475 
 
 3°9 
 
 0.319 
 
 0.744 
 
 0.780 
 
 30 
 
 398 
 
 284 
 
 0.471 
 
 0.976 
 
 0.989 
 
 
 40 
 
 473 
 
 
 0.322 
 
 0.748 
 
 0.783 
 
 40 
 
 397 
 
 283 
 
 0.473 
 
 0.981 
 
 0-993 
 
 
 5° 
 
 472 
 
 308 
 
 0.324 
 
 0.751 
 
 0.786 
 
 SO 
 
 396 
 
 283 
 
 0.476 
 
 0.985 
 
 0-997 
 
 
 42 00 
 
 471 
 
 308 
 
 0.327 
 
 0-755 
 
 0.789 
 
 52 00 
 
 394 
 
 282 
 
 0.478 
 
 0.989 
 
 l.OOI 
 
 
 10 
 
 470 
 
 307 
 
 0.329 
 
 0.759 
 
 0.792 
 
 10 
 
 393 
 
 282 
 
 0.481 
 
 0.993 
 
 1.005 
 
 
 20 
 
 468 
 
 307 
 
 0.332 
 
 0.762 
 
 0.796 
 
 20 
 
 392 
 
 281 
 
 0.484 
 
 0.998 
 
 i.oog 
 
 
 3° 
 
 467 
 
 306 
 
 0-334 
 
 0.766 
 
 0.799 
 
 30 
 
 391 
 
 281 
 
 0.486 
 
 1.002 
 
 1.013 
 
 
 40 
 
 466 
 
 306 
 
 0.337 
 
 0.770 
 
 0.802 
 
 40 
 
 
 281 
 
 0.489 
 
 1.006 
 
 1.017 
 
 
 SO 
 
 464 
 
 306 
 
 0.339 
 
 0.774 
 
 0.805 
 
 50 
 
 388 
 
 280 
 
 0.491 
 
 I.OIO 
 
 1.021 
 
 
 4300 
 
 463 
 
 305 
 
 0.342 
 
 0.777 
 
 0.808 
 
 5300 
 
 ^ll 
 
 280 
 
 0.494 
 
 1.015 
 
 1.025 
 
 
 10 
 
 462 
 
 305 
 
 0.344 
 
 0.781 
 
 0.812 
 
 10 
 
 386 
 
 279 
 
 0.497 
 
 1. 01 9 
 
 1.030 
 
 
 20 
 
 461 
 
 304 
 
 0.347 
 
 0.785 
 
 0.815 
 
 20 
 
 384 
 
 279 
 
 0.499 
 
 1.023 
 
 1.034 
 
 
 30 
 
 459 
 
 304 
 
 0.349 
 
 0.788 
 
 0.818 
 
 30 
 
 383 
 
 279 
 
 0.502 
 
 1.028 
 
 1.038 
 
 
 40 
 
 458 
 
 303 
 
 0.352 
 
 0.792 
 
 0.821 
 
 40 
 
 382 
 
 278 
 
 0.505 
 
 1.032 
 
 1.042 
 
 
 50 
 
 457 
 
 303 
 
 0-354 
 
 0.796 
 
 0.824 
 
 SO 
 
 381 
 
 278 
 
 0.507 
 
 1.036 
 
 1.046 
 
 
 4400 
 
 455 
 
 303 
 
 0-357 
 
 0.800 
 
 0.828 
 
 5400 
 
 379 
 
 277 
 
 0.510 
 
 1. 041 
 
 1.050 
 
 
 10 
 
 454 
 
 302 
 
 0-359 
 
 0.803 
 
 0.831 
 
 10 
 
 378 
 
 277 
 
 0.512 
 
 1.045 
 
 i-oSS 
 
 
 20 
 
 453 
 
 302 
 
 0.362 
 
 0.807 
 
 0.834 
 
 20 
 
 377 
 
 277 
 
 0.515 
 
 1.049 
 
 1.059 
 
 
 3° 
 
 452 
 
 301 
 
 0.364 
 
 0.81 1 
 
 0.838 
 
 30 
 
 376 
 
 276 
 
 0.518 
 
 1.054 
 
 1.063 
 
 
 40 
 
 450 
 
 301 
 
 0.367 
 
 0.815 
 
 0.841 
 
 40 
 
 375 
 
 276 
 
 0.520 
 
 1.058 
 
 1.067 
 
 
 5° 
 
 449 
 
 300 
 
 0.370 
 
 0.818 
 
 0.844 
 
 50 
 
 373 
 
 275 
 
 0.523 
 
 1.063 
 
 1.072 
 
 
 4500 
 
 448 
 
 300 
 
 0.372 
 
 0.822 
 
 0.848 
 
 5500 
 
 372 
 
 275 
 
 0.526 
 
 1.067 
 
 1.076 
 
 
 10 
 
 446 
 
 300 
 
 0-375 
 
 0.826 
 
 0.851 
 
 10 
 
 371 
 
 275 
 
 0.528 
 
 1.072 
 
 1.080 
 
 
 20 
 
 445 
 
 299 
 
 0.377 
 
 0.830 
 
 0.854 
 
 20 
 
 370 
 
 274 
 
 0-531 
 
 1.076 
 
 1.084 
 
 
 30 
 
 444 
 
 299 
 
 0.380 
 
 0.833 
 
 0.858 
 
 30 
 
 369 
 
 274 
 
 0-534 
 
 1.0S1 
 
 1.089 
 
 
 40 
 
 443 
 
 298 
 
 0.382 
 
 0.837 
 
 0.861 
 
 40 
 
 3^J 
 
 273 
 
 0-537 
 
 1.085 
 
 1.093 
 
 
 S° 
 
 441 
 
 298 
 
 0.385 
 
 0.841 
 
 0.865 
 
 50 
 
 366 
 
 273 
 
 0-539 
 
 1.090 
 
 1.098 
 
 
 4600 
 
 440 
 
 297 
 
 0.387 
 
 0.845 
 
 0.868 
 
 5600 
 
 365 
 
 273 
 
 0.542 
 
 1.094 
 
 1. 102 
 
 
 10 
 
 439 
 
 297 
 
 0.390 
 
 0.849 
 
 0.872 
 
 10 
 
 364 
 
 272 
 
 0.545 
 
 1.099 
 
 1.106 
 
 
 20 
 
 437 
 
 297 
 
 0.392 
 
 0.853 
 
 0.87 s 
 
 20 
 
 363 
 
 272 
 
 0-547 
 
 1. 104 
 
 I. HI 
 
 
 30 
 
 436 
 
 296 
 
 0-395 
 
 0.856 
 
 °il^ 
 
 30 
 
 361 
 
 271 
 
 0.550 
 
 1.108 
 
 1.115 
 
 
 40 
 
 435 
 
 296 
 
 0-397 
 
 0.860 
 
 0.882 
 
 40 
 
 360 
 
 271 
 
 0-553 
 
 1.113 
 
 1.120 
 
 
 5° 
 
 434 
 
 295 
 
 0.400 
 
 0.864 
 
 0.885 
 
 SO 
 
 359 
 
 271 
 
 0.556 
 
 1. 118 
 
 1.124 
 
 
 4700 
 
 432 
 
 295 
 
 0.402 
 
 0.868 
 
 0.889 
 
 57 00 
 
 358 
 
 270 
 
 0-558 
 
 1.122 
 
 1.129 
 
 
 10 
 
 431 
 
 294 
 
 0.405 
 
 0.872 
 
 0.892 
 
 lo 
 
 357 
 
 270 
 
 0.561 
 
 1. 127 
 
 1-134 
 
 
 20 
 
 430 
 
 294 
 
 0.407 
 
 0.876 
 
 0.896 
 
 20 
 
 356 
 
 269 
 
 0.564 
 
 1.132 
 
 1.138 
 
 
 30 
 
 428 
 
 294 
 
 0.410 
 
 0.880 
 
 0.900 
 
 30 
 
 354 
 
 269 
 
 0.567 
 
 I-I37 
 
 1.143 
 
 
 40 
 
 427 
 
 293 
 
 0.412 
 
 0.884 
 
 0.903 
 
 40 
 
 353 
 
 269 
 
 0.569 
 
 1. 141 
 
 1.147 
 
 
 50 
 
 426 
 
 293 
 
 0.415 
 
 0.888 
 
 0.907 
 
 SO 
 
 352 
 
 268 
 
 0-572 
 
 1. 146 
 
 1.152 
 
 
 48 00 
 
 425 
 
 292 
 
 0.417 
 
 0.891 
 
 0.910 
 
 5800 
 
 35' 
 
 268 
 
 0.575 
 
 1.151 
 
 I -1 57 
 
 
 10 
 
 423 
 
 292 
 
 0.420 
 
 0.895 
 
 0.914 
 
 10 
 
 350 
 
 267 
 
 0.578 
 
 1. 156 
 
 1. 162 
 
 
 20 
 
 422 
 
 291 
 
 0.422 
 
 0.899 
 
 0.918 
 
 20 
 
 349 
 
 267 
 
 0.581 
 
 1. 161 
 
 1.166 
 
 
 3° 
 
 421 
 
 291 
 
 0.425 
 
 0.903 
 
 0.921 
 
 30 
 
 347 
 
 267 
 
 0.583 
 
 1.166 
 
 1.171 
 
 
 40 
 
 420 
 
 291 
 
 0.427 
 
 0.907 
 
 0.925 
 
 40 
 
 346 
 
 266 
 
 0.586 
 
 1. 170 
 
 1.176 
 
 
 50 
 
 418 
 
 290 
 
 0.430 
 
 0.911 
 
 0.929 
 
 so 
 
 345 
 
 266 
 
 0.589 
 
 1.175 
 
 1. 181 
 
 
 4900 
 
 417 
 
 290 
 
 0.432 
 
 0.91 s 
 
 0.932 
 
 5900 
 
 344 
 
 266 
 
 0.592 
 
 1.180 
 
 1.185 
 
 
 10 
 
 416 
 
 289 
 
 0.435 
 0.438 
 
 0.919 
 
 0.936 
 
 10 
 
 343 
 
 265 
 
 0-595 
 
 1.185 
 
 1.190 
 
 
 20 
 
 414 
 
 289 
 
 0.923 
 
 0.940 
 
 20 
 
 342 
 
 265 
 
 0.598 
 
 1.190 
 
 1.195 
 
 
 3° 
 
 413 
 
 289 
 
 0.440 
 
 0.927 
 
 0-943 
 
 30 
 
 341 
 
 264 
 
 0.600 
 
 1-195 
 
 1.200 
 
 
 40 
 
 412 
 
 288 
 
 0-443 
 
 0.931 
 
 0-947 
 
 40 
 
 339 
 
 264 
 
 0.603 
 
 1.200 
 
 1.205 
 
 
 SO 
 
 411 
 
 288 
 
 0.445 
 
 0-93S 
 
 0.951 
 
 50 
 
 338 
 
 264 
 
 0.606 
 
 1.205 
 
 1.210 
 
 
 5000 
 
 409 
 
 287 
 
 0.448 
 
 0-939 
 
 0-955 
 
 6000 
 
 337 
 
 263 
 
 0.609 
 
 1.210 
 
 1.215 
 
 
 Smithsonian Tables. 
 
 72 
 
Table 15. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANCULATION. 
 
 UNIT = THE ENGLISH FOOT. 
 
 [Derivation and use of table explained on p. Ix.] 
 
 9 
 
 "1 
 
 61 = Cl 
 
 02 
 
 *2 
 
 ^a 
 
 
 
 11 
 
 h = <:i 
 
 02 
 
 h 
 
 <-2 
 
 6o°oo' 
 
 7-99337 
 
 7.99263 
 
 0.609 
 
 I.2I0 
 
 1.215 
 
 7o°oo' 
 
 7.99278 
 
 7-99244 
 
 0.809 
 
 1-575 
 
 1-576 
 
 lO 
 
 336 
 
 263 
 
 0.612 
 
 I.216 
 
 1.220 
 
 10 
 
 277 
 
 243 
 
 0.813 
 
 1-583 
 
 1.584 
 
 20 
 
 335 
 
 263 
 
 0.615 
 
 1. 221 
 
 1.225 
 
 20 
 
 277 
 
 243 
 
 0.817 
 
 1.590 
 
 1.591 
 
 3° 
 
 334 
 
 262 
 
 0.618 
 
 1.226 
 
 1.230 
 
 30 
 
 276 
 
 243 
 
 0.821 
 
 1-598 
 
 1-S99 
 
 40 
 
 333 
 
 262 
 
 0.621 
 
 I.23I 
 
 1-235 
 
 40 
 
 27 s 
 
 242 
 
 0.825 
 
 1.605 
 
 
 5° 
 
 332 
 
 261 
 
 0.624 
 
 1.236 
 
 1.240 
 
 50 
 
 274 
 
 242 
 
 0.829 
 
 1-613 
 
 1.614 
 
 6i 00 
 
 331 
 
 261 
 
 0.627 
 
 1. 241 
 
 1-245 
 
 71 00 
 
 273 
 
 242 
 
 0-833 
 
 1.621 
 
 1.621 
 
 10 
 
 329 
 
 261 
 
 0.630 
 
 1.247 
 
 1-251 
 
 10 
 
 273 
 
 242 
 
 0.837 
 
 1.629 
 
 1.629 
 
 20 
 
 328 
 
 260 
 
 0-633 
 
 1.252 
 
 1.256 
 
 20 
 
 272 
 
 241 
 
 0.841 
 
 1.636 
 
 1-637 
 
 3° 
 
 327 
 
 260 
 
 0.636 
 
 1.257 
 
 1.261 
 
 30 
 
 271 
 
 241 
 
 0.845 
 
 1.644 
 
 1.645 
 
 40 
 
 326 
 
 260 
 
 0.639 
 
 1.263 
 
 1.266 
 
 40 
 
 270 
 
 241 
 
 0.849 
 
 1.652 
 
 1-653 
 
 5° 
 
 325 
 
 259 
 
 0.642 
 
 1.268 
 
 1.272 
 
 SO 
 
 269 
 
 241 
 
 0.854 
 
 1.660 
 
 1.661 
 
 62 00 
 
 324 
 
 259 
 
 0.645 
 
 1-273 
 
 1.277 
 
 72 00 
 
 269 
 
 240 
 
 0.858 
 
 1.669 
 
 1.669 
 
 10 
 
 323 
 
 259 
 
 0.648 
 
 1.279 
 
 1.282 
 
 10 
 
 268 
 
 240 
 
 0.862 
 
 1.677 
 
 1.677 
 
 20 
 
 322 
 
 258 
 
 0.651 
 
 1.284 
 
 1.288^ 
 
 20 
 
 267 
 
 240 
 
 0.866 
 
 1.685 
 
 1.686 
 
 30 
 
 321 
 
 258 
 
 0.654 
 
 1.290 
 
 1.293 
 
 30 
 
 266 
 
 240 
 
 0.871 
 
 1.694 
 
 1.694 
 
 40 
 
 320 
 
 257 
 
 0.657 
 
 1.295 
 
 I.29S 
 
 40 
 
 266 
 
 239 
 
 °iP 
 
 1.702 
 
 1.702 
 
 5° 
 
 319 
 
 257 
 
 0.660 
 
 1-301 
 
 1.304 
 
 50 
 
 265 
 
 239 
 
 0.880 
 
 1.710 
 
 1.711 
 
 6300 
 
 318 
 
 257 
 
 0.663 
 
 1.306 
 
 1.309 
 
 7300 
 
 264 
 
 239 
 
 0.884 
 
 1.719 
 
 1.720 
 
 10 
 
 317 
 
 256 
 
 0.666 
 
 1.312 
 
 I-315 
 
 10 
 
 264 
 
 239 
 
 0.889 
 
 1.728 
 
 1.728 
 
 20 
 
 316 
 
 256 
 
 0.669 
 
 1-318 
 
 1.320 
 
 20 
 
 263 
 
 238 
 
 0.893 
 
 1-737 
 
 1-737 
 
 3° 
 
 31s 
 
 256 
 
 0.672 
 
 1-323 
 
 1.326 
 
 30 
 
 262 
 
 238 
 
 0.898 
 
 I-74S 
 
 1.746 
 
 40 
 
 314 
 
 255 
 
 0.676 
 
 1.329 
 
 1-332 
 
 40 
 
 261 
 
 238 
 
 0-903 
 
 1-754 
 
 I-7SS 
 
 5° 
 
 313 
 
 255 
 
 0.679 
 
 1-335 
 
 1-337 
 
 50 
 
 261 
 
 238 
 
 0.907 
 
 1-763 
 
 1.764 
 
 6400 
 
 312 
 
 255 
 
 0.682 
 
 I-34I 
 
 1-343 
 
 7400 
 
 260 
 
 238 
 
 0.912 
 
 1.772 
 
 1-773 
 
 10 
 
 3" 
 
 254 
 
 0.685 
 
 1.346 
 
 1-349 
 
 10 
 
 259 
 
 237 
 
 0.917 
 
 1.782 
 
 1.782 
 
 20 
 
 310 
 
 254 
 
 0.688 
 
 1-352 
 
 1-355 
 
 20 
 
 259 
 
 237 
 
 0.922 
 
 1.791 
 
 1.791 
 
 30 
 
 
 254 
 
 0.692 
 
 1-358 
 
 1.360 
 
 30 
 
 258 
 
 237 
 
 0.927 
 
 1.800 
 
 1.801 
 
 40 
 
 308 
 
 253 
 
 0.695 
 
 1-363 
 
 1-366 
 
 40 
 
 257 
 
 237 
 
 0.931 
 
 1.810 
 
 1.810 
 
 50 
 
 307 
 
 253 
 
 0.698 
 
 1.370 
 
 1-372 
 
 50 
 
 257 
 
 236 
 
 0.936 
 
 1.820 
 
 1.820 
 
 6500 
 
 306 
 
 253 
 
 0.701 
 
 1-376 
 
 1-378 
 
 7500 
 
 256 
 
 236 
 
 0.941 
 
 1.829 
 
 1.830 
 
 10 
 
 3°S 
 
 252 
 
 0.705 
 
 1.382 
 
 1.384 
 
 10 
 
 25s 
 
 236 
 
 0.946 
 
 1.839 
 
 1.839 
 
 20 
 
 304 
 
 252 
 
 0.708 
 
 1.388 
 
 1-390 
 
 20 
 
 255 
 
 236 
 
 0.952 
 
 1.849 
 
 1.849 
 
 30 
 
 303 
 
 252 
 
 0.7 1 1 
 
 1-394 
 
 1-396 
 
 30 
 
 254 
 
 236 
 
 0-957 
 
 1.859 
 
 1.859 
 
 40 
 
 302 
 
 251 
 
 0.715 
 
 1.400 
 
 1.402 
 
 40 
 
 254 
 
 235 
 
 0.962 
 
 1.869 
 
 1.869 
 
 5° 
 
 301 
 
 251 
 
 0.718 
 
 1.406 
 
 1.408 
 
 50 
 
 253 
 
 235 
 
 0.967 
 
 1.879 
 
 1.880 
 
 6600 
 
 300 
 
 251 
 
 0.721 
 
 I -41 3 
 
 1-414 
 
 7600 
 
 252 
 
 235 
 
 0-973 
 
 1.890 
 
 1.890 
 
 10 
 
 
 250 
 
 0.725 
 
 1.419 
 
 1. 42 1 
 
 10 
 
 252 
 
 235 
 
 0.978 
 
 1.900 
 
 1.901 
 
 20 
 
 298 
 
 250 
 
 0.728 
 
 1.425 
 
 1-427 
 
 20 
 
 251 
 
 23s 
 
 0.984 
 
 1.911 
 
 1.911 
 
 30 
 
 297 
 
 250 
 
 0.732 
 
 1.432 
 
 1-433 
 
 30 
 
 250 
 
 234 
 
 0.989 
 
 1.922 
 
 1.922 
 
 40 
 
 296 
 
 249 
 
 0-73S 
 
 1.438 
 
 1.440 
 
 40 
 
 250 
 
 234 
 
 0-995 
 
 1-933 
 
 1-933 
 
 5° 
 
 29s 
 
 249 
 
 0-739 
 
 1.444 
 
 1.446 
 
 SO 
 
 249 
 
 234 
 
 1. 000 
 
 1.944 
 
 1.944 
 
 67 00 
 
 294 
 
 249 
 
 0.742 
 
 1-451 
 
 1.452 
 
 7700 
 
 249 
 
 234 
 
 1.006 
 
 1-955 
 
 1-955 
 
 10 
 
 293 
 
 249 
 
 0.746 
 
 1-457 
 
 1-459 
 
 10 
 
 248 
 
 234 
 
 1.012 
 
 1.966 
 
 1.966 
 
 20 
 
 292 
 
 
 0.749 
 
 1.464 
 
 1-465 
 
 20 
 
 248 
 
 233 
 
 1.018 
 
 1.978 
 
 1-978 
 
 30 
 
 291 
 
 248 
 
 0-753 
 
 1.470 
 
 1-472 
 
 30 
 
 247 
 
 233 
 
 1.024 
 
 1.989 
 
 1.989 
 
 40 
 
 290 
 
 248 
 
 0.756 
 
 1-477 
 
 1.478 
 
 40 
 
 247 
 
 233 
 
 1-030 
 
 2.001 
 
 2.001 
 
 50 
 
 289 
 
 247 
 
 0.760 
 
 1.484 
 
 1-485 
 
 50 
 
 246 
 
 233 
 
 1.036 
 
 2.013 
 
 2.013 
 
 6800 
 
 289 
 
 247 
 
 0.763 
 
 1.491 
 
 1.492 
 
 78 00 
 
 245 
 
 233 
 
 1.042 
 
 2.025 
 
 2.025 
 
 10 
 
 288 
 
 247 
 
 0.767 
 
 1-497 
 
 1-499 
 
 10 
 
 245 
 
 233 
 
 1.048 
 
 2.037 
 
 2.037 
 
 20 
 
 287 
 
 246 
 
 0.771 
 
 1-504 
 
 1-505 
 
 20 
 
 244 
 
 232 
 
 1.054 
 
 2.050 
 
 2.050 
 
 30 
 
 286 
 
 246 
 
 0.774 
 
 1.511 
 
 1. 512 
 
 30 
 
 244 
 
 232 
 
 1. 061 
 
 2.062 
 
 2.062 
 
 40 
 
 285 
 
 246 
 
 0.778 
 
 1.518 
 
 1-519 
 
 40 
 
 243 
 
 232 
 
 1.067 
 
 2.075 
 
 2-075 
 2.088 
 
 SO 
 
 284 
 
 246 
 
 0.782 
 
 1.525 
 
 1.526 
 
 SO 
 
 243 
 
 232 
 
 1-074 
 
 2.088 
 
 6900 
 
 283 
 
 245 
 
 0.786 
 
 1-532 
 
 1-533 
 
 7900 
 
 242 
 
 232 
 
 1. 08 1 
 
 2.101 
 
 2.101 
 
 10 
 
 282 
 
 245 
 
 0.789 
 
 1-539 
 
 1.540 
 
 10 
 
 242 
 
 232 
 
 1.087 
 
 2.114 
 
 2.114 
 2.128 
 
 20 
 
 282 
 
 245 
 
 0-793 
 
 1-546 
 
 1-547 
 
 20 
 
 242 
 
 231 
 
 1.094 
 
 2.128 
 
 3° 
 
 281 
 
 244 
 
 0-797 
 
 1-553 
 
 1-554 
 
 30 
 
 241 
 
 231 
 
 I.IOI 
 
 2.142 
 2.156 
 
 2.142 
 2.156 
 
 40 
 
 280 
 
 244 
 
 0.801 
 
 1.561 
 
 1.562 
 
 40 
 
 241 
 
 231 
 
 1. 108 
 
 5° 
 
 279 
 
 244 
 
 0.805 
 
 1.568 
 
 1.569 
 
 50 
 
 240 
 
 231 
 
 1. 116 
 
 2.170 
 
 2.170 
 
 70 00 
 
 278 
 
 244 
 
 0.809 
 
 1-575 
 
 1-576 
 
 80 00 
 
 240 
 
 231 
 
 1-123 
 
 2.184 
 
 2.184 
 
 Smithsonian Tables. 
 
 73 
 
Table 16. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION. 
 
 UNIT = THE METRE, 
 [Derivation and use of table explained on p. Ix.] 
 
 •t' 
 
 01 
 
 h = ci 
 
 32 
 
 ^2 
 
 "^2 
 
 1> 
 
 fll 
 
 *1=^1 
 
 «2 
 
 h 
 
 C2 
 
 o°oo' 
 
 8.51268 
 
 8.50973 
 
 00 
 
 00 
 
 1.404 
 
 io°oo' 
 
 8.51254 
 
 8.50968 
 
 0.653 
 
 0.958 
 
 1.430 
 
 10 
 
 268 
 
 973 
 
 8.871 
 
 9.169 
 
 1.404 
 
 10 
 
 254 
 
 9^f 
 
 0.660 
 
 0.965 
 
 1-431 
 
 20 
 
 268 
 
 973 
 
 9.172 
 
 9.470 
 
 1.404 
 
 20 
 
 253- 
 
 968 
 
 0.668 
 
 0-973 
 
 1.432 
 
 30 
 
 268 
 
 973 
 
 9-348 
 
 9.646 
 
 1.404 
 
 30 
 
 253 
 
 9^i 
 
 0.675 
 
 0.980 
 
 1-433 
 
 40 
 
 268 
 
 973 
 
 9-473 
 
 Hll 
 
 1-404 
 
 40 
 
 2S3 
 
 968 
 
 0.682 
 
 0.987 
 
 1-434 
 
 5° 
 
 268 
 
 973 
 
 9.570 
 
 9.868 
 
 1.404 
 
 SO 
 
 252 
 
 967 
 
 0.689 
 
 0-99S 
 
 1-43S 
 
 I 00 
 
 267 
 
 973 
 
 9.649 
 
 9-947 
 
 1.404 
 
 II 00 
 
 252 
 
 967 
 
 0.695 
 
 1.002 
 
 1.436 
 
 10 
 
 267 
 
 973 
 
 9.716 
 
 0.014 
 
 1.404 
 
 10 
 
 251 
 
 967 
 
 0.702 
 
 1.009 
 
 1.436 
 
 20 
 
 267 
 
 973 
 
 9-774 
 
 0.072 
 
 1.404 
 
 20 
 
 251 
 
 967 
 
 0.709 
 
 1.015 
 
 1-437 
 
 30 
 
 267 
 
 973 
 
 9.825 
 
 0.123 
 
 1.405 
 
 30 
 
 250 
 
 967 
 
 0.715 
 
 1.022 
 
 1-438 
 
 40 
 
 267 
 
 973 
 
 9.871 
 
 0.169 
 
 1.405 
 
 40 
 
 250 
 
 9^7 
 
 0.722 
 
 1.029 
 
 1-439 
 
 SO 
 
 267 
 
 973 
 
 9.912 
 
 0.21 1 
 
 1-405 
 
 SO 
 
 249 
 
 966 
 
 0.728 
 
 1-035 
 
 1.440 
 
 2 00 
 
 267 
 
 972 
 
 9-950 
 
 0.248 
 
 1.405 
 
 12 00 
 
 249 
 
 966 
 
 0.734 
 
 1.042 
 
 1-441 
 
 10 
 
 267 
 
 972 
 
 9.985 
 
 0.283 
 
 1.405 
 
 10 
 
 248 
 
 966 
 
 0.740 
 
 1.048 
 
 1.442 
 
 20 
 
 267 
 
 972 
 
 0.017 
 
 0-31 s 
 
 1.405 
 
 20 
 
 248 
 
 966 
 
 0.746 
 
 1.055 
 
 1.444 
 
 30 
 
 266 
 
 972 
 
 0.047 
 
 0.346 
 
 1.406 
 
 30 
 
 247 
 
 966 
 
 0.752 
 
 1. 061 
 
 1.445 
 
 40 
 
 266 
 
 972 
 
 0.075 
 
 0-374 
 
 1.406 
 
 40 
 
 246 
 
 966 
 
 0.758 
 
 1.067 
 
 1.446 
 
 SO 
 
 266 
 
 972 
 
 O.IOI 
 
 0.400 
 
 1.406 
 
 50 
 
 246 
 
 96s 
 
 0.764 
 
 1-073 
 
 1-447 
 
 300 
 
 266 
 
 972 
 
 0.126 
 
 0.425 
 
 1.406 
 
 1300 
 
 245 
 
 965 
 
 0.770 
 
 1.080 
 
 1.448 
 
 10 
 
 266 
 
 972 
 
 0.150 
 
 0.449 
 
 1.407 
 
 10 
 
 245 
 
 96s 
 
 0.776 
 
 1.086 
 
 1.449 
 
 20 
 
 266 
 
 972 
 
 0.172 
 
 0.471 
 
 1.407 
 
 20 
 
 244 
 
 965 
 
 0.781 
 
 1.092 
 
 1.450 
 
 30 
 
 266 
 
 972 
 
 0.193 
 
 0.492 
 
 1.407 
 
 30 
 
 244 
 
 965 
 
 0.787 
 
 1.097 
 
 1-451 
 
 40 
 
 266 
 
 972 
 
 0.214 
 
 0-513 
 
 1.408 
 
 40 
 
 243 
 
 964 
 
 0.792 
 
 1-103 
 
 1.452 
 
 SO 
 
 266 
 
 972 
 
 0.233 
 
 0.532 
 
 1.408 
 
 SO 
 
 242 
 
 964 
 
 0.798 
 
 1. 109 
 
 1-454 
 
 400 
 
 265 
 
 972 
 
 0.252 
 
 o-SSi 
 
 1.408 
 
 1400 
 
 242 
 
 964 
 
 0.803 
 
 1. 115 
 
 1-455 
 
 10 
 
 26s 
 
 972 
 
 0.269 
 
 0.569 
 
 1.409 
 
 10 
 
 241 
 
 964 
 
 0.809 
 
 1. 120 
 
 1.456 
 
 20 
 
 265 
 
 972 
 
 0.286 
 
 0.586 
 
 1.409 
 
 20 
 
 241 
 
 964 
 
 0.814 
 
 1. 126 
 
 1-457 
 
 30 
 
 265 
 
 972 
 
 0-303 
 
 0.602 
 
 1.409 
 
 30 
 
 240 
 
 963 
 
 0.819 
 
 1.132 
 
 1.458 
 
 40 
 
 26s 
 
 972 
 
 0.319 
 
 0.618 
 
 I.4I0 
 
 40 
 
 239 
 
 963 
 
 0.824 
 
 1-137 
 
 1.460 
 
 SO 
 
 264 
 
 972 
 
 0-334 
 
 0.634 
 
 I.4I0 
 
 SO 
 
 239 
 
 963 
 
 0.830 
 
 1-143 
 
 1. 461 
 
 Soo 
 
 264 
 
 972 
 
 0.349 
 
 0.649 
 
 I.4II 
 
 15 00 
 
 238 
 
 963 
 
 0.835 
 
 1.148 
 
 1.462 
 
 10 
 
 264 
 
 971 
 
 0-363 
 
 0.663 
 
 I.4II 
 
 10 
 
 237 
 
 963 
 
 0.840 
 
 i-i S3 
 
 1-463 
 
 20 
 
 264 
 
 971 
 
 0-377 
 
 0.677 
 
 I.4II 
 
 20 
 
 237 
 
 962 
 
 0.84s 
 
 I.I 59 
 
 1.465 
 
 30 
 
 264 
 
 971 
 
 0.390 
 
 0.691 
 
 I.4I2 
 
 30 
 
 236 
 
 962 
 
 0.850 
 
 1. 164 
 
 1.466 
 
 40 
 
 263 
 
 971 
 
 0.404 
 
 0.704 
 
 I.4I2 
 
 40 
 
 23s 
 
 962 
 
 0.854 
 
 1. 169 
 
 1.467 
 
 SO 
 
 263 
 
 971 
 
 0.416 
 
 0.717 
 
 I-4I3 
 
 SO 
 
 23s 
 
 962 
 
 0.859 
 
 1-174 
 
 1.469 
 
 600 
 
 263 
 
 971 
 
 0.428 
 
 0.729 
 
 1-413 
 
 i6oo 
 
 234 
 
 961 
 
 0.864 
 
 1.179 
 
 1.470 
 
 10 
 
 263 
 
 971 
 
 0.440 
 
 0.741 
 
 I.4I4 
 
 10 
 
 233 
 
 961 
 
 0.869 
 
 1.185 
 
 1.471 
 
 20 
 
 262 
 
 971 
 
 0.452 
 
 0-753 
 
 1-415 
 
 20 
 
 233 
 
 961 
 
 0-873 
 
 1. 190 
 
 1-473 
 
 30 
 
 262 
 
 971 
 
 0.464 
 
 0.764 
 
 I.4IS 
 
 30 
 
 232 
 
 961 
 
 0.878 
 
 1.195 
 
 1-474 
 
 40 
 
 262 
 
 971 
 
 0.475 
 
 0.776 
 
 I.4I6 
 
 40 
 
 231 
 
 961 
 
 0.883 
 
 1.200 
 
 1-475 
 
 SO 
 
 261 
 
 971 
 
 0.485 
 
 0.787 
 
 I.4I6 
 
 SO 
 
 231 
 
 960 
 
 0.887 
 
 1.205 
 
 1-477 
 
 700 
 
 261 
 
 970 
 
 0.496 
 
 0-797 
 
 I.4I7 
 
 17 00 
 
 230 
 
 960 
 
 0.892 
 
 1.210 
 
 1-478 
 
 10 
 
 261 
 
 970 
 
 0.506 
 
 0.808 
 
 I.4I7 
 
 10 
 
 229 
 
 960 
 
 0.896 
 
 1. 214 
 
 1.4S0 
 
 20 
 
 260 
 
 970 
 
 0.516 
 
 0.818 
 
 I.4I8 
 
 20 
 
 228 
 
 960 
 
 0.901 
 
 1. 219 
 
 1. 481 
 
 30 
 
 260 
 
 970 
 
 0.526 
 
 0.828 
 
 1-419 
 
 30 
 
 228 
 
 959 
 
 0.905 
 
 1.224 
 
 1.482 
 
 40 
 
 260 
 
 970 
 
 0.536 
 
 °Pl 
 
 1-419 
 
 40 
 
 227 
 
 959 
 
 0.910 
 
 1.229 
 
 1.484 
 
 SO 
 
 2S9 
 
 970 
 
 0.545 
 
 0.848 
 
 1.420 
 
 SO 
 
 226 
 
 959 
 
 0.914 
 
 1.234 
 
 1.485 
 
 800 
 
 259 
 
 970 
 
 0-SS5 
 
 °-il2 
 
 1. 42 1 
 
 1800 
 
 225 
 
 959 
 
 0.918 
 
 1.238 
 
 1.487 
 
 10 
 
 259 
 
 970 
 
 0.564 
 
 0.866 
 
 I.42I 
 
 10 
 
 225 
 
 958 
 
 0.922 
 
 1.248 
 
 1-489 
 
 20 
 
 258 
 
 970 
 
 0-573 
 
 0.875 
 
 1.422 
 
 20 
 
 224 
 
 958 
 
 0.927 
 
 1.490 
 
 30 
 
 258 
 
 969 
 
 0.581 
 
 0.884 
 
 1-423 
 
 30 
 
 223 
 
 958 
 
 0.931 
 
 1.252 
 
 1-491 
 
 40 
 
 258 
 
 ^^ 
 
 0.590 
 
 0.893 
 
 1.424 
 
 40 
 
 223 
 
 958 
 
 0-93S 
 
 1-257 
 
 1-493 
 
 SO 
 
 2S7 
 
 969 
 
 0.598 
 
 0.902 
 
 1.424 
 
 50 
 
 222 
 
 957 
 
 0-939 
 
 1. 261 
 
 1-495 
 
 9 00 
 
 257 
 
 969 
 
 0.607 
 
 0.910 
 
 1.425 
 
 1900 
 
 221 
 
 957 
 
 0-943 
 
 1.266 
 
 1.496 
 
 10 
 
 256 
 
 969 
 
 0.615 
 
 0.918 
 
 1.426 
 
 10 
 
 220 
 
 957 
 
 0.947 
 
 1.271 
 
 1.498 
 
 20 
 
 256 
 
 969 
 
 0.623 
 
 0.927 
 
 1.427 
 
 20 
 
 219 
 
 957 
 
 0-951 
 
 I-27S 
 
 1-499 
 
 30 
 
 256 
 
 969 
 
 0.630 
 
 0-935 
 
 1.428 
 
 30 
 
 2X8 
 
 955 
 
 0.955 
 
 1.279 
 
 1. 501 
 
 40 
 
 ZSS 
 
 969 
 
 0.638 
 
 0.942 
 
 1.428 
 
 40 
 
 218 
 
 956 
 
 0.959 
 
 1.284 
 
 1.502 
 
 SO 
 
 2SS 
 
 968 
 
 0.646 
 
 0.950 
 
 1.429 
 
 SO 
 
 217 
 
 956 
 
 0.963 
 
 1.288 
 
 1-504 
 
 1000 
 
 254 
 
 968 
 
 0.653 
 
 0.958 
 
 1.430 
 
 20 00 
 
 216 
 
 955 
 
 0.967 
 
 1.293 
 
 1.506 
 
 Smithsonian Tables. 
 
 74 
 
Table 16. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANCULATION. 
 
 UNIT = THE METRE. 
 
 [Derivation and use of table explained on p. Ix.] 
 
 <!> 
 
 ai 
 
 *i=a 
 
 oa 
 
 h 
 
 <^2 
 
 * 
 
 «i 
 
 i5i = ri 
 
 aa 
 
 h 
 
 ^2 
 
 
 20°00' 
 
 8.51216 
 
 8.50955 
 
 0.967 
 
 1.293 
 
 1.506 
 
 30°oo' 
 
 8-S1157 
 
 8.50936 
 
 1. 167 
 
 1.528 
 
 1.625 
 
 
 10 
 
 21S 
 
 9S5 
 
 0.971 
 
 1.297 
 
 1.507 
 
 10 
 
 156 
 
 936 
 
 1. 170 
 
 I-S32 
 
 1.627 
 
 
 20 
 
 214 
 
 9SS 
 
 0.975 
 
 1.301 
 
 1.509 
 
 20 
 
 15s 
 
 935 
 
 1-173 
 
 1-535 
 
 1.630 
 
 
 30 
 
 214 
 
 9SS 
 
 0.979 
 
 1.306 
 
 I.5H 
 
 30 
 
 154 
 
 935 
 
 1. 176 
 
 1-539 
 
 1.632 
 
 
 40 
 
 213 
 
 9S4 
 
 0.983 
 
 1.310 
 
 1.512 
 
 40 
 
 153 
 
 934 
 
 1.178 
 
 1-543 
 
 1-635 
 
 
 5° 
 
 212 
 
 9S4 
 
 0.987 
 
 I-3H 
 
 I.514 
 
 SO 
 
 152 
 
 934 
 
 1. 181 
 
 1.546 
 
 1-637 
 
 
 21 00 
 
 211 
 
 9S4 
 
 0.990 
 
 1-319 
 
 1.516 
 
 31 00 
 
 151 
 
 934 
 
 1.184 
 
 1.550 
 
 1.639 
 
 
 10 
 
 210 
 
 953 
 
 0.994 
 
 1-323 
 
 I.518 
 
 10 
 
 149 
 
 933 
 
 1.187 
 
 I -554 
 
 1.642 
 
 
 20 
 
 209 
 
 9S3 
 
 0.998 
 
 1-327 
 
 I.519 
 
 20 
 
 148 
 
 933 
 
 1. 190 
 
 1-557 
 
 1.644 
 
 
 30 
 
 208 
 
 953 
 
 1.002 
 
 I-33I 
 
 1.521 
 
 30 
 
 147 
 
 933 
 
 1-193 
 
 1.561 
 
 1.646 
 
 
 40 
 
 207 
 
 953 
 
 1.005 
 
 1-336 
 
 1-523 
 
 40 
 
 146 
 
 932 
 
 1-195 
 
 1.564 
 
 1.649 
 
 
 5° 
 
 207 
 
 952 
 
 1.009 
 
 1.340 
 
 1.524 
 
 SO 
 
 145 
 
 932 
 
 1.198 
 
 1.568 
 
 1.651 
 
 
 2200 
 
 206 
 
 952 
 
 1-013 
 
 1.344 
 
 1.526 
 
 3200 
 
 144 
 
 931 
 
 1.201 
 
 1.572 
 
 1.654 
 
 
 10 
 
 20s 
 
 952 
 
 1.016 
 
 1.348 
 
 1.528 
 
 10 
 
 143 
 
 931 
 
 1.204 
 
 1-575 
 
 1.656 
 
 
 20 
 
 204 
 
 951 
 
 1. 020 
 
 1-352 
 
 1-530 
 
 20 
 
 141 
 
 931 
 
 1.207 
 
 1-579 
 
 1.659 
 
 
 30 
 
 203 
 
 951 
 
 1.023 
 
 1-356 
 
 I-S32 
 
 30 
 
 140 
 
 930 
 
 1.209 
 
 1.582 
 
 1.661 
 
 
 40 
 
 202 
 
 951 
 
 1.027 
 
 
 1-534 
 
 40 
 
 139 
 
 930 
 
 1.212 
 
 1.586 
 
 1.664 
 
 
 SO 
 
 201 
 
 9SI 
 
 1.030 
 
 1.364 
 
 I-S3S 
 
 50 
 
 138 
 
 929 
 
 1.215 
 
 1.590 
 
 1.666 
 
 
 2300 
 
 200 
 
 950 
 
 1.034 
 
 1.368 
 
 1-537 
 
 3300 
 
 137 
 
 929 
 
 1.218 
 
 1-593 
 
 1.669 
 
 
 10 
 
 199 
 
 950 
 
 1-037 
 
 1.372 
 
 1-539 
 
 10 
 
 136 
 
 929 
 
 1.220 
 
 I-S97 
 
 1.671 
 
 
 20 
 
 198 
 
 950 
 
 1.041 
 
 1-376 
 
 1.541 
 
 20 
 
 134 
 
 928 
 
 1.223 
 
 1.600 
 
 1.674 
 
 
 30 
 
 197 
 
 949 
 
 1.044 
 
 1.380 
 
 1-543 
 
 30 
 
 133 
 
 928 
 
 . 1.226 
 
 1.604 
 
 1.676 
 
 
 40 
 
 197 
 
 949 
 
 1.048 
 
 1.384 
 
 I-54S 
 
 40 
 
 132 
 
 927 
 
 1.229 
 
 1.607 
 
 1.679 
 
 
 5° 
 
 196 
 
 949 
 
 1.051 
 
 1-388 
 
 1-547 
 
 50 
 
 131 
 
 927 
 
 1.231 
 
 1.611 
 
 1.682 
 
 
 2400 
 
 19s 
 
 948 
 
 1.055 
 
 1.392 
 
 1.549 
 
 3400 
 
 130 
 
 927 
 
 1-234 
 
 1.615 
 1.618 
 
 1.684 
 
 
 10 
 
 194 
 
 948 
 
 1.058 
 
 1.396 
 
 1.550 
 
 10 
 
 128 
 
 926 
 
 1-237 
 
 1.687 
 
 
 20 
 
 193 
 
 948 
 
 1. 061 
 
 1.400 
 
 1.552 
 
 20 
 
 127 
 
 ' 926 
 
 1.239 
 
 1.622 
 
 1.689 
 
 
 30 
 
 192 
 
 947 
 
 1.065 
 1.068 
 
 1.404 
 
 1-554 
 
 30 
 
 126 
 
 925 
 
 1.242 
 
 1.625 
 
 1.692 
 
 
 40 
 
 191 
 
 947 
 
 1.408 
 
 1.556 
 
 40 
 
 125 
 
 925 
 
 1.245 
 
 1.629 
 
 1.695 
 
 
 50 
 
 190 
 
 947 
 
 1. 07 1 
 
 1.412 
 
 1.558 
 
 50 
 
 124 
 
 925 
 
 1.248 
 
 1.632 
 
 1.697 
 
 
 2500 
 
 189 
 
 946 
 
 1.075 
 1.078 
 
 1.416 
 
 1.560 
 
 3500 
 
 122 
 
 924 
 
 1.250 
 
 1.636 
 
 1.700 
 
 
 10 
 
 188 
 
 946 
 
 1.420 
 
 1.562 
 
 10 
 
 121 
 
 924 
 
 1-253 
 
 1.639 
 
 1.702 
 
 
 20 
 
 187 
 
 946 
 
 1.081 
 
 1.424 
 
 1.564 
 
 20 
 
 120 
 
 923 
 
 1.256 
 
 1.643 
 
 1.705 
 
 
 30 
 
 186 
 
 94S 
 
 1.084 
 
 1.427 
 
 1.566 
 
 30 
 
 119 
 
 923 
 
 1.258 
 
 1.647 
 
 1.708 
 
 
 40 
 
 '?s 
 
 945 
 
 1.088 
 
 1-431 
 
 1.568 
 
 40 
 
 118 
 
 923 
 
 1.261 
 
 1.650 
 
 1.7 1 1 
 
 
 SO 
 
 184 
 
 945 
 
 1.091 
 
 I-43S 
 
 I-S70 
 
 SO 
 
 116 
 
 922 
 
 1.264 
 
 1.654 
 
 1-713 
 
 
 2600 
 
 'f3 
 
 944 
 
 1.094 
 
 1-439 
 
 1.572 
 
 3600 
 
 "5 
 
 922 
 
 1.266 
 
 1.657 
 
 1. 7 16 
 
 
 to 
 
 182 
 
 944 
 
 1.097 
 
 1-443 
 
 1-575 
 
 10 
 
 114 
 
 921 
 
 1.269 
 
 1.661 
 
 1.719 
 
 
 20 
 
 181 
 
 944 
 
 1. 100 
 
 1.447 
 
 1-577 
 
 20 
 
 "3 
 
 921 
 
 1.271 
 
 1.664 
 
 1.721 
 
 
 30 
 
 180 
 
 943 
 
 1. 104 
 
 1.450 
 
 1-579 
 
 30 
 
 III 
 
 921 
 
 1.274 
 
 1.668 
 
 1.724 
 
 
 40 
 
 179 
 
 943 
 
 1.107 
 
 1.454 
 
 1.581 
 
 40 
 
 110 
 
 920 
 
 1.277 
 
 1.672 
 
 1.727 
 
 
 SO 
 
 178 
 
 943 
 
 I.IIO 
 
 1.458 
 
 1-583 
 
 SO 
 
 109 
 
 920 
 
 1.279 
 
 1.675 
 
 1-730 
 
 
 27 00 
 
 177 
 
 942 
 
 1. 113 
 
 1.462 
 
 1-58S 
 
 3700 
 
 108 
 
 919 
 
 1.282 
 
 1.679 
 
 1-732 
 
 
 10 
 
 176 
 
 942 
 
 1. 116 
 
 1.465 
 
 1.587 
 
 10 
 
 106 
 
 919 
 
 1.285 
 
 1.682 
 
 I-73S 
 
 
 20 
 
 17s 
 
 942 
 
 1.119 
 
 1.469 
 
 1.589 
 
 20 
 
 105 
 
 919 
 
 1.287 
 
 1.686 
 
 1-738 
 
 
 30 
 
 174 
 
 941 
 
 1.122 
 
 1-473 
 
 1-591 
 
 30 
 
 104 
 
 918 
 
 1.290 
 
 1.689 
 
 1.741 
 
 
 40 
 
 172 
 
 941 
 
 1. 125 
 
 1-477 
 
 
 40 
 
 103 
 
 918 
 
 1.292 
 
 1.693 
 
 1.744 
 
 
 SO 
 
 171 
 
 941 
 
 1.128 
 
 1.480 
 
 1.596 
 
 SO 
 
 102 
 
 917 
 
 1.295 
 
 1.697 
 
 1-747 
 
 
 2800 
 
 170 
 
 940 
 
 1.131 
 
 1.484 
 
 1.598 
 
 3800 
 
 100 
 
 917 
 
 1.298 
 
 1.700 
 
 1.749 
 
 
 10 
 
 169 
 
 940 
 
 1-134 
 
 1.488 
 
 1.600 
 
 10 
 
 099 
 
 916 
 
 1.300 
 
 1.704 
 
 1-752 
 
 
 20 
 
 168 
 
 940 
 
 I-I37 
 
 1.492 
 
 1.602 
 
 20 
 
 098 
 
 916 
 
 1-303 
 
 1.707 
 
 1-755 
 
 
 30 
 
 167 
 
 939 
 
 1.140 
 
 1.495 
 
 1.605 
 
 30 
 
 097 
 
 916 
 
 1-30S 
 
 1.7U 
 
 1.758 
 
 
 40 
 
 166 
 
 939 
 
 I-I43 
 
 1.499 
 
 1.607 
 
 40 
 
 095 
 
 91 S 
 
 1.308 
 
 1.715 
 1.718 
 
 1.761 
 
 
 SO 
 
 i6s 
 
 
 1.146 
 
 1-503 
 
 1.609 
 
 SO 
 
 094 
 
 915 
 
 1.310 
 
 1.764 
 
 
 2900 
 
 Z64 
 
 938 
 
 1.149 
 
 1.506 
 
 1.611 
 
 3900 
 
 093 
 
 914 
 
 1-313 
 
 1.722 
 
 1.767 
 
 
 10 
 
 163 
 
 938 
 
 1.152 
 
 1.510 
 
 1.614 
 
 10 
 
 092 
 
 914 
 
 1.316 
 
 1.725 
 
 1.770 
 
 
 20 
 
 162 
 
 937 
 
 1.155 
 
 1.514 
 
 i.6i6 
 
 20 
 
 090 
 
 914 
 
 1-318 
 
 1.729 
 
 1-773 
 
 
 30 
 
 161 
 
 937 
 
 1.158 
 
 1.517 
 
 1.618 
 
 30 
 
 089 
 
 913 
 
 1-321 
 
 1-733 
 
 1.776 
 
 
 40 
 
 i6o 
 
 937 
 
 1. 161 
 
 1.521 
 
 1.620 
 
 40 
 
 088 
 
 913 
 
 1-323 
 
 1-736 
 
 1.779 
 
 
 SO 
 
 158 
 
 936 
 
 1. 164 
 
 1.525 
 
 1-623 
 
 SO 
 
 086 
 
 912 
 
 1.326 
 
 1.740 
 
 1.781 
 
 
 3000 
 
 IS7 
 
 936 
 
 1.167 
 
 1.528 
 
 1-625 
 
 4000 
 
 085 
 
 912 
 
 1.328 
 
 1-743 
 
 1.784 
 
 
 Gmithsonian Tables. 
 
 75 
 
Table 1 6. 
 LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION. 
 
 UNIT=THE METRE. 
 [Derivation and use of table explained on p. Ix.] 
 
 I- 
 
 ai 
 
 h=ci 
 
 02 
 
 ii 
 
 C2 
 
 * 
 
 01 
 
 *i = ^i 
 
 02 
 
 h 
 
 £2 
 
 40°oo' 
 
 8.51085 
 
 8.50912 
 
 1.328 
 
 1-743 
 
 1.784 
 
 So°oo' 
 
 8.51008 
 
 8.50886 
 
 1.480 
 
 1.971 
 
 1.987 
 
 10 
 
 084 
 
 911 
 
 '•33' 
 
 1-747 
 
 1.787 
 
 10 
 
 007 
 
 886 
 
 1.482 
 
 1.975 
 
 1.990 
 
 20 
 
 083 
 
 911 
 
 1-333 
 
 1.751 
 
 1.790 
 
 20 
 
 006 
 
 885 
 
 1.485 
 
 1.980 
 
 1.994 
 
 3° 
 
 081 
 
 911 
 
 I '336 
 
 I-7S4 
 
 1-793 
 
 30 
 
 005 
 
 ff5 
 
 1.487 
 
 ■•9?^ 
 
 1.998 
 
 40 
 
 080 
 
 910 
 
 1-338 
 
 1.758 
 
 1.797 
 
 40 
 
 003 
 
 f§5 
 
 1.490 
 
 1.988 
 
 2.002 
 
 SO 
 
 079 
 
 910 
 
 1-341 
 
 1.762 
 
 1.800 
 
 SO 
 
 002 
 
 884 
 
 1.492 
 
 1.992 
 
 2.006 
 
 41 oo 
 
 078 
 
 909 
 
 1-344 
 
 1.765 
 
 1.803 
 
 51 00 
 
 001 
 
 884 
 
 1.49s 
 1.498 
 
 1.996 
 
 2.010 
 
 10 
 
 076 
 
 909 
 
 1.346 
 
 1.769 
 
 1.806 
 
 10 
 
 000 
 
 883 
 
 2.000 
 
 2.014 
 
 20 
 
 07s 
 
 908 
 
 1-349 
 
 1.772 
 
 1.809 
 
 20 
 
 8.50998 
 
 883 
 
 1.500 
 
 2.004 
 
 2.017 
 
 3° 
 
 074 
 
 908 
 
 I-3SI 
 
 1.776 
 
 I.8l2 
 
 30 
 
 997 
 
 882 
 
 1-S03 
 
 2.008 
 
 2.021 
 
 40 
 
 072 
 
 908 
 
 I-3S4 
 
 1.780 
 
 I.8I5 
 
 40 
 
 996 
 
 882 
 
 1.50S 
 1.508 
 
 2.013 
 
 2.025 
 
 SO 
 
 071 
 
 907 
 
 I-3S6 
 
 1-783 
 
 I.8I8 
 
 SO 
 
 994 
 
 882 
 
 2.017 
 
 2.029 
 
 42 00 
 
 070 
 
 907 
 
 I-3S9 
 
 1.787 
 
 1. 82 1 
 
 52 00 
 
 993 
 
 881 
 
 1.510 
 
 2.021 
 
 2.033 
 
 10 
 
 069 
 
 906 
 
 1.361 
 
 1.791 
 
 1.824 
 
 10 
 
 992 
 
 881 
 
 1-S13 
 
 2.025 
 
 2.037 
 
 20 
 
 067 
 
 906 
 
 1.364 
 
 1.794 
 
 1.828 
 
 20 
 
 991 
 
 880 
 
 1.516 
 
 2.030 
 
 2.041 
 
 30 
 
 066 
 
 90s 
 
 1.366 
 
 1.798 
 
 1. 83 1 
 
 30 
 
 990 
 
 880 
 
 1.518 
 
 2.034 
 
 2.045 
 
 40 
 
 065 
 
 90s 
 
 1.369 
 
 1.802 
 
 1.834 
 
 40 
 
 988 
 
 880 
 
 1. 521 
 
 2.038 
 
 2.049 
 
 5° 
 
 063 
 
 90s 
 
 I-37I 
 
 1.805 
 
 1-837 
 
 SO 
 
 987 
 
 879 
 
 I-S23 
 
 2.042 
 
 2.053 
 
 4300 
 
 062 
 
 904 
 
 1-374 
 
 1.809 
 
 1.840 
 
 S3 00 
 
 986 
 
 879 
 
 1.526 
 
 2.047 
 
 2.057 
 
 10 
 
 o6i 
 
 904 
 
 1-376 
 
 1.813 
 
 1.843 
 
 10 
 
 985 
 
 878 
 
 1.529 
 
 2.051 
 
 2.062 
 
 20 
 
 060 
 
 903 
 
 1-379 
 
 1.817 
 
 1.847 
 
 20 
 
 983 
 
 878 
 
 I-S3' 
 
 2.05s 
 
 2.066 
 
 30 
 
 058 
 
 903 
 
 1.381 
 
 1.820 
 
 1.850 
 
 30 
 
 982 
 
 877 
 
 I-S34 
 
 2.060 
 
 2.070 
 
 40 
 
 057 
 
 902 
 
 1.384 
 
 1.824 
 
 I-8S3 
 
 40 
 
 981 
 
 877 
 
 I-S37 
 
 2.064 
 
 2.074 
 
 SO 
 
 056 
 
 902 
 
 1.386 
 
 1.828 
 
 1.856 
 
 SO 
 
 980 
 
 877 
 
 I-S39 
 
 2.068 
 
 2.078 
 
 4400 
 
 054 
 
 902 
 
 1.389 
 
 1.832 
 
 1.860 
 
 54 00 
 
 978 
 
 876 
 
 1.542 
 
 2.073 
 
 2.082 
 
 10 
 
 053 
 
 901 
 
 I -391 
 
 1.83s 
 
 1.863 
 
 10 
 
 977 
 
 876 
 
 I-S44 
 
 2.077 
 
 2.086 
 
 20 
 
 052 
 
 901 
 
 '-394 
 
 1.839 
 
 1.866 
 
 20 
 
 976 
 
 87s 
 
 1-S47 
 
 2.081 
 
 2.091 
 
 30 
 
 051 
 
 900 
 
 1.396 
 
 1.843 
 
 1.870 
 
 30 
 
 97S 
 
 87s 
 
 i-SSO 
 
 2.086 
 
 2.09s 
 
 40 
 
 049 
 
 goo 
 
 1-399 
 
 1.847 
 
 '■f73 
 
 40 
 
 973 
 
 87s 
 
 I-SS2 
 
 2.090 
 
 2.099 
 
 SO 
 
 048 
 
 899 
 
 1. 401 
 
 1.850 
 
 1.876 
 
 SO 
 
 972 
 
 874 
 
 i-SSS 
 
 2.095 
 
 2.104 
 
 4500 
 
 047 
 
 899 
 
 1.404 
 
 1.854 
 
 1.880 
 
 5500 
 
 971 
 
 874 
 
 '-558 
 
 2.099 
 
 2.108 
 
 10 
 
 04S 
 
 899 
 
 1.407 
 
 1-858 
 
 1.883 
 
 10 
 
 970 
 
 873 
 
 1.560 
 
 2.104 
 
 2.112 
 
 20 
 
 044 
 
 898 
 
 1.409 
 
 1.862 
 
 1.886 
 
 20 
 
 969 
 
 873 
 
 1-563 
 
 2.108 
 
 2.1 16 
 
 30 
 
 043 
 
 898 
 
 1.412 
 
 1.86s 
 
 1.890 
 
 30 
 
 967 
 
 873 
 
 1.566 
 
 2.113 
 
 2.I2I 
 
 40 
 
 042 
 
 897 
 
 1-414 
 
 1.869 
 
 1.893 
 
 40 
 
 966 
 
 872 
 
 1.568 
 
 2.117 
 
 2.125 
 
 SO 
 
 040 
 
 897 
 
 1.417 
 
 1-873 
 
 1.897 
 
 SO 
 
 965 
 
 872 
 
 1.571 
 
 2.122 
 
 2.130 
 
 4600 
 
 039 
 
 ^96 
 
 1.419 
 
 'ir 
 
 1.900 
 
 56 00 
 
 964 
 
 871 
 
 I-S74 
 
 2.126 
 
 2.134 
 
 lO 
 
 038 
 
 l^. 
 
 1.422 
 
 1.881 
 
 1.903 
 
 10 
 
 963 
 
 871 
 
 1-577 
 
 2.131 
 
 2.138 
 
 20 
 
 036 
 
 896 
 
 1.424 
 
 1.885 
 
 1.907 
 
 20 
 
 961 
 
 871 
 
 1-S79 
 
 2.136 
 
 2.143 
 
 30 
 
 03s 
 
 89s 
 
 1.427 
 
 1.888 
 
 1.910 
 
 30 
 
 960 
 
 870 
 
 1.582 
 
 2.140 
 
 2.147 
 
 40 
 
 034 
 
 89s 
 
 1.429 
 
 1.892 
 
 1.914 
 
 40 
 
 9S9 
 
 870 
 
 \-$^ 
 
 2.145 
 
 2.152 
 
 so 
 
 033 
 
 894 
 
 1.432 
 
 1.896 
 
 1.917 
 
 SO 
 
 958 
 
 869 
 
 2.150 
 
 2.156 
 
 4700 
 
 031 
 
 894 
 
 1-434 
 
 1.900 
 
 1.921 
 
 5700 
 
 9S7 
 
 869 
 
 1.590 
 
 2.154 
 
 2.161 
 
 10 
 
 030 
 
 893 
 
 1-437 
 
 1.904 
 
 1.924 
 
 10 
 
 956 
 
 869 
 
 I-S93 
 
 2.159 
 
 2.166 
 
 20 
 
 029 
 
 893 
 
 1-439 
 
 1.908 
 
 1.928 
 
 20 
 
 9S4 
 
 868 
 
 1.596 
 
 2.164 
 
 2.170 
 
 30 
 
 027 
 026 
 
 893 
 
 1.442 
 
 1.912 
 
 1.932 
 
 30 
 
 9S3 
 
 868 
 
 1.599 
 
 2.169 
 
 2.I7S 
 
 40 
 
 892 
 
 1.444 
 
 1.916 
 
 I -935 
 
 40 
 
 952 
 
 867 
 
 1.601 
 
 2.173 
 2.178 
 
 2.179 
 
 SO 
 
 025 
 
 892 
 
 1.447 
 
 1.920 
 
 1-939 
 
 SO 
 
 9SI 
 
 867 
 
 1.604 
 
 2.184 
 
 48 00 
 
 024 
 
 891 
 
 1.449 
 
 1.923 
 
 1.942 
 
 5800 
 
 950 
 
 867 
 
 1.607 
 
 2.183 
 2.188 
 
 2.189 
 
 10 
 
 022 
 
 891 
 
 1.452 
 
 1.927 
 
 1.946 
 
 10 
 
 949 
 
 866 
 
 1.610 
 
 2.193 
 2.1^ 
 
 20 
 
 021 
 
 890 
 
 1.454 
 
 1-931 
 
 1.950 
 
 20 
 
 947 
 
 866 
 
 1.613 
 
 2.193 
 
 30 
 
 020 
 
 890 
 
 I-4S7 
 
 J-93S 
 
 I-9S3 
 
 30 
 
 946 
 
 866 
 
 1.615 
 
 2.197 
 
 2.203 
 2.208 
 
 40 
 
 019 
 
 «f 
 
 •■459 
 
 1-939 
 
 I-9S7 
 
 40 
 
 94S 
 
 865 
 
 1.618 
 
 2.202 
 
 SO 
 
 017 
 
 889 
 
 1.462 
 
 1-943 
 
 1. 961 
 
 SO 
 
 944 
 
 86^ 
 
 1.621 
 
 2.207 
 
 2.213 
 
 4900 
 
 016 
 
 889 
 888 
 888 
 
 1.464 
 
 1.947 
 
 1.964 
 
 59 00 
 
 943 
 
 864 
 
 1.624 
 
 2.212 
 
 2.217 
 
 10 
 
 oiS 
 
 1.467 
 
 1.951 
 
 1.968 
 
 10 
 
 942 
 
 864 
 
 1.627 
 
 2.217 
 
 2.222 
 
 20 
 
 013 
 
 1.469 
 
 I-9SS 
 
 1.972 
 
 20 
 
 941 
 
 864 
 
 1.630 
 
 2.222 
 
 2.227 
 
 30 
 40 
 SO 
 
 012 
 on 
 010 
 
 888 
 887 
 887 
 
 1.472 
 I-47S 
 1-477 
 
 1.959 
 1.963 
 1.967 
 
 I-97S 
 1.979 
 
 1.983 
 
 30 
 40 
 SO 
 
 939 
 938 
 937 
 
 863 
 863 
 863 
 
 1.632 
 
 2.227 
 2.232 
 
 2.2-;7 
 
 2.232 
 2.237 
 2.242 
 
 5000 
 
 008 
 
 886 
 
 1.480 1.97 1 
 
 1.987 
 
 6000 
 
 936 
 
 862 
 
 1.641 2.242 
 
 2.247 
 
 
 
 Smithsonian Tables. 
 
 76 
 
Table 16. 
 LOCARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- 
 TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION. 
 
 UNIT = THE METRE. 
 
 [Derivation and use of tabic explained on p. Ix.] 
 
 
 
 "1 
 
 il = ci 
 
 02 
 
 h 
 
 Ci 
 
 t- 
 
 fli 
 
 h=ci 
 
 aa 
 
 h 
 
 ^2 
 
 6o°oo' 
 
 8.50936 
 
 8.50862 
 
 1.641 
 
 2.242 
 
 2.247 
 
 70°oo' 
 
 8.50877 
 
 5.50842 
 
 1.841 
 
 2.607 
 
 2.608 
 
 10 
 
 93S 
 
 862 
 
 1.644 
 
 2.247 
 
 2.252 
 
 10 
 
 876 
 
 842 
 
 1.84s 
 
 2.615 
 
 2.616 
 
 20 
 
 934 
 
 861 
 
 1.647 
 
 2.253 
 
 2.257 
 
 20 
 
 875 
 
 842 
 
 1.849 
 
 2.622 
 
 2.623 
 
 3° 
 
 933 
 
 861 
 
 1.650 
 
 2.258 
 
 2.262 
 
 30 
 
 875 
 
 842 
 
 1-853 
 
 2.630 
 
 2.631 
 
 40 
 
 932 
 
 861 
 
 1-653 
 
 2.263 
 2,268 
 
 2.267 
 
 40 
 
 874 
 
 841 
 
 1-857 
 
 2.637 
 
 2.638 
 
 5° 
 
 931 
 
 860 
 
 1.656 
 
 2.272 
 
 50 
 
 873 
 
 841 
 
 1.861 
 
 2.645 
 
 2.646 
 
 6i 00 
 
 929 
 
 860 
 
 1.659 
 
 2.273 
 
 2.277 
 
 71 00 
 
 872 
 
 841 
 
 1.865 
 
 2-653 
 
 2-653 
 
 lO 
 
 928 
 
 860 
 
 1.662 
 
 2.279 
 
 2.283 
 
 10 
 
 871 
 
 841 
 
 1.869 
 
 2.661 
 
 2.661 
 
 20 
 
 927 
 
 859 
 
 1.665 
 
 2.284 
 
 2.288 
 
 20 
 
 871 
 
 840 
 
 1-873 
 
 2.668 
 
 2.669 
 
 3° 
 
 926 
 
 859 
 
 1.668 
 
 2.289 
 
 2.293 
 
 3° 
 
 870 
 
 840 
 
 1.877 
 
 2.676 
 
 2.677 
 
 40 
 
 925 
 
 fsf 
 
 1.67 1 
 
 2.29s 
 
 2.298 
 
 40 
 
 869 
 
 840 
 
 1.881 
 
 2.684 
 
 2.685 
 
 5° 
 
 924 
 
 858 
 
 1.674 
 
 2.300 
 
 2.303 
 
 50 
 
 868 
 
 840 
 
 1.886 
 
 2.692 
 
 2-693 
 
 62 00 
 
 923 
 
 858 
 
 1.677 
 
 2.305 
 
 2.309 
 
 72 00 
 
 868 
 
 839 
 
 1.890 
 
 2.701 
 
 2.701 
 
 10 
 
 922 
 
 fS7 
 
 1. 680 
 
 2.31 1 
 
 2.314 
 
 10 
 
 867 
 
 839 
 
 1.894 
 
 2.709 
 
 2.709 
 
 20 
 
 921 
 
 857 
 
 1.683 
 
 2.316 
 
 2.320 
 
 20 
 
 866 
 
 839 
 
 1.898 
 
 2.717 
 
 2.718 
 
 3° 
 
 920 
 
 ^57 
 
 1.686 
 
 2.322 
 
 2.325 
 
 30 
 
 86s 
 
 ^39 
 
 1.903 
 
 2.725 
 
 2.726 
 
 40 
 
 919 
 
 856 
 
 1.689 
 
 2.327 
 
 2.330 
 
 40 
 
 l^^ 
 
 ^3f 
 
 1.907 
 
 2.734 
 
 2.734 
 
 5° 
 
 918 
 
 856 
 
 1.692 
 
 2.333 
 
 2-336 
 
 5° 
 
 864 
 
 838 
 
 I 912 
 
 2.742 
 
 2.742 
 
 6300 
 
 917 
 
 856 
 
 1.69s 
 1.698 
 
 2-338 
 
 2.341 
 
 7300 
 
 ff3 
 
 838 
 
 1.916 
 
 2-751 
 
 2.751 
 
 10 
 
 916 
 
 855 
 
 2.344 
 
 2-347 
 
 10 
 
 862 
 
 838 
 
 1.921 
 
 2.760 
 
 2.760 
 
 20 
 
 91 S 
 
 85s 
 
 1.701 
 
 2.350 
 
 2.352 
 
 20 
 
 862 
 
 837 
 
 1.925 
 
 2.769 
 
 2.769 
 
 30 
 
 913 
 
 85s 
 
 1.704 
 
 2-3SS 
 
 2.358 
 
 30 
 
 861 
 
 837 
 
 1-930 
 
 2.777 
 
 2.778 
 
 40 
 
 912 
 
 854 
 
 1.708 
 
 2.361 
 
 2.364 
 
 40 
 
 860 
 
 837 
 
 1-935 
 
 2.786 
 
 2-787 
 
 5° 
 
 911 
 
 854 
 
 1.711 
 
 2.367 
 
 2.369 
 
 50 
 
 860 
 
 837 
 
 1-939 
 
 2.795 
 
 2.796 
 
 6400 
 
 910 
 
 854 
 
 1.714 
 
 2.373 
 
 2.375 
 
 7400 
 
 ^59 
 
 |3f 
 
 1.944 
 
 2.804 
 
 2.805 
 
 10 
 
 909 
 
 853 
 
 1.717 
 
 2.378 
 
 2.381 
 
 10 
 
 858 
 
 836 
 
 1.949 
 
 2.814 
 
 2.814 
 
 20 
 
 908 
 
 8S3 
 
 1.720 
 
 2.384 
 
 2.387 
 
 20 
 
 858 
 
 836 
 
 1.954 
 
 2.823 
 
 2.823 
 
 30 
 
 907 
 
 853 
 
 1.724 
 
 2.390 
 
 2.392 
 
 30 
 
 ^57 
 
 836 
 
 1.958 
 
 2.832 
 
 2-833 
 
 40 
 
 906 
 
 852 
 
 1.727 
 
 2.396 
 
 2.398 
 
 40 
 
 ^5^ 
 
 836 
 
 'i:g 
 
 2.842 
 
 2.842 
 
 5° 
 
 905 
 
 852 
 
 1730 
 
 2.402 
 
 2.404 
 
 5° 
 
 856 
 
 835 
 
 2.851 
 
 2.852 
 
 6500 
 
 904 
 
 852 
 
 1-733 
 
 2.408 
 
 2.410 
 
 7500 
 
 ^55 
 
 83s 
 
 1-973 
 
 2.861 
 
 2.861 
 
 10 
 
 903 
 
 851 
 
 1-737 
 
 2.414 
 
 2.416 
 
 10 
 
 854 
 
 f3S 
 
 1.978 
 
 2.871 
 
 Hi' 
 
 20 
 
 902 
 
 851 
 
 1.740 
 
 ?„i\?.o 
 
 2.422 
 
 20 
 
 854 
 
 835 
 
 1.984 
 
 2.881 
 
 2.881 
 
 3° 
 
 901 
 
 851 
 
 1-743 
 
 2.426 
 
 2.428 
 
 3° 
 
 i" 
 
 834 
 
 1.989 
 
 2-891 
 
 2.891 
 
 40 
 
 900 
 
 850 
 
 1-747 
 
 2.432 
 
 2-434 
 
 40 
 
 852 
 
 834 
 
 1.994 
 
 2.901 
 
 2.901 
 
 50 
 
 900 
 
 850 
 
 1.750 
 
 2.438 
 
 2.440 
 
 5° 
 
 852 
 
 834 
 
 1.999 
 
 2.911 
 
 2.912 
 
 6600 
 
 
 850 
 
 I -7 S3 
 
 2.445 
 
 2.446 
 
 7600 
 
 ^5' 
 
 834 
 
 2.005 
 
 2.922 
 
 2.922 
 
 10 
 
 898 
 
 849 
 
 I-7S7 
 
 2.451 
 
 2-453 
 
 10 
 
 ?5' 
 
 834 
 
 2-010 
 
 2.932 
 
 2-933 
 
 20 
 
 897 
 
 849 
 
 1.760 
 
 2.457 
 
 2.459 
 
 20 
 
 850 
 
 833 
 
 2.015 
 
 2.943 
 
 2.943 
 
 30 
 
 896 
 
 849 
 
 1.764 
 
 2.464 
 
 2.465 
 
 30 
 
 849 
 
 833 
 
 2.021 
 
 2-954 
 
 2-954 
 
 40 
 
 89s 
 
 848 
 
 1.767 
 
 2.470 
 
 2472 
 
 40 
 
 849 
 
 833 
 
 2.027 
 
 2.96s 
 
 2.965 
 
 5° 
 
 8^ 
 
 848 
 
 I -77 1 
 
 2.476 
 
 2.478 
 
 50 
 
 848 
 
 833 
 
 2.032 
 
 2.976 
 
 2.976 
 
 67 00 
 
 893 
 
 848 
 
 1-774 
 
 2.483 
 
 2.484 
 
 7700 
 
 848 
 
 833 
 
 2.038 
 
 2.987 
 
 2.987 
 
 10 
 
 892 
 
 847 
 
 1.778 
 
 2.489 
 
 2.491 
 
 10 
 
 847 
 
 832 
 
 2-044 
 
 2.998 
 
 2.998 
 
 20 
 
 891 
 
 847 
 
 1.781 
 
 2.496 
 
 2.497 
 
 20 
 
 847 
 
 832 
 
 2.050 
 
 3.010 
 
 3-010 
 
 30 
 
 890 
 
 847 
 
 1.785 
 1.788 
 
 2.502 
 
 2.504 
 
 30 
 
 846 
 
 832 
 
 2-056 
 
 3.021 
 
 3.021 
 
 40 
 
 889 
 
 847 
 
 2.509 
 
 2.510 
 
 40 
 
 845 
 
 832 
 
 2.062 
 
 3-033 
 
 3-033 
 
 50 
 
 888 
 
 846 
 
 1.792 
 
 2.516 
 
 2.517 
 
 50 
 
 845 
 
 832 
 
 2.068 
 
 3-045 
 
 3-045 
 
 6800 
 
 887 
 
 846 
 
 I-79S 
 
 2.522 
 
 2.524 
 
 78 00 
 
 844 
 
 832 
 
 2.074 
 
 3-057 
 
 3-057 
 
 10 
 
 887 
 
 846 
 
 1.799 
 
 2.529 
 
 2-531 
 
 10 
 
 844 
 
 831 
 
 2.080 
 
 3-069 
 
 3.069 
 
 20 
 
 886 
 
 845 
 
 1.803 
 
 2.536 
 
 2-537 
 
 20 
 
 843 
 
 831 
 
 2.086 
 
 3.082 
 
 3-082 
 
 30 
 
 88s 
 
 845 
 
 1.806 
 
 2-543 
 
 2.544 
 
 30 
 
 843 
 
 !3' 
 
 2-093 
 
 3-094 
 
 3-094 
 
 40 
 
 884 
 
 845 
 
 1.810 
 
 2.550 
 
 2-55' 
 
 40 
 
 842 
 
 P' 
 
 2.099 
 
 3-107 
 
 3-107 
 
 SO 
 
 883 
 
 844 
 
 1.814 
 
 2-557 
 
 2.558 
 
 SO 
 
 842 
 
 831 
 
 2.106 
 
 3.120 
 
 3.120 
 
 6900 
 
 882 
 
 844 
 
 1.818 
 
 2.564 
 
 2.565 
 
 7900 
 
 841 
 
 831 
 
 2.113 
 
 3-133 
 
 3-133 
 
 10 
 
 881 
 
 844 
 
 1.821 
 
 2.571 
 
 2.572 
 
 10 
 
 841 
 
 830 
 
 2.II9 
 
 3.146 
 
 3-146 
 
 20 
 
 880 
 
 844 
 
 1.825 
 
 2.578 
 
 2.579 
 
 20 
 
 840 
 
 830 
 
 2.126 
 
 3.160 
 
 3-160 
 
 30 
 
 880 
 
 843 
 
 1.829 
 
 2-585 
 
 2.586 
 
 30 
 
 840 
 
 830 
 
 2-133 
 
 3-'^^ 
 
 3-'^t 
 
 40 
 
 879 
 
 843 
 
 1-833 
 
 2-593 
 
 2.594 
 
 40 
 
 839 
 
 830 
 
 2.140 
 
 3.188 
 
 3.188 
 
 50 
 
 878 
 
 843 
 
 1-837 
 
 2.600 
 
 2.601 
 
 50 
 
 839 
 
 830 
 
 2.148 
 
 3.202 
 
 3.202 
 
 7000 
 
 877 
 
 842 
 
 1.841 
 
 2.607 
 
 2.608 
 
 80 00 
 
 839 
 
 830 
 
 2.155 
 
 3.216 
 
 3.216 
 
 Smithsonian Tables. 
 
 77 
 
Table 1 7. 
 
 LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN. 
 
 [Derivation of table explained on p. xlvi.] 
 
 Latitude 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Interval. 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 lo" 
 
 1007.66 
 
 1007.66 
 
 vxyj.iyj 
 
 1007.68 
 
 1007.71 
 
 ■ 20 
 
 2015.31 
 
 2015.32 
 
 2015.34 
 
 2015.37 
 
 2015.41 
 
 30 
 
 3022.97 
 
 3022.98 
 
 3023.01 
 
 3023.06 
 
 3023.12 
 
 40 
 
 4030.63 
 
 4030.64 
 
 4030.68 
 
 4030.74 
 
 4030.83 
 
 50 
 
 5038.28 
 
 5038.30 
 
 5038.35 
 
 5038.42 
 
 5038.54 
 
 6a 
 
 6045.94 
 
 6045.96 
 
 6046.02 
 
 6046. 11 
 
 6046.24 
 
 lO* 
 
 60450.4 
 
 60459.6 
 
 60460.2 
 
 60461.1 
 
 60462.4 
 
 20 
 
 120918.8 
 
 120919.2 
 
 120920.4 
 
 120922.2 
 
 120924.8 
 
 30 
 
 181378.3 
 
 181378.8 
 
 181380.6 
 
 181383.3 
 
 181387,3 
 
 40 
 
 241837.7 
 
 241838.4 
 
 241840.8 
 
 241844.4 
 
 241849,7 
 
 50 
 
 302297.1 
 
 302298.0 
 
 302301.0 
 
 302305.5 
 
 302312,1 
 
 60 
 
 362756.5 
 
 362757.6 
 
 362761.2 
 
 362766.6 
 
 362774.5 
 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 10" 
 
 1007.73 
 
 1007.77 
 
 1007.81 
 
 1007.86 
 
 1007,91 
 
 20 
 
 2015.47 
 
 2015.54 
 
 2015.62 
 
 2015.71 
 
 2015,82 
 
 30 
 
 3023.20 
 
 3023.31 
 
 3023.43 
 
 3023.56 
 
 3023.72 
 
 40 
 
 4030.94 
 
 4031.08 
 
 4031.24 
 
 4031.42 
 
 4031.63 
 
 50 
 
 5038.67 
 
 5038.84 
 
 5089.04 
 
 5039.28 
 
 5039,54 
 
 60 
 
 6046.41 
 
 6046.61 
 
 6046.85 
 
 6047.13 
 
 6047.45 
 
 10' 
 
 60464. z 
 
 60466.1 
 
 60468.5 
 
 60471.3 
 
 60474.5 
 
 20 
 
 I2092S.2 
 
 120932.3 
 
 I20937.t 
 
 120942.6 
 
 120949,0 
 
 30 
 
 181392.3 
 
 18139S.4 
 
 181405.6 
 
 181413.9 
 
 181423.4 
 
 40 
 
 241856.4 
 
 241864.6 
 
 241874.2 
 
 241885.2 
 
 241897.9 
 
 50 
 
 302320.5 
 
 302330.7 
 
 302342.7 
 
 302356.5 
 
 302372.4 
 
 60 
 
 362784.6 
 
 362796.8 
 
 362811.2 
 
 362827.8 
 
 362846.9 
 
 
 10° 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 to" 
 
 1007.97 
 
 1008.03 
 
 1008.10 
 
 1008.18 
 
 1008.26 
 
 20 
 
 2015.93 
 
 2016.06 
 
 2016. 20 
 
 2016.35 
 
 2016.51 
 
 .30 
 
 3023.90 
 
 3024.09 
 
 3024.30 
 
 3024.52 
 
 3024.77 
 
 40 
 
 4031.86 
 
 4032.12 
 
 4032.40 
 
 4032.70 
 
 4033.02 
 
 50 
 
 5039-83 
 
 5040.15 
 
 5040.50 
 
 5040.88 
 
 5041.28 
 
 60 
 
 6047.80 
 
 6048.18 
 
 6048.60 
 
 6049.05 
 
 6049,54 
 
 lol 
 
 60478.0 
 
 60481.8 
 
 60486.0 
 
 60490.5 
 
 60495.4 
 
 20 
 
 120955.9 
 
 120963.6 
 
 120972.0 
 
 120981.0 
 
 120990.7 
 
 30 
 
 181433.9 
 
 181445.4 
 
 181458.0 
 
 181471.5 
 
 181486.1 
 
 40 
 
 241911.8 
 
 241927.2 
 
 241944.0 
 
 241962.0 
 
 241981.4 
 302476.8 
 
 1° 
 
 302389.8 
 
 302409.0 
 
 302430.0 
 
 302452,5 
 
 60 
 
 362867.8 
 
 362890.8 
 
 362916.0 
 
 362943.0 
 
 362972.2 
 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 19° 
 
 10" 
 
 1008.34 
 
 1008.44 
 
 1008.53 
 
 1008.63 
 
 1008.74 
 
 20 
 
 2016.69 
 
 2016.87 
 
 2017.06 
 
 2017.27 
 
 2017.48 
 
 30 
 
 3025.03 
 
 3025.30 
 
 3025.60 
 
 3025.90 
 
 3026.23 
 
 40 
 
 4033.37 
 
 4033-74 
 
 4034-13 
 
 4034-54 
 
 403497 
 
 so 
 
 5041.72 
 
 5042. iS 
 
 5042.66 
 
 5043.18 
 
 5043.71 
 
 60 
 
 6050.06 
 
 6050.61 
 
 6051.19 
 
 6051.81 
 
 6052.45 
 
 to/ 
 
 60500.6 
 
 60506.1 
 
 60511.9 
 
 605 18. I 
 
 60524,5 
 
 20 
 
 121001.2 
 
 I2rOI2.2 
 
 121023,8 
 
 121036.2 
 
 121049,0 
 
 30 
 
 181501.7 
 
 I8I5I8.3 
 
 181535-8 
 
 181554.3 
 
 181573,6 
 
 40 
 
 242002.3 
 
 242024.4 
 
 242047.7 
 
 242072.4 
 
 242098, 1 
 
 50 
 
 302502.9 
 
 302530.5 
 
 302559.6 
 
 302590.5 
 
 302622,6 
 
 60 
 
 363003.5 
 
 363036.6 
 
 363071.5 
 
 363108.6 
 
 363147.1 
 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 10" 
 
 1008.86 
 
 1008.97 
 
 1000.10 
 2018.19 
 
 loog,22 
 
 1009,35 
 
 20 
 
 2017.71 
 
 2017.95 
 
 2018.44 
 
 2018,70 
 
 30 
 
 3026.56 
 
 3026.92 
 
 3027.28 
 
 3027.66 
 
 3028,06 
 
 40 
 
 4035.42 
 
 4035.89 
 
 4036.38 
 
 4036.88 
 
 4037,41 
 
 50 
 
 5044-28 
 
 5044.86 
 
 5045.48 
 
 5046.10 
 
 5046,76 
 
 60 
 
 6053.13 
 
 6053.84 
 
 6054.57 
 
 6055,33 
 
 6056,11 
 
 lol 
 
 60531.3 
 
 60538.4 
 
 60545.7 
 
 60553,3 
 
 60561.1 
 
 20 
 
 121062.6 
 
 121076.8 
 
 121091,4 
 
 121 106,5 
 
 181659,8 
 
 121122,3 
 
 30 
 
 181593.9 
 
 181615.1 
 
 181637.1 
 
 181683,4 
 
 40 
 
 242125.2 
 
 242153.5 
 
 242182,8 
 
 242213,0 
 
 242244,5 
 
 50 
 
 302656.5 
 
 302691.9 
 
 302728.5 
 
 302766,3 
 
 302805,6 
 
 60 
 
 363187.8 
 
 363230.3 
 
 363274.2 
 
 363319,6 
 
 363366,7 
 
 Smithsonian Tables. 
 
 78 
 
Table 17. 
 LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN. 
 
 [Derivation of table explained on p. xlvi.] 
 
 Latitude 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Interval. 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 lO" 
 
 1009.49 
 
 1009.63 
 
 1009.77 
 
 1009.92 
 
 1010.07 
 
 20 
 
 2018.97 
 
 2019.25 
 
 2019.54 
 
 2019.83 
 
 2020. 13 
 
 30 
 
 3028.46 
 
 3028.88 
 
 3029.31 
 
 3029.75 
 
 3030.20 
 
 40 
 
 4037.95 
 
 4038.51 
 
 4039.08 
 
 4039.67 
 
 4040.27 
 
 SO 
 
 5047.44 
 
 5048.13 
 
 5048.85 
 
 5049.58 
 
 5050.33 
 
 60 
 
 6056.92 
 
 6057.75 
 
 6058.62 
 
 6059.50 
 
 6060.40 
 
 10' 
 
 60569.2 
 
 60577.6 
 
 60586.2 
 
 60595.0 
 
 60604.0 
 
 20 
 
 12113S.5 
 
 121155.2 
 
 121172.3 
 
 121190.0 
 
 12120S.0 
 
 30 
 
 181707.7 
 
 181732.7 
 
 181758.5 
 
 181785.0 
 
 1S1812.0 
 
 40 
 
 242276.9 
 
 242310.3 
 
 242344.7 
 
 242379.9 
 
 242416.0 
 
 50 
 
 302846.1 
 
 302SS7.9 
 
 302930.9 
 
 302974.9 
 
 303019.9 
 
 60 
 
 363415.4 
 
 363465-5 
 
 363517.1 
 
 363569.9 
 
 363623.9 
 
 
 30° 
 
 31° 
 
 32° 
 
 33° 
 
 34° 
 
 10" 
 
 1010.22 
 
 1010.38 
 
 1010.54 
 
 1010.70 
 
 1010.86 
 
 20 
 
 2020.44 
 
 2020.75 
 
 2021.07 
 
 2021.40 
 
 2021.73 
 
 30 
 
 3030.66 
 
 3031.13 
 
 3031.61 
 
 3032.10 
 
 3032.59 
 
 40 
 
 4040.88 
 
 4041.51 
 
 4042.15 
 
 4042.80 
 
 4043.46 
 
 50 
 
 5051.10 
 
 5051.89 
 
 5052.68 
 
 5053.50 
 
 5054.32 
 
 60 
 
 6061.32 
 
 6062.26 
 
 6063.22 
 
 6064.20 
 
 6065.19 
 
 10/ 
 
 60613.2 
 
 60622.6 
 
 60632.2 
 
 60642.0 
 
 60651.9 
 
 20 
 
 121226.4 
 
 121245.3 
 
 121264.4 
 
 121283.9 
 
 121303.8 
 
 30 
 
 181839-7 
 
 181867.9 
 
 181896.6 
 
 181925.9 
 
 181955.7 
 
 40 ' 
 
 242452-9 
 
 242490.5 
 
 242528.8 
 
 242567-9 
 
 242607.6 
 
 50 
 
 303066.1 
 
 303113.2 
 
 303161.1 
 
 303209.9 
 
 303259-4 
 
 60 
 
 363679-3 
 
 363735-8 
 
 363793.3 
 
 363851.8 
 
 363911-3 
 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 10" 
 
 1011.03 
 
 I0II.20 
 
 1011.37 
 
 1011.55 
 
 1011.72 
 
 20 
 
 2022.06 
 
 2022.40 
 
 2022.7S 
 
 Z023.09 
 
 2023.44 
 
 30 
 
 3033.10 
 
 3033-61 
 
 3034.12 
 
 3034.64 
 
 3035.17 
 
 40 
 
 4044.13 
 
 4044.81 
 
 4045.50 
 
 4046.19 
 
 4046.S9 
 
 50 
 
 5055.16 
 
 5056.01 
 
 S056.87 
 
 5057-74 
 
 5058.61 
 
 60 
 
 6066.19 
 
 6067.21 
 
 6068.24 
 
 6069.29 
 
 6070.34 
 
 lO* 
 
 60661.9 
 
 60672.1 
 
 60682.4 
 
 60692.9 
 
 60703.4 
 
 20 
 
 121323.9 
 
 I21344-3 
 
 121364.9 
 
 121385.7 
 
 121406.7 
 
 30 
 
 181985.8 
 
 I 820 I 6.4 
 
 182047.3 
 
 182078.6 
 
 182110.1 
 
 40 
 
 242647.S 
 
 242688.S 
 
 242729.7 
 
 242771.4 
 
 242813.4 
 
 50 
 
 303309.7 
 
 303360.6 
 
 303412.2 
 
 303464.3 
 
 303516.8 
 
 60 
 
 363971.7 
 
 364032.8 
 
 364094.6 
 
 364157. I 
 
 364220.2 
 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 10" 
 
 1011.90 
 
 1012.08 
 
 1012.25 
 
 1012.43 
 
 1012.61 
 
 20 
 
 2023.80 
 
 2024.15 
 
 2024.51 
 
 2024.S7 
 
 2025.23 
 
 30 
 
 3035-7° 
 
 3036.23 
 
 3036.77 
 
 3037.30 
 
 3037-84 
 
 40 
 
 4047.60 
 
 4048.31 
 
 4049.02 
 
 4049.74 
 
 4050.46 
 
 50 
 
 5059.50 
 
 5060.38 
 
 5061.28 
 
 5062.17 
 
 5063.07 
 
 60 
 
 6071.39 
 
 607Z.46 
 
 6073.53 
 
 6074.61 
 
 6075.69 
 
 lO* 
 
 60713.9 
 
 60724.6 
 
 60735.3 
 
 60746.1 
 
 60756.9 
 
 20 
 
 121427.9 
 
 121449.2 
 
 I21470.6 
 
 121492.2 
 
 121513.7 
 
 30 
 
 182141.8 
 
 182173.8 
 
 182206.0 
 
 182238.2 
 
 182270.6 
 
 40 
 
 242855.8 
 
 242S98.4 
 
 242941.3 
 
 242984-3 
 
 243027.4 
 
 5° 
 
 303569.7 
 
 303623.0 
 
 303676.6 
 
 303730.4 
 
 303784-3 
 
 60 
 
 364283.7 
 
 364347.6 
 
 3644II.9 
 
 364476.5 
 
 364541.2 
 
 
 45° 
 
 46- 
 
 47° 
 
 48° 
 
 49° 
 
 10" 
 
 1012.79 
 
 1012.97 
 
 1013.15 
 
 1013.33 
 
 1013.51 
 
 20 
 
 2025.59 
 
 2025.95 
 
 2026.31 
 
 2026.67 
 
 2027.02 
 
 30 
 
 3038.3S 
 
 3038.92 
 
 3039.46 
 
 3040.00 
 
 3040.54 
 
 40 
 
 4051.1S 
 
 4051.90 
 
 4052.62 
 
 4053-34 
 
 4054.05 
 
 50 
 
 5063.97 
 
 5064.87 
 
 5065.77 
 
 5066.67 
 
 5067.55 
 
 60 
 
 6076.77 
 
 6077.8s 
 
 6078.93 
 
 6080.00 
 
 6o8i.o8 
 
 lO* 
 
 60767.7 
 
 60778.5 
 
 60789.3 
 
 60800.0 
 
 60810.8 
 
 20 
 
 1ZJS35-3 
 
 121556.9 
 
 182367.8 
 
 121600. 1 
 
 121621.5 
 
 30 
 
 182303.0 
 
 182335.4 
 
 182400.1 
 
 182432.3 
 
 40 
 
 243070.6 
 
 243113.9 
 
 243157.0 
 
 243200.Z 
 
 243243.0 
 
 50 
 
 303838.3 
 
 303892.4 
 
 303946.3 
 
 304000.1 
 
 304053.8 
 
 6a 
 
 364606.0 
 
 364670,8 
 
 364735-5 
 
 364800.2 
 
 364864.5 
 
 
 
 
 
 
 
 Smithsonian Tables. 
 
 79 
 
Table 17. 
 
 LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN. 
 
 [Derivation of table explained on p. xlvi.] 
 
 Latitude 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 
 Interval. 
 
 So° 
 
 51° 
 
 S2° 
 
 53° 
 
 54° 
 
 55° 
 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 
 lo" 
 
 loi3.6g 
 
 1013.87 
 
 1014.04 
 
 1014.22 
 
 1014.39 
 
 1014.56 
 
 
 2Q 
 
 2027.38 
 
 2027.74 
 
 2028.09 
 
 2028.44 
 
 2028.78 
 
 2029.12 
 
 
 3° 
 
 3041.07 
 
 3041.60 
 
 3042.13 
 
 3042.65 
 
 3043-17 
 
 3043.68 
 
 
 40 
 
 4054.76 
 
 4055.47 
 
 4056.17 
 
 4056.87 
 
 4057.56 
 
 4058.24 
 
 
 50 
 
 5068.46 
 
 5069.34 
 
 5070.22 
 
 5071.09 
 
 5071.96 
 
 5072.80 
 
 
 60 
 
 6082.15 
 
 6083.21 
 
 6084.26 
 
 6085.31 
 
 6086.35 
 
 6087.37 
 
 
 lol 
 
 60821.5 
 
 60832.1 
 
 60842.6 
 
 60853.1 
 
 60863.5 
 
 60873.7 
 
 
 20 
 
 121642.9 
 
 121664.2 
 
 121685.2 
 
 121706.2 
 
 121726.9 • 
 
 121747.3 
 
 
 30 
 
 182464.4 
 
 182496.2 
 
 182527.7 
 
 182559.2 
 
 182590.4 
 
 182621.0 
 
 
 40 
 
 243285.8 
 
 243328.3 
 
 243370.3 
 
 243412.3 
 
 243453.8 
 
 243494.6 
 
 
 50 
 
 304107.3 
 
 304160.4 
 
 304212.9 
 
 304265.4 
 
 304317-3 
 
 304368.3 
 
 
 60 
 
 364928.8 
 
 364992.5 
 
 365055.5 
 
 365118.5 
 
 365180.8 
 
 365242.0 
 
 
 
 56° 
 
 57° 
 
 58O 
 
 59° 
 
 60° 
 
 61° 
 
 
 10" 
 
 1014.73 
 
 1014.90 
 
 1015.06 
 
 1015.22 
 
 1015.38 
 
 1015.53 
 
 
 20 
 
 2029.46 
 
 2029.79 
 
 2030.12 
 
 2030.44 
 
 2030.76 
 
 Z031.07 
 
 
 30 
 
 3044.19 
 
 3044.69 
 
 3045.18 
 
 3045-66 
 
 3046.14 
 
 3046.60 
 
 
 40 
 
 4058.92 
 
 4059.58 
 
 4060.24 
 
 4060.88 
 
 4061.52 
 
 4062.14 
 
 
 50 
 
 5073.65 
 
 5074.48 
 
 5075.30 
 
 5076.10 
 
 5076.90 
 
 5077.67 
 
 
 60 
 
 6088.38 
 
 6089.38 
 
 6090.36 
 
 6091.33 
 
 6092.27 
 
 6093.20 
 
 
 10' 
 
 60883.8 
 
 60893.8 
 
 60903.6 
 
 60913.3 
 
 60922.7 
 121845.5 
 
 60932.0 
 
 
 20 
 
 121767.6 
 
 121787.5 
 
 121807.Z 
 
 121826.5 
 
 121864.1 
 
 
 30 
 
 182651.4 
 
 182681.3 
 
 182710.8 
 
 182739.8 
 
 182768.2 
 
 182796.1 
 
 
 40 
 
 243535-2 
 
 2435750 
 
 243614.4 
 
 243653.0 
 
 243691.0 
 
 243728.2 
 
 
 50 
 
 304419.0 
 
 304468.8 
 
 304518.0 
 
 304566.3 
 
 304613.7 
 
 304660.2 
 
 
 60 
 
 365302.S 
 
 365362.6 
 
 365421.6 
 
 365479.6 
 
 365536.4 
 
 365592..2 
 
 
 
 62° 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 
 10" 
 
 1015.69 
 
 1015.83 
 
 1015.98 
 
 1016.12 
 
 1016.26 
 
 1016.39 
 2032.78 
 
 
 20 
 
 2031.37 
 
 2031.67 
 
 2031.96 
 
 2032.24 
 
 2032.51 
 
 
 • 30 
 
 3047.06 
 
 3047.50 
 
 3047.94 
 
 3048.36 
 
 3048.77 
 
 3049.16 
 
 
 40 
 
 4062.74 
 
 4063.34 
 
 4063.92 
 
 4064.48 
 
 4065.02 
 
 4065.55 
 
 
 50 
 
 5078.43 
 
 5079.17 
 
 5079.90 
 
 5080.60 
 
 5081.28 
 
 5081.94 
 
 
 60 
 
 6094.12 
 
 6095.00 
 
 6095.87 
 
 6096.71 
 
 6097.54 
 
 6098.33 
 
 
 lO* 
 
 60941.2 
 
 60950.0 
 
 60958.7 
 
 60967.1 
 
 60975.4 
 
 60983.3 
 
 
 20 
 
 121882.3 
 
 12 1900.1 
 
 121917.5 
 182876.2 
 
 121934.3 
 
 121950.7 
 
 121966.6 
 
 
 30 
 
 182823.5 
 
 182850.1 
 
 182901.4 
 243868.6 
 
 182926.1 
 
 182949.8 
 
 
 40 
 
 243764.6 
 
 243800.2 
 
 243835-0 
 
 304876.8 
 
 243933.1 
 
 
 50 
 
 304705.8 
 
 304750.2 
 
 304793-7 
 
 304835-7 
 
 304916.4 
 
 
 60 
 
 365647.0 
 
 365700.2 
 
 365752.4 
 
 365802.8 
 
 365852.2 
 
 365899.7 
 
 
 
 68° 
 
 69° 
 
 70° 
 
 71° 
 
 72° 
 
 73° 
 
 
 10" 
 
 1016.52 
 
 1016.64 
 
 1016.76 
 
 1016.87 
 
 1016.98 
 
 IOZ7.O9 
 
 
 20 
 
 2033.03 
 
 2033.28 
 
 2033-52 
 
 2033-75 
 
 2033.96 . 
 
 2034-17 
 
 
 30 
 
 3049.55 
 
 3049.92 
 
 3050.2S 
 
 3050.62 
 
 3050.95 
 
 3051.26 
 
 
 40 
 
 4066.07 
 
 4066.56 
 
 406704 
 
 4067.49 
 
 4067.93 
 
 4068.34 
 
 
 50 
 
 5082.58 
 
 5083.20 
 
 5083.80 
 
 5084.36 
 
 5084.91 
 6101.89 
 
 5085.43 
 
 
 60 
 
 6099. 10 
 
 6199.84 
 
 6100.55 
 
 6101.24 
 
 6102.52 
 
 
 10' 
 
 60991.0 
 
 61998.4 
 
 61005.5 
 
 61012.4 
 
 61018.9 
 122037,8 
 
 61025.2 
 
 
 20 
 
 121982.0 
 
 121996.8 
 
 122011.1 
 
 122024.8 
 
 122050.3 
 
 
 30 
 
 182973.1 
 
 182995.2 
 
 183016.6 
 
 183037.1 
 
 183056.8 
 
 183075,5 
 
 
 40 
 
 243964.1 
 
 243993-6 
 
 244022.2 
 
 244049.5 
 
 244075.7 
 
 244100.6 
 
 
 SO 
 
 304955.1 
 
 304992.0 
 
 305027.7 
 
 305061.9 
 
 305094.6 
 
 305125.8 
 
 
 60 
 
 365946.1 
 
 365990.4 
 
 366033.2 
 
 366074.3 
 
 366113.5 
 
 366151.O 
 
 
 
 74° 
 
 75° 
 
 76° 
 
 77° 
 
 78° 
 
 79° 
 
 
 io" 
 
 1017.1S 
 
 1017.28 
 
 1017.37 
 
 1017.45 
 
 1017.53 
 
 1017.60 
 
 
 20 
 
 2034.37 
 
 2034.56 
 
 2034-73 
 
 2034.90 
 
 2035.05 
 3052.58 
 
 2035.19 
 
 
 30 
 
 3051.56 
 
 3051.84 
 
 3052.10 
 
 3052.35 
 
 3052.79 
 
 
 40 
 
 4068.74 
 
 4069.12 
 
 4069.46 
 
 4069.80 
 
 4070.10 
 
 4070.38 
 
 
 s. 
 
 5085.92 
 
 5086.40 
 
 5086.83 
 
 5087.24 
 
 5087.63 
 
 5087.98 
 
 
 60 
 
 6103.11 
 
 6103.67 
 
 6104.20 
 
 6104.69 
 
 6105.16 
 
 6105.58 
 
 
 lo/ 
 
 6103 1. 1 
 
 61036.7 
 
 61042.0 
 
 61046.9 
 
 61051.6 
 
 61055.8 
 
 
 20 
 
 122062.2 
 
 122073.5 
 
 122083.9 
 
 122093.9 
 
 122103.x 
 
 122111.5 
 
 
 30 
 
 183093.3 
 
 183110.2 
 
 183125.9 
 
 183140.8 
 
 183154.7 
 
 183167.3 
 
 
 40 
 
 244124.4 
 
 244147.0 
 
 244167.8 
 
 244187.8 
 
 244206.2 
 
 244223.0 
 
 
 r 
 
 305155.5 
 
 305183-7 
 
 305209.8 
 
 305234-7 
 
 305257.8 
 
 305278.8 
 
 
 60 
 
 366186.6 
 
 366220.4 
 
 366251.8 
 
 366281.6 
 
 366309.4 
 
 366334.6 
 
 
 Smith SON 
 
 AN Tables 
 
 
 
 
 
 
 
 80 
 
Table 18. 
 LENGTHS OF TERRESTRIAL ARCS OF PARALLEL. 
 
 [Derivation of table explained on p. xlix.] 
 
 Longitude 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Interval. 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 id" 
 
 1014.52 
 
 1014.37 , 
 
 1013.91 
 
 1013.14 
 
 1012.07 
 
 20 
 
 2029.05 
 
 2028.74 
 
 2027. 82 
 
 2026.29 
 
 Z024.14 
 
 3° 
 
 3043.57 
 
 3043- 11 
 
 3041.73 
 
 3039.43 
 
 3036.21 
 
 40 
 
 4058.10 
 
 4057.48 
 
 4055.64 
 
 4052 57 
 
 4048.28 
 
 SO 
 
 5072.62 
 
 5071.86 
 
 S069.55 
 
 S065.72 
 
 5060.35 
 
 60 
 
 6087.14 
 
 6086.23 
 
 6083.46 
 
 6078.S6 
 
 6072.42 
 
 10' 
 
 60S71.4 
 
 60862.3 
 
 60834.6 
 
 60788.6 
 
 60724.2 
 
 20 
 
 121742.9 
 
 121724.5 
 
 121669.2 
 
 121577.2 
 
 121448.4 
 
 30 
 
 182614.3 
 
 182586.8 
 
 182503 .8 
 
 182365.7 
 
 182172.6 
 
 40 
 
 243485.8 
 
 243449.0 
 
 243338.4 
 
 243154.3 
 
 242896.8 
 
 SO 
 
 304357-2 
 
 304311-3 
 
 304173.0 
 
 303942.9 
 
 303621.0 
 
 60 
 
 365228.5 
 
 365173-6 
 
 365007.6 
 
 364731.5 
 
 364345-2 
 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 10" 
 
 1010.69 
 
 1D09.00 
 
 1007.01 
 
 1004.72 
 
 1002.12 
 
 20 
 
 2021.38 
 
 201S.01 
 
 2014.03 
 
 2009.43 
 
 2004.23 
 
 30 
 
 3032.07 
 
 3027.01 
 
 3021 04 
 
 30I4-IS 
 
 3006.35 
 
 40 
 
 4042.76 
 
 4036.02 
 
 4028.05 
 
 4018.87 
 
 4008.47 
 
 50 
 
 5053-45 
 
 5045.02 
 
 5035.06 
 
 5023-58 
 
 5010.58 
 
 60 
 
 6064.14 
 
 6054.02 
 
 6042.08 
 
 6028.30 
 
 6012.70 
 
 lO* 
 
 60641.4 
 
 60540.2 
 
 60420.8 
 
 60283.0 
 
 60127.0 
 
 20 
 
 121282.8 
 
 121080.5 
 
 120S41.6 
 
 120566.0 
 
 120254.0 
 
 30 
 
 181924.2 
 
 181620.7 
 
 1S1262.3 
 
 180849.1 
 
 18038 1. 1 
 
 40 
 
 242565.6 
 
 242161.0 
 
 241683.1 
 
 24II32.I 
 
 240508.1 
 
 SO 
 
 303207.0 
 
 302701.2 
 
 302103.9 
 
 301415.1 
 
 300635.1 
 
 60 
 
 36384S.4 
 
 363241.4 
 
 362524.7 
 
 361698.1 
 
 360762,1 
 
 
 10° 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 10" 
 
 999.21 
 
 996.01 
 
 992.50 
 
 98869 
 
 984.58 
 
 20 
 
 1998.43 
 
 1992.01 
 
 1985.00 
 
 1977-38 
 
 1969.17 
 
 30 
 
 2997.64 
 
 2988.02 
 
 2977.50 
 
 2966-07 
 
 2953.75 
 
 40 
 
 3996-85 
 
 3984.03 
 
 3970.00 
 
 395476 
 
 393S.34 
 
 50 
 
 4996.06 
 
 4980.04 
 
 4962.50 
 
 4943-46 
 
 4922.92 
 
 60 
 
 5995-28 
 
 5976.04 
 
 5955-00 
 
 5932.15 
 
 5907.50 
 
 lo/ 
 
 59952-8 
 
 59760.4 
 
 S9SSO.O 
 
 59321-s 
 
 59075-0 
 
 20 
 
 I 19905.6 
 
 119520.8 
 
 119100.0 
 
 118042.9 
 
 118150.1 
 
 30 
 
 179858.3 
 
 179281.3 
 
 178650.0 
 
 177964.4 
 
 177225. I 
 
 40 
 
 239811.1 
 
 239041.7 
 
 238200.0 
 
 237285.8 
 
 236300.2 
 
 5° 
 
 299763.9 
 
 298802.1 
 
 297750.0 
 
 296607.3 
 
 295375.2 
 
 60 
 
 359716-7 
 
 358562.5 
 
 357300.0 
 
 3559Z8.8 
 
 3S4450.2 
 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 19° 
 
 10" 
 
 980.18 
 
 975-47 
 
 970.48 
 
 965.18 
 
 959.60 
 
 20 
 
 1960.35 
 
 1950-95 
 
 1940.95 
 
 1930.36 
 
 1919.19 
 
 30 
 
 2940-53. 
 
 2926.42 
 
 2911.42 
 
 2895.5s 
 
 2878.79 
 
 40 
 
 3920.71 
 
 3901.90 
 
 3881.90 
 
 3860.73 
 
 3838.3S 
 
 5° 
 
 4900.S8 
 
 4877-37 
 
 4852.38 
 
 4825.91 
 
 4797.98 
 
 60 
 
 5881.06 
 
 5852-84 
 
 5822.85 
 
 5791.09 
 
 5757-58 
 
 lo/ 
 
 58810.6 
 
 58528.4 
 
 58228.5 
 
 57910.9 
 
 57575-8 
 
 20 
 
 117621.2 
 
 117056.9 
 
 I 16457.0 
 
 115821.8 
 
 11S151-S 
 
 30 
 
 176431.9 
 
 175585-3 
 
 174685.5 
 
 173732.8 
 
 172727.3 
 
 40 
 
 235242.5 
 
 234113.8 
 
 232914.0 
 
 231643.7 
 
 230303.0 
 
 50 
 
 294053.1 
 
 292642.2 
 
 291142.S 
 
 289554.6 
 
 287878.8 
 
 60 
 
 352863.7 
 
 351170.6 
 
 349371.0 
 
 347465.5 
 
 345454-6 
 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 10" 
 
 953.72 
 
 947.5s 
 
 941.10 
 
 934.36 
 
 927-33 
 
 20 
 
 1907.44 
 
 1895. 10 
 
 1882.19 
 
 1S68.71 
 
 1854-67 
 
 30 
 
 2861.15 
 
 2842.66 
 
 2823.29 
 
 2803.07 
 
 2782.00 
 
 40 
 
 3814.87 
 
 3790.21 
 
 3764.3S 
 
 3737-43 
 
 3709-33 
 
 50 
 
 4768-59 
 
 4737-76 
 
 4705.48 
 
 4671.78 
 
 4636.66 
 
 60 
 
 5722.31 
 
 5685.31 
 
 5646.58 
 
 5606.14 
 
 5564.00 
 
 10' 
 
 57223.1 
 
 56853.1 
 
 56465.8 
 
 56061.4 
 
 55640.0 
 
 20 
 
 114446.2 
 
 I 13706.2 
 
 112931.5 
 
 112122.8 
 
 11T280.0 
 
 30 
 
 171669.2 
 
 170559.4 
 
 169397-3 
 
 168184.3 
 
 166919.9 
 
 40 
 
 22S892.3 
 
 227412.5 
 
 225863.0 
 
 224245.7 
 
 222559.9 
 
 5° 
 
 286115.4 
 
 2S4265.6 
 
 282328.8 
 
 280307.1 
 
 278199-9 
 
 60 
 
 343338.S 
 
 34«"8.7 
 
 338794.6 
 
 336368.5 
 
 333839-9 
 
 Smithsonian Tables. 
 
 81 
 
Table 18. 
 LENGTHS OF TERRESTRIAL ARCS OF PARALLEL. 
 
 [Derivation of table explained on p. xlix.] 
 
 Longitude Latitude, 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude, 
 
 
 Interval. 
 
 250 
 
 26° 
 
 27O 
 
 28° 
 
 29° 
 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 
 lo" 
 
 920.03 
 
 912.44 
 
 904.58 
 
 896.44 
 
 888.03 
 
 
 20 
 
 1840.0s 
 
 1824.88 
 
 1809.16 
 
 1792.88 
 
 1776.06 
 
 
 30 
 
 2760.08 
 
 2737-33 
 
 2713-74 
 
 2689.32 
 
 2664.09 
 
 
 40 
 
 3680.11 
 
 3649-77 
 
 3618.32 
 
 3585.76 
 
 3552.12 
 
 
 50 
 
 4600.14 
 
 4562.21 
 
 4522.89 
 
 4482.20 
 
 4440,15 
 
 
 60 
 
 SS20.17 
 
 5474-65 
 
 5427-47 
 
 5378.64 
 
 5328.18 
 
 
 10' 
 
 SS201.7 
 
 S4746-S 
 
 54274-7 
 
 53786.4 
 
 53281.8 
 
 
 20 
 
 110403.3 - 
 
 109493.0 
 
 108549.5 
 
 107572.9 
 
 106563.5 
 
 
 30 
 
 16S605.0 
 
 164239.5 
 
 162S24.2 
 
 161359.3 
 
 159845.3 
 
 
 40 
 
 220806.6 
 
 218986.1 
 
 217099.0 
 
 215"45.7 
 
 213127.1 
 
 
 so 
 
 276008.3 
 
 273732.5 
 
 271373.7 
 
 268932.2 
 
 266408.8 
 
 
 60 
 
 331209.9 
 
 328479.1 
 
 325648.4 
 
 322718.6 
 
 319690.5 
 
 
 
 30° 
 
 31° 
 
 32° 
 
 33° 
 
 34° 
 
 
 10" 
 
 879-35 
 
 870.40 
 
 861.18 
 
 851.71 
 
 841.97 
 
 
 20 
 
 1758.70 
 
 1740.80 
 
 1722.37 
 
 1703.41 
 
 1683.94 
 
 
 30 
 
 2638.04 
 
 2611.20 
 
 2583.55 
 
 2S55-'2 
 
 2525.91 
 3367.88 
 
 
 40 
 
 3517-39 
 
 3481.59 
 
 3444-74 
 
 3406.83 
 
 
 50 
 
 4396-74 
 
 4351-99 
 
 4305.92 
 
 4258.53 
 
 4209.8s 
 
 
 60 
 
 5276.09 
 
 5222.39 
 
 5167.10 
 
 5110.24 
 
 5051,82 
 
 
 lO* 
 
 52760.9 
 
 52223.9 
 
 51671,0 
 
 51102,4 
 
 50518,2 
 
 
 20 
 
 IOSS2I.8 
 
 104447.8 
 
 103342.1 
 
 102204.8 
 
 101036.4 
 
 
 30 
 
 158282.5 
 
 156671.8 
 
 155013.1 
 
 153307.3 
 
 151554-6 
 
 
 40 
 
 2 1 1043. s 
 
 208895.7 
 
 206684.2 
 
 204409.7 
 
 202072.8 
 
 
 Si 
 
 263804.4 
 
 261119.6 
 
 258.355-2 
 
 255512.1 
 
 252591.0 
 
 
 60 
 
 316565.3 
 
 313343.5 
 
 310026.3 
 
 306614.S 
 
 303109.2 
 
 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 
 10" 
 
 831.98 
 
 821.73 
 
 811.23 
 
 800.48 
 
 789.49 
 
 
 20 
 
 1663.95 
 
 1643.46 
 
 1622.46 
 
 1600.97 
 
 1578.98 
 
 
 30 
 
 2495-93 
 
 2465.19 
 
 2433,69 
 
 2401.45 
 
 2368.48 
 
 
 40 
 
 3327-91 
 
 3286.91 
 
 3244-92 
 
 3201.93 
 
 3157-97 
 
 
 50 
 
 4<59.88 
 
 4108.64 
 
 4056-15 
 
 4002.42 
 
 3947-46 
 
 
 60 
 
 4991.85 
 
 4930.37 
 
 4867.38 
 
 4802.90 
 
 4736.95 
 
 
 tol 
 
 499-8.6 
 
 49303-7 
 
 48673.8 
 
 48029.0 
 
 47369-5 
 
 
 20 
 
 99837-2 
 
 98607.4 
 
 97347-6 
 
 96058.0 
 
 94739.1 
 
 
 30 
 
 149755-8 
 
 147911.2 
 
 146021.4 
 
 144087,0 
 
 142108.5 
 
 
 40 
 
 199674-3 
 
 197214.9 
 
 194695.2 
 
 192116.0 
 
 189478.2 
 
 
 1° 
 
 249592.9 
 
 246518.6 
 
 243369.0 
 
 240145.0 
 
 236847.7 
 
 
 60 
 
 2995 n-S 
 
 295822.3 
 
 292042.8 
 
 288174.0 
 
 284217.2 
 
 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 
 lo" 
 
 778.26 
 
 766.79 
 
 755.08 
 
 743.15 
 
 730.98 
 
 
 20 
 
 1556.52 
 
 1533.58 
 
 1510.17 
 
 1486.29 
 
 1461.96 
 
 
 30 
 
 2334.78 
 
 2300.37 
 
 2265.25 
 
 2229.44 
 
 2192.9s 
 
 
 40 
 
 3113.04 
 
 3067.15 
 
 3020.33 
 
 2972.59 
 
 2923-93 
 
 
 so 
 
 3891.30 
 
 3833.94 
 
 3775-42 
 
 3715.73 
 
 3654.91 
 
 
 60 
 
 4669.55 
 
 4600.73 
 
 4530.50 
 
 4458.88 
 
 4385.89 
 
 
 lof 
 
 46695.6 
 
 46007.3 
 
 45305-0 
 
 44588.8 
 
 43858.9 
 
 
 20 
 
 93391.2 
 
 92014.7 
 
 90610.0 
 
 89177.6 
 
 87717-9 
 
 
 30 
 
 140086.7 
 
 138022.0 
 
 135915-0 
 
 133766.4 
 
 131576.8 
 
 
 40 
 
 186782.3 
 
 184029.3 
 
 181220.0 
 
 178355.2 
 
 175435.8 
 
 
 so 
 
 2334779 
 
 230035.7 
 
 226525.0 
 
 222944.0 
 
 219294.7 
 
 
 60 
 
 280173.5 
 
 276044.0 
 
 271830.1 
 
 267532.8 
 
 263153.6 
 
 
 
 4S° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 
 10" 
 
 718.59 
 
 705.99 
 
 693.16 
 
 680.12 
 
 666.87 
 
 
 20 
 
 1437-19 
 
 1411.97 
 
 1386.32 
 
 1360.24 
 
 "333.75 
 
 
 30 
 
 ^JS5-7f 
 
 2117.96 
 
 2079.48 
 
 2040.36 
 
 2000.62 
 
 
 40 
 
 2874-38 
 
 2823.94 
 
 2772.64 
 
 2720.49 
 
 2667.50 
 
 
 g 
 
 3592.97 
 
 3529.93 
 
 3465-80 
 
 3400.61 
 
 3334-37 
 
 
 4311.56 
 
 4235-91 
 
 4158-96 
 
 4080.73 
 
 4001.25 
 
 
 10' 
 20 
 
 43115-6 
 86231.3 
 
 42359-1 
 84718.2 
 
 41589-6 
 83179-2 
 
 40807.3 
 81614.6 
 
 40012.5 
 80024.9 
 
 
 30 
 
 129346.9 
 
 127077-3 
 
 124768.7 
 
 122421.9 
 
 120037.4 
 
 
 40 
 
 172462.S 
 
 169436.5 
 
 166358.3 
 
 163229.2 
 
 160049.9 
 
 
 S 
 
 215578.2 
 
 211795.5 
 
 207947.9 
 
 204036.4 
 
 200062.3 
 240074.8 
 
 
 60 
 
 258693.8 
 
 254154-7 
 
 249S37-S 
 
 244843,7 
 
 
 Smithsoni 
 
 AN Tables 
 
 
 
 
 
 
 82 
 
Table 18. 
 LENGTHS OF TERRESTRIAL ARCS OF PARALLEL. 
 
 [Derivation of table explained on p. xlix.] 
 
 Longitude 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Latitude. 
 
 Interval. 
 
 So° 
 
 51° 
 
 52° 
 
 53° 
 
 54° 
 
 55° 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 lO" 
 
 653.42 
 
 639-77 
 
 625.92 
 
 611.88 
 
 597-65 
 
 583.23 
 
 20 
 
 1306.8s 
 
 1279.54 
 
 1251.84 
 
 1223.76 
 
 1195.30 
 
 1166.47 
 
 30 
 
 1960.27 
 
 1919.31 
 
 1877-76 
 
 1835-63 
 
 1792.94 
 
 1749.70 
 
 40 
 
 2613.69 
 
 2559.08 
 3198.85 
 
 2503.68 
 
 2447-51 
 
 2390-59 
 
 2332-93 
 
 50 
 
 3267.12 
 
 3129.60 
 
 3059-39 
 
 2988.28 
 
 2916.16 
 
 60 
 
 3920.54 
 
 3838.62 
 
 3755-52 
 
 3671.27 
 
 3585-89 
 
 3499-40 
 
 10' 
 
 39205.4 
 
 38386.2 
 
 37555-2 
 
 36712.7 
 
 35858-9 
 
 34994-0 
 
 20 
 
 78410.8 
 
 76772.4 
 
 75110-4 
 
 73425-4 
 
 71717.8 
 
 69988-a 
 
 30 
 
 I 17616. I 
 
 115158.6 
 
 112665.6 
 
 110138.0 
 
 107576.6 
 
 104981.9 
 
 40 
 
 156821.5 
 
 I53S44-8 
 
 150220.8 
 
 146850.7 
 
 143435-5 
 
 '39975-9 
 
 1° 
 
 196026.9 
 
 191931.0 
 
 187776.0 
 
 183563.4 
 
 179294.4 
 
 174969.9 
 
 60 
 
 235232.3 
 
 230317.2 
 
 225331.2 
 
 220276-1 
 
 215153-3 
 
 209963.9 
 
 
 56° 
 
 57° 
 
 58° 
 
 59° 
 
 60° 
 
 61° 
 
 10" 
 
 568.64 
 
 553.87 
 
 538.93 
 
 523-82 
 
 508-55 
 
 493-13 
 
 20 
 
 1137-28 
 
 H07.74 
 
 1077.86 
 
 1047.65 
 
 1017.11 
 
 986.26 
 
 30 
 
 1703.92 
 
 l66i.6i 
 
 1616.79 
 
 1571-47 
 
 1525.66 
 
 1479.38 
 
 40 
 
 2274.56 
 
 2215.4S 
 
 2155-72 
 
 2095.29 
 
 2034.22 
 
 1972.52 
 
 50 
 
 2843.20 
 
 2769-35 
 
 2694.64 
 
 2619.12 
 
 2542.77 
 
 2465.64 
 
 6a 
 
 3411.83 
 
 3323-22 
 
 3233-57 
 
 3142-94 
 
 3051-33 
 
 2958.77 
 
 10' 
 
 34118.3 
 
 33232.2 
 
 32335-7 
 
 31429.4 
 
 30513-3 
 
 29587-7 
 
 20 
 
 68236.7 
 
 66464.4 
 
 64671.5 
 
 62858.8 
 
 61026.6 
 
 59175.5 
 
 30 
 
 102355.0 
 
 99696.6 
 
 97007.2 
 
 942S8.1 
 
 91539-9 
 
 88763.2 
 
 40 
 
 136473-4 
 
 132928.8 
 
 129343.0 
 
 125717-5 
 
 122053.2 
 
 118351.0 
 
 50 
 
 170591.7 
 
 166161.0 
 
 161678.7 
 
 ■57146-9 
 
 152566.5 
 
 147938.7 
 
 60 
 
 204710.0 
 
 199393-2 
 
 194014.4 
 
 188576.3 
 
 183079.8 
 
 177526.4 
 
 
 62° 
 
 63° 
 
 64° 
 
 65° 
 
 65° 
 
 67° 
 
 10" 
 
 477-55 
 
 461.83 
 
 445.96 
 
 429-95 
 
 413-82 
 
 397-55 
 
 20 
 
 955.10 
 
 923.65 
 
 891.92 
 
 859.91 
 
 827.63 
 
 795.10 
 
 30 
 
 1432.66 
 
 1385.48 
 
 1337-88 
 
 1289.86 
 
 1241.44 
 
 1192.64 
 
 40 
 
 1910.21 
 
 1847.31 
 
 1783.84 
 
 1719.81 
 
 1655.26 
 
 1590.19 
 
 50 
 
 2387-76 
 
 2309.14 
 
 2229.80 
 
 2149.76 
 
 2069.08 
 
 1987.74 
 
 6a 
 
 2865.31 
 
 2770.96 
 
 2675.75 
 
 2579-72 
 
 2482.89 
 
 2385.29 
 
 la* 
 
 28653.1 
 
 27709.6 
 
 26757.5 
 
 25797-2 
 
 24828.9 
 
 23852.9 
 
 2a 
 
 57306.2 
 
 55419.2 
 
 S35'5-i 
 
 51594.4 
 
 49657-8 
 
 47705.8 
 
 3a 
 
 85959-4 
 
 83128.9 
 
 80272.6 
 
 7739I-S 
 
 74486.7 
 
 71558.6 
 
 40 
 
 114612.5 
 
 110838.5 
 
 107030.2 
 
 103 188.7 
 
 99315.6 
 
 95411.5 
 
 50 
 
 143265.6 
 
 138548. 1 
 
 133787-7 
 
 128985.9 
 
 124144- 5 
 
 I 19264.4 
 
 60 
 
 171918.7 
 
 166257.7 
 
 160545.2 
 
 154783.1 
 
 148973.4 
 
 143117-3 
 
 
 68° 
 
 69° 
 
 70° 
 
 71° 
 
 72° 
 
 73° 
 
 10" 
 
 381.16 
 
 364.65 
 
 348.03 
 
 331-30 
 
 314-47 
 
 297-54 
 
 20 
 
 762.32 
 
 729-30 
 
 696.06 
 
 662.60 
 
 628.94 
 
 595-08 
 
 30 
 
 "43-47 
 
 1093.95 
 
 1044.09 
 
 993.90 
 
 943-41 
 
 892.62 
 
 40 
 
 1524.63 
 
 1458.60 
 
 1392- " 
 
 1325-20 
 
 1257-88 
 
 1190.16 
 
 SO 
 
 1905.79 
 
 1823.25 
 
 1740.14 
 
 1656.5a 
 
 1572-34 
 
 1487.70 
 
 60 
 
 2286.95 
 
 2187.90 
 
 2088.17 
 
 1987.81 
 
 1886.81 
 
 1785.23 
 
 10' 
 
 22869.5 
 
 21879.0 
 
 20881.7 
 
 19878.1 
 
 18868. 1 
 
 17852.3 
 
 20 
 
 45739-0 
 
 43758-0 
 
 41763-5 
 
 39756.1 
 
 37736.3 
 
 35704-7 
 
 30 
 
 68608.4 
 
 65637.0 
 
 62645.2 
 
 59634.2 
 
 56604.4 
 
 53557-0 
 
 40 
 
 91477.9 
 
 87516.0 
 
 83527-0 
 
 79512.2 
 
 75472.6 
 
 71409.4 
 
 SO 
 
 "4347-4 
 
 109395.0 
 
 104408.7 
 
 99390.3 
 
 94340.7 
 
 89261.7 
 
 60 
 
 137216.9 
 
 131274.0 
 
 125290.4 
 
 119268.4 
 
 113208.8 
 
 107114.0 
 
 
 74° 
 
 75° 
 
 76° 
 
 77° 
 
 78° 
 
 79° 
 
 to" 
 
 280.52 
 
 263.41 
 
 246.22 
 
 228.96 
 
 211.62 
 
 '21^^ 
 
 20 
 
 561.04 
 
 526.82 
 
 492.44 
 
 457-91 
 
 423.24 
 
 388.43 
 
 30 
 
 841.56 
 
 790.23 
 
 738.66 
 
 686.86 
 
 634-85 
 
 582.64 
 
 40 
 
 1122.08 
 
 1053.64 
 
 984.88 
 
 915-82 
 
 846.47 
 
 776.86 
 
 SO 
 
 1402.60 
 
 1317-06 
 
 1231.10 
 
 1144.78 
 
 1058-09 
 
 971.08 
 
 60 
 
 1683.11 
 
 1580.47 
 
 1477-33 
 
 1373-73 
 
 1269.71 
 
 1165.29 
 
 10' 
 
 16831.1 
 
 15804.7 
 
 14773-3 
 
 13737-3 
 
 J2697.1 
 
 11652.9 
 
 20 
 
 33662.3 
 
 31609.3 
 
 29546.5 
 
 27474.6 
 
 25394-2 
 
 23305.8 
 
 30 
 
 50493.4 
 
 47414.0 
 
 443 19-8 
 
 41211.9 
 
 38091.2 
 
 34958-7 
 
 40 
 
 67324.6 
 
 63218.6 
 
 59093-0 
 
 54949.2 
 
 50788.3 
 
 46611.6 
 
 SO 
 
 84155-7 
 
 79023-3 
 
 73866.3 
 
 68686.5 
 
 63485-4 
 
 58264.5 
 
 60 
 
 I009S6.8 
 
 94828.0 
 
 88639.6 
 
 82423.8 
 
 761S2.5 
 
 69917.4 
 
 Smithsonian Tables. 
 
 83 
 
Table 1 9. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ubhii - 
 
 [Derivation of table explained on pp. liii — Ivi.] 
 
 •is 
 
 o"oo' 
 
 30 
 
 45 
 I 00 
 
 IS 
 
 30 
 4S 
 
 'S 
 
 30 
 
 45 
 300 
 
 IS 
 
 30 
 
 45 
 
 400 
 
 15 
 30 
 
 45 
 500 
 
 IS 
 30 
 45 
 
 6 00 
 
 15 
 
 30 
 45 
 
 7 00 
 
 15 
 30 
 45 
 
 800 
 
 IS 
 
 30 
 45 
 
 900 
 
 15 
 30 
 
 45 
 
 •ogS 
 •CS"I3 
 
 V s V p. 
 
 Inches. 
 
 ^•353 
 8.706 
 
 I3-°S9 
 17.412 
 
 8.706 
 I3-°S9 
 
 17.412 
 
 8.706 
 13059 
 
 17413 
 
 ^•353 
 8.706 
 
 13.060 
 17-413 
 
 4-353 
 8.707 
 13.060 
 
 17413 
 
 4-353 
 
 8.707 
 
 13.060 
 
 17-414 
 
 4-354 
 
 8.707 
 
 13.061 
 
 17-414 
 
 4-354 
 8-707 
 13.061 
 
 17-415 
 
 ^•354 
 8.708 
 
 13.062 
 17.416 
 
 1354 
 8.708 
 13.062 
 
 17.417 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR- 
 
 15' longitude. 
 
 Inches, 
 
 4-383 
 4-383 
 4-383 
 4.382 
 
 4.382 
 
 4.382 
 4-381 
 4-381 
 
 4-380 
 
 4-379 
 4-379 
 4-378 
 
 4-377 
 
 4-376 
 4-375 
 4-373 
 
 4-37 z 
 
 4-371 
 4-369 
 4.368 
 
 4.366 
 
 4-364 
 4-363 
 4.361 
 
 4-359 
 
 4-357 
 4-355 
 4-353 
 
 4-35° 
 
 4-348 
 4-346 
 4-343 
 
 4-340 
 
 4-338 
 4-335 
 4-332 
 
 4-329 
 
 4.326 
 
 4-323 
 4.320 
 
 4-317 
 
 Inches. 
 
 .000 
 .000 
 .000 
 .000 
 
 .000 
 
 .000 
 
 .000 
 
 .000 
 .000 
 .000 
 
 .000 
 
 .000 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 
 .002 
 
 .002 
 
 .002 
 .002 
 .002 
 
 .002 
 
 3c/ longitude. 
 
 Inches. 
 
 8.766 
 8.766 
 
 8-765 
 8-765 
 
 8.764 
 
 8.764 
 8.763 
 
 8.762 
 8.760 
 
 8-759 
 8-757 
 8-755 
 
 8-753 
 
 8.751 
 8.749 
 8.747 
 
 8-744 
 
 8.742 
 
 8.739 
 8.736 
 
 8.732 
 
 8.729 
 
 8-725 
 8.722 
 
 8.718 
 
 8.714 
 8.710 
 8.70s 
 
 8.701 
 
 8.696 
 8.691 
 8.686 
 
 8.681 
 
 8.67s 
 8.670 
 8.664 
 
 8.658 
 
 8.652 
 8.646 
 8.640 
 
 8-633 
 
 Inches. 
 
 .000 
 .000 
 .000 
 .001 
 
 .001 
 
 .001 
 .001 
 .001 
 
 .001 
 
 .001 
 .001 
 
 .002 
 
 .002 
 
 .002 
 
 .002 
 .002 
 
 -003 
 
 .003 
 .003 
 .003 
 
 ■003 
 
 .003 
 .004 
 .004 
 
 .004 
 
 .004 
 .004 
 .004 
 
 .005 
 
 .005 
 .005 
 .005 
 
 .005 
 
 .005 
 .006 
 .006 
 
 .006 
 
 .006 
 '.006 
 .006 
 
 .006 
 
 45' longitude. 
 
 13.148 
 13.148 
 13.148 
 13-147 
 
 13.146 
 
 13-145 
 13-144 
 13.142 
 
 13.141 
 
 13-138 
 13-136 
 13-133 
 
 13-130 
 
 13-127 
 13.124 
 13.120 
 
 13.116 
 
 13-112 
 13.108 
 13.104 
 
 13.099 
 
 13.094 
 13.088 
 13.082 
 
 13.076 
 
 13.071 
 13.064 
 13-058 
 
 13-051 
 
 13.044 
 13-036 
 13.029 
 
 13.021 
 
 13-013 
 13.005 
 12.996 
 
 12.987 
 
 12.979 
 12.969 
 12.960 
 
 12.950 
 
 Inches. 
 
 .000 
 .000 
 .001 
 .001 
 
 .001 
 
 .002 
 .002 
 -003 
 
 -003 
 
 .003 
 .004 
 .004 
 
 .004 
 
 .005 
 .005 
 .006 
 
 .006 
 
 .006 
 .007 
 .007 
 
 .007 
 
 .008 
 .008 
 .008 
 
 .009 
 
 .009 
 .010 
 .010 
 
 .010 
 
 .oil 
 ,011 
 .oil 
 
 .012 
 
 .012 
 -013 
 .013 
 
 .013 
 
 .014 
 .014 
 .014 
 
 .015 
 
 1° longitude. 
 
 Inches. 
 
 17-531 
 17-531 
 17-530 
 17-530 
 
 17.528 
 
 17-527 
 17-525 
 17-523 
 
 17.521 
 
 17.518 
 
 17-514 
 17.511 
 
 17.507 
 
 17-503 
 17.498 
 17.494 
 
 17.488 
 
 17-483 
 17-478 
 17.472 
 
 17-465 
 
 17-458 
 17-451 
 17-443 
 
 17-435 
 
 17.428 
 17.419 
 17.410 
 
 17.401 
 
 17.392 
 17.382 
 17-372 
 
 17.362 
 
 17-351 
 17-340 
 17.328 
 
 17.316 
 
 17-305 
 17.292 
 17.280 
 
 17.266 
 
 Inches. 
 
 .000 
 .001 
 .001 
 .002 
 
 .003 
 
 .003 
 .004 
 .005 
 
 .005 
 
 .006 
 .007 
 .007 
 
 .008 
 
 .008 
 .009 
 .009 
 
 .010 
 
 .011 
 .012 
 .013 
 
 -013 
 
 .014 
 .014 
 
 .015 
 .016 
 
 .017 
 .017 
 .018 
 
 .019 
 
 .019 
 
 .020 
 .020 
 
 .021 
 
 .022 
 .022 
 -023 
 
 .024 
 
 .024 
 .025 
 .026 
 
 .026 
 
 Smithsonian Tables. 
 
 84 
 
Table 19. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE mUi - 
 
 [Derivation of table explained on pp. liii — Ivi.] 
 
 Latitude of 
 parallel. 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 15' longitude. 
 
 30^ longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Indies. 
 
 Inches. 
 
 Inches. 
 
 I0°oo' 
 
 
 4.317 
 
 .002 
 
 8-633 
 
 .006 
 
 12.950 
 
 .015 
 
 17.266 
 
 .026 
 
 IS 
 3° 
 45 
 
 4-354 
 8.709 
 
 13-063 
 
 4-313 
 4.310 
 4.306 
 
 .002 
 .002 
 .002 
 
 8.626 
 8.620 
 8.613 
 
 .007 
 .007 
 .007 
 
 12.940 
 12.930 
 12.919 
 
 .015 
 .015 
 .016 
 
 17-253 
 17.240 
 17.226 
 
 .027 
 .027 
 .028 
 
 II 00 
 
 IS 
 30 
 4S 
 
 17.418 
 
 4-303 
 
 4.299 
 4.295 
 4.292 
 
 .002 
 
 .002 
 .002 
 .002 
 
 8.606 
 
 8.598 
 
 8-591 
 8.583 
 
 .007 
 
 .007 
 .007 
 .008 
 
 12.908 
 
 1 218^6 
 12.875 
 
 .016 
 
 .016 
 .017 
 .017 
 
 17.2II 
 
 17.196 
 17.182 
 17.166 
 
 .029 
 
 .029 
 .030 
 .031 
 
 4-355 
 
 8.709 
 
 13.064 
 
 12 00 
 
 IS 
 
 3° 
 4S 
 
 17.419 
 
 4.288 
 
 4.284 
 4.280 
 
 4-275 
 
 .002 
 
 .002 
 .002 
 .002 
 
 8-575 
 
 8.567 
 
 8-559 
 8.551 
 
 .008 
 
 .008 
 .008 
 .008 
 
 12.863 
 
 12.851 
 12.839 
 12.826 
 
 .017 
 
 .018 
 .018 
 .019 
 
 17.150 
 
 17-134 
 I7.I18 
 17.102 
 
 •031 
 
 -03Z 
 .032 
 
 -033 
 
 4-355 
 8.710 
 13.065 
 
 1300 
 
 IS 
 30 
 4S 
 
 17.420 
 
 4-271 
 
 4.267 
 4.262 
 4.258 
 
 .002 
 
 .002 
 .002 
 .002 
 
 8.542 
 8.534 
 8.516 
 
 .008 
 
 .009 
 .009 
 .009 
 
 12.813 
 
 12.800 
 12.787 
 12.774 
 
 .019 
 
 .019 
 .020 
 .020 
 
 17.084 
 
 17.067 
 17.050 
 17.032 
 
 •034 
 
 -034 
 -03s 
 ■035 
 
 4-355 
 
 8.7 1 1 
 
 13.066 
 
 1400 
 
 IS 
 
 30 
 
 4S 
 
 17.421 
 
 4-253 
 
 4.249 
 4.244 
 4-239 
 
 .002 
 
 .002 
 .002 
 .002 
 
 8.507 
 
 8.498 
 8.488 
 8.479 
 
 .009 
 
 .009 
 .009 
 ,009 
 
 12.760 
 
 12.746 
 12.732 
 12.718 
 
 .020 
 
 .021 
 .021 
 .021 
 
 17.013 
 
 16.995 
 16.976 
 16.957 
 
 •036 
 
 .036 
 
 -037 
 -038 
 
 4-356 
 
 8.7 II 
 
 13.067 
 
 1500 
 
 IS 
 3° 
 
 45 
 
 17-423 
 
 4-234 
 
 4.229 
 4.224 
 4.219 
 
 .002 
 
 .002 
 .002 
 .002 
 
 8.469 
 
 8.459 
 8.449 
 8.439 
 
 .010 
 
 .010 
 .010 
 .010 
 
 12.703 
 
 12.688 
 12.673 
 12.658 
 
 .022 
 
 .022 
 .022 
 .022 
 
 16.938 
 
 16.918 
 16.898 
 16.877 
 
 .038 
 
 ■039 
 •039 
 .040 
 
 4-356 
 
 8.712 
 
 13.068 
 
 1600 
 
 IS 
 30 
 
 45 
 
 17.424 
 
 4.214 
 
 4.209 
 4.204 
 4.198 
 
 .003 
 
 .003 
 .003 
 .003 
 
 8.428 
 
 8.417 
 8.407 
 8.396 
 
 .010 
 
 .010 
 .010 
 .Oil 
 
 12.642 
 
 12.626 
 12.610 
 12.594 
 
 .023 
 
 .023 
 .023 
 .024 
 
 16.856 
 
 16.835 
 16.814 
 16.792 
 
 .041 
 
 .041 
 .042 
 .042 
 
 4-356 
 
 8-713 
 
 13.069 
 
 17 00 
 
 IS 
 30 
 45 
 
 17.426 
 
 4.192 
 
 4.187 
 4.181 
 4-175 
 
 -003 
 
 .003 
 .003 
 .003 
 
 8-385 
 
 8.374 
 8.362 
 
 8.351 
 
 .oil 
 
 .oil 
 .oil 
 .oil 
 
 12-577 
 
 12.561 
 12.544 
 12.526 
 
 .024 
 
 .024 
 .025 
 .025 
 
 16.770 
 
 16.748 
 16.725 
 16.702 
 
 •043 
 
 .043 
 .044 
 .044 
 
 4-357 
 8.714 
 13.071 
 
 i8oo 
 
 IS 
 30 
 45 
 
 17.427 
 
 4.170 
 
 4.164 
 4.158 
 4.152 
 
 .003 
 
 .003 
 .003 
 .003 
 
 8.339 
 
 8.327 
 8.316 
 8.303 
 
 .oil 
 
 .oil 
 .012 
 
 .012 
 
 12.509 
 
 12.491 
 
 12.473 
 12.455 
 
 .025 
 
 .026 
 .026 
 .026 
 
 16.679 
 
 16.631 
 16.606 
 
 •045 
 
 .045 
 .046 
 .046 
 
 4.357 
 
 8-715 
 
 13.072 
 
 1900 
 
 IS 
 30 
 45 
 
 17.429 
 
 4.145 
 
 4-139 
 4-133 
 4.127 
 
 .003 
 
 .003 
 ■003 
 .003 
 
 8.291 
 
 8.278 
 8.266 
 8-253 
 
 .012 
 
 .012 
 .OI2 
 .012 
 
 12.436 
 
 12.418 
 12.399 
 12.380 
 
 .026 
 
 .027 
 .027 
 .027 
 
 16.582 
 
 16.557 
 16.532 
 16.506 
 
 •047 
 
 .048 
 .048 
 .048 
 
 8.716 
 13-073 
 
 2000 
 
 17.431 
 
 4.120 
 
 -003 
 
 8.240 
 
 .012 
 
 12.360 
 
 .028 
 
 16.480 
 
 .049 
 
 
 Smithsonian Tables. 
 
 8S 
 
Table 1 9. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE mW- 
 
 [Derivation of table explained on pp. liii-lvi,] 
 
 ■s 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 15' longitude. 
 
 30* longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 20°00' 
 
 15 
 3° 
 45 
 
 8.717 
 13-075 
 
 4.120 
 4-114 
 4-107 
 4.100 
 
 .003 
 .003 
 .003 
 .003 
 
 8.240 
 8.227 
 8.214 
 8.200 
 
 .012 
 .012 
 •013 
 •013 
 
 12.360 
 
 12.349 
 12.321 
 12.301 
 
 .028 
 .028 
 .028 
 .029 
 
 16.480 
 16.454 
 16.428 
 16.401 
 
 .049 
 •050 
 .050 
 .051 
 
 21 00 
 
 15 
 3° 
 
 45 
 
 17433 
 
 4.094 
 
 4.087 
 4.080 
 4-073 
 
 ■003 
 
 -003 
 ■003 
 .003 
 
 8.187 
 
 8.173 
 8-159 
 8-145 
 
 .013 
 
 .013 
 .013 
 -013 
 
 12.280 
 
 12.260 
 12.239 
 12.218 
 
 .029 
 
 .029 
 .029 
 .030 
 
 16.374 
 
 16.346 
 16.318 
 16.291 
 
 .051 
 
 .052 
 .052 
 
 •053 
 
 ^359 
 8.718 
 
 13.076 
 
 22 00 
 15 
 
 3° 
 45 
 
 17-435 
 
 4.066 
 
 4-058 
 4.051 
 4.044 
 
 .003 
 
 .003 
 .003 
 .003 
 
 8.I3I 
 8.117 
 
 8.102 
 
 8.088 
 
 .013 
 
 .013 
 .014 
 .014 
 
 12.197 
 
 12.175 
 12.154 
 12.132 
 
 .030 
 
 -030 
 •030 
 .031 
 
 16.262 
 
 16.234 
 16.205 
 16.176 
 
 •053 
 
 .054 
 ■054 
 •055 
 
 4-359 
 13.078 
 
 2300 
 
 IS 
 30 
 45 
 
 17-437 
 
 4.036 
 
 4.029 
 4.021 
 4.014 
 
 .003 
 
 .003 
 .003 
 .004 
 
 8.073 
 
 8.058 
 8.043 
 8.028 
 
 .014 
 
 .014 
 .014 
 .014 
 
 12.109 
 
 12.087 
 12.064 
 12.041 
 
 .031 
 
 .031 
 .031 
 ■032 
 
 16.146 
 
 16.116 
 16.086 
 16.055 
 
 ■055 
 
 •055 
 .056 
 .056 
 
 4.360 
 
 8.720 
 
 13.080 
 
 2400 
 
 15 
 30 
 45 
 
 17-439 
 
 4.006 
 
 3-998 
 3-990 
 3.982 
 
 .004 
 
 .004 
 .004 
 .004 
 
 8.012 
 
 7-997 
 7-981 
 7-965 
 
 .014 
 
 .014 
 .014 
 
 -015 
 
 12.018 
 
 11.995 
 II.971 
 11.948 
 
 -032 
 
 .032 
 .032 
 -033 
 
 16.024 
 
 15-993 
 15.962 
 
 15-930 
 
 •057 
 
 .058 
 .058 
 
 4-360 
 
 8.721 
 
 13.081 
 
 2500 
 
 15 
 30 
 45 
 
 17.442 
 
 3-974 
 
 3-966 
 3-958 
 3-950 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7-949 
 
 7-933 
 7-916 
 7.900 
 
 .015 
 
 .015 
 .015 
 .015 
 
 11-923 
 
 H.899 
 
 11.874 
 H.850 
 
 ■033 
 
 •033 
 -033 
 •034 
 
 15-898 
 
 15.865 
 
 15-832 
 15.800 
 
 •059, 
 •059 
 
 :o°l^ 
 
 4.361 
 8.722 
 13-083 
 
 2600 
 
 15 
 30 
 
 45 
 
 17.444 
 
 3-942 
 
 3-933 
 3-925 
 3-916 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7.883 
 
 7.866 
 7.849 
 7-833 
 
 .015 
 
 •015 
 .015 
 •015 
 
 11.825 
 11.800 
 
 11-774 
 11.749 
 
 •034 
 
 •034 
 •034 
 -035 
 
 15.767 
 
 15-733 
 15.699 
 
 15-665 
 
 .060 
 
 .061 
 .061 
 .061 
 
 4.362 
 
 8.723 
 
 13-085 
 
 27 00 
 
 15 
 30 
 
 45 
 
 17.446 
 
 3.908 
 3-899 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7.816 
 
 7.798 
 7.780 
 7-763 
 
 .015 
 
 .016 
 .016 
 .016 
 
 11.723 
 
 11.697 
 11.671 
 1 1.644 
 
 -035 
 
 •035 
 -03s 
 .036 
 
 15-631 
 
 15-596 
 15.561 
 15.526 
 
 .062 
 .062 
 •063 
 
 4.362 
 
 8.724 
 
 13-087 
 
 2800 
 
 17-449 
 
 3-873 
 
 .004 
 
 7-745 
 
 .016 
 
 11.618 
 
 •036 
 
 15.490 
 
 .064 
 
 IS 
 30 
 45 
 
 4-363 
 
 8.726 
 
 13.088 
 
 3-863 
 3854 
 3-845 
 
 .004 
 .004 
 .004 
 
 7-727 
 7.709 
 7.691 
 
 .016 
 .016 
 .016 
 
 11-591 
 11-563 
 11.536 
 
 .036 
 .036 
 .036 
 
 15-454 
 15.418 
 15-382 
 
 .064 
 .064 
 .065 
 
 2900 
 
 IS 
 30 
 45 
 
 17-451 
 
 3-836 
 
 3.827 
 
 3-817 
 3.808 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7-673 
 7-654 
 7-6i6 
 
 .016 
 
 .016 
 .016 
 .016 
 
 11.509 
 
 11.481 
 
 11-453 
 11.425 
 
 -036 
 
 -037 
 •037 
 •037 
 
 15-345 
 
 15-308 
 15-270 
 15-233 
 
 .065 
 .066 
 
 4-363 
 
 8.727 
 
 13.091 
 
 3000 
 
 17-454 
 
 3-799 
 
 .004 
 
 7-598 
 
 .017 
 
 11.396 
 
 ■037 
 
 15-195 
 
 .066 
 
 
 Smithsonian Tables. 
 
 86 
 
Table 1 9. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE aiiaSaa - 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Latitude ol 
 parallel. 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 
 15' longitude. 
 
 30' longitude. 
 
 45' longitude. 
 
 \° longitude. 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 30°00' 
 
 •s 
 
 3° 
 45 
 
 4364 
 8.728 
 13.092 
 
 3-799 
 3-789 
 3-779 
 3-770 
 
 .004 
 .004 
 .004 
 .004 
 
 7-598 
 7-578 
 
 7-559 
 7-540 
 
 .017 
 .017 
 .017 
 .017 
 
 11.396 
 11.367 
 
 "-338 
 11.309 
 
 -037 
 ■037 
 .038 
 .038 
 
 15-195 
 15.156 
 I5.I18 
 15.079 
 
 .066 
 .067 
 .067 
 .067 
 
 
 31 00 
 
 '5 
 30 
 45 
 
 17-457 
 
 3.760 
 
 3-750 
 3-740 
 3-730 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7.520 
 
 7.500 
 7480 
 7.460 
 
 .017 
 
 .017 
 .017 
 .017 
 
 11.280 
 
 11.250 
 II. 221 
 II. 191 
 
 .038 
 
 .038 
 .038 
 .038 
 
 1 5.040 
 
 15.001 
 14.961 
 14.921 
 
 .068 
 
 .c68 
 .068 
 .068 
 
 
 4-365 
 8.730 
 
 •3-095 
 
 
 3200 
 
 '15 
 3° 
 45 
 
 17.460 
 
 3.720 
 
 3-7 10 
 3-700 
 3.690 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7.441 
 
 7.420 
 7.400 
 7-379 
 
 .017 
 
 .017 
 .017 
 .017 
 
 II. 161 
 
 II. 130 
 II. 100 
 11.069 
 
 •039 
 
 -039 
 -039 
 -039 
 
 14.881 
 
 14.840 
 
 14-799 
 14.758 
 
 .069 
 
 .069 
 .069 
 .070 
 
 
 4.366 
 
 8.73' 
 
 i3-°97 
 
 
 3300 
 
 IS 
 30 
 45 
 
 17.462 
 
 3-679 
 3.648 
 
 .004 
 
 .004 
 .004 
 .004 
 
 7-359 
 
 7-338 
 7-317 
 7.296 
 
 .017 
 
 .018 
 .oi8 
 .018 
 
 1 1.038 
 
 11.007 
 10.975 
 10.943 
 
 -039 
 
 -039 
 
 .040 
 .040 
 
 14.718 
 
 14.676 
 
 H-633 
 14.591 
 
 .070 
 
 .070 
 .070 
 .071 
 
 
 4.366 
 
 8-733 
 13.099 
 
 
 3400 
 
 15 
 30 
 45 
 
 17.465 
 
 3-637 
 
 3.626 
 3.616 
 3-605 
 
 .004 
 
 .004 
 .0,04 
 .004 
 
 7-275 
 
 7-253 
 7-231 
 7.210 
 
 .018 
 
 .018 
 .018 
 .018 
 
 10.912 
 
 10.879 
 10847 
 10.815 
 
 .040 
 
 .040 
 .040 
 .040 
 
 14.549 
 
 14.506 
 14.463 
 14.420 
 
 .071 
 
 .071 
 .071 
 .072 
 
 
 4-367 
 8-734 
 13.101 
 
 
 3500 
 
 15 
 3° 
 45 
 
 17.468 
 
 3-594 
 
 3-583 
 3 572 
 3-561 
 
 .004 
 
 .004 
 .004 ■ 
 
 .005 
 
 7.188 
 
 7.166 
 7.144 
 7.122 
 
 .018 
 
 .018 
 .018 
 .018 
 
 10.782 
 
 10749 
 10.716 
 10.683 
 
 .040 
 
 .041 
 .041 
 .041 
 
 14-376 
 
 14332 
 14.288 
 14.244 
 
 .072 
 
 .072 
 .072 
 -073 
 
 
 4.368 
 
 8-735 
 
 13-103 
 
 
 3600 
 
 '5 
 30 
 45 
 
 17-471 
 
 3-550 
 
 3-539 
 3-527 
 3-516 
 
 .005 
 
 .005 
 .005 
 .005 
 
 7.100 
 
 7-077 
 7.054 
 7.032 
 
 .018 
 
 .018 
 .018 
 .018 
 
 10.650 
 
 10.616 
 10.582 
 10.547 
 
 .041 
 
 .041 
 .041 
 .041 
 
 14.200 
 
 14.154 
 14.109 
 14.063 
 
 -073 
 
 -073 
 -073 
 •073 
 
 
 4-368 
 
 8.736 
 
 13-105 
 
 
 3700 
 
 15 
 30 
 45 
 
 17-473 
 
 3-504 
 3-470 
 
 .005 
 
 .005 
 .005 
 .005 
 
 7.009 
 
 6.986 
 6.963 
 6-939 
 
 .018 
 
 .018 
 .018 
 .018 
 
 10.513 
 
 10.479 
 10.444 
 10.409 
 
 .041 
 
 .041 
 .042 
 .042 
 
 14.018 
 
 13-972 
 13-925 
 13-879 
 
 -074 
 
 .074 
 .074 
 -074 
 
 
 4-369 
 
 8.738 
 
 13.108 
 
 
 3800 
 
 15 
 30 
 45 
 
 17-477 
 
 3-458 
 
 3-446 
 3-434 
 3.422 
 
 .005 
 
 .005 
 .005 
 .005 
 
 6.916 
 
 6.892 
 6.869 
 6.845 
 
 .019 
 
 .019 
 .019 
 .019 
 
 10.374 
 
 10-339 
 10-303 
 10.267 
 
 .042 
 
 .042 
 .042 
 .042 
 
 13-832 
 
 1.3-785 
 13-737 
 13.690 
 
 •074 
 
 .074 
 
 •075 
 .075 
 
 
 4-370 
 
 8.740 
 
 13.110 
 
 
 3900 
 
 15 
 30 
 
 45 
 
 17.480 
 
 3-4" 
 
 3-398 
 3-386 
 3-374 
 
 .005 
 
 .005 
 -005 
 .005 
 
 6.821 
 
 6-797 
 6-773 
 6.748 
 
 .019 
 
 .019 
 .019 
 .019 
 
 10.232 
 
 10.195 
 10.159 
 10.123 
 
 .042 
 
 .042 
 .042 
 .042 
 
 13.642 
 
 13-594 
 13-545 
 13-497 
 
 .075 
 
 -075 
 -075 
 .075 
 
 
 4-371 
 
 8.741 
 
 13.112 
 
 
 4000 
 
 17-483 
 
 3-362 
 
 .005 
 
 6.724 
 
 .019 
 
 10.086 
 
 .042 
 
 13-448 
 
 -075 
 
 
 Smithsonian Tables. 
 
 87 
 
Table 19. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE nxiW- 
 [Derivation of table explained on p. liii-Ivi.] 
 
 ■s 
 
 11 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR- 
 
 
 15' longitude. 
 
 so' longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 Inches* 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 40°oo' 
 IS 
 3° 
 45 
 
 4-371 
 
 8743 
 
 13-114 
 
 3-362 
 
 3-350 
 3-337 
 3-325 
 
 .005 
 .005 
 .005 
 .005 
 
 6.724 
 6.699 
 
 6.650 
 
 .019 
 .019 
 .019 
 .019 
 
 10.086 
 10.049 
 10.012 
 
 9-975 
 
 .042 
 .042 
 •043 
 •043 
 
 13.448 
 13-399 
 
 13-349 
 13-300 
 
 •075 
 
 .075 
 .076 
 .076 
 
 
 41 00 
 
 IS 
 30 
 45 
 
 17.486 
 
 3-312 
 
 3-300 
 3-287 
 3-275 
 
 .005 
 
 .005 
 .005 
 -005 
 
 6.625 
 6.600 
 
 6-57S 
 6-549 
 
 .019 
 
 .019 
 .019 
 .019 
 
 9-937 
 
 9.900 
 9.862 
 9.824 
 
 •043 
 
 •043 
 -043 
 •043 
 
 13.250 
 
 13.200 
 13-149 
 13-098 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-372 
 8.744 
 
 13-117 
 
 
 42 00 
 
 IS 
 
 30 
 4S 
 
 17.489 
 
 3.262 
 
 3-249 
 3-236 
 3-223 
 
 .005 
 
 .005 
 .005 
 .005 
 
 6.524 
 
 6.498 
 6.472 
 6-447 
 
 .019 
 
 .019 
 .019 
 .019 
 
 9.786 
 
 9-747 
 9.709 
 9.670 
 
 •043 
 
 -043 
 •043 
 •043 
 
 13.048 
 
 12.996 
 12.945 
 12.893 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-373 
 
 8.746 
 
 13.119 
 
 
 4300 
 
 IS 
 30 
 4S 
 
 17.492 
 
 3-210 
 
 3-197 
 3.184 
 3-170 
 
 .005 
 
 .005 
 .005 
 .005 
 
 6.421 
 
 6-394 
 6.368 
 6.342 
 
 .019 
 
 .019 
 .019 
 .019 
 
 9.631 
 
 9.592 
 9-552 
 9-513 
 
 •043 
 
 •043 
 -043 
 •043 
 
 12.S42 
 12.789 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-374 
 8.747 
 13.121 
 
 
 4400 
 
 IS 
 30 
 
 4S 
 
 17-495 
 
 3-158 
 
 3-144 
 3-131 
 3.118 
 
 .005 
 
 .005 
 .005 
 .005 
 
 6.316 
 
 6.289 
 6.262 
 6.235 
 
 .019 
 
 .019 
 .019 
 .019 
 
 9-473 
 
 9-433 
 9-393 
 9-3S3 
 
 -043 
 
 •043 
 ■043 
 ■043 
 
 12.631 
 
 12.578 
 12.524 
 12.471 
 
 .077 
 
 .077 
 .077 
 .077 
 
 
 4-375 
 8.749 
 13.124 
 
 
 4500 
 
 IS 
 30 
 45 
 
 17.498 
 
 3-104 
 
 3-091 
 3-077 
 3-063 
 
 .005 
 
 .005 
 .005 
 .005 
 
 6.209 
 
 6.i8i 
 
 6-154 
 6.127 
 
 .019 
 
 .019 
 .019 
 .019 
 
 9-313 
 
 9.272 
 
 9-231 
 9.190 
 
 •043 
 
 -043 
 ■043 
 •043 
 
 12.417 
 12.367 
 
 12.308 
 
 12.254 
 
 .077 
 
 .077 
 .077 
 .077 
 
 
 4-375 
 
 8-751' 
 
 13.126 
 
 
 4600 
 
 17.501 
 
 3.050 
 
 .005 
 
 6.100 
 
 .019 
 
 9-150 
 
 •043 
 
 12.200 
 
 .077 
 
 
 IS 
 30 
 45 
 
 4-376 
 8.752 
 13.128 
 
 3-036 
 3.022 
 3.008 
 
 .005 
 .005 
 .005 
 
 6.072 
 6.044 
 6.017 
 
 .019 
 .019 
 .019 
 
 9.108 
 9.067 
 9.025 
 
 •043 
 -043 
 -043 
 
 12.144 
 12.089 
 12.033 
 
 .077 
 .077 
 .077 
 
 
 4700 
 
 IS 
 30 
 45 
 
 17.504 
 
 2.994 
 
 2.980 
 2.966 
 2.952 
 
 .005 
 
 .005 
 .005 
 .005 
 
 S-989 
 
 5.961 
 
 S-933 
 5.904 
 
 .019 
 
 .019 
 .019 
 .019 
 
 8.983 
 8.857 
 
 -043 
 
 •043 
 -043 
 •043 
 
 11.978 
 
 11.922 
 11.865 
 11.809 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-377 
 
 8-754 
 
 13-131 
 
 
 48 00 
 
 IS 
 30 
 
 45 
 
 17.508 
 
 2.938 
 
 2.924 
 2.909 
 2.89s 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.876 
 
 S-848 
 5.819 
 5.790 
 
 .019 
 
 .019 
 .019 
 .019 
 
 8.814 
 
 8.771 
 8.728 
 8.686 
 
 •043 
 
 •043 
 •043 
 •043 
 
 11.752 
 
 111638 
 II.581 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-378 
 
 8-7SS 
 
 13-133 
 
 
 4900 
 
 15 
 30 
 45 
 
 17.511 
 
 2.881 
 
 2.866 
 2.852 
 2.837 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.762 
 
 S-733 
 5-704 
 5-675 
 
 .019 
 
 .019 
 .019 
 .019 
 
 8.643 
 
 8,599 
 
 8-555 
 8.512 
 
 -043 
 
 •043 
 ■043 
 .042 
 
 11.524 
 
 11.465 
 11.407 
 
 11-349 
 
 .076 
 
 .076 
 .076 
 .076 
 
 
 4-378 
 
 8-757 
 
 13-135 
 
 
 50 00 
 
 17-514 
 
 2.823 
 
 •005 
 
 5.646 
 
 .019 
 
 8.468 
 
 .042 
 
 11.291 
 
 .076 
 
 
 
 
 Smithsonian Tables. 
 
 88 
 
Table 19. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE T^isTS- 
 
 [Derivation of table explained on p. liii-lvi.] 
 
 "S . 
 
 9 n 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 is' longitude. 
 
 30^ longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inckts. 
 
 /nckgs. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 So°oo' 
 IS 
 3° 
 4S 
 
 13-137 
 
 2.823 
 2.808 
 
 2.793 
 2.779 
 
 .005 
 .005 
 .005 
 .005 
 
 5.646 
 5.616 
 
 5- 587 
 5-557 
 
 .019 
 .019 
 .019 
 .019 
 
 8.468 
 8.424 
 8.380 
 8-336 
 
 .042 
 .042 
 .042 
 .042 
 
 II. 291 
 11.232 
 II.I74 
 II. 114 
 
 .076 
 -075 
 -07 s 
 -075 
 
 SI 00 
 
 IS 
 30 
 45 
 
 17-517 
 
 2.764 
 
 2.749 
 2-734 
 
 2.719 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.528 
 
 5-498 
 5.468 
 
 S-438 
 
 .019 
 
 .019 
 .019 
 .019 
 
 8.291 
 
 8.247 
 8.202 
 8-157 
 
 .042 
 
 .042 
 .042 
 .042 
 
 11.055 
 
 10.996 
 10.936 
 10.876 
 
 .075 
 
 •075 
 .075 
 .075 
 
 4.380 
 
 8.760 
 
 13.140 
 
 5200 
 
 IS 
 30 
 4S 
 
 17.520 
 
 2.704 
 
 2.689 
 2.674 
 2.659 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.408 
 
 5-378 
 5-347 
 5-317 
 
 .019 
 
 .019 
 .019 
 .018 
 
 8.II2 
 
 - 8.067 
 
 8.021 
 
 7.976 
 
 .042 
 
 .042 
 .041 
 .041 
 
 10.816 
 
 10.756 
 10.695 
 10.634 
 
 .074 
 
 •074 
 .074 
 .074 
 
 t.fe\ 
 13.142 
 
 S3 00 
 
 IS 
 30 
 45 
 
 17-523 
 
 2.643 
 2.628 
 
 2.613 
 2-597 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.287 
 
 5.256 
 5-225 
 5-195 
 
 .018 
 
 .018 
 .018 
 .018 
 
 7-930 
 
 7.792 
 
 .041 
 
 .041 
 .041 
 .041 
 
 10-573 
 
 10.512 
 10.451 
 10.389 
 
 .074 
 
 .074 
 -073 
 -073 
 
 4-381 
 8.763 
 
 13-144 
 
 54 00 
 
 IS 
 30 
 45 
 
 17.526 
 
 2.582 
 
 2.566 
 2.551 
 2-535 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.164 
 
 5-133 
 5.102 
 5.070 
 
 .018 
 
 .018 
 .018 
 .018 
 
 7-745 
 7.699 
 7.606 
 
 .041 
 
 .041 
 .041 
 .041 
 
 10.327 
 
 10.266 
 10.203 
 10.141 
 
 •073 
 
 ■073 
 •073 
 .072 
 
 4.382 
 
 8.764 
 
 13147 
 
 SSOo 
 
 IS 
 
 30 
 45 
 
 17.529 
 
 2.520 
 
 2488 
 2.472 
 
 .005 
 
 .004 
 .004 
 .004 
 
 5-039 
 
 5.008 
 4.976 
 4-945 
 
 .018 
 
 .018 
 .018 
 .018 
 
 7-559 
 
 7-512 
 7-465 
 7-417 
 
 .041 
 
 .040 
 .040 
 .040 
 
 10.078 
 
 10.016 
 
 9-953 
 9.890 
 
 .072 
 
 .072 
 .072 
 .071 
 
 13-149 
 
 S6oo 
 
 IS 
 30 
 45 
 
 17-532 
 
 2.456 
 
 2.441 
 2.425 
 2.409 
 
 .004 
 
 .004 
 .004 
 .004 
 
 4-913 
 
 4.881 
 
 4-849 
 4.817 
 
 .018 
 
 .018 
 .018 
 .018 
 
 7-370 
 
 7.322 
 7.274 
 7.226 
 
 .040 
 
 .040 
 .040 
 .040 
 
 9.826 
 
 9763 
 9.699 
 
 9-635 
 
 .071 
 
 .071 
 .071 
 .070 
 
 4-384 
 
 8.767 
 
 13-151 
 
 5700 
 
 IS 
 
 30 
 45 
 
 17-535 
 
 2-393 
 
 2.377 
 2.361 
 
 2-344 
 
 .004 
 
 .004 
 .004 
 .004 
 
 4-785 
 
 4-753 
 4.721 
 
 4.689 
 
 .018 
 
 .017 
 .017 
 .017 
 
 7.178 
 
 7-130 
 7.082 
 
 7-033 
 
 -039 
 
 •039 
 -039 
 •039 
 
 9-S7I 
 
 9.507 
 9-442 
 9-378 
 
 .070 
 
 .070 
 .070 
 .069 
 
 tr4 
 13-153 
 
 S8oo 
 
 17-537 
 
 2.328 
 
 .004 
 
 4.656 
 
 .017 
 
 6.985 
 
 •039 
 
 9-313 
 
 .069 
 
 15 
 30 
 45 
 
 4-385 
 8.770 
 
 13-155 
 
 2.312 
 2.296 
 2.279 
 
 .004 
 .004 
 .004 
 
 4.624 
 4.591 
 4-559 
 
 .017 
 .017 
 .017 
 
 6.936 
 
 6.8H7 
 6.838 
 
 •039 
 [038 
 
 9.248 
 9.183 
 9.117 
 
 ;o68 
 
 5900 
 
 15 
 30 
 45 
 
 17-540 
 
 2.263 
 
 2.246 
 2.230 
 2.214 
 
 .004 
 
 .004 
 .004 
 .004 
 
 4.526 
 
 4-493 
 4.460 
 4.427 
 
 .017 
 
 .017 
 .017 
 .017 
 
 6.789 
 
 6.740 
 6.690 
 6.641 
 
 .038 
 
 .038 
 .038 
 .038 
 
 9-052 
 
 8.986 
 8.920 
 8.854 
 
 .068 
 .068 
 
 .067 
 
 .067 
 
 4.386 
 
 8.772 
 
 13-157 
 
 6000 
 
 17-543 
 
 2.197 
 
 .004 
 
 4-394 
 
 .017 
 
 6.591 
 
 -037 
 
 8.788 
 
 .067 
 
 Smithsonian Tables. 
 
 89 
 
Table 19. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE sWinnF- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 11 
 
 Meridional dis- 
 tances Irum 
 even degree 
 Ijarallels. 
 
 CO-ORDIN.'VTES OF DEVELOPED PARALLEL FOR — 
 
 15' longitude. 
 
 30^ longitude. 
 
 45' longitude. 
 
 1° longitude. 1 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 htches. 
 
 Inches. 
 
 IncJies 
 
 Inches. 
 
 Inches, 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 6o°oo' 
 15 
 3° 
 4S 
 
 4-386' 
 
 8.773 
 
 I3-'S9 
 
 2.197 
 2.180 
 2.164 
 2.147 
 
 .004 
 .004 
 .004 
 .004 
 
 4-394 
 4.361 
 
 4.327 
 4.294 
 
 .017 
 .017 
 .016 
 .016 
 
 6.591 
 6.541 
 6.491 
 6.441 
 
 -037 
 •037 
 -037 
 •037 
 
 8.788 
 8.722 
 
 15^ 
 
 .067 
 .066 
 .066 
 .066 
 
 61 00 
 
 IS 
 30 
 45 
 
 17.546 
 
 2.130 
 
 2.II4 
 2.097 
 2.080 
 
 .004 
 
 .004 
 .004 
 .004 
 
 4.261 
 
 4.227 
 
 4-194 
 4.160 
 
 .016 
 
 .016 
 .016 
 .016 
 
 6.391 
 
 6.340 
 6.290 
 6.240 
 
 •037 
 
 .036 
 ■036 
 •036 
 
 8.521 
 
 8.454 
 8-387 
 8.320 
 
 .06s 
 
 .064 
 
 .064 
 
 4.387 
 
 8.774 
 
 13.161 
 
 62 00 
 
 IS 
 
 30 
 45 
 
 17.548 
 
 2.063 
 
 2.046 
 2.029 
 2.012 
 
 .004 
 
 .004 
 .004 
 .004 
 
 4.126 
 
 4.092 
 4.058 
 4.024 
 
 .016 
 
 .016 
 .016 
 .016 
 
 6.189 
 
 6.138 
 6.088 
 6.036 
 
 .036 
 
 •036 
 •035 
 •03s 
 
 8.252 
 
 8.184 
 8.117 
 8.048 
 
 .064 
 
 .063 
 .063 
 
 4.388 
 
 8.776 
 
 13.163 
 
 6300 
 
 IS 
 30 
 45 
 
 17.551 
 
 1.99s 
 
 1-978 
 1. 96 1 
 
 1-944 
 
 .004 
 
 .004 
 .004 
 .004 
 
 3-99° 
 3-956 
 |8%^ 
 
 .015 
 
 .015 
 .015 
 
 .015 
 
 5-985 
 5.883 
 
 5-83" 
 
 •03s 
 
 •03s 
 •034 
 •034 
 
 7.980 
 
 7.912 
 7.844 
 7-775 
 
 .062 
 
 .062 
 .061 
 .061 
 
 4.388 
 8.777 
 13.165 
 
 6400 
 
 IS 
 30 
 45 
 
 17-554 
 
 1.926 
 
 ;i9^ 
 1.875 
 
 .004 
 
 .004 
 .004 
 .004 
 
 3-853 
 
 3.819 
 3-784 
 3-749 
 
 .015 
 
 .015 
 .015 
 .015 
 
 5.780 
 
 5.676 
 5.624 
 
 •034 
 
 •034 
 •034 
 •033 
 
 7.706 
 7-499 
 
 .060 
 
 .060 
 .060 
 .059 
 
 13.167 
 
 6500 
 
 IS 
 30 
 
 45 
 
 17.556 
 
 1.857 
 
 1.840 
 1.823 
 1.805 
 
 .004 
 
 .004 
 .004 
 .004 
 
 3-715 
 
 3.680 
 
 3-645 
 3.610 
 
 .015 
 
 .015 
 .014 
 .014 
 
 5-572 
 
 5.520 
 5.468 
 5-415 
 
 •033 
 
 -033 
 •033 
 -C32 
 
 7.430 
 
 7.360 
 7.290 
 7.220 
 
 •059 
 
 .059 
 .058 
 .058 
 
 4.390 
 8-779 
 13.169 
 
 6600 
 
 15 
 30 
 
 45 
 
 17-559 
 
 1.788 
 
 1.770 
 1-753 
 
 1-735 
 
 .004 
 
 .004 
 .004 
 .003 
 
 3-575 
 
 3-540 
 3-505 
 3-470 
 
 .014 
 
 .014 
 .014 
 .014 
 
 5-363 
 
 5-310 
 5.258 
 5-205 
 
 .032 
 
 •032 
 .032 
 .031 
 
 7-151 
 
 7.080 
 7.010 
 6.940 
 
 •057 
 
 •057 
 .056 
 .056 
 
 8780 
 13.171 
 
 67 00 
 
 IS 
 30 
 
 45 
 
 17.561 
 
 1.717 
 
 1.700 
 1.682 
 1.664 
 
 .003 
 
 .003 
 .003 
 .003 
 
 3-435 
 
 3.400 
 3-364 
 3-329 
 
 .014 
 
 .014 
 .014 
 .013 
 
 5.152 
 
 5-099 
 5-046 
 
 4-993 
 
 .031 
 
 .031 
 .031 
 .030 
 
 6.870 
 
 6.799 
 6.728 
 6.658 
 
 •05s 
 
 .055 
 •054 
 •054 
 
 8.782 
 13.172 
 
 6800 
 
 15 
 30 
 45 
 
 17.563 
 
 1.647 
 
 1.629 
 1.6(1 
 1-593 
 
 -003 
 
 •003 
 .003 
 .003 
 
 3-293 
 
 3-258 
 3.222 
 3.186 
 
 .013 
 
 .013 
 .013 
 •013 
 
 4.940 
 , 4.886 
 
 .030 
 
 ■030 
 .029 
 .029 
 
 6.586 
 
 6.515 
 6.444 
 6.373 
 
 .053 
 
 •053 
 .052 
 .052 
 
 4.391 
 8.783 
 
 13-174 
 
 69 00 
 
 15 
 30 
 45 
 
 17.565 
 
 1-575 
 
 I-5S7 
 1.540 
 1.522 
 
 .003 
 
 .003 
 -003 
 .003 
 
 3.151 
 
 3-"5 
 3-079 
 3.043 
 
 .013 
 
 .013 
 .013 
 .012 
 
 4.726 
 
 4-672 
 4.618 
 4.564 
 
 .029 
 
 .029 
 .028 
 .028 
 
 6.301 
 
 6.230 
 6.158 
 
 .051 
 
 .051 
 .051 
 .050 
 
 1392 
 8-784 
 13.176 
 
 7000 
 
 17.568 
 
 1.504 
 
 .003 
 
 3.007 
 
 .012 
 
 4.510 
 
 .028 
 
 6.014 
 
 .049 
 
 
 Smithsonian Tables. 
 
 90 
 
CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Table 19> 
 SCALE Wffinf 
 
 ■g 
 
 Meridional dis 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 
 15' longitude. 
 
 30* longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 70°oo' 
 IS 
 3° 
 4S 
 
 13-177 
 
 1.504 
 1.486 
 1.467 
 1-449 
 
 .003 
 •003 
 .003 
 .003 
 
 3-007 
 2.971 
 
 2.93s 
 
 .012 
 .012 
 .012 
 .012 
 
 4.510 
 
 4-456 
 4.402 
 
 4-348 
 
 .028 
 .028 
 .027 
 .027 
 
 6.014 
 5.942 
 5-870 
 
 5-797 
 
 •049 
 -049 
 .048 
 .048 
 
 
 71 00 
 
 IS 
 30 
 4S 
 
 17.570 
 
 I-43I 
 
 I -41 3 
 1-395 
 1-377 
 
 .003 
 
 .003 
 .003 
 .003 
 
 2.862 
 
 2.826 
 2.790 
 2-753 
 
 .012 
 .012 
 
 .on 
 ■oil 
 
 4.294 
 
 4-239 
 4.185 
 4.130 
 
 .027 
 
 .026 
 .026 
 .026 
 
 5-725 
 
 ^So 
 S-S07 
 
 .047 
 
 .047 
 .046 
 .046 
 
 
 8.786 
 13-179 
 
 
 7200 
 
 17.572 
 
 1-358 
 
 .003 
 
 2.717 
 
 .oil 
 
 4-075 
 
 .025 
 
 5-434 
 
 .045 
 
 
 IS 
 30 
 4S 
 
 13.180 
 
 1-340 
 1.322 
 1.304 
 
 .003 
 .003 
 -003 
 
 2.681 
 
 2.644 
 2.607 
 
 .011 
 .oil 
 .oil 
 
 4.021 
 3.966 
 
 3-9 1 1 
 
 .025 
 .025 
 .024 
 
 5.288 
 5-215 
 
 •045 
 .044 
 
 .044 
 
 
 7300 
 
 IS 
 30 
 4S 
 
 17-573 
 
 1.285 
 
 1.267 
 1.249 
 1.230 
 
 -003 
 
 .003 
 -003 
 .003 
 
 2.571 
 
 2-S34 
 2.497 
 2.461 
 
 .011 
 .oil 
 
 .010 
 .010 
 
 3-856 
 
 3.801 
 3-746 
 3.691 
 
 .024 
 
 .024 
 .024 
 .023 
 
 5.142 
 
 5.068 
 4.994 
 4.921 
 
 •043 
 
 .043 
 .042 
 .041 
 
 
 4-394 
 
 8.788 
 
 13.181 
 
 
 7400 
 
 . IS 
 30 
 4S 
 
 17-575 
 
 I.2I2 
 I-I93 
 
 1..7S 
 1.156 
 
 .003 
 
 -003 
 .002 
 .002 
 
 2.424 
 
 2.387 
 2-350 
 Z-313 
 
 .010 
 
 .010 
 .010 
 
 .010 
 
 3636 
 
 3.580 
 
 3-525 
 3-470 
 
 -023 
 
 .023 
 .022 
 .022 
 
 4.848 
 
 4-774 
 4.700 
 4.626 
 
 .041 
 
 .040 
 .040 
 -039 
 
 
 13-183 
 
 
 7500 
 
 17-577 
 
 I -138 
 
 .002 
 
 2.276 
 
 .010 
 
 3-414 
 
 .022 
 
 4-552 
 
 .038 
 
 
 IS 
 
 30 
 45 
 
 4-395 
 
 8.789 
 
 13.184 
 
 1.119 
 
 I.IOI 
 
 1.082 
 
 .002 
 .002 
 .002 
 
 2.239 
 2.202 
 2.165 
 
 .009 
 
 .C09 
 
 .009 
 
 3-358 
 3-303 
 3-247 
 
 .021 
 .021 
 .021 
 
 4.478 
 4.404 
 4-329 
 
 .038 
 
 •037 
 •037 
 
 
 7600 
 
 IS 
 
 30 
 45 
 
 17-579 
 
 1.064 
 
 1.045 
 1.026 
 1.008 
 
 .002 
 
 .002 
 .002 
 .002 
 
 2.127 
 
 2.090 
 2.053 
 2.016 
 
 .009 
 
 .009 
 
 .009 
 
 .009 
 
 3-191 
 
 3-135 
 3-079 
 3-023 
 
 .020 
 
 .020 
 .020 
 .019 
 
 4.255 
 
 4.180 
 4.106 
 4.031 
 
 .036 
 
 .036 
 -035 
 -034 
 
 
 4-395 
 13-185 
 
 
 7700 
 
 IS 
 30 
 45 
 
 17.580 
 
 0.989 
 
 0.970 
 0.952 
 0-933 
 
 .002 
 
 .002 
 .002 
 .002 
 
 1.978 
 1.941 
 
 .008 
 
 .008 
 .008 
 .008 
 
 2.967 
 
 2.911 
 
 2-855 
 2-799 
 
 .019 
 
 .019 
 .018 
 .018 
 
 3-956 
 
 3-882 
 3-807 
 3-732 
 
 •034 
 
 -033 
 -033 
 .032 
 
 
 4-395 
 
 8.791 
 
 13.186 
 
 
 7800 
 
 IS 
 30 
 45 
 
 17.582 
 
 0.914 
 
 0.895 
 0.877 
 0.858 
 
 .002 
 
 .002 
 .002 
 .002 
 
 1.828 
 
 1.791 
 
 1-753 
 1.716 
 
 .008 
 
 .008 
 .008 
 .008 
 
 2-743 
 
 2.686 
 2.630 
 2-573 
 
 .018 
 
 .017 
 .017 
 .017 
 
 3-657 
 
 3-582 
 3-506 
 3-43' 
 
 .031 
 
 .031 
 .030 
 .030 
 
 
 4-396 
 8.791 
 13.187 
 
 
 7900 
 
 IS 
 
 30 
 45 
 
 17-583 
 
 0.839 
 
 0.820 
 0.801 
 0.782 
 
 .002 
 
 .002 
 .002 
 .002 
 
 1.678 
 
 1.640 
 1.603 
 1-565 
 
 .007 
 
 .007 
 .007 
 
 .007 
 
 2.517 
 
 2.461 
 2.404 
 2.348 
 
 .016 
 
 .016 
 .016 
 .015 
 
 3-356 
 
 3.281 
 3.205 
 3-130 
 
 .029 
 
 .028 
 .028 
 .027 
 
 
 4-396 
 8.792 
 13.188 
 
 
 80 00 
 
 17-584 
 
 0.764 
 
 .002 
 
 1.527 
 
 .007 
 
 2.291 
 
 .015 
 
 3-OS4 
 
 .026 
 
 
 
 
 Smithsonian Tables. 
 
 91 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iT^innr- 
 [Derivation of table explained on pp. liii — Ivi.] 
 
 ■g _ 
 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 tf^ T^ TN T IkT A fl T> t^ f^ v^ 
 
 
 
 
 
 
 
 
 
 ORDINATES OF 
 
 
 ^§ 
 
 
 
 
 
 
 
 
 DEVELOPED 
 
 
 •S c 
 
 "S c 5 c 
 
 s' 
 
 10' 
 
 IS' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 2^ 
 
 s^«a 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 Iftches. 
 
 Inches, 
 
 Itiches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 4) 
 
 
 
 
 o°oo' 
 
 
 2.922 
 
 5.844 
 
 8.765 
 
 11.687 
 
 14.609 
 
 17-531 
 
 ■Ii 
 
 0° 
 
 1° 
 
 
 lO 
 
 ■■5.804" 
 
 2.922 
 
 5-843 
 
 8.76s 
 
 11.687 
 
 14.608 
 
 17-530 
 
 3-2 
 
 
 
 
 20 
 
 1 1.608 
 
 2.922 
 2.922 
 
 S-843 
 S-843 
 
 8.765 
 8.76s 
 
 11.686 
 
 14.608 
 14.608 
 
 17-530 
 17-530 
 
 
 
 
 
 30 
 
 17.412 
 
 11.686 
 
 
 Inches. 
 
 Inches. 
 
 
 40 
 
 23.216 
 
 2.922 
 
 S-843 
 
 8.764 
 
 11.686 
 
 14.608 
 
 17-529 
 17-528 
 
 5' 
 10 
 
 
 0.000 
 .000 
 
 
 SO 
 
 29.020 
 
 2.921 
 
 S-843 
 
 8.764 
 
 11.686 
 
 14.607 
 
 .000 
 
 
 I 00 
 
 
 2.921 
 
 S-843 
 
 8.764 
 
 11.685 
 
 14.606 
 
 17.528 
 
 IS 
 
 20 
 
 .000 
 .000 
 
 .000 
 .001 
 .001 
 tOOI 
 
 
 10 
 
 5.840 
 
 2.921 
 
 5.842 
 
 8.763 
 
 11.684 
 
 14.606 
 
 17.527 
 
 25 
 30 
 
 .000 
 
 •000 
 
 
 , 20 
 
 1 1.608 
 
 2.921 
 
 S.842 
 
 8.763 
 
 11.684 
 
 14.604 
 
 17.525 
 
 
 30 
 
 17.412 
 
 2.921 
 
 5.841 
 
 8.762 
 
 11.683 
 
 14.604 
 
 17.524 
 
 
 
 
 40 
 
 23.216 
 
 2.920 
 
 5.841 
 
 8.761 
 
 11.682 
 
 14.602 
 
 17.522 
 
 
 
 
 
 S° 
 
 29.020 
 
 2.920 
 
 5.840 
 
 8.761 
 
 11.681 
 
 14.601 
 
 17.521 
 
 
 
 
 
 
 
 
 
 2 00 
 
 5-804' 
 
 2.920 
 
 5.840 
 S-839 
 
 8.760 
 8.759 
 
 11.680 
 11.678 
 
 14.600 
 14.598 
 
 17.520 
 17.518 
 
 
 2° 
 
 3° 
 
 
 10 
 
 2.920 
 
 
 
 
 
 20 
 
 11.608 
 
 2.919 
 
 S-f39 
 
 8.758 
 
 11.677 
 
 14.596 
 
 17.516 
 
 5 
 
 0.000 
 
 0.000 
 
 
 30 
 
 17.412 
 
 2.919 
 
 S-838 
 
 8-757 
 
 11.676 
 
 14.594 
 
 17.513 
 
 10 
 
 .000 
 
 .000 
 
 
 40 
 
 23.216 
 
 2.918 
 
 S-i37 
 
 8.756 
 
 11.674 
 
 14.592 
 
 17.511 
 
 15 
 
 .001 
 
 .001 
 
 
 SO 
 
 29.020 
 
 2.918 
 
 5.836 
 
 8-755 
 
 11.673 
 
 14.591 
 
 17.509 
 
 20 
 
 .001 
 
 .002 
 
 
 300 
 
 
 2.918 
 
 5-836 
 
 8-753 
 
 ,1.671 
 
 14.589 
 
 17.507 
 
 25 
 
 3° 
 
 .002 
 .003 
 
 .003 
 .004 
 
 
 10 
 
 ■ 'i&^^ 
 
 2.917 
 
 5-835 
 
 8.752 
 
 11.669 
 
 14.586 
 
 17-504 
 
 
 
 
 
 20 
 
 11.608 
 
 2.917 
 2.916 
 
 5-834 
 5-832 
 
 8.750 
 8.749 
 
 11.667 
 11.665 
 
 14.584 
 14.581 
 
 17.501 
 
 
 
 
 
 30 
 
 17413 
 
 17.497 
 
 
 
 
 
 40 
 
 23.217 
 
 2.916 
 
 5-831 
 
 8.747 
 
 I.. 663 
 
 14.578 
 
 17.494 
 
 
 4° 
 
 f 
 
 
 s° 
 
 4 00 
 
 29.021 
 
 2.915 
 2.915 
 
 5.830 
 
 8.746 
 8.744 
 
 11.661 
 11.659 
 
 14.576 
 
 14-574 
 
 17.491 
 17.488 
 
 
 
 
 5 
 
 0.000 
 
 0.000 
 
 
 10 
 
 ■■ ■5.804' 
 
 2.914 
 
 5.828 
 
 8.742 
 
 11.656 
 
 14-570 
 
 17.484 
 
 10 
 
 .001 
 
 .001 
 
 
 20 
 
 11.609 
 
 2.913 
 
 5.827 
 
 §•740 
 
 11.654 
 
 14.567 
 
 17.480 
 
 15 
 
 .001 
 
 .002 
 
 
 30 
 
 17413 
 
 2.913 
 
 5.825 
 
 8.738 
 
 11.651 
 
 14.564 
 
 17.476 
 
 20 
 
 .002 
 
 .003 
 
 
 40 
 
 23.217 
 
 2.912 
 
 5.824 
 
 8.736 
 
 11.648 
 
 14.560 
 
 i-^A^ 
 
 25 
 
 .004 
 
 .005 
 
 
 SO 
 
 29.022 
 
 2.911 
 
 5-823 
 
 8-734 
 
 11.646 
 
 14-557 
 
 30 
 
 .005 
 
 .007 
 
 
 Soo 
 10 
 
 '"iko^ 
 
 2.911 
 2.910 
 
 5.822 
 5.820 
 
 8.732 
 8.730 
 
 11.643 
 11.640 
 
 14-554 
 14.550 
 
 17.465 
 17.459 
 
 
 
 
 
 
 
 
 
 20 
 
 11.609 
 
 2.909 
 
 5.818 
 
 8.727 
 
 11.636 
 
 14.546 
 
 17455 
 
 
 6° 
 
 7° 
 
 
 30 
 
 17.414 
 
 2.QO0 
 
 5.817 
 
 8.725 
 8.722 
 
 "-633 
 11.630 
 
 14.542 
 14-538 
 
 17.450 
 17.445 
 
 
 
 
 
 40 
 
 23.218 
 
 2.908 
 
 5-815 
 
 
 
 
 
 SO 
 
 29.022 
 
 2.907 
 
 5-813 
 
 8.720 
 
 11.627 
 
 14-534 
 
 17-440 
 
 5 
 10 
 
 0.000 
 .001 
 
 0.000 
 .001 
 
 
 600 
 
 
 2.906 
 
 5.812 
 
 8.718 
 
 11.624 
 
 14-530 
 
 17.43s 
 
 IS 
 
 .002 
 
 .002 
 
 
 10 
 20 
 
 " 5-805' 
 11.609 
 
 2.905 
 2.904 
 
 5-810 
 5.808 
 
 8-71S 
 8.712 
 
 11.620 
 ii.6i6 
 
 14.524 
 14.520 
 
 17.429 
 17.424 
 
 20 
 
 25 
 
 .004 
 .006 
 
 .008 
 
 .004 
 .006 
 
 
 30 
 
 17.414 
 
 2.903 
 
 5.806 
 
 8.709 
 
 11.612 
 
 14-515 
 
 17.418 
 
 30 
 
 •009 
 
 
 40 
 
 23.219 
 
 2.902 
 
 5.804 
 
 8.706 
 
 11.608 
 
 14.510 
 
 i7-4'3 
 
 
 
 
 
 SO 
 
 29.024 
 
 2.901 
 
 5.802 
 
 8.703 
 
 11.604 
 
 14.506 
 
 17.407 
 
 
 
 
 
 
 
 
 
 700 
 10 
 
 "'s-Sos' 
 
 2.900 
 
 5.800 
 5-798 
 
 8.701 
 8.697 
 
 11.601 
 11.596 
 
 14-501 
 
 17.401 
 
 
 8° 
 
 
 
 2.099 
 
 14.496 
 
 17.395 
 
 
 
 
 
 20 
 
 11.610 
 
 2.898 
 
 S-796 
 
 8.694 
 
 11.592 
 
 14.490 
 14.484 
 
 17.387 
 
 5 
 
 0.000 
 
 
 
 30 
 
 17.415 
 
 2.897 
 
 5-794 
 
 ff§° 
 
 11.587 
 
 17.381 
 
 10 
 
 .001 
 
 
 
 40 
 
 23.220 
 
 2,896 
 
 5-791 
 
 8.687 
 
 11.583 
 
 14-478 
 
 17.374 
 
 15 
 
 •003 
 .005 
 
 
 
 5° 
 
 29.025 
 
 2.895 
 
 5-789 
 
 8.684 
 
 11.578 
 
 14-473 
 
 17.368 
 
 20 
 
 
 
 800 
 
 
 2.894 
 
 S-787 
 
 8.680 
 
 11.574 
 
 14.468 
 
 17.361 
 
 25 
 
 30 
 
 .007 
 010 
 
 
 
 Smithson 
 
 IAN TaBL 
 
 ES. 
 
 
 
 
 
 
 
 
 
 
 92 
 
Table 20. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE nhtf 
 
 [Derivation of table explained on pp. liU — Ivi.] 
 
 ■S3 
 
 8°oo' 
 
 lO 
 
 5.80s 
 
 20 
 
 11.610 
 
 3° 
 
 17.416 
 
 40 
 
 23.221 
 
 5° 
 
 29.026 
 
 900 
 
 
 10 
 
 5.806 
 
 20 
 
 ii.6ti 
 
 30 
 
 17.417 
 
 40 
 
 23.222 
 
 SO 
 
 29.028 
 
 10 00 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 11 00 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 12 00 
 10 
 20 
 
 3° 
 
 40 
 
 so 
 1300 
 
 10 
 20 
 
 30 
 40 
 
 so 
 1400 
 
 10 
 20 
 
 30 
 
 40 
 
 50 
 
 15 00 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 1600 
 
 
 Inches. 
 
 5.806 
 II.612 
 17.417 
 23.223 
 29.029 
 
 5.806 
 II.612 
 17.419 
 23.225 
 29.031 
 
 5.807 
 II.613 
 17.420 
 23.226 
 29.033 
 
 5.807 
 II.614 
 17.421 
 23.228 
 29.03s 
 
 5.808 
 II.615 
 17.422 
 23.230 
 29.038 
 
 5.808 
 II.616 
 17.424 
 23.232 
 29.040 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 s' 
 
 longitude. 
 
 2.894 
 2.892 
 2.891 
 2.890 
 
 2.888 
 2.887 
 
 2.886 
 2.885 
 2.883 
 2.882 
 2.881 
 2.879 
 
 2.878 
 2.876 
 
 2.875 
 2.873 
 2.872 
 2.870 
 
 2.869 
 2.867 
 2.865 
 2.864 
 2.862 
 2.860 
 
 2.858 
 
 2-857 
 2.855 
 2.853 
 2.851 
 2.849 
 
 2.847 
 2.846 
 2.844 
 2.842 
 2.840 
 2.838 
 
 2.836 
 2.834 
 2.831 
 2.829 
 2.827 
 2.825 
 
 2.823 
 2.821 
 2.818 
 2.816 
 2.814 
 2.812 
 
 2.809 
 
 10' 
 
 longitude. 
 
 Inches, 
 
 S787 
 5.784 
 5.782 
 
 S-779 
 S777 
 S-77S 
 
 S-772 
 S-769 
 5-767 
 5.764 
 S-761 
 S-7S8 
 
 S-7SS 
 S-752 
 S-749 
 5.746 
 
 S-743 
 5.740 
 
 S-737 
 S-734 
 S-730 
 5.727 
 
 5-724 
 5.720 
 
 S-7I7 
 S-713 
 5.709 
 5.706 
 5.702 
 5.698 
 
 5.695 
 5.691 
 5.687 
 5.683 
 5.679 
 S-675 
 
 5.671 
 5.667 
 
 5-654 
 5.650 
 
 5.646 
 5.641 
 
 5-637 
 5.632 
 5.628 
 5.623 
 
 5.619 
 
 IS' 
 
 longitude. 
 
 8.680 
 8.677 
 8.673 
 8.669 
 
 8.666 
 8.662 
 
 8.658 
 8.654 
 8.650 
 8.646 
 8.642 
 8.637 
 
 8-633 
 8.628 
 8.624 
 8.619 
 8.614 
 8.610 
 
 8.606 
 8.601 
 8.596 
 8.590 
 
 8.580 
 
 8-S7S 
 8.570 
 8.564 
 8-559 
 8-SS3 
 8.548 
 
 8.542 
 
 8-536 
 8.530 
 8.524 
 8.519 
 8-S13 
 
 8.507 
 8.500 
 8.494 
 8.488 
 8.481 
 8.475 
 
 8.469 
 8.462 
 
 8-455 
 8.448 
 8.441 
 8-435 
 
 8.428 
 
 20' 
 
 longitude. 
 
 Inches, 
 
 569 
 564 
 
 559 
 554 
 549 
 
 544 
 539 
 533 
 528 
 522 
 516 
 
 5" 
 504 
 498 
 492 
 ,486 
 .480 
 
 ■474 
 .468 
 461 
 454 
 447 
 ,440 
 
 434 
 ,426 
 419 
 412 
 404 
 397 
 
 390 
 ,382 
 
 374 
 366 
 358 
 350 
 
 ■342 
 '334 
 ,326 
 
 317 
 308 
 300 
 
 292 
 282 
 274 
 264 
 
 255 
 ,246 
 
 11.237 
 
 25' 
 longitude. 
 
 Inches. 
 
 14.468 
 14.461 
 
 14-455 
 14.448 
 14.442 
 14.436 
 
 14.430 
 14.424 
 14.416 
 14.410 
 14.402 
 14.396 
 
 14.388 
 14.380 
 
 14-373 
 14.366 
 
 14-358 
 14-350 
 
 14.342 
 
 14-334 
 14.326 
 14.318 
 14.309 
 14.300 
 
 14.292 
 14.282 
 14.274 
 14.264 
 14.256 
 14.246 
 
 14-237 
 14.228 
 14.218 
 14.208 
 14.198 
 14.188 
 
 14.178 
 14.168 
 14.157 
 14.146 
 14.136 
 14.125 
 
 14.114 
 14.103 
 14.092 
 14.080 
 14.069 
 14.058 
 
 14.046 
 
 30' 
 
 longitude. 
 
 7-361 
 7-353 
 7-346 
 7-338 
 7-331 
 7-324 
 
 7-317 
 7-308 
 7-300 
 7.291 
 7.283 
 7-275 
 
 7.266 
 7.257 
 7.248 
 
 7-239 
 7.229 
 7.220 
 
 7.211 
 7.201 
 7.191 
 7.181 
 7.171 
 7.161 
 
 7.150 
 
 7-139 
 7.128 
 7.117 
 7.107 
 7-095 
 
 7.084 
 
 7-073 
 7.061 
 
 7-049 
 7-038 
 7.026 
 
 7-014 
 7.001 
 6.988 
 
 6-975 
 6.963 
 6.950 
 
 6-937 
 6.924 
 6.910 
 6.897 
 6.883 
 6.870 
 
 16.856 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 3-s 
 
 Inches. 
 
 0.000 
 .COl 
 .003 
 .005 
 .007 
 .010 
 
 0.000 
 .001 
 -003 
 .006 
 .009 
 ■013 
 
 0.000 
 .002 
 .004 
 .007 
 .011 
 .016 
 
 14° 
 
 0.000 
 .002 
 .004 
 .008 
 .012 
 .018 
 
 16° 
 
 0.001 
 .002 
 .005 
 .009 
 .014 
 .020 
 
 Inches. 
 
 0.000 
 .001 
 .003 
 .005 
 .008 
 .012 
 
 0.000 
 
 .002 
 .004 
 .006 
 
 .010 
 
 .014 
 
 13° 
 
 0.000 
 .002 
 
 .004 
 .007 
 .012 
 .017 
 
 15° 
 
 O.OOI 
 .002 
 .005 
 .009 
 .013 
 .019 
 
 Smithsonian Tables. 
 
 93 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jit^tsz- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 "Set) 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 "S 
 
 •»i|. 
 
 
 
 
 
 
 
 ORDINATES OF 
 DEVELOPED 
 
 
 u 
 
 .2 «-aii 
 
 
 
 
 
 
 
 
 
 11 11 
 
 S' 
 
 10' 
 
 15' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 2'^ 
 
 |S V P. 
 
 longitude. 
 
 * 
 
 longitude 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 Inches, 
 
 Inches. 
 
 Inches. 
 
 Indus, 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Il- 
 
 
 
 
 i6°oo' 
 
 
 2.809 
 
 5.619 
 
 8.428 
 
 11.237 
 
 14.046 
 
 16.856 
 
 ls 
 
 16° 
 
 17° 
 
 
 10 
 
 V-809 
 
 2.807 
 
 5.614 
 
 8.421 
 
 11.228 
 
 14.034 
 
 16.841 
 
 
 
 
 
 
 11.617 
 17.426 
 
 2.804 
 2.802 
 
 5.609 
 5.604 
 
 8.414 
 8.406 
 
 11.218 
 11.208 
 
 
 16.827 
 16.813 
 16.798 
 
 
 
 
 
 zo 
 3° 
 
 14.022 
 14.010 
 
 
 Inches. 
 
 Inches. 
 
 
 40 
 
 23-234 
 
 2.800 
 
 S-599 
 
 8.399 
 
 11.199 
 11.189 
 
 13998 
 
 s' 
 
 10 
 
 0.00 1 
 
 O.OOI 
 
 
 SO 
 
 29.043 
 
 2.797 
 
 S-S9S 
 
 8-392 
 
 13.986 
 
 16.784 
 
 .002 
 
 .002 
 
 
 17 00 
 
 11.618 
 
 2.795 
 
 S-S90 
 
 8.385 
 
 ' 11.180 
 
 13-974 
 
 16.769 
 
 '5 
 20 
 
 .005 
 .009 
 .014 
 .020 
 
 .COS 
 
 .010 
 
 
 10 
 
 2.792 
 
 5-S8s 
 
 \^v 
 
 11.170 
 
 13.962 
 
 16.754 
 
 25 
 30 
 
 .015 
 
 .021 
 
 
 20 
 
 2.790 
 
 5.580 
 
 8.369 
 
 11.159 
 
 13-949 
 
 16.739 
 
 
 30 
 
 17.427 
 
 2.787 
 
 S-S7S 
 
 8.362 
 
 11.149 
 
 13-936 
 
 16.724 
 
 
 
 
 40 
 
 23-236 
 
 2.785 
 
 5-570 
 
 8.354 
 
 U.I39 
 
 13-924 
 
 16.709 
 
 
 
 
 
 S° 
 
 29.046 
 
 2.782 
 
 5-564 
 
 8-347 
 
 11.129 
 
 13.911 
 
 16.693 
 
 
 
 
 
 
 
 
 18 00 
 
 ' s'sio' 
 
 2.780 
 
 5-559 
 
 8.339 
 8.331 
 
 11.119 
 
 13.898 
 
 16.678 
 
 
 18° 
 
 19° 
 
 
 10 
 
 2-777 
 
 S-5S4 
 
 11.108 
 
 13.885 
 
 16.662 
 
 
 
 
 
 20 
 
 11.619 
 
 2.774 
 
 5-549 
 
 8-323 
 
 11.097 
 
 13-872 
 
 16.646 
 
 5 
 
 0.001 
 
 O.OOI 
 
 
 30 
 
 17.429 
 
 2.772 
 
 5-543 
 
 8.315 
 
 11.087 
 
 13-859 
 
 16.630 
 
 10 
 
 .002 
 
 ■'^ 
 
 
 40 
 
 23-^39 
 
 2.769 
 
 S-S38 
 
 8.307 
 
 11.076 
 
 13-845 
 
 16.614 
 
 15 
 
 .006 
 
 
 50 
 
 29.049 
 
 2.766 
 
 5-533 
 
 8.299 
 
 11.065 
 
 13-832 
 
 16.598 
 
 20 
 25 
 
 .010 
 .016 
 
 .010 
 
 .016 
 
 
 19 00 
 
 
 2.764 
 
 S-527 
 
 8.291 
 
 11.054 
 
 13.818 
 
 16.582 
 
 30 
 
 .022 
 
 .024 
 
 
 10 
 
 5.810 
 
 2.761 
 
 5.522 
 
 8.282 
 
 11.043 
 
 13.804 
 
 16.565 
 
 
 
 
 
 20 
 
 11.621 
 
 2.758 
 
 5-516 
 
 8.274 
 
 11.032 
 
 13-790 
 
 16.548 
 
 
 
 
 
 3° 
 
 17-431 
 
 2-7SS 
 
 5-5'o 
 
 8.266 
 
 11.021 
 
 13-776 
 
 16.531 
 
 
 
 
 
 40 
 
 23.242 
 
 2.752 
 
 5- 505 
 
 8-257 
 
 11.009 
 
 13.762 
 
 16.514 
 
 
 20° 
 
 21° 
 
 
 50 
 2000 
 
 29.052 
 
 2.750 
 2.747 
 
 5-499 
 5-493 
 
 8.249 
 8.240 
 
 10.998 
 10.987 
 
 13-748 
 13-734 
 
 16.497 
 16.480 
 
 
 
 
 
 5 
 
 0.00 1 
 
 O.OOI 
 
 
 10 
 
 5.811 
 
 2-743 
 
 5.487 
 
 8.231 
 
 10.975 
 
 13-719 
 
 16.462 
 
 lO 
 
 .003 
 
 .006 
 
 
 20 
 
 H.622 
 
 2.741 
 
 5.482 
 
 8.222 
 
 10.963 
 
 13-704 
 
 16.445 
 
 15 
 
 .006 
 
 
 30 
 
 17-433 
 
 2-738 
 
 5.476 
 
 8.213 
 
 10.951 
 
 13.689 
 
 16,427 
 
 20 
 
 .oil 
 
 .oil 
 
 
 40 
 
 23.244 
 
 2-73S 
 
 5.470 
 
 8.204 
 
 10.939 
 
 13-674 
 
 16.409 
 
 25 
 
 .017 
 
 .018 
 
 
 SO 
 
 29.055 
 
 2.732 
 
 5-464 
 
 8.196 
 
 10.928 
 
 13.660 
 
 16.391 
 
 30 
 
 .025 
 
 .026 
 
 
 21 00 
 10 
 
 ■■■s-8;2" 
 
 2.729 
 2.726 
 
 S-458 
 5-4S2 
 
 8.187 
 8.177 
 
 10.916 
 10.903 
 
 13-645 
 13.629 
 
 '6-373 
 '6-355 
 
 
 
 
 
 
 
 
 
 20 
 
 U.623 
 
 2-723 
 
 5-445 
 
 8.168 
 
 10.891 
 
 13.614 
 
 16.336 
 
 
 22° 
 
 23° 
 
 
 30 
 40 
 
 '7-435 
 
 2.720 
 
 S-439 
 5-433 
 
 8.159 
 8.150 
 
 10.878 
 
 10.866 
 
 13-598 
 13-583 
 
 16.318 
 16.300 
 
 
 
 
 
 23-247 
 
 2.717 
 
 
 
 
 
 SO 
 
 29.058 
 
 2.714 
 
 5-427 
 
 8.141 
 
 10.854 
 
 13-568 
 
 16.281 
 
 5 
 10 
 
 0.00 [ 
 ■003 
 
 6.001 
 .003 
 
 
 22 00 
 
 
 2.710 
 
 5-421 
 
 8.131 
 
 10.842 
 
 '3-552 
 
 16.262 
 
 'S 
 
 .007 
 
 .007 
 
 
 ID 
 
 "5-812' 
 
 2.707 
 
 5.414 
 
 8.122 
 
 10.829 
 
 '3-536 
 
 16.243 
 
 20 
 
 .012 
 .018 
 
 .012 
 
 
 20 
 
 11.625 
 
 2.704 
 
 5.408 
 
 8.II2 
 
 10.816 
 
 13.520 
 
 16.223 
 
 25 
 
 ■oig 
 .028 
 
 
 30 
 
 17-437 
 
 2.701 
 
 5.401 
 
 8.102 
 
 10.802 
 
 13-503 
 
 16.204 
 
 30 
 
 .027 
 
 
 40 
 
 23.250 
 
 2.697 
 
 m 
 
 8.092 
 
 10.790 
 
 '3-487 
 
 16.184 
 
 
 
 
 
 5° 
 
 29.062 
 
 2.694 
 
 8.083 
 
 10.777 
 
 '3-471 
 
 16.165 
 
 
 
 
 
 
 
 
 2300 
 
 "^■^'l 
 
 2.691 
 
 5.382 
 
 8.073 
 8.063 
 
 10C764 
 
 '3-455 
 
 16.145 
 
 
 24° 
 
 
 
 10 
 
 s^:^^ 
 
 10.750 
 
 13-438 
 
 16.125 
 
 
 
 
 
 20 
 
 11.626 
 
 2.6S4 
 
 8.053 
 
 10.737 
 
 13.421 
 
 16.105 
 
 5 
 
 O.OOI 
 
 
 
 30 
 
 17-439 
 
 2.681 
 
 s-362 
 
 8.042 
 
 10.723 
 
 13.404 
 
 16.085 
 
 10 
 
 -003 
 
 
 
 40 
 
 23-252 
 
 2.677 
 
 5-355 
 
 8.032 
 
 10.710 
 
 i3-3'<7 
 
 16.064 
 
 15 
 
 .007 
 
 
 
 50 
 
 29.066 
 
 2.674 
 
 5-348 
 
 8.022 
 
 10.696 
 
 13-371 
 
 16.045 
 
 20 
 
 25 
 
 .013 
 
 .020 
 
 
 
 24 00 
 
 
 2.671 
 
 S-34I 
 
 8.012 
 
 10.683 
 
 13-354 
 
 16.024 
 
 30 
 
 .028 
 
 
 
 Smithsom 
 
 AN TaBL 
 
 ES. 
 
 
 
 
 
 
 i^MMl 
 
 
 
 
 94 
 
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TTsWlf 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Table 20- 
 
 a ■a 
 
 '5 « 
 
 .2 "■oi! 
 .■gSc-3 
 
 £stn. 
 
 ABSCISSAS OF DEVELUPKD PARALLEL. 
 
 longitude. 
 
 10' 
 longitude. 
 
 15' 
 
 longitude. 
 
 20 
 longitude. 
 
 25' 
 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 24''00' 
 lO 
 20 
 
 3° 
 40 
 
 5° 
 
 25 00 
 10 
 20 
 
 30 
 40 
 
 50 
 
 2600 
 10 
 20 
 
 30 
 
 40 
 
 50 
 
 27 00 
 10 
 20 
 
 30 
 40 
 
 ■ 50 
 
 2800 
 10 
 20 
 
 3° 
 40 
 
 SO 
 
 29 00 
 10 
 20 
 30 
 40 
 50 
 
 3000 
 10 
 20 
 30 
 40 
 
 50 
 
 31 00 
 10 
 20 
 30 
 40 
 50 
 
 3200 
 
 Inches 
 
 S.814 
 11.628 
 17.442 
 23.256 
 29.069 
 
 5.81S 
 11.629 
 17.444 
 23.259 
 29.074 
 
 5.816 
 II.631 
 17.446 
 23.262 
 29.077 
 
 5.816 
 
 I '-633 
 17.449 
 23.265 
 29.082 
 
 5-817 
 U.634 
 17.451 
 23.268 
 29.086 
 
 5.818 
 11.636 
 
 17454 
 23.272 
 29.090 
 
 5.819 
 11.638 
 
 17457 
 23.276 
 29.094 
 
 5.820 
 11.640 
 17.460 
 23.280 
 29.100 
 
 Inches. 
 
 2.671 
 2.667 
 2.664 
 2.660 
 2.657 
 2.653 
 
 2.650 
 2.646 
 2.642 
 2.639 
 2.635 
 2.631 
 
 2.628 
 2.624 
 2.620 
 2.616 
 2.613 
 2.609 
 
 2.605 
 
 2.6oi 
 
 2.597 
 
 2-593 
 2.589 
 2.586 
 
 2.582 
 2.578 
 2.574 
 2.570 
 2.566 
 2.562 
 
 2.558 
 2-553 
 2.549 
 2-545 
 2.541 
 
 2-537 
 
 2-533 
 2.528 
 2.524 
 2.520 
 2.515 
 2.51 1 
 
 2.507 
 2.502 
 2.498 
 
 2.493 
 2.489 
 2.485 
 
 2.480 
 
 Inches. 
 
 5-341 
 5-334 
 5-327 
 5-320 
 
 5-313 
 5.306 
 
 5-299 
 5.292 
 
 5-285 
 5-278 
 5-270 
 5.263 
 
 5.256 
 5.248 
 5.240 
 
 5-233 
 5.225 
 5.218 
 
 5.210 
 5.203 
 
 5-195 
 5.187 
 
 5-'79 
 5 171 
 
 5-163 
 5-155 
 5-147 
 5-139 
 5-131 
 5-123 
 
 5.115 
 5.107 
 5.098 
 5.090 
 5.082 
 5-073 
 
 5.065 
 5.056 
 5.048 
 5-039 
 S-031 
 5.022 
 
 5.014 
 5.005 
 4.996 
 4.987 
 4.978 
 4.969 
 
 4.960 
 
 Inches. 
 
 8.012 
 8.002 
 7.991 
 7.981 
 7-970 
 7.960 
 
 7-949 
 7-938 
 7-927 
 7.916 
 7.905 
 7.894 
 
 7.883 
 7.872 
 7.861 
 7.849 
 7.838 
 7.827 
 
 7.816 
 7.804 
 
 7-792 
 7.780 
 7.768 
 7-757 
 
 7-745 
 7-733 
 7.721 
 7.709 
 7.697 
 7.685 
 
 7-673 
 7.660 
 7.648 
 
 7-635 
 7.622 
 7.610 
 
 7-598 
 7-585 
 7-572 
 7-559 
 7.546 
 
 7-533 
 
 7-520 
 7-507 
 7-494 
 7.480 
 7.467 
 7-454 
 
 7.441 
 
 0.683 
 0.669 
 0.655 
 0.641 
 0.627 
 0.613 
 
 0.599 
 0.584 
 0.570 
 
 0-555 
 0.540 
 0.526 
 
 0.51 1 
 0.496 
 0.481 
 0.466 
 0.451 
 0.436 
 
 0.421 
 0.405 
 0.390 
 0.3-4 
 0.358 
 0.342 
 
 0.327 
 0.311 
 0.294 
 0.278 
 0.262 
 0.246 
 
 0.230 
 0.213 
 
 0.197 
 0.180 
 0.163 
 0.146 
 
 0.130 
 0.1 13 
 0.096 
 0.078 
 0.061 
 0.044 
 
 0.027 
 0.009 
 9.992 
 
 9-974 
 9.956 
 
 9-938 
 9.921 
 
 Inches. 
 
 3-354 
 3-336 
 3-319 
 3-301 
 3.284 
 3.266 
 
 3.249 
 3-231 
 
 3-212 
 3194 
 3-176 
 
 3-157 
 
 3-139 
 3.120 
 3.IOI 
 3.082 
 3-063 
 3-045 
 
 3.026 
 3.006 
 2.987 
 2.967 
 
 2.947 
 2.928 
 
 2.909 
 2.889 
 
 2.868 
 2.848 
 2.828 
 2.808 
 
 2.788 
 2.767 
 2.746 
 
 2.725 
 2.704 
 2.683 
 
 2.662 
 2.641 
 2.620 
 2.598 
 2-577 
 2-555 
 
 2-534 
 2.512 
 2.490 
 2.467 
 2.445 
 2.423 
 
 [2.401 
 
 Inches. 
 
 6.024 
 6.003 
 5.982 
 5-961 
 5-940 
 
 5-9 1 9 
 
 5.898 
 5-877 
 5.854 
 
 5-833 
 5.81 1 
 5.788 
 
 5-767 
 5-744 
 5-721 
 5-698 
 5.6/6 
 
 5-654 
 
 5-631 
 5.608 
 
 5-584 
 5.560 
 
 5-537 
 S-514 
 
 5.490 
 5.466 
 5442 
 5.418 
 5-394 
 5-369 
 
 5-345 
 5-320 
 
 5-295 
 5.270 
 
 5-245 
 5.220 
 
 5-195 
 5.169 
 
 5-143 
 5-118 
 
 5.066 
 
 5.040 
 5.014 
 4.987 
 4.960 
 
 4-934 
 4.908 
 
 24° 
 
 Inches. 
 
 0.00 1 
 •003 
 .007 
 
 .013 
 .020 
 .028 
 
 26° 
 
 O.OOI 
 
 .003 
 .008 
 
 .013 
 
 .021 
 •030 
 
 28° 
 
 .004 
 .008 
 .014 
 .022 
 .032 
 
 30" 
 
 .004 
 .008 
 .015 
 •023 
 -033 
 
 32° 
 
 O.OOI 
 
 .004 
 .009 
 
 .015 
 
 .024 
 
 -034 
 
 25° 
 
 Inches. 
 
 O.OOI 
 .003 
 .007 
 .013 
 .020 
 .029 
 
 27" 
 
 O.OOI 
 
 .003 
 .008 
 .014 
 .022 
 
 .031 
 
 29° 
 
 .004 
 
 .008 
 .014 
 .023 
 
 .032 
 
 31° 
 
 O.OOI 
 
 .004 
 
 .008 
 .015 
 
 .023 
 
 -034 
 
 Smithsonian Tables. 
 
 95 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ahti - 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 .2 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 •s . 
 
 1^ tf» 
 
 
 
 
 
 
 
 ORDINATES OF 
 
 
 
 
 
 
 
 
 
 ■S*" 
 
 .2 m-aS 
 
 
 
 
 
 
 
 DEVELOPED 
 
 
 .^1 
 
 •ISg-S 
 
 S' 
 
 10' 
 
 15' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 3S- 
 
 |ssa 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 Itiches. 
 
 Inches. 
 
 Iftches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 %i 
 
 
 
 
 32°oo' 
 
 
 2.480 
 
 4.960 
 
 7.441 
 
 9.921 
 
 12.401 
 
 14.881 
 
 II 
 
 Q a 
 
 32° 
 
 33° 
 
 
 10 
 20 
 
 5.821 
 11.642 
 
 2.476 
 2.471 
 
 4.951 
 4.942 
 
 7.427 
 7-413 
 
 9-903 
 
 12.379 
 12.355 
 
 14.854 
 14.827 
 
 J 
 
 
 
 
 
 
 
 
 3° 
 
 17.462 
 
 2.467 
 
 4-933 
 
 7.400 
 
 9:866 
 
 12.333 
 
 14.800 
 
 
 Inches. 
 
 Inches. 
 
 
 40 
 
 23.283 
 
 2.462 
 
 4.924 
 
 7.386 
 
 9.848 
 
 "•3i° 
 
 14.772 
 
 5' 
 10 
 
 0.001 
 
 0.001 
 
 
 5° 
 
 29.104 
 
 2.458 
 
 4.915 
 
 7-373 
 
 9.830 
 
 12.288 
 
 14-745 
 
 .004 
 
 .004 
 
 
 3300 
 
 
 24S3 
 
 4.906 
 
 7-359 
 
 9.812 
 
 12.265 
 
 14.717 
 
 IS 
 
 ' 20 
 
 .009 
 
 .MS 
 .024 
 
 .034 
 
 .009 
 .016 
 
 
 ID 
 
 5.822 
 
 2.448 
 
 4.896 
 
 7-345 
 
 9-793 
 
 12.241 
 
 14.689 
 
 25 
 30 
 
 .624 
 .035 
 
 
 20 
 
 11.643 
 
 2.444 
 
 4.887 
 
 7-33' 
 
 9-774 
 
 12.218 
 
 14.661 
 
 
 30 
 
 17.465 
 
 2439 
 
 4.878 
 
 7-316 
 
 9-755 
 
 12.194 
 
 14.633 
 
 
 40 
 
 23.287 
 
 2434 
 2.429 
 
 4.868 
 4.859 
 
 7.302 
 7.288 
 
 9-736 
 9.7:8 
 
 12.171 
 
 14.605 
 14.576 
 
 
 
 
 
 29>io9 
 
 12.147 
 
 
 
 
 
 3400 
 
 
 2.425 
 
 4.850 
 
 7-274 
 
 9.699 
 
 12.124 
 
 14.549 
 
 
 34° 
 
 35° 
 
 
 10 
 
 5.823 
 
 2.420 
 
 4.840 
 
 7.260 
 
 9.680 
 
 12.100 
 
 14.520 
 
 
 
 
 
 20 
 
 11.64s 
 17.468 
 
 241 S 
 
 4.830 
 
 7.246 
 
 9.661 
 
 12.076 
 
 14.491 
 
 5 
 
 0.001 
 
 O.OOI 
 
 
 30 
 
 2.410 
 
 4.821 
 
 7-231 
 
 9.642 
 
 1 2.052 
 
 14.462 
 
 10 
 
 .004 
 
 .004 
 
 
 40 
 
 23.291 
 
 2.406 
 
 4.811 
 
 7.217 
 
 9.622 
 
 12.028 
 
 14.434 
 
 15 
 
 .009 
 
 .009 
 
 
 5° 
 
 29.113 
 
 2.401 
 
 4.802 
 
 7.203 
 
 9.604 
 
 12.004 
 
 14.405 
 
 20 
 25 
 
 .016 
 .025 
 
 .016 
 .025 
 
 
 35 00 
 
 
 2.396 
 
 4.792 
 
 7.188 
 
 9.584 
 
 11.980 
 
 14.376 
 
 30 
 
 .036 
 
 .036 
 
 
 10 
 
 ■■5.824" 
 
 2.391 
 
 4.782 
 
 7-174 
 
 9-565 
 
 11.956 
 
 14.347 
 
 
 
 
 
 20 
 
 11.647 
 
 2.386 
 
 4-773 
 
 7-159 
 
 9-545 
 
 11.932 
 
 '4-318 
 
 
 
 
 
 30 
 
 17.471 
 
 2.381 
 
 4-763 
 
 7.144 
 
 9.526 
 
 11.907 
 
 14.288 
 
 
 
 
 
 
 40 
 
 23.294 
 
 2.377 
 
 4-753 
 
 7-13° 
 
 9.506 
 
 "■1^3 
 
 14.259 
 
 
 36° 
 
 37° 
 
 
 50 
 3600 
 
 29.118 
 
 2.372 
 2.367 
 
 4-743 
 4-733 
 
 7-115 
 7.099 
 
 9.486 
 9.466 
 
 11.858 
 11-833 
 
 14-230 
 14.200 
 
 
 
 
 
 5 
 
 0.001 
 
 0.001 
 
 
 10 
 
 ■■■s.82;' 
 
 2.362 
 
 4-723 
 
 7.08s 
 
 9-446 
 
 11.808 
 
 14.170 
 
 10 
 
 .004 
 
 .004 
 
 
 20 
 
 11.649 
 
 2-357 
 
 4-713 
 
 7.070 
 
 9.426 
 
 11.783 
 
 14-139 
 
 15 
 
 .009 
 
 .009 
 
 
 30 
 
 17473 
 
 2-351 
 
 4-703 
 
 7-055 
 
 9.406 
 
 "•757 
 
 14.109 
 
 20 
 
 .016 
 
 .016 
 
 
 40 
 
 23.297 
 
 2.346 
 
 4-693 
 
 7-039 
 
 9-386 
 
 11.732 
 
 14.078 
 
 25 
 
 .025 
 
 .026 
 
 
 5° 
 
 29.122 
 
 2.341 
 
 4.683 
 
 7.024 
 
 9-366 
 
 11.707 
 
 14.048 
 
 30 
 
 .036 
 
 .037 
 
 
 37 00 
 
 
 2-336 
 
 4-673 
 
 7.009 
 
 9-345 
 9-325 
 
 11.682 
 
 14.018 
 
 
 
 
 
 10 
 
 's.826^ 
 
 2-335 
 
 4.662 
 
 6.994 
 
 11.656 
 
 13.987 
 
 
 38° 
 
 __o 
 
 
 20 
 
 I I. 651 
 
 2.326 
 
 4-652 
 
 6.978 
 
 9-3°4 
 
 11.630 
 
 J3.956 
 
 
 39 
 
 
 30 
 
 17477 
 
 2.321 
 
 4.642 
 
 6.963 
 
 9.284 
 
 11.605 
 
 13-925 
 
 
 
 
 
 
 
 
 
 40 
 
 23.302 
 
 2.316 
 
 4.631 
 
 6-947 
 
 9.263 
 
 11.579 
 
 13.894 
 
 5 
 10 
 
 0.001 
 .004 
 
 0*OOI 
 
 
 50 
 
 29.128 
 
 2.31 1 
 
 4.621 
 
 6.932 
 
 9.242 
 
 "■553 
 
 13.864 
 
 .004 
 
 
 38 00 
 10 
 
 ■■ ■5:827 ■ 
 
 2.305 
 2.300 
 
 4.611 
 4.600 
 
 6.916 
 
 9.222 
 9.200 
 
 11.527 
 11.501 
 
 13-832 
 13.801 
 
 15 
 20 
 
 25 
 
 .009 
 .017 
 .026 
 
 .009 
 
 
 20 
 
 "•653 
 
 2.295 
 
 4-59° 
 
 6.884 
 
 9.179 
 
 11.474 
 
 13-769 
 
 .037 
 
 
 30 
 
 17.480 
 
 2.290 
 
 4-579 
 
 6.869 
 
 9.158 
 
 11.448 
 
 '3-737 
 
 30 
 
 .037 
 
 
 40 
 50 
 
 23.306 
 
 2.284 
 2.279 
 
 4.568 
 4-558 
 
 6.853 
 6.837 
 
 9-137 
 9.116 
 
 11.421 
 
 13-705 
 13-673 
 
 
 
 
 
 29.133 
 
 "-395 
 
 
 
 
 
 3900 
 10 
 
 "■5.828' 
 
 2.274 
 2.268 
 
 4-548 
 4-537 
 
 6.821 
 6.805 
 
 9.09s 
 9-073 
 
 U.369 
 11.342 
 
 13.642 
 13.610 
 
 
 40° 
 
 
 
 
 
 
 
 20 
 
 11.655 
 17483 
 
 2.263 
 2.258 
 
 4.526 
 
 6.789 
 
 9.052 
 
 11-315 
 
 13-577 
 
 S 
 
 0.001 
 
 
 
 30 
 
 4-515 
 
 6-773 
 
 9.030 
 
 11.288 
 
 13.545 
 
 10 
 
 .004 
 
 
 
 40 
 
 23.310 
 
 2.252 
 
 4-504 
 
 6-756 
 
 9.008 
 
 11.261 
 
 13.513 
 
 15 
 
 .009 
 
 
 
 5° 
 
 29.138 
 
 2.247 
 
 4-493 
 
 6.740 
 
 8.987 
 
 11.234 
 
 13.480 
 
 20 
 25 
 
 .017 
 
 .026 
 
 
 
 40 CO 
 
 
 2.241 
 
 4-483 
 
 6.724 
 
 8.965 
 
 11.207 
 
 13.448 
 
 30 
 
 .038 
 
 
 
 Smithsonian Tables. 
 
 96 
 

 
 
 
 
 
 
 
 
 Table 20. 
 
 / 
 
 :0-ORDINATES FOR 
 
 PROJECTION OF MAPS. SCALE 
 
 1 
 
 
 \ 
 
 18646ff' 
 
 
 
 
 [Derivation o£ table explained on pp. liii-lvi.] 
 
 
 *K 
 
 2 is 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 /"^TiTXTXT A T" 
 
 
 o . 
 
 
 
 
 
 
 
 
 UKUlJNALco \js 
 
 ■S3 
 
 otJ-S" 
 
 
 
 
 
 
 
 DEVELOPKD 
 
 11 
 
 lisi 
 
 5' 
 
 10' 
 
 15' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 S^ 
 
 lasa 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 
 
 40°oo' 
 
 
 Z.2i,\ 
 
 4-483 
 
 6.724 
 
 8.965 
 
 11.207 
 
 13.448 
 
 = 
 
 40° 
 
 41° 
 
 10 
 
 'I829 
 
 2.236 
 
 4.472 
 
 6.707 
 
 8.943 
 
 II.179 
 
 13-415 
 
 J- 
 
 
 
 20 
 
 11.657 
 
 2.230 
 
 4.461 
 
 6.691 
 
 !•§" 
 
 II. 152 
 
 13-382 
 
 
 
 
 
 
 3° 
 
 17.486 
 
 2.225 
 
 4.450 
 
 ^•^7^ 
 
 f-f99 
 
 II. 124 
 
 13-349 
 
 
 Inches. 
 
 Inches. 
 
 40 
 
 23-314 
 
 2.219 
 
 4-439 
 
 6.658 
 
 f-|77 
 
 11.097 
 
 13-316 
 
 5' 
 10 
 
 0.00 1 
 
 O.OOI 
 
 5° 
 
 29.143 
 
 2.214 
 
 4.428 
 
 6.641 
 
 8.855 
 
 11.069 
 
 13-283 
 
 .004 
 
 .004 
 
 41 00 
 
 
 2.208 
 
 4.417 
 
 f-f"l 
 
 ^■?34 
 
 11. 042 
 
 13-250 
 
 15 
 20 
 
 .009 
 .017 
 .026 
 
 .009 
 .017 
 .026 
 
 10 
 
 V-Sso' 
 
 2.203 
 
 4-406 
 
 6.608 
 
 8.81 1 
 
 II.014 
 
 13.217 
 
 25 
 30 
 
 20 
 
 11.659 
 
 2.197 
 
 4-394 
 
 6.591 
 
 8.788 
 
 10.985 
 
 13-183 
 
 ■038 
 
 .038 
 
 30 
 
 17.489 
 
 2.192 
 
 4-383 
 
 6-575 
 6-558 
 6-541 
 
 8.766 
 
 10.958 
 
 13-149 
 
 40 
 
 23-319 
 
 2.186 
 2.180 
 
 4-372 
 4.360 
 
 8-744 
 8.721 
 
 10.929 
 
 13-115 
 13.081 
 
 
 
 
 5° 
 
 29.149 
 
 10.901 
 
 
 
 
 4200 
 10 
 
 "5-831 
 
 2-175 
 2.169 
 
 4-349 
 4-338 
 
 6.524 
 6.507 
 
 8.698 
 8.676 
 
 10.873 
 10.844 
 
 13.048 
 13-013 
 
 
 42° 
 
 43° 
 
 
 
 
 20 
 
 1 1. 661 
 
 2.163 
 
 4.326 
 
 6.490 
 
 8.653 
 
 10.816 
 
 12.979 
 
 5 
 
 O.OOl 
 
 O.OOI 
 
 30 
 
 17.492 
 
 2.157 
 
 4-315 
 
 6.472 
 
 8.630 
 
 10.787 
 
 12-945 
 
 10 
 
 .004 
 
 .004 
 
 40 
 
 23-323 
 
 2.152 
 
 4-303 
 
 6.455 
 
 8.607 
 
 10.759 
 
 12.910 
 
 15 
 
 .010 
 
 .010 
 
 SO 
 
 29.154 
 
 2.146 
 
 4.292 
 
 6.438 
 
 8.584 
 
 10.730 
 
 12.876 
 
 20 
 25 
 
 .017 
 .026 
 
 .017 
 
 .027 
 
 4300 
 
 
 2.140 
 
 4.281 
 
 6.421 
 
 8.561 
 
 10.702 
 
 12.842 
 
 30 
 
 .038 
 
 .038 
 
 10 
 
 " S-832' 
 
 2-13S 
 
 4.269 
 
 6.403 
 
 8.538 
 
 10.672 
 
 12.807 
 
 
 
 
 20 
 30 
 
 11.663 
 17-495 
 
 2.129 
 2.123 
 
 4.257 
 4.246 
 
 6.386 
 6.368 
 
 8.514 
 
 10.643 
 10.614 
 
 12.772 
 12.737 
 
 
 
 
 
 
 
 40 
 
 23-327 
 
 2.117 
 
 4-234 
 
 6-351 
 
 8.468 
 
 10.585 
 
 12.701 
 
 
 44° 
 
 45° 
 
 5° 
 
 29.159 
 
 2.111 
 
 4.222 
 
 6-333 
 
 8-444 
 
 10.556 
 
 12.667 
 
 
 
 
 4400 
 
 
 2.105 
 
 4.210 
 
 6.316 
 
 8.421 
 
 10.526 
 
 12.631 
 
 5 
 
 O.OOI 
 
 0.001 
 
 10 
 
 5-833 
 
 2.099 
 
 4.199 
 
 6.298 
 
 8-397 
 
 10.496 
 
 12.596 
 
 10 
 
 .004 
 
 .004 
 
 20 
 
 11.666 
 
 2.093 
 
 4.187 
 
 • 6.280 
 
 8-373 
 
 10.467 
 
 12.560 
 
 IS 
 
 .010 
 
 .010 
 
 3° 
 
 17.498 
 
 2.087 
 
 4-I7S 
 
 6.262 
 
 8.350 
 
 10-437 
 
 12.524 
 
 20 
 
 .017 
 
 .017 
 
 40 
 
 23-331 
 
 2.08l 
 
 4.163 
 
 6.244 
 
 8.326 
 
 10.407 
 
 12.489 
 
 25 
 
 .027 
 •038 
 
 ■038 
 
 5° 
 
 29.164 
 
 2.076 
 
 4.151 
 
 6.227 
 
 8.302 
 
 10.378 
 
 12.453 
 
 30 
 
 4500 
 10 
 
 
 2.070 
 2.064 
 
 4-139 
 4.127 
 
 6.209 
 6. 1 91 
 
 8.278 
 8.254 
 
 10.348 
 10.317 
 
 12.417 
 12.381 
 
 
 
 
 "V-834' 
 
 
 46° 
 
 47° 
 
 20 
 
 11.668 
 
 2.057 
 
 4.11S 
 
 6.172 
 6.154 
 
 8.230 
 8.206 
 
 10.288 
 10.257 
 
 12.345 
 12.308 
 
 
 30 
 
 17.501 
 
 2.051 
 
 4.103 
 
 
 
 
 40 
 
 23-335 
 
 2.045 
 
 4.091 
 
 6.136 
 
 8.i8i 
 
 10.226 
 
 12.272 
 
 5 
 10 
 
 0.001 
 
 O.OOI 
 
 5° 
 
 29.169 
 
 2.039 
 
 4.079 
 
 6.118 
 
 8.157 
 
 10.197 
 
 12.236 
 
 .004 
 
 .004 
 
 
 
 
 
 
 
 
 
 15 
 
 20 
 
 .OlO 
 
 .010 
 
 4600 
 
 
 2-033 
 
 4.067 
 
 6.100 
 
 ^•'33 
 
 10.166 
 
 12.199 
 
 .017 
 
 .017 
 
 10 
 
 "5-835' 
 
 2.027 
 
 4-054 
 
 6.081 
 
 8.108 
 
 10.136 
 
 12.163 
 
 25 
 
 .027 
 
 .027 
 
 20 
 
 11.670 
 
 2.021 
 
 4.042 
 
 6.063 
 
 8.084 
 
 10.104 
 
 12.125 
 
 30 
 
 .038 
 
 -038 
 
 30 
 
 17-504 
 
 2.015 
 
 4.030 
 
 6.044 
 
 8.059 
 
 10.074 
 
 12.089 
 
 
 
 
 40 
 50 
 
 23-339 
 29-174 
 
 2.009 
 2.003 
 
 4.017 
 
 4.005 
 
 6.026 
 6.008 
 
 8.034 
 8.010 
 
 10.043 
 10.013 
 
 12.052 
 12.015 
 
 
 
 
 
 48° 
 
 
 4700 
 10 
 
 "5-836' 
 
 1.996 
 
 3-992 
 3.980 
 
 S-989 
 5.970 
 
 7.985 
 7.960 
 
 9.981 
 9-951 
 
 11.978 
 11.941 
 
 
 
 
 
 
 20 
 
 11.672 
 
 1.984 
 
 3-968 
 
 5-951 
 
 7-935 
 
 9-gJ§ 
 
 11.903 
 
 S 
 
 0.001 
 
 
 30 
 
 17.508 
 
 1.978 
 
 3-955 
 
 5-933 
 
 7-§"° 
 
 9.888 
 
 1 1 .866 
 
 10 
 
 .004 
 
 
 40 
 
 23-344 
 
 I -97 1 
 
 3-943 
 
 5-9'4 
 
 7-885 
 
 9.857 
 
 11.828 
 
 IS 
 
 .010 
 
 
 50 
 
 29.180 
 
 1.965 
 
 3-930 
 
 5-895 
 
 7.860 
 
 9.826 
 
 11-791 
 
 20 
 25 
 
 30 
 
 .017 
 
 .026 
 
 
 48 00 
 
 
 1-959 
 
 3-917 
 
 5.876 
 
 7-835 
 
 9-794 
 
 11.752 
 
 .038 
 
 
 Smithsonian Tables. 
 
 97 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ttbW- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 48°00' 
 10 
 20 
 
 3° 
 40 
 
 5° 
 
 4900 
 10 
 
 20 
 
 30 
 40 
 
 5° 
 
 5000 
 10 
 20 
 
 3^ 
 40 
 
 5° 
 
 51 CO 
 
 10 
 
 20 
 
 3° 
 40 
 
 50 
 
 52 CO 
 10 
 
 20 
 
 3° 
 
 40 
 
 50 
 
 5300 
 
 10 
 
 20 
 
 30 
 40 
 
 5° 
 
 54 00 
 10 
 20 
 
 30 
 40 
 
 50 
 
 SSoo 
 10 
 20 
 
 30 
 40 
 
 5° 
 5600 
 
 if E S 
 
 >« V _=5 
 .a (J - a 
 
 b> c V h 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 s- 
 
 u a< 
 
 Inches, 
 
 5-837 
 11.674 
 17.51 1 
 
 23-34'^ 
 29.185 
 
 5.838 
 n.676 
 17-514 
 23-352 
 29.190 
 
 5' 
 
 longitude. 
 
 5-839 
 11.678 
 
 17-517 
 23-356 
 29.194 
 
 5.840 
 11.680 
 17.520 
 23.360 
 29.200 
 
 5-841 
 11.682 
 
 17-523 
 23-364 
 29.204 
 
 5.842 
 11.684 
 17.526 
 23.368 
 29.210 
 
 5-843 
 11.686 
 17.529 
 
 23-372 
 29.214 
 
 5-844 
 11.688 
 
 17-532 
 23-376 
 29.220 
 
 Inches. 
 
 1.959 
 1.952 
 1-946 
 1.9+0 
 
 1-933 
 1.927 
 
 1.921 
 1.914 
 1.908 
 1.901 
 1.895 
 1.888 
 
 1.882 
 1.875 
 1.869 
 1.863 
 1.856 
 
 1.842 
 1.836 
 1.829 
 1.823 
 1.816 
 1.809 
 
 1.803 
 1.796 
 1.789 
 1.782 
 1.776 
 1.769 
 
 1.762 
 
 1-755 
 1-748 
 1.742 
 
 •-73S 
 1.728 
 
 1.721 
 
 1-714 
 1.707 
 1.700 
 1.694 
 1.687 
 
 1.680 
 1.673 
 1.666 
 1.659 
 1.652 
 1.645 
 
 1.638 
 
 10' 
 longitude. 
 
 15' 
 
 longitude. 
 
 Inches. 
 
 3-917 
 3-905 
 3.892 
 
 3-879 
 3-867 
 3-854 
 
 3.841 
 3.828 
 3-815 
 3-803 
 3-790 
 3-777 
 
 3-764 
 3-753 
 3-737 
 3-724 
 
 3-711 
 3-698 
 
 3-685 
 3-672 
 3-658 
 3-645 
 3-632 
 3.618 
 
 3-605 
 3-592 
 3-578 
 3-565 
 3-55' 
 3-538 
 
 3-524 
 3-511 
 3-497 
 3-483 
 3-470 
 3-456 
 
 3-442 
 3-429 
 3-415 
 3.401 
 
 3-387 
 3-373 
 
 3-359 
 3-345 
 3-331 
 3-317 
 3-303 
 3.289 
 
 3-275 
 
 5.876 
 
 5.838 
 
 5-819 
 5.800 
 5.781 
 
 5-762 
 5-743 
 5-723 
 5.704 
 5.684 
 5-665 
 
 5.646 
 5.626 
 5.606 
 S-587 
 5-567 
 5-547 
 
 5-528 
 
 5-527 
 5.488 
 5.468 
 5.448 
 5.428 
 
 5.408 
 5-388 
 5-367 
 5-347 
 5-327 
 5-307 
 
 5.287 
 5.266 
 5.246 
 5-225 
 5-205 
 5.184 
 
 5.164 
 
 5-t43 
 5.122 
 5.101 
 5.080 
 5.060 
 
 5-039 
 5.018 
 
 4-997 
 4-976 
 4-955 
 4-934 
 
 4-913 
 
 20 
 longitude. 
 
 Inches, 
 
 7-835 
 7.810 
 
 7-784 
 7-759 
 7-733 
 7.708 
 
 7.682 
 7-657 
 7-631 
 7-605 
 
 7-579 
 7-553 
 
 7-527 
 7-501 
 7-475 
 7-449 
 7.422 
 
 7-396 
 
 7-370 
 7-343 
 7-317 
 7.290 
 7.264 
 7-237 
 
 7.210 
 7.184 
 7.156 
 7-130 
 7-103 
 7.076 
 
 7.049 
 7.022 
 6.994 
 6.967 
 6.940 
 6.912 
 
 6.885 
 6.857 
 6.830 
 6.802 
 
 6-774 
 6.746 
 
 6.719 
 6.691 
 6.663 
 
 6-579 
 6.551 
 
 25' 
 longitude. 
 
 Inches. 
 
 9-794 
 9.762 
 
 9-730 
 9-699 
 9.667 
 
 9-635 
 9-603 
 
 9-571 
 9-539 
 9-507 
 9-474 
 9-442 
 
 9-409 
 9-376 
 9-344 
 9-3 1' 
 9-278 
 9.245 
 
 9.212 
 9.179 
 9.146 
 
 9-1 '3 
 9.080 
 9.046 
 
 9.013 
 8.980 
 8.946 
 8.912 
 8.878 
 8.844 
 
 8.811 
 8.777 
 8.742 
 8.708 
 8.674 
 8.640 
 
 8.606 
 8.572 
 
 8-537 
 8.502 
 8.468 
 8-433 
 
 8.398 
 8.364 
 8.328 
 8.294 
 8.258 
 8.224 
 
 8.188 
 
 30' 
 
 longitude. 
 
 Inches. 
 
 11.752 
 11.714 
 11.677 
 11.638 
 11.600 
 n.562 
 
 11.523 
 11.485 
 1 1.446 
 11. 408 
 11.369 
 11-330 
 
 11.291 
 11.251 
 11.212 
 
 11-173 
 11.134 
 11.094 
 
 11.055 
 11.015 
 10.975 
 10.936 
 10895 
 10.855 
 
 10.816 
 10.775 
 10.734 
 10.694 
 10.654 
 10.613 
 
 10.573 
 10.533 
 10.491 
 10.450 
 
 r 0.409 
 10.368 
 
 10.327 
 10.286 
 10.244 
 10.202 
 10.161 
 10.120 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 10.078 
 
 10.036 
 
 9.994 
 
 9-952 
 9.910 
 9.868 
 
 25 
 9.826 30 
 
 48° 
 
 Inches. 
 
 0.00 1 
 .004. 
 .010 
 .017 
 .026 
 .038 
 
 49° 
 
 50° 
 
 0.001 
 .004 
 .009 
 .017 
 .026 
 -03S 
 
 52° 
 
 0.00 1 
 .004 
 .009 
 .017 
 .026 
 •037 
 
 54° 
 
 0.001 
 .004 
 .009 
 .016 
 .025 
 .036 
 
 56° 
 
 .004 
 .009 
 .016 
 .025 
 .036 
 
 Inches. 
 
 0.00 1 
 .004 
 .010 
 .017 
 .026 
 .038 
 
 51° 
 
 0.001 
 
 .004 
 .009 
 
 017 
 
 .026 
 
 •037 
 
 53° 
 
 .004 
 .009 
 .016 
 .026 
 •037 
 
 55° 
 
 0.00 1 
 .004 
 .009 
 .016 
 .025 
 .036 
 
 Smithsonian Tables. 
 
 98 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWinnr- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 5j !» 
 
 ."3 y c'rt 
 
 St 0, 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 longitude. 
 
 longitude. 
 
 '5' 
 
 longitude. 
 
 20' 
 longitude. 
 
 25' 
 
 longitude 
 
 30' 
 
 longitude. 
 
 ORDTNATES OF 
 DEVELOPED 
 PARALLEL. 
 
 56°oo' 
 
 lO 
 20 
 
 3° 
 40 
 
 SO 
 
 5700 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 58 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 59 00 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 6000 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 61 00 
 10 
 20 
 30 
 40 
 
 SO 
 
 62 00 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 6300 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 64 00 
 
 Inches. 
 
 S-845 
 11.690 
 
 I7-S3S 
 23.380 
 29.224 
 
 5.846 
 11.692 
 I7-S37 
 23-383 
 29.229 
 
 5.847 
 11.694 
 17.540 
 
 23-387 
 29.234 
 
 5.848 
 11.695 
 17-543 
 23-391 
 29.238 
 
 5.849 
 11.697 
 17.546 
 
 23-394 
 29.243 
 
 5.850 
 11.699 
 17-549 
 23-398 
 29.248 
 
 5.850 
 11.701 
 
 17-551 
 23.402 
 29.252 
 
 5-851 
 11.702 
 
 17-554 
 23-405 
 29256 
 
 Inches. 
 
 .638 
 .631 
 .624 
 .616 
 .609 
 .602 
 
 -59S 
 ■588 
 
 .566 
 -559 
 
 -552 
 •545 
 ■538 
 -530 
 
 .516 
 
 509 
 .501 
 -494 
 -487 
 -479 
 -472 
 
 .465 
 
 -457 
 .450 
 .442 
 
 -435 
 .428 
 
 .420 
 
 ■413 
 •405 
 ■398 
 -390 
 -383 
 
 •3^5 
 -338 
 
 .360 
 
 -353 
 
 •345 
 
 -338 
 
 -330 
 .322 
 
 -31 S 
 
 ■307 
 .300 
 .292 
 
 1.284 
 
 Inches. 
 
 3-27S 
 3.261 
 
 3-247 
 3-233 
 3.219 
 
 3.204 
 
 3.190 
 
 3-176 
 3.162 
 
 3-147 
 3-133 
 3-"9 
 
 3.104 
 3.090 
 
 3075 
 3.061 
 3.046 
 3-032 
 
 3-017 
 3-003 
 2988 
 
 2973 
 2.959 
 2.944 
 
 2.929 
 2.914 
 2.900 
 2.885 
 2.870 
 2.855 
 
 2.840 
 2.825 
 2.810 
 2.795 
 2.781 
 2.766 
 
 2.751 
 
 2.736 
 2.720 
 2.705 
 2.690 
 2.675 
 
 2.660 
 2.645 
 2.630 
 2.614 
 
 2-599 
 2.584 
 
 2.569 
 
 Inches. 
 
 4-913 
 4.892 
 4.870 
 4.849 
 4.828 
 4.807 
 
 4.785 
 4.764 
 4-742 
 4.721 
 4.699 
 4.678 
 
 4.656 
 
 4-634 
 4.613 
 4.591 
 4.569 
 4-547 
 
 4.526 
 4.504 
 4.482 
 4.460 
 4-438 
 4.416 
 
 4-394 
 4-372 
 4-349 
 4-327 
 4-3QS 
 4-283 
 
 4.261 
 4.238 
 4.216 
 
 4-103 
 4.171 
 4148 
 
 4.126 
 4.103 
 4.081 
 4.058 
 4-035 
 4013 
 
 3-990 
 3-967 
 3-944 
 3921 
 3-899 
 3.876 
 
 3853 
 
 Inches, 
 
 6.551 
 6.522 
 6.494 
 6.466 
 
 6.437 
 6.409 
 
 6.380 
 6.352 
 
 6-323 
 6.294 
 6.266 
 6.237 
 
 6.208 
 6.179 
 6.150 
 6.122 
 6.092 
 6.063 
 
 6.034 
 6.005 
 s-976 
 5.946 
 
 5-9 '7 
 5.888 
 
 5.858 
 5.829 
 
 5-799 
 S-770 
 5-740 
 5.710 
 
 5.681 
 5.651 
 5.621 
 
 5-591 
 5.561 
 
 S-531 
 
 S-501 
 5-47" 
 5.441 
 5.410 
 5.380 
 S-350 
 
 5-320 
 5.290 
 
 S-2S9 
 5.228 
 5.198 
 5.168 
 
 5- 1 37 
 
 Inches. 
 8.188 
 
 8.118 
 8.082 
 8.046 
 
 8.011 
 
 7.976 
 7-940 
 7-904 
 7.868 
 7832 
 7.796 
 
 7.760 
 
 7-724 
 7.688 
 7.652 
 7.6r6 
 7-579 
 
 7-543 
 7-506 
 7-470 
 7-433 
 7-396 
 7.360 
 
 7-323 
 7.286 
 7.249 
 7.212 
 
 7-I7S 
 7-138 
 
 7.101 
 7.064 
 7.026 
 6.988 
 6.952 
 6.914 
 
 6.877 
 6.839 
 6.801 
 6.763 
 6.726 
 6.688 
 
 6.650 
 6.612 
 
 6-574 
 6.536 
 6.498 
 6.460 
 
 6.422 
 
 Inches. 
 
 9.826 
 9.784 
 9-741 
 
 9.698 
 9.656 
 9.613 
 
 9- 57 1 
 9-527 
 9.485 
 9.442 
 9-398 
 9-356 
 
 9-313 
 9.269 
 9.226 
 g.182 
 9-139 
 9-095 
 
 9.052 
 9.008 
 
 8.963 
 8.920 
 8.876 
 8.831 
 
 8.788 
 
 8.743 
 8.699 
 8.654 
 8.610 
 8.566 
 
 8.521 
 8.476 
 8.431 
 8.386 
 8.342 
 8.297 
 
 8.252 
 8.207 
 8.i6i 
 8.116 
 8.071 
 8.026 
 
 7.9S0 
 
 7-934 
 7.889 
 
 7-843 
 7-797 
 7-751 
 
 7.706 
 
 56° 
 
 Inches. 
 
 0.00 1 
 .004 
 .009 
 .016 
 .025 
 •036 
 
 58° 
 
 O.OOI 
 
 .004 
 .009 
 
 .015 
 
 .024 
 •034 
 
 60° 
 
 O.OOI 
 
 .004 
 
 .008 
 .015 
 •023 
 •033 
 
 62° 
 
 O.OOI 
 
 .004 
 
 .008 
 
 .014 
 
 .022 
 .032 
 
 64° 
 
 0.001 
 .003 
 .008 
 .013 
 .021 
 .030 
 
 57° 
 
 Inches. 
 
 O.OOI 
 
 .C04 
 .009 
 .016 
 .024 
 
 •03s 
 
 59° 
 
 .004 
 .008 
 .015 
 .024 
 •034 
 
 61° 
 
 O.OOI 
 
 .004 
 
 .008 
 .014 
 .023 
 
 -033 
 
 63° 
 
 O.OOI 
 
 .003 
 .008 
 
 .014 
 .022 
 
 .031 
 
 Smithsonian Tables. 
 
 99 
 
Table 20. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jTsW- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 o 
 
 i, 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 r\r}T\j'K kT 
 
 
 
 
 bH S?^ 
 
 
 
 
 
 
 
 UKl^iJNAljLa \JS 
 
 
 .gTJ 
 
 .2 "i-ojS 
 
 
 
 
 
 
 
 DEVELOPED 
 
 
 |1 
 
 <u S > rt 
 
 S' 
 
 10' 
 
 IS' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 J"- 
 
 gSSo, 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 iTiches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 ■2^5 
 
 
 
 
 64°oo' 
 
 
 1.284 
 
 2.569 
 
 3-853 
 
 S-I37 
 
 6.422 
 
 y.yc6 
 
 Ss 
 
 64° 
 
 65° 
 
 
 lO 
 
 "'i.hs2 
 
 1.277 
 
 2-553 
 
 3-830 
 
 5.106 
 
 6-383 
 
 7.660 
 
 ^■2 
 
 
 
 
 20 
 
 n.704 
 
 I7-5S6 
 
 1.269 
 I.261 
 
 2-538 
 2.523 
 
 3.807 
 3-784 
 
 5-076 
 5.045 
 
 6-345 
 
 7.614 
 7.568 
 
 
 
 
 
 30 
 
 
 Ifickes. 
 
 Inches, 
 
 
 40 
 
 23.408 
 
 1.254 
 
 2.507 
 
 3-761 
 
 5.014 
 
 6.268 
 
 7-522 
 
 s' 
 
 10 
 
 0,001 
 
 0.001 
 .003 
 
 
 SO 
 
 29.260 
 
 1.246 
 
 2.492 
 
 3-738 
 
 4-984 
 
 6.230 
 
 7-476 
 
 .003 
 
 
 65 00 
 
 
 1.238 
 
 2.477 
 
 3-71S 
 
 4-953 
 
 6.192 
 
 7-430 
 
 IS 
 
 20 
 
 .008 
 .013 
 .021 
 
 .007 
 .013 
 •020 
 
 
 10 
 
 S-8S3' 
 
 I.231 
 
 2.461 
 
 3-^? 
 
 4.922 
 
 6-153 
 
 7-384 
 
 25 
 30 
 
 
 20 
 
 11.706 
 
 1.223 
 
 2.446 
 
 3.668 
 
 4.891 
 
 6.114 
 
 7-337 
 
 .030 
 
 .029 
 
 
 30 
 
 17-558 
 
 I.215 
 
 2.430 
 
 3-645 
 
 4.860 
 
 6,075 
 
 7.290 
 
 
 40 
 
 23.411 
 
 1.207 
 
 2.415 
 
 3.622 
 
 4.829 
 
 6.037 
 
 7-244 
 
 
 
 
 
 SO 
 
 29.264 
 
 1.200 
 
 2-399 
 
 3-599 
 
 4.798 
 
 5.998 
 
 7.198 
 
 
 
 
 
 
 
 
 
 6600 
 
 s-Sm' 
 
 I.I92 
 1.184 
 
 -384 
 
 3-575 
 
 4.767 
 4-736 
 
 5-959 
 
 7.151 
 
 
 66° 
 
 67° 
 
 
 10 
 
 3-552 
 
 5.920 
 
 7.104 
 
 
 
 
 
 20 
 
 11.707 
 
 1. 176 
 
 2.352 
 
 3-529 
 
 4-705 
 
 5.881 
 
 7-057 
 
 s 
 
 0.001 
 
 O.OOI 
 
 
 30 
 
 17.561 
 
 1.168 
 
 2-337 
 
 3- 50s 
 
 4-673 
 
 5.842 
 
 7.010 
 
 10 
 
 •003 
 
 -003 
 
 
 40 
 
 23.414 
 
 1.161 
 
 2.321 
 
 3-482 
 
 4.642 
 
 5-803 
 
 6.963 
 
 15 
 
 .007 
 
 .007 
 
 
 SO 
 
 29.268 
 
 I-IS3 
 
 2-30S 
 
 3-458 
 
 4.611 
 
 5.764 
 
 6.916 
 
 20 
 
 -013 
 
 .012 
 
 
 67 00 
 
 
 1.145 
 
 2.290 
 
 3-435 
 
 4-580 
 
 5-725 
 
 6.869 
 
 25 
 30 
 
 .020 
 .029 
 
 .019 
 .028 
 
 
 10 
 
 "s-Sm" 
 
 I-I37 
 
 2.274 
 
 34" 
 
 4.548 
 
 5-685 
 
 6.822 
 
 
 
 
 
 20 
 
 11.709 
 17-563 
 
 1.129 
 
 2.258 
 
 3-388 
 3-364 
 
 4-S17 
 4.485 
 
 5.646 
 5.607 
 
 6.775 
 6.728 
 
 
 
 
 
 30 
 
 1.121 
 
 2.243 
 
 
 
 
 
 40 
 
 23.418 
 
 1. 113 
 
 2.227 
 
 3-340 
 
 4-454 
 
 5567 
 
 6.680 
 
 
 68° 
 
 69° 
 
 
 SO 
 6800 
 
 29.272 
 
 1.106 
 1.098 
 
 2.21 1 
 2.105 
 
 3-317 
 3-293 
 
 4.422 
 4-391 
 
 5.528 
 5.489 
 
 6.634 
 6.586 
 
 
 
 
 
 5 
 
 0.001 
 
 0.001 
 
 
 10 
 
 5-855" 
 
 1.090 
 1.082 
 
 2.180 
 
 3.269 
 
 4-359 
 
 5-449 
 
 6-539 
 
 10 
 
 .003 
 
 .003 
 
 
 20 
 
 II.7I0 
 
 2.164 
 
 3.246 
 
 4.328 
 
 5.410 
 
 6.491 
 
 15 
 
 .007 
 
 .006 
 
 
 30 
 
 17.565 
 
 1.074 
 
 2.148 
 
 3.222 
 
 4.296 
 
 S-370 
 
 6-443 
 
 20 
 
 .012 
 
 .oil 
 
 
 40 
 
 23.420 
 
 1.066 
 
 2.132 
 
 3.198 
 
 4.264 
 
 S-330 
 
 6-396 
 
 25 
 
 .019 
 
 .018 
 
 
 SO 
 
 29.276 
 
 1.058 
 
 2.116 
 
 3-174 
 
 4.232 
 
 5-291 
 
 6-349 
 
 30 
 
 .027 
 
 .026 
 
 
 69 00 
 
 
 1.050 
 
 2.100 
 
 3-151 
 
 4.201 
 
 5.251 
 
 6.301 
 6.253 
 
 
 
 
 
 10 
 
 "'i.k'{e 
 
 1.042 
 
 2.084 
 
 3.127 
 
 4.169 
 
 5.211 
 
 
 MnO 
 
 utO 
 
 
 20 
 
 11.712 
 
 1.034 
 
 2.068 
 
 3-103 
 
 4-137 
 
 5.171 
 
 6.205 
 
 
 70° 
 
 71 
 
 
 30 
 
 17-567 
 
 1.026 
 
 2.052 
 
 3-079 
 
 4105 
 
 S-131 
 
 6.157 
 
 
 
 
 
 
 
 
 
 40 
 
 23-423 
 
 1.018 
 
 2.037 
 
 3-055 
 
 4-073 
 
 5.092 
 
 6.110 
 
 5 
 10 
 
 0,001 
 
 0.001 
 
 
 SO 
 
 29.279 
 
 l.OIO 
 
 2.021 
 
 3-031 
 
 4.041 
 
 5.052 
 
 6.062 
 
 .003 
 
 -003 
 
 
 7000 
 10 
 
 ■■ 5-856 ■ 
 
 1.002 
 .994 
 
 2.005 
 1.989 
 
 3.007 
 2.983 
 
 4.009 
 3-977 
 
 5.012 
 4-972 
 
 6.014 
 5.966 
 
 IS 
 20 
 
 25 
 
 .006 
 
 .011 
 
 .017 
 .024 
 
 .006 
 
 .010 
 .016 
 
 
 20 
 
 11-713 
 
 .986 
 
 1.972 
 
 2-959 
 
 3-945 
 
 4-931 
 
 S-917 
 
 .024 
 
 
 30 
 
 17.570 
 
 .978 
 
 1.956 
 
 2-935 
 
 3-913 
 
 4.891 
 
 5.869 
 
 30 
 
 
 40 
 SO 
 
 23.426 
 29.282 
 
 .970 
 .962 
 
 1.940 
 
 2.911 
 2.886 
 
 3-881 
 3.848 
 
 4.851 
 4.811 
 
 5.821 
 
 
 
 
 
 1.924 
 
 5-773 
 
 
 
 
 
 71 00 
 10 
 
 ■■ 5-857 ■ 
 
 -954 
 -946 
 
 1.892 
 
 2.862 
 2.838 
 
 3.816 
 3-784 
 
 4-771 
 4-730 
 
 5-676 
 
 
 72° 
 
 
 
 
 
 
 20 
 
 11.714 
 
 -938 
 
 1.876 
 
 2.814 
 
 3-752 
 
 4.690 
 
 5.628 
 
 5 
 
 0.001 
 
 
 
 30 
 
 17.572 
 
 ■93° 
 
 1.860 
 
 2.790 
 
 3.720 
 
 4-650 
 
 5-579 
 
 10 
 
 .003 
 
 
 
 40 
 
 23.429 
 
 .922 
 
 1.844 
 
 2.765 
 
 3.687 
 
 4.609 
 
 5-531 
 
 15 
 
 .006 
 
 
 
 SO 
 
 29.286 
 
 .914 
 
 1.828 
 
 2.741 
 
 3-655 
 
 4.569 
 
 5-483 
 
 20 
 25 
 
 .010 
 .016 
 
 
 
 72 00 
 
 
 .906 
 
 1.811 
 
 2.717 
 
 3-623 
 
 4-529 
 
 5-434 
 
 30 
 
 .023 
 
 
 
 Smithsonian Tables. 
 
Table 20. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ji^sini- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 ■"Eg 
 
 .2 »'S2 
 
 C S oj h 
 
 ^ S > SS 
 ^ j3 u p< 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 5 
 
 longitude. 
 
 longitude. 
 
 15' 
 
 longitude. 
 
 longitude. 
 
 25' 
 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 72"00' 
 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 7300 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 7400 
 10 
 20 
 30 
 40 
 50 
 
 7500 
 10 
 20 
 30 
 40 
 SO 
 
 7600 
 10 
 20 
 30 
 40 
 SO 
 
 77 00 
 
 10 
 20 
 30 
 40 
 
 SO 
 
 78 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 7900 
 
 10 
 20 
 
 30 
 40 
 
 so 
 8000 
 
 IncheS' 
 
 S.858 
 II.716 
 
 '7-573 
 23431 
 29.289 
 
 5.858 
 1 1.7 17 
 '7-575 
 23-434 
 29.292 
 
 5.859 
 11.718 
 '7-S77 
 23-436 
 29.295 
 
 5.860 
 n.719 
 17-578 
 23-438 
 29.298 
 
 5.860 
 11.720 
 17.580 
 23.440 
 29.300 
 
 5.860 
 11.721 
 17.582 
 23.442 
 29.302 
 
 5.861 
 11.722 
 
 '7-583 
 23.444 
 
 29.304 
 
 5.861 
 11.723 
 17-584 
 23-445 
 29.306 
 
 Inches. 
 .906 
 
 873 
 865 
 
 ,857 
 
 849 
 .841 
 832 
 824 
 .816 
 
 808 
 .800 
 79' 
 •783 
 
 767 
 
 759 
 ■750 
 742 
 
 734 
 ,726 
 ,717 
 
 ,709 
 ,701 
 .692 
 ,684 
 676 
 .668 
 
 ,659 
 ,651 
 643 
 ■634 
 ,626 
 ,618 
 
 609 
 601 
 
 593 
 584 
 576 
 568 
 
 559 
 55' 
 542 
 
 534 
 526 
 
 5'7 
 .509 
 
 Inches. 
 ,811 
 
 ■795 
 -779 
 763 
 746 
 730 
 
 7'4 
 ,697 
 681 
 ,665 
 ,648 
 632 
 
 616 
 599 
 
 566 
 550 
 
 534 
 
 517 
 
 484 
 468 
 
 451 
 
 435 
 
 418 
 402 
 
 368 
 352 
 
 335 
 
 319 
 302 
 28s 
 269 
 ,252 
 23s 
 
 219 
 
 202 
 
 169 
 152 
 '35 
 
 119 
 102 
 
 085 
 068 
 052 
 035 
 
 1.018 
 
 Inches. 
 
 2.717 
 2.693 
 2.668 
 2.644 
 2.620 
 
 2-595 
 
 2.571 
 2.546 
 2.522 
 2.497 
 
 2-473 
 2.448 
 
 2.424 
 2-399 
 2.374 
 2.350 
 
 2-325 
 2.300 
 
 2.276 
 2.251 
 2.226 
 2.201 
 2.177 
 2.152 
 
 2.127 
 2.102 
 2.078 
 2.053 
 2.028 
 2.003 
 
 1.978 
 
 I -953 
 1.928 
 1.903 
 1.878 
 1-853 
 
 1.828 
 1.803 
 1.778 
 
 1-753 
 1.728 
 
 1-703 
 
 1.678 
 
 '•653 
 1.628 
 1.602 
 I -577 
 '■552 
 
 1.527 
 
 Inches. 
 
 3-623 
 3-590 
 3-558 
 3-525 
 
 3-493 
 3.460 
 
 3.428 
 3-395 
 3-362 
 3-330 
 3-297 
 3-264 
 
 3-232 
 3- '99 
 3.160 
 
 3-'33 
 3.100 
 3.067 
 
 3-034 
 3.002 
 2.968 
 
 2-935 
 2.902 
 2.870 
 
 2.836 
 2.803 
 2.770 
 2-737 
 2-704 
 2.671 
 
 2.638 
 2.604 
 2.571 
 
 2-538 
 2.504 
 2.471 
 
 2.438 
 2.404 
 2.371 
 2.338 
 2.304 
 2.270 
 
 2-237 
 2.204 
 2.170 
 2-136 
 2.103 
 2.070 
 
 2.036 
 
 Inches. 
 
 4.529 
 4.488 
 
 4-447 
 4.407 
 4-366 
 4-325 
 
 4.285 
 4.244 
 4.203 
 4.162 
 4.121 
 4.081 
 
 4.040 
 3-999 
 3-957 
 3.916 
 
 3-875 
 3-834 
 
 3-793 
 3-752 
 3-7" 
 3.669 
 3.628 
 3-587 
 
 3-546 
 3-504 
 3-463 
 3.421 
 3-380 
 3-339 
 
 3-297 
 3.256 
 
 3-214 
 3-'72 
 
 3-'3' 
 3.089 
 
 3-047 
 3.005 
 2.964 
 2.922 
 2.880 
 
 2.797 
 2-755 
 2-713 
 2.671 
 2.629 
 2.587 
 
 2-54S 
 
 iTtches. 
 
 5-434 
 5.386 
 
 5-336 
 5.288 
 
 5-239 
 5.190 
 
 5.141 
 5.092 
 5.044 
 4-994 
 4-945 
 4.897 
 
 4-847 
 4.798 
 4.748 
 4.699 
 4.650 
 4.601 
 
 4.552 
 4.502 
 
 4-453 
 4-403 
 4-354 
 4-304 
 
 4-255 
 4.205 
 
 4-155 
 4.105 
 4.056 
 4.006 
 
 3-956 
 3-907 
 3.856 
 3.806 
 
 3-757 
 3.706 
 
 3.606 
 
 3-556 
 3.506 
 
 3-456 
 3.406 
 
 3-356 
 3-305 
 3-255 
 3-205 
 3-155 
 3-104 
 
 3-054 
 
 %% 
 
 72" 
 
 Inches. 
 
 0.001 
 .003 
 .006 
 .010 
 .016 
 .023 
 
 74° 
 
 0.001 
 .002 
 .005 
 .009 
 .014 
 .020 
 
 76° 
 
 O.OOI 
 .002 
 
 .005 
 
 .008 
 
 -013 
 .018 
 
 78° 
 
 0.000 
 .002 
 
 .004 
 .007 
 .oil 
 .016 
 
 73° 
 
 Inches. 
 
 0.001 
 .002 
 .005 
 .010 
 .015 
 .021 
 
 75° 
 
 0.001 
 .002 
 .005 
 .009 
 -013 
 .019 
 
 77° 
 
 0.000 
 .002 
 .004 
 .007 
 .012 
 .017 
 
 79° 
 
 0.000 
 .002 
 .004 
 .006 
 .010 
 .014 
 
 Smithsonian Tables. 
 
Table 21 . 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWtit- 
 
 [Derivatiun of table explained od pp. liii-lvi.] 
 
 o 
 
 1-1 * 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 
 15' longitude. 
 
 30' longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Incites. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 o°oo' 
 
 15 
 30 
 
 45 
 
 "'8.588' 
 17.176 
 25.764 
 
 8.647 
 8.646 
 8.646 
 8.646 
 
 .000 
 .000 
 .000 
 .000 
 
 17-293 
 17-293 
 17.292 
 17.291 
 
 .000 
 .001 
 .001 
 .001 
 
 25.940 
 
 25-939 
 25.938 
 
 25-937 
 
 .000 
 .001 
 .001 
 .002 
 
 34-586 
 34-585 
 34-584 
 34.582 
 
 .000 
 .001 
 .003 
 .004 
 
 
 I 00 
 
 15 
 30 
 45 
 
 34-352 
 
 8.645 
 
 8.644 
 8.643 
 8.642 
 
 .000 
 
 .000 
 
 .000 
 .001 
 
 17.291 
 
 17.289 
 17.287 
 17.28s 
 
 .001 
 
 .002 
 .002 
 .002 
 
 25.936 
 
 25-933 
 25.930 
 
 25-927 
 
 .003 
 
 .003 
 .004 
 .005 
 
 34-581 
 
 34-577 
 34-573 
 34-569 
 
 .005 
 
 .008 
 .009 
 
 
 8.588 
 17.176 
 25.764 
 
 
 2 00 
 
 '5 
 
 30 
 45 
 
 • 34-352 
 
 8.641 
 
 8.640 
 8.638 
 8.636 
 
 .001 
 
 .001 
 .001 
 .001 
 
 17-283 
 
 17.279 
 17.276 
 17-273 
 
 .003 
 
 .003 
 .003 
 .004 
 
 25-924 
 
 25-919 
 25.914 
 25.909 
 
 .006 
 
 .007 
 .007 
 .008 
 
 34-565 
 
 34-559 
 34-552 
 34-546 
 
 .Oil 
 
 .012 
 .014 
 .015 
 
 
 8.588 
 17.176 
 25.765 
 
 
 300 
 
 IS 
 30 
 45 
 
 34-353 
 
 8.635 
 
 8.633 
 8.630 
 8.628 
 
 .001 
 
 .001 
 .001 
 .001 
 
 17.270 
 
 17.265 
 17.260 
 17.256 
 
 .004 
 
 .004 
 .005 
 .005 
 
 25.904 
 
 25.898 
 25.891 
 25.884 
 
 .009 
 
 .009 
 .010 
 
 .oil 
 
 34-539 
 
 34-530 
 34-521 
 34-512 
 
 .016 
 
 .018 
 .019 
 .020 
 
 
 8.588 
 17.177 
 25-765 
 
 
 4 00 
 
 34-353 
 
 8.626 
 
 .001 
 
 17.251 
 
 .005 
 
 25-877 
 
 .012 
 
 34-502 
 
 .021 
 
 
 15 
 
 30 
 45 
 
 8.589 
 17.177 
 25.766 
 
 8.623 
 8.620 
 8.617 
 
 .OOI 
 .001 
 .002 
 
 17.245 
 17.240 
 17-234 
 
 .006 
 .005 
 .006 
 
 25.868 
 
 25-859 
 25.850 
 
 .012 
 
 •013 
 .014 
 
 34-491 
 34-479 
 34-467 
 
 ■023 
 .024 
 .025 
 
 
 5 00 
 
 34-354 
 
 8.614 
 
 .002 
 
 17.228 
 
 .007 
 
 25.842 
 
 .015 
 
 34-456 
 
 .026 
 
 
 15 
 30 
 45 
 
 8.589 
 
 17-177 
 25.766 
 
 8.610 
 8.607 
 8.603 
 
 .002 
 
 .002 
 .002 
 
 17.221 
 17.213 
 17.206 
 
 .007 
 .007 
 .008 
 
 25-831 
 25.820 
 25.809 
 
 .016 
 .016 
 .017 
 
 34-441 
 34-427 
 34.412 
 
 .028 
 .029 
 .030 
 
 
 6 00 
 
 15 
 30 
 
 45 
 
 34-355 
 
 8.600 
 
 8-595 
 8.587 
 
 .002 
 
 .002 
 .002 
 .002 
 
 17.199 
 
 17.191 
 17.182 
 17.174 
 
 .008 
 
 .008 
 .008 
 .009 
 
 25-799 
 
 25.786 
 
 25-773 
 25.760 
 
 .018 
 
 .019 
 .020 
 .021 
 
 34-398 
 
 34-381 
 34-364 
 34-347 
 
 .031 
 
 •033 
 •034 
 •035 
 
 
 8.589 
 17.178 
 25.767 
 
 
 7 00 
 
 15 
 30 
 45 
 
 34-356 
 
 8.583 
 8.578 
 
 .002 
 
 .002 
 .003 
 .003 
 
 17.165 
 
 17-155 
 17.145 
 17.136 
 
 .009 
 
 .009 
 .009 
 .010 
 
 25-748 
 
 25-7.-H3 
 25.718 
 25.704 
 
 .021 
 
 .022 
 .022 
 .023 
 
 34-330 
 
 34-3'0 
 34.291 
 34-272 
 
 •037 
 
 .038 
 .040 
 .041 
 
 
 8.589 
 17.179 
 25.768 
 
 
 800 
 
 34-358 
 
 8-563 
 
 .003 
 
 17.126 
 
 .010 
 
 25.689 
 
 .023 
 
 34-252 
 
 .042 
 
 
 15 
 
 30 
 45 
 
 8.590 
 17.180 
 25.769 
 
 ■8.S5S 
 8.552 
 8.546 
 
 .003 
 .003 
 .003 
 
 17-115 
 17.104 
 
 17-093 
 
 .010 
 .011 
 .011 
 
 25-673 
 25-656 
 
 25-639 
 
 .024 
 .024 
 .025 
 
 34-230 
 34.208 
 34.186 
 
 .044 
 .045 
 .046 
 
 
 900 
 
 15 
 30 
 45 
 
 34-359 
 
 8.541 
 
 8-535 
 8.528 
 8.522 
 
 .003 
 
 .003 
 .003 
 .003 
 
 17.082 
 
 17.069 
 17.057 
 17-045 
 
 .012 
 
 .OI2 
 .012 
 .013 
 
 25.622 
 
 25.604 
 25-585 
 25-567 
 
 .026 
 
 .027 
 .027 
 .028 
 
 34-163 
 
 34-138 
 34-114 
 34.089 
 
 -047 
 
 .048 
 .050 
 .051 
 
 
 8.590 
 17.180 
 25-771 
 
 
 10 00 
 
 34-361 
 
 8.516 
 
 .003 
 
 17.032 
 
 ■013 
 
 25-548 
 
 .029 
 
 34.064 
 
 .052 
 
 
 
 
 Smithsonian Tables. 
 
Table 21 . 
 CO-ORDINATES FOR PROJECTION OF IVIAPS. SCALE iifsW- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 ■^uf, 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR - 
 
 
 i3 "■ 
 
 llll 
 
 Ilia 
 
 
 
 
 
 
 
 
 
 15' longitude. 
 
 yj longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 Inches, 
 
 Inches, 
 
 Inches, 
 
 Inches, 
 
 Inches. 
 
 Inches, 
 
 Inches, 
 
 Inches. 
 
 io°oo' 
 IS 
 3° 
 
 45 
 
 8.591 
 17.181 
 25.772 
 
 8.516 
 8.509 
 8.502 
 8.496 
 
 -003 
 -003 
 .003 
 -003 
 
 17.032 
 17.019 
 17.005 
 16.991 
 
 -013 
 .013 
 .013 
 .014 
 
 25.548 
 25.528 
 25.507 
 25-487 
 
 .029 
 .030 
 -031 
 .032 
 
 34.064 
 
 34-037 
 34-010 
 33-982 
 
 -052 
 -054 
 
 -055 
 .056 
 
 II 00 
 
 15 
 3° 
 
 45 
 
 34-363 
 
 8.489 
 8.481 
 
 .004 
 
 .004 
 .004 
 .004 
 
 16.977 
 
 16.962 
 16.947 
 16-933 
 
 .014 
 
 .014 
 .015 
 .015 
 
 25.466 
 
 25-444 
 25.421 
 
 25-399 
 
 .032 
 
 •033 
 -033 
 -034 
 
 33-955 
 
 33-925 
 33-895 
 33-865 
 
 •057 
 .058 
 
 8.591 
 17.183 
 25774 
 
 12 00 
 
 IS 
 3° 
 45 
 
 34-365 
 
 8.459 
 
 8.451 
 8.443 
 8.434 
 
 .004 
 
 .004 
 .004 
 .004 
 
 16.918 
 
 16.901 
 16.885 
 16.869 
 
 .015 
 
 .016 
 .016 
 .016 
 
 25-376 
 
 25-352 
 25.328 
 
 25-304 
 
 •03s 
 
 -03 s 
 •.036 
 .036 
 
 33-835 
 
 33803 
 33-770 
 33-738 
 
 .061 
 
 .063 
 .064 
 .065 
 
 8.592 
 17.184 
 25.776 
 
 1300 
 
 15 
 
 30 
 45 
 
 34-368 
 
 8.426 
 
 8.418 
 8.409 
 8.400 
 
 .004 
 
 .004 
 .004 
 .004 
 
 16.853 
 
 16.83s 
 16.818 
 16.800 
 
 .017 
 
 .017 
 .017 
 .018 
 
 25.279 
 
 25-253 
 25.227 
 25.201 
 
 •037 
 
 .038 
 
 -039 
 .040 
 
 33-706 
 
 33-636 
 33.601 
 
 .066 
 
 .067 
 .069 
 .070 
 
 8.592 
 17.185 
 25.778 
 
 1400 
 
 IS 
 30 
 45 
 
 34-370 
 
 8.391 
 8.382 
 
 8-363 
 
 .004 
 
 -005 
 .005 
 .005 
 
 16.783 
 
 16.764 
 
 16.745 
 16.726 
 
 .018 
 
 .018 
 .018 
 .019 
 
 25.174 
 
 25.146 
 25.118 
 25.090 
 
 .040 
 
 .041 
 .041 
 .042 
 
 33-566 
 
 33-528 
 33-490 
 33-453 
 
 .071 
 
 .072 
 
 -073 
 .074 
 
 8-593 
 17.186 
 25.780 
 
 1500 
 
 15 
 30 
 45 
 
 34-373 
 
 8-354 
 
 8-344 
 8.334 
 8.324 
 
 •005 
 
 .005 
 .005 
 •005 
 
 16.708 
 
 16.688 
 16.668 
 16.647 
 
 .019 
 
 .019 
 .019 
 .020 
 
 25.061 
 
 25-031 
 25.001 
 24.971 
 
 .042 
 
 -043 
 -044 
 •045 
 
 33-415 
 
 33-375 
 33-335 
 33-295 
 
 •075 
 
 .077 
 .078 
 ■079 
 
 8.594 
 17.188 
 25.782 
 
 1600 
 
 15 
 30 
 45 
 
 34-376 
 
 8-314 
 
 8.303 
 8.292 
 8.282 
 
 .005 
 
 .005 
 .005 
 .005 
 
 16.627 
 16.606 
 
 16.564 
 
 .020 
 
 .020 
 .020 
 .021 
 
 24.941 
 
 24.909 
 24.877 
 24-845 
 
 .045 
 
 •045 
 .046 
 .046 
 
 33-255 
 
 33.212 
 33-170 
 33-127 
 
 .080 
 
 .081 
 .082 
 .083 
 
 8-595 
 17.190 
 25.784 
 
 1700 
 
 IS 
 30 
 45 
 
 34-379 
 
 8.271 
 
 8.260 
 8.249 
 8-237 
 
 .005 
 .005 
 .006 
 
 16.542 
 
 16.520 
 16.497 
 16.475 
 
 .021 
 
 .021 
 .021 
 
 .022 
 
 24.813 
 
 24-779 
 24.746 
 24.712 
 
 -047 
 
 .048 
 .049 
 .050 
 
 33-084 
 
 33-039 
 32-994 
 32-949 
 
 .084 
 .085 
 
 8.596 
 17.191 
 25.787 
 
 1800 
 
 IS 
 30 
 45 
 
 34-382 
 
 8.226 
 
 8.214 
 8.202 
 8.190 
 
 .006 
 
 .006 
 .006 
 .006 
 
 16.452 
 
 16.428 
 16.404 
 16.381 
 
 .022 
 
 .022 
 .023 
 .023 
 
 24.678 
 
 24.642 
 24.607 
 24.571 
 
 .050 
 
 .051 
 
 -051 
 .052 
 
 32.904 
 
 32-856 
 32.809 
 32.761 
 
 .089 
 
 .090 
 .091 
 .092 
 
 8.596 
 
 17-193 
 25.790 
 
 1900 
 
 IS 
 30 
 45 
 
 34-386 
 
 8.178 
 
 8.166 
 
 8-IS3 
 8.141 
 
 .006 
 
 .006 
 .006 
 .006 
 
 16-357 
 
 16.332 
 16.307 
 16.282 
 
 -023 
 
 .023 
 .024 
 .024 
 
 24-535 
 
 24.498 
 24.460 
 24.422 
 
 .052 
 
 -053 
 -054 
 .055 
 
 32-714 
 
 32.664 
 32.614 
 32-563 
 
 -093 
 
 -094 
 .095 
 .096 
 
 8-597 
 17.195 
 25.792 
 
 20 00 
 
 34-39° 
 
 8.128 
 
 .006 
 
 16.257 
 
 .024 
 
 24.385 
 
 -05s 
 
 32-513 
 
 .097 
 
 
 Smithsonian Tables. 
 
 103 
 
Table 21. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWnrr- 
 [Derivation of table explained on pp. liii-lvi,] 
 
 ii 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 
 CO-ORDINATES 
 
 OF DEVELOPED PARALLEL FOR — 
 
 1 
 
 15' longitude. 
 
 so' longitude. 
 
 45' longitude. 
 
 1° longitude. 1 
 
 c 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 20°00' 
 
 ■S 
 
 30 
 4S 
 
 ■■'8.598' 
 17.197 
 
 25-795 
 
 8.128 
 8.II5 
 8.102 
 8.089 
 
 .006 
 .006 
 .006 
 .006 
 
 16.257 
 16.230 
 16.204 
 16.178 
 
 .024 
 .024 
 .025 
 .025 
 
 24.385 
 24.346 
 24.306 
 24.267 
 
 .056 
 .056 
 .057 
 
 32.513 
 32.461 
 32.408 
 32.356 
 
 •099 
 .100 
 
 21 00 
 
 IS 
 30 
 4S 
 
 34-394 
 
 8.076 
 
 8.062 
 8.048 
 8.035 
 
 .006 
 
 .006 
 .006 
 .007 
 
 16.152 
 
 16.124 
 16.097 
 16.069 
 
 .025 
 
 .025 
 .026 
 .026 
 
 24.227 
 
 24.186 
 24.145 
 24.104 
 
 •057 
 
 .058 
 .058 
 .059 
 
 32.303 
 
 32.24S 
 32.193 
 32.138 
 
 .101 
 .102 
 
 -103 
 
 .104 
 
 8-599 
 17.199 
 
 25.798 
 
 22 00 
 IS 
 
 30 
 
 4S 
 
 34-398 
 
 8.021 
 
 8.006 
 
 7-992 
 7-978 
 
 .007 
 
 .007 
 .007 
 .007 
 
 16.042 
 
 16.013 
 15.984 
 15-955 
 
 .026 
 
 .026 
 .027 
 .027 
 
 24.062 
 
 24.019 
 23.976 
 23-933 
 
 •059 
 
 .060 
 .060 
 .061 
 
 32.083 
 
 32.026 
 31.968 
 31.911 
 
 .105 
 .106 
 
 8.600 
 17.201 
 25.801 
 
 2300 
 IS 
 
 30 
 
 4S 
 
 34.402 
 
 7-963 
 7.948 
 
 7-933 
 7.918 
 
 .007 
 
 .007 
 .007 
 .007 
 
 15.927 
 
 15-897 
 15.867 
 IS-837 
 
 .027 
 
 .027 
 .028 
 .028 
 
 23.890 
 
 23.845 
 23.800 
 23-756 
 
 .061 
 
 .062 
 .062 
 .063 
 
 31.853 
 
 31.794 
 31-734 
 31-674 
 
 .109 
 .109 
 
 .110 
 
 .111 
 
 8.602 
 17.203 
 25.804 
 
 2400 
 
 IS 
 3° 
 45 
 
 34.406 
 
 7.904 
 
 7.888 
 7.872 
 7-857 
 
 .007 
 
 .007 
 .007 
 .007 
 
 15.807 
 
 15.776 
 15-745 
 15-713 
 
 .028 
 
 .028 
 .029 
 .029 
 
 23.711 
 
 23.664 
 23.617 
 23-570 
 
 •063 
 
 .064 
 .064 
 •065 
 
 31.614 
 
 31.552 
 31.489 
 
 31-427 
 
 .112 
 
 .113 
 
 .114 
 
 •115 
 
 8.603 
 17.205 
 25.808 
 
 2500 
 IS 
 
 30 
 
 4S 
 
 34-410 
 
 7.841 
 
 7.825 
 7.809 
 7-793 
 
 .007 
 
 .007 
 .007 
 .007 
 
 15.682 
 
 15.650 
 15.617 
 15-585 
 
 .029 
 
 .029 
 .029 
 .030 
 
 23-524 
 
 23-475 
 23.426 
 
 23-378 
 
 .065 
 
 .065 
 .066 
 .067 
 
 31-365 
 
 31-30° 
 31-235 
 31.170 
 
 .116 
 
 .117 
 .117 
 .118 
 
 8.604 
 17.207 
 25.811 
 
 2600 
 IS 
 
 30 
 
 45 
 
 34-415 
 
 7.776 
 
 7.760 
 
 7-743 
 7.726 
 
 .007 
 .008 
 
 15-553 
 
 15.519 
 15.486 
 15-452 
 
 .030 
 
 .030 
 .030 
 .030 
 
 23-329 
 
 23-279 
 23.229 
 
 23-179 
 
 .067 
 !o68 
 
 31.106 
 
 31-039 
 30.972 
 30.905 
 
 .119 
 
 .120 
 .121 
 .121 
 
 8.605 
 17.210 
 25.814 
 
 27 00 
 
 IS 
 
 30 
 
 45 
 
 34-419 
 
 7.709 
 
 7.692 
 
 7-675 
 7.657 
 
 .008 
 
 .008 
 .008 
 .008 
 
 15.419 
 
 15-384 
 15-350 
 15-315 
 
 .031 
 
 .031 
 .031 
 .031 
 
 23.128 
 
 23.076 
 23.024 
 22.972 
 
 .069 
 
 .069 
 .070 
 .070 
 
 30.838 
 
 30-769 
 30.699 
 30.630 
 
 .122 
 
 .123 
 .124 
 .124 
 
 8.606 
 17.212 
 25.818 
 
 28 00 
 
 15 
 30 
 
 45 
 
 34.424 
 
 7.640 
 
 7.622 
 7.604 
 7.586 
 
 .008 
 
 .008 
 .008 
 .008 
 
 15.280 
 
 15.244 
 15.208 
 15-173 
 
 .031 
 
 .031 
 .032 
 •032 
 
 22.920 
 
 22.866 
 22.813 
 22.759 
 
 .070 
 
 .071 
 .071 
 .072 
 
 30.560 
 
 30-489 
 30-417 
 30-345 
 
 .125 
 
 .126 
 .127 
 .127 
 
 8.607 
 17.215 
 25.822 
 
 29 00 
 
 IS 
 
 30 
 45 
 
 34-43° 
 
 7.568 
 
 7-55° 
 7-531 
 7-513 
 
 .008 
 
 .008 
 .008 
 .008 
 
 15-137 
 
 15.100 
 15-063 
 15.026 
 
 .032 
 
 .032 
 •032 
 ■033 
 
 22.705 
 
 22.650 
 22.594 
 22.539 
 
 .072 
 
 .072 
 .073 
 ■073 
 
 30.274 
 
 30.200 
 30.125 
 30.051 
 
 .123 
 
 .129 
 .130 
 .130 
 
 8.609 
 17.217 
 25.826 
 
 3000 
 
 34-435 
 
 7-494 
 
 .008 
 
 14.989 
 
 ■033 
 
 22.483 
 
 .074 
 
 29.978 
 
 131 
 
 
 Smithsonian Tables. 
 
 104 
 
Table 21 , 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi, j 
 
 SCALE 
 
 144T44' 
 
 Latitude of 
 parallel. 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 
 CO-ORDINATES 
 
 3F DEVELOPED PARALLEL FOR — 
 
 
 is' longitude. 
 
 30' longitude. 
 
 45' longitude. 
 
 1° longitude. | 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Incites. 
 
 30°00' 
 
 30 
 45 
 
 8.610 
 17.220 
 25-830 
 
 7-494 
 7-475 
 7-456 
 7-437 
 
 .008 
 .008 
 .008 
 .008 
 
 14.989 
 14-951 
 
 14-913 
 14-874 
 
 •033 
 -033 
 •033 
 -033 
 
 22.483 
 22.426 
 22.369 
 22.312 
 
 -074 
 .074 
 
 -074 
 -075 
 
 29.978 
 29.902 
 29.825 
 29.749 
 
 ■131 
 •131 
 .132 
 
 -133 
 
 31 00 
 
 IS 
 3° 
 4S 
 
 34.440 
 
 7.418 
 
 7-398 
 7-379 
 7-3S9 
 
 .008 
 
 .008 
 .008 
 .008 
 
 14.836 
 
 14.797 
 14.758 
 14.718 
 
 ■°33 
 
 -033 
 •034 
 -034 
 
 22.254 
 
 22.19s 
 22.137 
 22.078 
 
 -075 
 
 .075 
 .076 
 .076 
 
 29.672 
 
 29.594 
 29.515 
 29-437 
 
 •133 
 
 -134 
 •13s 
 -13s 
 
 8.61 1 
 17.213 
 25.834 
 
 32 00 
 
 "5 
 3° 
 45 
 
 34.446 
 
 7-340 
 
 7-319 
 7.299 
 7.279 
 
 .008 
 
 .008 
 .009 
 .009 
 
 14.679 
 
 14.639 
 14.598 
 14.558 
 
 •034 
 
 -034 
 -034 
 •034 
 
 22.019 
 
 2i'.898 
 , 21.837 
 
 .076 
 
 .077 
 
 -077 
 .077 
 
 29-358 
 
 29.278 
 29.197 
 29.116 
 
 -136 
 
 .136 
 -137 
 •137 
 
 8.613 
 17.225 
 25.838 
 
 33 °o 
 
 15 
 30 
 45 
 
 34.451 
 
 7.259 
 
 7.238 
 7-217 
 7-197 
 
 .009 
 
 .009 
 .009 
 .009 
 
 14.518 
 
 14-476 
 14-435 
 14-393 
 
 -034 
 
 -035 
 -03s 
 -035 
 
 21.777 
 
 21.714 
 21.652 
 21.590 
 
 .078 
 
 .078 
 .078 
 .078 
 
 29.036 
 
 28.953 
 28.869 
 28.786 
 
 .138 
 
 .138 
 •139 
 -139 
 
 8.614 
 17.228 
 25.842 
 
 3400 
 
 IS 
 30 
 45 
 
 34.456 
 
 7.176 
 
 7-1 54 
 7-133 
 7.112 
 
 .oog 
 
 .009 
 .009 
 .009 
 
 14-352 
 
 14.309 
 14.266 
 14.224 
 
 -03s 
 
 -°35 
 -03s 
 -035 
 
 21.527 
 
 21.464 
 21.400 
 21.336 
 
 .079 
 
 .079 
 .079 
 
 28.703 
 
 28.618 
 
 ^^533 
 28.448 
 
 .140 
 
 .141 
 .141 
 .142 
 
 8.615 
 17.231 
 25.846 
 
 3500 
 
 IS 
 3° 
 45 
 
 34.462 
 
 7.091 
 
 7.069 
 7-047 
 7-025 
 
 .009 
 
 .009 
 .009 
 .009 
 
 14.181 
 
 14.138 
 14.094 
 14-050 
 
 •035 
 
 .036 
 .036 
 .036 
 
 21.272 
 
 21.207 
 21.141 
 21.076 
 
 .080 
 
 .080 
 .080 
 .080 
 
 28.362 
 
 28.275 
 28.188 
 28.101 
 
 .142 
 
 .142 
 ■143 
 -143 
 
 8.617 
 
 17.234 
 25.851 
 
 3600 
 
 15 
 30 
 4S 
 
 34.468 
 
 7.003 
 6.981 
 6.936 
 
 .009 
 
 .009 
 .009 
 .009 
 
 14-007 
 
 13.962 
 13-917 
 13-873 
 
 .036 
 
 .036 
 -036 
 •036 
 
 21.010 
 
 20.943 
 20.876 
 20.809 
 
 .081 
 
 .081 
 .081 
 .081 
 
 28.014 
 
 27.924 
 27-835 
 27-745 
 
 -144 
 
 -144 
 •144 
 •145 
 
 8.618 
 17.237 
 25.855 
 
 3700 
 
 IS 
 30 
 
 45 
 
 34-474 
 
 6-914 
 
 6.891 
 6.868 
 6.845 
 
 .009 
 
 .009 
 .009 
 .009 
 
 13.828 
 
 13.782 
 
 13-736 
 13.690 
 
 .036 
 
 .036 
 .036 
 -037 
 
 20.742 
 
 20.673 
 20.604 
 20.536 
 
 .082 
 
 .082 
 .082 
 .082 
 
 27.655 
 
 27.564 
 27-472 
 27.381 
 
 •145 
 
 .145 
 .146 
 .146 
 
 8.620 
 17.240 
 25.860 
 
 3800 
 
 IS 
 3° 
 45 
 
 34.480 
 
 6.822 
 
 6.799 
 6-775 
 6-752 
 
 .009 
 
 .009 
 .009 
 .009 
 
 13-645 
 
 13-598 
 13-551 
 I3-S°4 
 
 -037 
 
 -037 
 -037 
 •037 
 
 20.467 
 
 20.397 
 20.326 
 20.256 
 
 .082 
 
 .083 
 .083 
 -083 
 
 27.289 
 
 27.196 
 27.102 
 27.008 
 
 .147 
 
 .147 
 .147 
 
 •147 
 
 8.621 
 
 17-243 
 25.864 
 
 3900 
 
 15 
 30 
 45 
 
 34-485 
 
 6.729 
 
 6.705 
 6.681 
 6.657 
 
 .009 
 
 .009 
 .009 
 .009 
 
 13-457 
 
 13.409 
 13-361 
 13-314 
 
 -037 
 
 -037 
 -037 
 -037 
 
 20.186 
 
 20.114 
 20.042 
 19.970 
 
 .083 
 
 .083 
 .083 
 .084 
 
 26.914 
 
 26.819 
 26.723 
 26.627 
 
 .148 
 
 .148 
 .148 
 .148 
 
 8.623 
 17.246 
 25.868 
 
 4000 
 
 34-491 
 
 6.633 
 
 .009 
 
 13.266 
 
 -037 
 
 19.899 
 
 .084 
 
 26.532 
 
 .149 
 
 
 Smithsonian Tables. 
 
 I°S 
 
Table 21. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE m^if 
 [Derivation of table explained on pp. liii-lvi.] 
 
 40°00' 
 IS 
 30 
 45 
 
 41 00 
 
 IS 
 30 
 45 
 
 42 00 
 
 15 
 30 
 45 
 
 4300 
 
 IS 
 30 
 45 
 
 4400 
 
 IS 
 30 
 45 
 
 45 00 
 
 15 
 30 
 45 
 
 4600 
 
 IS' 
 
 30 
 
 45 
 
 4700 
 
 15 
 30 
 45 
 
 48 00 
 
 IS 
 30 
 45 
 
 4900 
 
 IS 
 30 
 45 
 
 5000 
 
 •■as 8 
 
 Inches. 
 
 8.624 
 17.249 
 25-873 
 
 34-497 
 
 8.625 
 17.250 
 25-875 
 
 34.500 
 
 8.627 
 
 25.882 
 34-Sio 
 
 8.629 
 
 17-257 
 25.886 
 
 34-515 
 
 8.630 
 17.261 
 25.891 
 
 34-522 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 15' longitude. 
 
 8.632 
 17.264 
 25.896 
 
 34-528 
 
 8-633 
 17.267 
 25.901 
 
 34-534 
 
 8.635 
 17.270 
 25.905 
 
 34-54° 
 
 8.637 
 
 17-273 
 25.910 
 
 34-546 
 
 8.638 
 17.276 
 25.914 
 
 34-552 
 
 Inches. 
 
 6.633 
 6.608 
 6.584 
 6.560 
 
 6-535 
 
 6.510 
 6.485 
 6.460 
 
 6-435 
 
 6.410 
 6.385 
 6-359 
 
 6-334 
 
 6.308 
 6.282 
 6.256 
 
 6.230 
 
 6.203 
 6.177 
 6.151 
 
 6.124 
 
 6.097 
 6.071 
 6.044 
 
 6.017 
 
 5.990 
 5.962 
 5-935 
 
 5.908 
 
 5.880 
 5.852 
 5.824 
 
 5-796 
 
 5-768 
 5-740 
 5.712 
 
 5.684 
 
 5.626 
 
 5-598 
 
 5-569 
 
 Inches. 
 
 .009 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 
 ;oo9 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 so' longitude. 
 
 Inches. 
 
 13.266 
 13.217 
 13.168 
 13.119 
 
 13.070 
 
 13.020 
 12.970 
 12.920 
 
 12.871 
 
 12.820 
 12.769 
 12.718 
 
 12.667 
 
 12.615 
 12.563 
 12.512 
 
 12.460 
 
 12.407 
 
 12.354 
 12.301 
 
 12.249 
 
 12.195 
 12.141 
 12.088 
 
 12.034 
 
 11-979 
 11.925 
 11.870 
 
 11.815 
 
 11.760 
 11.704 
 11.648 
 
 11-593 
 
 11.536 
 11.480 
 11.424 
 
 11.367 
 
 11.310 
 11.253 
 11.195 
 
 11.138 
 
 Inches. 
 
 •037 
 •037 
 ■037 
 •037 
 
 037 
 
 •037 
 -037 
 •037 
 
 •037 
 
 •037 
 .038 
 .038 
 
 .038 
 
 .038 
 .038 
 .038 
 
 .038 
 
 .038 
 .038 
 .038 
 
 ■038 
 
 ■038 
 •038 
 .038 
 
 .038 
 
 .038 
 .038 
 .038 
 
 ■038 
 
 .038 
 .038 
 .038 
 
 ■038 
 
 .038 
 .038 
 •037 
 
 •037 
 
 •037 
 -037 
 •037 
 
 •037 
 
 45' longitude. 
 
 Inches. 
 
 19.899 
 19.825 
 
 19-752 
 19.679 
 
 19.605 
 
 19-53° 
 19.456 
 
 19.381 
 19.306 
 
 19.230 
 19.154 
 19-077 
 
 19.001 
 
 18.923 
 18.845 
 18.767 
 
 18.689 
 
 18.610 
 18.531 
 18.452 
 
 18.373 
 
 18.292 
 18.212 
 18.131 
 
 18.051 
 
 17.969 
 17.887 
 17.805 
 
 17.723 
 
 17.640 
 17556 
 17-473 
 
 17-389 
 
 17.30S 
 17.220 
 
 17-135 
 17.051 
 
 16.965 
 16.879 
 16.793 
 
 16.707 
 
 Inches. 
 
 .084 
 .084 
 .084 
 .084 
 
 .084 
 
 .084 
 .084 
 .084 
 
 .085 
 
 .085 
 .085 
 .085 
 
 .085 
 
 .085 
 .085 
 .085 
 
 .085 
 
 .085 
 .085 
 .085 
 
 .085 
 
 .085 
 .085 
 .085 
 
 .085 
 
 .085 
 •085 
 .085 
 
 .085 
 
 .085 
 ■085 
 .085 
 
 .085 
 
 .085 
 .084 
 .084 
 
 .084 
 
 .084 
 .084 
 .084 
 
 .084 
 
 1° longitude. 
 
 Inches, 
 
 26.532 
 26.434 
 26.336 
 26.238 
 
 26.140 
 
 26.041 
 
 25-941 
 25.841 
 
 25.741 
 
 25.640 
 
 25.538 
 25.436 
 
 25.335 
 
 25-231 
 25.127 
 25.023 
 
 24.919 
 
 24.814 
 24.708 
 24.603 
 
 24-497 
 
 24.390 
 24.283 
 24.175 
 
 24.068 
 
 23-959 
 23.849 
 23.740 
 
 23.631 
 
 23.520 
 23.408 
 23.297 
 
 23.186 
 
 23-073 
 22.960 
 22.847 
 
 22.734 
 
 22.620 
 22.505 
 22.391 
 
 22.276 
 
 Inches. 
 .149 
 •149 
 •149 
 .149 
 
 .150 
 
 .150 
 .150 
 .150 
 
 .150 
 
 .150 
 •151 
 ■151 
 
 •151 
 
 .151 
 .151 
 -151 
 
 .151 
 
 •151 
 ■151 
 .151 
 
 •151 
 
 .151 
 .151 
 .151 
 
 .151 
 
 •151 
 .151 
 
 ■151 
 .151 
 
 .151 
 .151 
 .151 
 
 .150 
 
 .150 
 .150 
 .150 
 
 .150 
 
 .150 
 .150 
 .150 
 
 •150 
 
 Smithsonian Tabues. 
 
 106 
 
Table 21 , 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 SCALE 
 
 iiPtif 
 
 So°oo' 
 IS 
 3° 
 45 
 
 51 00 
 
 "S 
 3° 
 45 
 
 5200 
 
 15 
 30 
 45 
 
 5300 
 
 IS 
 30 
 45 
 
 54 00 
 
 IS 
 
 30 
 45 
 
 5500 
 
 15 
 
 30 
 45 
 
 5600 
 
 IS 
 30 
 45 
 
 57(30 
 
 IS 
 30 
 
 45 
 
 5800 
 
 15 
 30 
 45 
 
 59 00 
 
 IS 
 30 
 45 
 
 60 00 
 
 =3eS 
 
 5 So? 
 S S S 
 
 Inches. 
 
 8.640 
 17.279 
 25.919 
 
 34-558 
 
 8.641 
 17.282 
 25.924 
 
 34-565 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 is' longitude. 
 
 8-643 
 17.285 
 25.928 
 
 34-571 
 
 8.644 
 17.288 
 25.932 
 
 34-576 
 
 8.646 
 17.291 
 25-937 
 
 34-582 
 
 8.647 
 17.294 
 25-941 
 
 34-588 
 
 8.648 
 17.297 
 25.946 
 
 34-594 
 
 8.650 
 17.300 
 25.950 
 
 34.600 
 
 8.651 
 17-303 
 25-954 
 
 34.605 
 
 8-653 
 17-305 
 25.958 
 
 34.611 
 
 Inches. 
 
 5-569 
 5-S40 
 5.511 
 5.482 
 
 5-453 
 
 5-423 
 5-394 
 5-364 
 
 5-334 
 
 5-305 
 5-275 
 5-245 
 
 5-215 
 
 5.185 
 5-154 
 5-124 
 
 5.094 
 
 5.063 
 
 5-032 
 5.002 
 
 4-971 
 
 4.940 
 4.909 
 4.878 
 
 4.846 
 
 4.81S 
 4.784 
 4-752 
 
 4.720 
 
 4.689 
 4.657 
 4.625 
 
 4-593 
 
 4.561 
 4.529 
 4-497 
 
 4.464 
 
 4-432 
 4-399 
 4-367 
 
 4-334 
 
 30' longitude. 
 
 Inches. 
 
 .009 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .009 
 .009 
 .009 
 
 .009 
 
 .008 
 .008 
 .008 
 
 .008 
 
 .008 
 .008 
 .008 
 
 .008 
 
 Inches. 
 
 II.138 
 11.080 
 11.022 
 10.963 
 
 10.905 
 
 10.846 
 10.787 
 10.728 
 
 10.669 
 
 10.609 
 10.549 
 10.490 
 
 10.430 
 
 10.369 
 10.309 
 10.248 
 
 10.187 
 
 10.126 
 10.064 
 10.003 
 
 9.942 
 
 9.879 
 9.817 
 
 9-755 
 9-693 
 
 9.630 
 9.567 
 9-504 
 
 9.441 
 
 9-377 
 9-314 
 9.250 
 
 9.186 
 
 9.122 
 9.058 
 8-993 
 
 8.929 
 
 8.864 
 8-799 
 8.734 
 
 8.669 
 
 45' longitude. 
 
 Inches. 
 
 -037 
 -037 
 •037 
 -037 
 
 •037 
 
 -037 
 •037 
 •037 
 
 -037 
 
 .036 
 .036 
 .036 
 
 .036 
 
 •036 
 .036 
 ■036 
 
 .036 
 
 .036 
 .036 
 .036 
 
 .036 
 
 •035 
 •035 
 
 -035 
 
 •035 
 
 •035 
 -035 
 -03s 
 
 •03s 
 
 •035 
 -034 
 -034 
 
 •034 
 
 -034 
 •034 
 -034 
 
 -033 
 
 -033 
 -033 
 •033 
 
 -033 
 
 Inches. 
 
 16.707 
 16.620 
 16.532 
 16.445 
 
 16.358 
 
 16.269 
 16.181 
 16.092 
 
 16.004 
 
 15-914 
 15.824 
 
 15-734 
 15.645 
 
 15-554 
 15-463 
 15-372 
 
 15.281 
 
 15.189 
 15.097 
 15.004 
 
 14.912 
 
 14.819 
 14.726 
 14-633 
 
 14-539 
 
 14.445 
 14-351 
 14.256 
 
 14.162 
 
 14.066 
 13-970 
 13-875 
 
 13-779 
 
 13-683 
 13.586 
 13.490 
 
 13-393 
 
 13.296' 
 13.198 
 13.100 
 
 Inches. 
 
 .084 
 .084 
 .084 
 .083 
 
 •083 
 
 .083 
 .083 
 .083 
 
 .083 
 
 .082 
 .082 
 .082 
 
 .082 
 
 .082 
 .081 
 .081 
 
 .081 
 
 .081 
 .080 
 .080 
 
 .080 
 
 .080 
 .079 
 .079 
 
 .079 
 
 .079 
 .078 
 .078 
 
 .078 
 
 .077 
 .077 
 .077 
 
 .076 
 
 .076 
 .076 
 .075 
 
 •075 
 
 .075 
 
 •075 
 .074 
 
 1° longitude. 
 
 13.003 .074 
 
 Inches. 
 
 22.276 
 22.160 
 22.043 
 21.927 
 
 21.692 
 
 21.574 
 21.456 
 
 21.338 
 
 21.218 
 21.099 
 20.979 
 
 20.860 
 
 20.738 
 20.617 
 20.496 
 
 20.374 
 
 20.252 
 20.129 
 20.000 
 
 19.883 
 
 19-759 
 19.634 
 19.510 
 
 19.386 
 
 19.260 
 
 19-134 
 . 19.008 
 
 18.882 
 
 18.754 
 18.627 
 18.500 
 
 18.372 
 
 18.244 
 18.115 
 17.986 
 
 17.858 
 
 17.728 
 
 17-597 
 17.467 
 
 17-337 
 
 Inches. 
 
 .150 
 
 -149 
 .149 
 .149 
 
 .148 
 
 .148 
 .148 
 .147 
 
 .147 
 
 .146 
 .146 
 .14s 
 
 .145 
 
 .145 
 .144 
 .144 
 
 .144 
 
 ■143 
 •143 
 .142 
 
 .142 
 
 .141 
 .141 
 .140 
 
 .140 
 
 .140 
 •139 
 -139 
 
 .138 
 
 .138 
 -137 
 -137 
 
 .136 
 
 •135 
 -13s 
 -134 
 
 -134 
 
 •133 
 -133 
 .132 
 
 •131 
 
 Smithsonian Tables. 
 
 107 
 
Table 21. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWnnr- 
 [Derivation of table explained on pp. liii-lvi*] 
 
 •S3 
 S'fi 
 
 J, 
 
 Sag 
 
 .2 mT3,SJ 
 C g lU K 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR - 
 
 15' longitude. 
 
 30^ longitude. 
 
 45' longitude. 
 
 1° longitude. 
 
 6o°oo' 
 IS 
 3° 
 45 
 
 61 00 
 
 IS 
 30 
 45 
 
 62 CO 
 
 IS 
 30 
 
 45 
 
 6300 
 
 IS 
 30 
 45 
 
 64 00 
 
 IS 
 30 
 
 45 
 
 65 00 
 
 IS 
 30 
 45 
 
 6600 
 
 IS 
 30 
 45 
 
 67 00 
 
 15 
 
 30 
 45 
 
 6800 
 
 15 
 30 
 
 45 
 
 6900 
 
 15 
 30 
 45 
 
 7000 
 
 Inches. 
 
 8.654 
 17.308 
 25.962 
 
 34.616 
 
 8.655 
 17.311 
 25.966 
 
 34.621 
 
 8.657 
 
 17.313 
 25.970 
 
 34.626 
 
 8.658 
 17.316 
 25-974 
 
 34.632 
 
 8.659 
 17.318 
 25.977 
 
 34.636 
 
 8.660 
 17.321 
 25.981 
 
 34.641 
 
 8.661 
 
 17-323 
 25.984 
 
 34.646 
 
 8.663 
 
 '7-325 
 25.988 
 
 34-650 
 
 8.664 
 17.327 
 25.991 
 
 34.655 
 
 8.665 
 
 17.329 
 25.994 
 
 34-659 
 
 Inches, 
 
 4-334 
 4.301 
 4.269 
 4-236 
 
 4.203 
 
 4.170 
 4.136 
 4.103 
 
 4.070 
 
 4-036 
 4.003 
 3-970 
 
 3936 
 
 3.902 
 3.868 
 3-835 
 
 3.801 
 3-767 
 
 3.664 
 
 3-630 
 3-596 
 3-561 
 
 3-527 
 
 3-492 
 3-458 
 3-423 
 
 3-388 
 
 3-353 
 3-318 
 3-283 
 
 3.248 
 
 2.966 
 
 Inches. 
 
 .008 
 .008 
 .008 
 .008 
 
 .008 
 
 .008 
 .008 
 .008 
 
 .008 
 
 .008 
 .008 
 .008 
 
 .008 
 
 .008 
 .007 
 .007 
 
 .007 
 
 .007 
 .007 
 .007 
 
 .007 
 
 .007 
 .007 
 .007 
 
 .007 
 
 .007 
 .007 
 .007 
 
 .007 
 
 .007 
 .007 
 .007 
 
 .007 
 
 3-213 
 3-178 
 3-143 
 
 .007 
 .006 
 
 .006 
 
 3.108 
 
 .006 
 
 3-072* 
 
 3-037 
 
 3.002 
 
 .006 
 
 .006 
 .006 
 
 .006 
 
 Inches. 
 
 8.669 
 8.603 
 8.537 
 8.471 
 
 8.406 
 
 8-339 
 8.273 
 8.207 
 
 8.140 
 
 8.073 
 8.006 
 
 7-939 
 7.872 
 
 7.804 
 
 7-737 
 7.669 
 
 7.602 
 
 7-533 
 7-465 
 7-397 
 
 7-329 
 
 7.260 
 7-191 
 7-123 
 
 7.054 
 
 6.984 
 
 6.91 5 
 6.846 
 
 6.776 
 
 6.706 
 6.637 
 6.567 
 
 6.497 
 
 6.427 
 6.356 
 6.286 
 
 6.216 
 
 6.145 
 6.074 
 6.003 
 
 S-932 
 
 Inches, 
 
 -033 
 •032 
 .032 
 .032 
 
 .032 
 
 .032 
 •032 
 .031 
 
 .031 
 
 .031 
 .031 
 .031 
 
 .031 
 
 .030 
 .030 
 .030 
 
 .030 
 
 .029 
 .029 
 .029 
 
 .029 
 
 .028 
 .028 
 .028 
 
 .028 
 
 .028 
 .027 
 .027 
 
 .027 
 
 .027 
 .020 
 .026 
 
 .026 
 
 .026 
 .025 
 -025 
 
 .025 
 
 .025 
 .024 
 .024 
 
 .024 
 
 Inches. 
 
 13-003 
 12.904 
 12.806 
 
 12.707 
 
 12.608 
 
 12.509 
 12.410 
 12.310 
 
 12.210 
 
 12.110 
 12.009 
 11.909 
 
 11.808 
 
 11.707 
 11.605 
 11.504 
 
 11.402 
 
 11.300 
 II.I9S 
 11.096 
 
 10.993 
 
 10.890 
 10.787 
 10.684 
 
 10.581 
 10.477 
 
 10-373 
 10.269 
 
 10.165 
 
 10.060 
 
 9-955 
 9-850 
 
 9.746 
 
 ■ 9.640 
 
 9-535 
 9-429 
 
 9-323 
 
 9.217 
 9.111 
 9.005 
 
 8.899 
 
 Inches. 
 
 .074 
 .074 
 -073 
 -073 
 
 .072 
 
 .072 
 .072 
 .071 
 
 .071 
 
 .071 
 .070 
 .070 
 
 .069 
 
 .069 
 .068 
 .068 
 
 .067 
 
 .067 
 .066 
 .066 
 
 .065 
 
 .065 
 .064 
 .064 
 
 .063 
 
 ■063 
 .062 
 .062 
 
 .061 
 
 .061 
 .060 
 .060 
 
 -059 
 
 •059 
 .058 
 .058 
 
 ■0S7 
 
 •057 
 •056 
 •056 
 
 •055 
 
 Inches, 
 
 17^337 
 17.206 
 
 17.074 
 16.943 
 
 16.81 1 
 
 16.679 
 16.546 
 16.413 
 
 16.280 
 
 16.146 
 16.012 
 15.878 
 
 15-744 
 
 15.609 
 15-474 
 15-338 
 
 15.203 
 
 15.067 
 14.930 
 14-794 
 
 14.658 
 
 14.520 
 
 14.383 
 14.245 
 
 14.108 
 
 13.969 
 
 13-830 
 13.692 
 
 13-553 
 
 13-413 
 13-273 
 13-134 
 
 12.994 
 
 12.854 
 
 12.713 
 12.572 
 
 12.431 
 
 12.290 
 12.148 
 12.006 
 
 11.865 
 
 Inches, 
 
 •131 
 •131 
 ■130" 
 .129 
 
 .128 
 
 .128 
 .127 
 .126 
 
 .I2S 
 
 .125 
 .124 
 .123 
 
 .122 
 
 .122 
 .121 
 .120 
 
 .119 
 
 .119 
 .118 
 .117 
 
 .116 
 
 .lis 
 
 .114 
 •113 
 
 .111 
 .III 
 .110 
 
 .109 
 
 .108 
 .107 
 .106 
 
 .105 
 
 .104 
 .103 
 .102 
 
 .101 
 
 .100 
 .099 
 .098 
 
 .097 
 
 Smithsonian Tables. 
 
 108 
 
Table 21. 
 CO-ORDINATES FOR PROJECTION OF IVIAPS. SCALE tztW- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 
 
 is' longitude. 
 
 so' longitude. 
 
 4S' longitude. 
 
 1° longitude. 1 
 
 X 
 
 y 
 
 A 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 Inches^ 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 70°oo' 
 IS 
 3° 
 4S 
 
 8.666" 
 
 17-331 
 25.997 
 
 2.966 
 2.930 
 2.895 
 2.859 
 
 .006 
 .006 
 .006 
 .006 
 
 S-932 
 5.861 
 
 S-790 
 5.718 
 
 .024 
 .024 
 -023 
 .023 
 
 8.899 
 
 8-578 
 
 .055 
 
 •OSS 
 -054 
 -053 
 
 11.865 
 11.722 
 11.580 
 
 11-437 
 
 .097 
 .096 
 .095 
 .094 
 
 71 00 
 
 IS 
 30 
 
 45 
 
 34-663 
 
 Z.824 
 
 2.788 
 2.752 
 2.716 
 
 .006 
 
 .006 
 .006 
 .C06 
 
 5.647 
 
 S-576 
 S-S04 
 S-432 
 
 .023 
 
 .023 
 .022 
 .022 
 
 8.471 
 
 8-363 
 8.256 
 8.148 
 
 .052 
 
 .052 
 .051 
 .051 
 
 11.294 
 
 II. 151 
 11.008 
 10.864 
 
 •093 
 
 .092 
 .091 
 .090 
 
 8.667 
 
 17-333 
 26.000 
 
 7200 
 
 IS 
 
 30 
 45 
 
 34-667 
 
 2.680 
 
 2.644 
 2.608 
 2.572 
 
 .006 
 
 .006 
 .005 
 .005 
 
 5-360 
 5.288 
 
 5.2r6 
 5-144 
 
 .022 
 
 .022 
 .021 
 .021 
 
 8.040 
 
 7.932 
 7.824 
 
 7-716 
 
 .050 
 
 .050 
 
 •049 
 .049 
 
 10.720 
 
 10.576 
 10.432 
 10.288 
 
 .089 
 
 .088 
 .087 
 .086 
 
 8.668 
 
 17-335 
 26.003 
 
 7300 
 
 IS 
 30 
 45 
 
 34.670 
 
 2-536 
 
 2.500 
 2.463 
 2.427 
 
 .005 
 
 .005 
 .005 
 .005 
 
 5.072 
 
 4.999 
 4.927 
 4.854 
 
 .021 
 
 .021 
 .020 
 .020 
 
 7.608 
 
 7-499 
 7-390 
 7.281 
 
 .048 
 
 .048 
 .047 
 .046 
 
 10.144 
 
 9-998 
 9-854 
 9.708 
 
 .085 
 
 .084 
 -083 
 .081 
 
 8.668 
 26.006 
 
 7400 
 
 IS 
 30 
 45 
 
 34-674 
 
 2.391 
 
 2-354 
 2.318 
 2.281 
 
 .005 
 
 .005 
 .005 
 .005 
 
 4.782 
 
 4-636 
 4-563 
 
 .020 
 
 .020 
 .019 
 .019 
 
 7-172 
 7.063 
 6.844 
 
 -045 
 
 ■044 
 -044 
 •043 
 
 9-563 
 
 9.417 
 9.272 
 9.126 
 
 .080 
 
 ■079 
 .078 
 .077 
 
 8.669 
 
 17-339 
 26.008 
 
 7500 
 
 IS 
 30 
 45 
 
 34-677 
 
 2.245 
 
 2.208 
 2.172 
 2-135 
 
 .COS 
 
 .004 
 .004 
 .004 
 
 4.490 
 
 4-417 
 4-343 
 4.270 
 
 .019 
 
 .019 
 .018 
 .018 
 
 6-735 
 
 6.625 
 6.515 
 6.405 
 
 -043 
 
 .042 
 .042 
 .041 
 
 8.980 
 
 8.540 
 
 .076 
 
 -074 
 •073 
 .072 
 
 8.670 
 17-340 
 26.010 
 
 7600 
 
 15 
 30 
 45 
 
 34.680 
 
 2.098 
 
 2.062 
 2.025 
 1.988 
 
 .004 
 
 .004 
 .004 
 .C04 
 
 4-197 
 
 4.123 
 4.050 
 3-976 
 
 .018 
 
 .018 
 .017 
 .017 
 
 6.296 
 
 6.185 
 6.075 
 5-964 
 
 .040 
 
 .040 
 
 -039 
 .038 
 
 8-394 
 
 8.247 
 8.100 
 7.952 
 
 .071 
 .067 
 
 8.671 
 17-342 
 26.013 
 
 7700 
 
 IS 
 3° 
 45 
 
 34.684 
 
 1.951 
 
 1-914 
 1.877 
 1.840 
 
 .004 
 
 .004 
 .004 
 .004 
 
 3-903 
 3.829 
 
 .017 
 
 .017 
 .016 
 .016 
 
 5.854 
 
 5-743 
 5.632 
 
 5-522 
 
 •037 
 
 .037 
 .036 
 .036 
 
 7.805 
 
 7.658 
 7.510 
 7.362 
 
 .066 
 
 .065 
 .064 
 .063 
 
 8.672 
 
 17-343 
 26.015 
 
 7800 
 
 15 
 30 
 45 
 
 34.686 
 
 1.804 
 
 1.766 
 
 1-729 
 1.692 
 
 .004 
 
 .004 
 .004 
 .004 
 
 3.607 
 
 3-533 
 3-4S9 
 3-385 
 
 .015 
 
 .015 
 
 •015 
 .014 
 
 5.41 1 
 
 5.188 
 S-077 
 
 •035 
 
 •034 
 -034 
 •033 
 
 7-214 
 7.066 
 
 6.91 8 
 6.769 
 
 .062 
 
 .060 
 .059 
 .058 
 
 8.672 
 
 17-344 
 26.017 
 
 7900 
 
 IS 
 
 30 
 45 
 
 34.689 
 
 1.655 
 
 1.618 
 1. 581 
 
 1.544 
 
 .004 
 
 .003 
 .003 
 -003 
 
 3-310 
 
 3-236 
 3.162 
 3.087 
 
 .014 
 
 .014 
 .013 
 .013 
 
 4.966 
 
 4-854 
 4-742 
 4.631 
 
 .032 
 
 •031 
 .030 
 -030 
 
 6.621 
 
 6.472 
 6-323 
 6.174 
 
 -057 
 
 -05s 
 .054 
 
 -053 
 
 8.673 
 17-346 
 26.018 
 
 8000 
 
 34.691 
 
 1.506 
 
 .003 
 
 3013 
 
 ■013 
 
 4.519 
 
 .029 
 
 6.026 
 
 .052 
 
 Smithsonian Tables. 
 
 109 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ittW- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 ■SI 
 3 2- ■ 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 5' 
 longitude. 
 
 10' 
 
 longitude. 
 
 IS' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 
 longitude. 
 
 o°oo' 
 
 10 
 20 
 30 
 40 
 
 SO 
 I 00 
 
 10 
 20 
 30 
 40 
 
 SO 
 200 
 
 10 
 20 
 30 
 40 
 
 SO 
 
 300 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 400 
 
 10 
 20 
 30 
 40 
 
 SO 
 5 00 
 
 10 
 20 
 30 
 40 
 
 50 
 
 600 
 
 10 
 
 20 
 30 
 40 
 50 
 
 700 
 
 Inchis. 
 
 Inches. 
 5.764 
 
 S-764 
 5-764 
 5.764 
 
 5-764 
 S-764 
 
 5.764 
 
 5-763 
 S-763 
 S-762 
 5.762 
 5.761 
 
 5.761 
 
 5.760 
 
 5-759 
 S-759 
 S-758 
 5-757 
 
 S-756 
 
 5-756 
 5-754 
 S-7S3 
 5-7 52 
 5-751 
 
 5-750 
 
 5-749 
 S-748 
 S-746 
 S-745 
 5-744 
 
 5-743 
 
 5-741 
 5-739 
 S-738 
 5-736 
 5-735 
 
 S-733 
 
 S-731 
 5-729 
 5-727 
 5-726 
 5-724 
 
 5.722 
 
 Inches. 
 
 11.529 
 
 11.528 
 11.528 
 11.528 
 n.528 
 11-527 
 
 11-527 
 
 11.526 
 11.525 
 11.524 
 11.524 
 11.523 
 
 11.522 
 
 11.520 
 11-519 
 11.517 
 II. 516 
 
 11.514 
 
 11-513 
 
 II. 511 
 11.509 
 11-507 
 "•505 
 11.503 
 
 II. 501 
 
 11.498 
 11.496 
 
 11-493 
 11.490 
 11.488 
 
 11.485 
 
 11.482 
 
 11-479 
 11.476 
 11.472 
 11.469 
 
 11.466 
 
 11.462 
 11.458 
 
 11-455 
 11.451 
 
 11.447 
 11.443 
 
 Inches. 
 
 17.293 
 
 17.293 
 17.292 
 17.292 
 17.291 
 17.291 
 
 17.291 
 
 17.288 
 17.287 
 17.285 
 17.284 
 
 17.283 
 
 17.281 
 17.278 
 17.276 
 
 17-274 
 17.272 
 
 17.270 
 
 17.267 
 17.264 
 17.260 
 
 17-257 
 17.254 
 
 17.251 
 
 17.247 
 17.243 
 17.240 
 17.236 
 17-232 
 
 17.228 
 
 17-223 
 17.218 
 17.213 
 17.209 
 17.204 
 
 17.199 
 
 '7-^93 
 17.188 
 
 17.182 
 
 17.177 
 
 17-171 
 
 17.165 
 
 Ittches. 
 23.058 
 
 23.057 
 23.056 
 23.056 
 
 23-055 
 23054 
 
 23-054 
 
 23.052 
 23.050 
 23.049 
 23.047 
 23-045 
 
 23.044 
 
 23.041 
 23.038 
 
 23-035 
 23.032 
 23-029 
 
 23.026 
 
 23.022 
 23.018 
 23.014 
 23.010 
 23.006 
 
 23.002 
 
 22.996 
 22.991 
 22.986 
 22.981 
 22.976 
 
 22.970 
 
 22.964 
 22.958 
 22.951 
 22.945 
 22.938 
 
 22.932 
 
 22.924 
 22.917 
 22.910 
 22.902 
 22.894 
 
 22.887 
 
 Inches. 
 28.822 
 
 28.821 
 28.821 
 28.820 
 28.819 
 28.818 
 
 28.818 
 
 28.816 
 28.813 
 28.811 
 28.809 
 28.807 
 
 28.805 
 
 28.801 
 
 28.797 
 28.794 
 28.790 
 28.786 
 
 28.783 
 
 28.778 
 
 28.773 
 28.767 
 28.762 
 28.757 
 
 28.752 
 
 28.746 
 28.739 
 28.733 
 28.726 
 28.720 
 
 28.713 
 28.705 
 
 28.689 
 28.681 
 28.673 
 
 28.665 
 
 28.656 
 28.646 
 28.637 
 28.628 
 28.618 
 
 28.609 
 
 Inches. 
 34.586 
 
 34.585 
 34-585 
 34-583 
 34^-583 
 34.582 
 
 34-581 
 
 34-579 
 34-576 
 34-573 
 34-571 
 34.568 
 
 34-565 
 
 34.561 
 34-556 
 34-552 
 34-548 
 34-543 
 
 34-539 
 
 34-533 
 34-527 
 34-520 
 34-514 
 34-508 
 
 34-502 
 
 34-495 
 34-487 
 34-479 
 34-471 
 34.463 
 
 34-456 
 
 34.446 
 34.436 
 34.427 
 34.417 
 34.408 
 
 34.398 
 
 34.387 
 34-375 
 34-364 
 34-353 
 34-342 
 
 34-330 
 
 
 0° 
 
 1° 
 
 11-451 
 22.901 
 
 34-352 
 45.803 
 
 57-254 
 68.704 
 
 5' 
 10 
 
 15 
 
 20 
 
 25 
 30 
 
 Inches. 
 
 0.000 
 .000 
 .000 
 .000 
 .000 
 .000 
 
 Inches. 
 
 0.000 
 .000 
 .001 
 .001 
 .002 
 .003 
 
 11.451 
 22.901 
 
 34.352 
 45.803 
 
 57.254 
 68.704 
 
 
 2° 
 
 3° 
 
 n.451 
 22.902 
 
 34.353 
 45.804 
 
 57-254 
 68.705 
 
 5 
 10 
 
 15 
 20 
 
 2S 
 30 
 
 0.000 
 .001 
 
 .001 
 .002 
 .004 
 .005 
 
 0.000 
 
 .001 
 .002 
 
 .003 
 
 11.451 
 22.902 
 
 34-353 
 45.804 
 
 57.255 
 68.706 
 
 
 4° 
 
 5° 
 
 5 
 10 
 
 15 
 20 
 
 25 
 
 30 
 
 0.000 
 .001 
 .003 
 .005 
 .007 
 .011 
 
 0.000 
 .001 
 
 .006 
 
 .009 
 
 .013 
 
 11.451 
 22.903 
 
 34-354 
 45-805 
 57-256 
 
 68.708 
 
 11.452 
 22.903 
 
 57.258 
 68.710 
 
 6° 
 
 t 
 
 5 
 10 
 
 IS 
 20 
 
 25 
 30 
 
 0.000 
 .002 
 .004 
 .007 
 .oil 
 .016 
 
 0.000 
 .002 
 
 :S 
 
 .013 
 .018 
 
 11.452 
 22.904 
 
 34-356 
 45.808 
 57.260 
 
 68.712 
 
 
 Smithsonian Tables. 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE fghTS- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 "C c « *- 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 5' 
 longitude. 
 
 lO' 
 
 longitude. 
 
 IS' 
 
 longitude. 
 
 20' 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 7°00 
 
 10 
 20 
 
 3° 
 
 40 
 
 5° 
 
 800 
 
 10 
 20 
 
 30 
 40 
 
 so 
 900 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 10 
 20 
 30 
 40 
 
 SO 
 
 1300 
 
 10 
 
 20 
 
 30 
 
 40 
 
 SO 
 1400 
 
 Inches. 
 68.712 
 
 11.452 
 22.905 
 
 34-3S8 
 45.810 
 57.262 
 
 68.715 
 
 "•4S3 
 22.906 
 
 34-3S9 
 45.81Z 
 57.265 
 
 68.718 
 
 11.454 
 22.907 
 
 33-361 
 45.814 
 57.268 
 
 68.722 
 
 11.454 
 22.909 
 34-263 
 
 4S-817 
 57.272 
 
 68.726 
 
 11.455 
 22.910 
 
 34-365 
 45.820 
 
 57-275 
 68.730 
 
 11.456 
 22.91Z 
 
 34-367 
 45.823 
 57.279 
 
 68.735 
 
 11.457 
 22.913 
 34-370 
 45.827 
 57.284 
 
 68.740 
 
 Inches. 
 
 5-722 
 5-720 
 
 5-7 1 7 
 5-715 
 5-7 13 
 5.7 1 1 
 
 5.709 
 
 5.706 
 5-704 
 5.701 
 
 5.696 
 
 5.694 
 
 5.691 
 5.688 
 5.686 
 5.683 
 5.680 
 
 5-677 
 
 5-674 
 5.671 
 5.668 
 
 5.662 
 
 5.659 
 
 5.656 
 5.652 
 
 5.646 
 5.642 
 
 5-639 
 
 5.636 
 s-632 
 5.628 
 5.625 
 5.621 
 
 5.618 
 
 5.614 
 5.610 
 5.606 
 5.602 
 5-598 
 
 5-594 
 
 Inches. 
 
 11-443 
 
 "•439 
 "-43S 
 11.430 
 11.426 
 11.422 
 
 11.417 
 
 11.412 
 11.407 
 11.403 
 11.398 
 "•393 
 
 11.388 
 
 11.382 
 "•377 
 "-37 1 
 11.366 
 11.360 
 
 "-3SS 
 
 "-349 
 "-343 
 "-337 
 "■331 
 11.324 
 
 11.318 
 
 11.312 
 11.305 
 11.298 
 11.292 
 11.285 
 
 11.278 
 
 11.271 
 11.264 
 11.257 
 11.250 
 11.242 
 
 n.235 
 
 11.227 
 11.220 
 
 11.212 
 11.204 
 II. 196 
 
 II.188 
 
 Inches. 
 
 17.165 
 
 17.159 
 17.152 
 17.146 
 
 17-139 
 17.132 
 
 17.126 
 
 17.119 
 17.111 
 17.104 
 17.096 
 17.089 
 
 17.082 
 
 17-073 
 17.065 
 17.057 
 17.049 
 17.040 
 
 17.032 
 
 17.023 
 17.014 
 17.005 
 16.996 
 16.987 
 
 16.978 
 
 16.968 
 16.958 
 16.948 
 16.938 
 16.928 
 
 16.918 
 
 16.907 
 16.896 
 16.885 
 16.874 
 16.864 
 
 16.853 
 
 16.841 
 16.829 
 16.818 
 16.806 
 16.794 
 
 16.783 
 
 Inches* 
 
 22.887 
 
 22.878 
 22.869 
 22.861 
 22.852 
 22.843 
 
 22.834 
 
 22.825 
 22.815 
 22.805 
 22.795 
 22.786 
 
 22.776 
 
 22.764 
 22.754 
 22.742 
 22.732 
 22.720 
 
 22.710 
 
 22.698 
 22.685 
 22.673 
 22.661 
 22.649 
 
 22.637 
 
 22.624 
 22.610 
 22.597 
 22.584 
 22.570 
 
 22.557 
 
 22.542 
 22.528 
 22.514 
 22.499 
 22.485 
 
 22.470 
 
 22.45s 
 22.439 
 22.424 
 22.408 
 22.392 
 
 22.377 
 
 Inches. 
 28.609 
 
 28.598 
 28.587 
 28.576 
 28.565 
 28.554 
 
 28.543 
 
 28.531 
 28.519 
 28.507 
 28.494 
 28.482 
 
 28.470 
 
 28.456 
 28.442 
 28.428 
 28.415 
 28.401 
 
 28.387 
 
 28.372 
 
 28.357 
 28.342 
 28.327 
 28.311 
 
 28.296 
 
 28.280 
 28.263 
 28.246 
 28.230 
 28.213 
 
 28.196 
 
 28.178 
 28.160 
 28.142 
 28.124 
 28.106 
 
 z8.o88 
 
 28.069 
 28.049 
 28.030 
 28.010 
 27.991 
 
 27.971 
 
 Inches. 
 
 34-330 
 
 34-317 
 34-304 
 34.291 
 34.278 
 34.265 
 
 34.252 
 
 34-237 
 34.222 
 34-208 
 
 34-193 
 34-178 
 
 34-163 
 
 34-147 
 34-130 
 34-"4 
 34-097 
 34-081 
 
 34.064 
 
 34.046 
 34.028 
 34.010 
 33-992 
 33-973 
 
 33-955 
 
 33-935 
 33-915 
 33-895 
 33-875 
 33-855 
 
 33-835 
 
 33-814 
 33-792 
 33-770 
 33-749 
 33-727 
 
 33-706 
 
 33.682 
 33659 
 33-635 
 33^6i:2 
 
 33-589 
 33-565 
 
 Inches. 
 
 0.000 
 .002 
 .005 
 .008 
 .013 
 .018 
 
 0.001 
 .003 
 .006 
 .010 
 .016 
 -023 
 
 O.COI 
 
 .003 
 .007 
 .013 
 .020 
 
 .028 
 
 13° 
 
 .004 
 
 .008 
 .015 
 
 .023 
 
 -033 
 
 Inches. 
 
 O.OOI 
 .002 
 .005 
 .009 
 .014 
 .021 
 
 .003 
 .006 
 .Oil 
 .018 
 .026 
 
 0.001 
 •003 
 .008 
 .014 
 .021 
 •031 
 
 14° 
 
 0.001 
 .004 
 .009 
 .016 
 .025 
 ■035 
 
 Smithsonian Tables. 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 SCALE ^ aioff - 
 
 
 ^a- 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 
 •ST! 
 
 .2 »-oi^ 
 
 
 
 
 
 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 
 S' 
 
 10' 
 
 >5' 
 
 20' 
 
 25' 
 
 30' 
 
 
 s°- 
 
 gSuo, 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 
 iTiches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 11 
 
 
 
 
 i4°oo' 
 
 lO 
 
 20 
 
 30 
 40 
 
 50 
 
 68.740 
 
 5-594 
 5.582 
 
 5- 578 
 5-573 
 
 II.188 
 
 II.180 
 II. 172 
 1 1. 163 
 II.155 
 II.147 
 
 16.783 
 
 16.770 
 16.758 
 16.745 
 
 16.733 
 16.720 
 
 22.377 
 
 22.360 
 
 22.344 
 22.327 
 22.310 
 22.294 
 
 27.971 
 
 27.950 
 27-930 
 
 271867 
 
 33-565 
 
 33-540 
 33-515 
 33-490 
 33-465 
 33-440 
 
 B 
 ►J'" 
 
 14° 
 
 15° 
 
 
 11.458 
 22.915 
 
 34.373 
 45.830 
 57.288 
 
 
 5' 
 10 
 
 15 
 
 Inches. 
 
 O.OOl 
 .004 
 .009 
 
 Inches. 
 
 0.001 
 .004 
 .009 
 
 
 15 00 
 
 10 
 20 
 
 30 
 40 
 
 50 
 1600 
 
 10 
 20 
 
 30 
 40 
 
 50 
 
 68.746 
 
 5-569 
 
 5-565 
 5.560 
 
 5-556 
 
 5-55" 
 5-547 
 
 5.542 
 
 5.538 
 5.533 
 5.528 
 5.524 
 5-519 
 
 11-138 
 
 11.130 
 II. 121 
 II. 112 
 II.I03 
 11.094 
 
 11.085 
 
 11.076 
 11.066 
 11.057 
 11.047 
 11.038 
 
 16.708 
 
 16.694 
 16.681 
 16.667 
 16.654 
 16.641 
 
 16.628 
 
 16.613 
 16.599 
 16.585 
 
 16.556 
 
 22.277 
 
 22.259 
 22.241 
 22.223 
 22.206 
 22.188 
 
 22.170 
 
 22.151 
 22.132 
 22.113 
 22.094 
 22.075 
 
 27.846 
 
 27.824 
 27.802 
 27.779 
 27-757 
 
 27-735 
 27-713 
 
 27.689 
 27.665 
 27.642 
 27.618 
 27.594 
 
 33.415 
 
 33-389 
 33-362 
 33-335 
 33-308 
 33-282 
 
 33-255 
 
 33.227 
 33-198 
 33-170 
 33.142 
 33-"3 
 
 20 
 25 
 30 
 
 .aib 
 .025 
 -03s 
 
 .017 
 .026 
 -038 
 
 
 11.459 
 22.917 
 34-376 
 45.834 
 
 57-293 
 68.752 
 
 
 5 
 10 
 
 15 
 20 
 
 16° 
 
 ^f 
 
 
 11.460 
 22.919 
 
 45.838 
 57-298 
 
 
 0.001 
 .004 
 .010 
 .018 
 
 O.OOI 
 
 .005 
 .011 
 .019 
 
 
 17 00 
 
 10 
 
 20 
 
 30 
 40 
 
 50 
 
 68.758 
 
 5-509 
 5-504 
 5-499 
 5.494 
 5.489 
 
 11.028 
 
 II.O18 
 1 1.008 
 
 10.978 
 
 16.542 
 
 16.527 
 16.512 
 16.497 
 16.482 
 16.467 
 
 22.056 
 
 22.036 
 22.016 
 21.996 
 21.976 
 21.956 
 
 27.571 
 
 27.546 
 27.521 
 
 27-495 
 27.470 
 
 27-445 
 
 33-085 
 
 33.055 
 33-025 
 32-994 
 32-964 
 32-934 
 
 25 
 30 
 
 .028 
 .040 
 
 .029 
 .042 
 
 
 1 1. 461 
 22.921 
 34-382 
 45-843 
 57-3°4 
 
 
 
 
 
 
 1800 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 68.764 
 
 5.484 
 
 5.479 
 
 5.468 
 
 5.463 
 5.458 
 
 10.968 
 
 10.957 
 10.947 
 10.936 
 10.926 
 
 10.915 
 
 16.452 
 
 16.436 
 16.420 
 16.404 
 16.389 
 16-373 
 
 21.936 
 
 21.915 
 21.894 
 21.872 
 21.852 
 21.830 
 
 27.420 
 
 27-394 
 27-367 
 27-341 
 27-315 
 27.288 
 
 32-904 
 
 32-872 
 32.840 
 32.809 
 32-777 
 32-746 
 
 
 18° 
 
 19° 
 
 
 11.462 
 22.924 
 34-386 
 45.848 
 57-310 
 
 5 
 10 
 
 15 
 20 
 
 2S 
 
 O.OOI 
 
 .005 
 
 .Oil 
 .020 
 .031 
 
 O.OOI 
 
 .005 
 
 .012 
 .021 
 -032 
 
 
 1900 
 
 68.771 
 
 5-452 
 
 10.905 
 
 16-357 
 
 21.809 
 
 27.262 
 
 32-714 
 
 30 
 
 •044 
 
 .046 
 
 
 10 
 20 
 
 3° 
 40 
 
 50 
 2000 
 
 11.463 
 22.926 
 34-390 
 45-853 
 57-316 
 
 68.779 
 
 5-447 
 5-441 
 5-436 
 5-430 
 5-424 
 
 5.419 
 
 10.893 
 10.882 
 10.871 
 10.860 
 10.849 
 
 10.838 
 
 16.340 
 16.324 
 16.307 
 16.290 
 16.274 
 
 16.257 
 
 21.787 
 21.765 
 21.742 
 21.720 
 21.698 
 
 21.676 
 
 27.234 
 27.206 
 27.178 
 27.150 
 27.123 
 
 27.095 
 
 32.680 
 
 32-647 
 32.614 
 32.580 
 32-547 
 
 32-S13 
 
 
 
 
 
 
 20° 
 
 21° 
 
 
 5 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 0.001 
 .005 
 .012 
 .022 
 
 -034 
 .049 
 
 0.001 
 .006 
 .013 
 .022 
 
 •03s 
 .051 
 
 
 10 
 
 20 
 30 
 40 
 
 s° 
 
 11.464 
 22.929 
 34-394 
 45-858 
 57-322 
 
 5-413 
 
 5-407 
 5.401 
 
 5-396 
 5-390 
 
 10.826 
 10.814 
 10.803 
 
 10.791 
 10.779 
 
 16.239 
 16.222 
 16.204 
 16.187 
 16.169 
 
 21.652 
 21.629 
 21.605 
 21.582 
 21.558 
 
 27.065 
 27.036 
 27.007 
 26.978 
 26.948 
 
 32.478 
 32-443 
 32-408 
 32-373 
 32.338 
 
 
 21 00 
 
 68.787 
 
 5-384 
 
 10.768 
 
 16.151 
 
 21-535 
 
 26.919 
 
 32.303 
 
 
 
 
 
 
 
 Smithsonian Tables. 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vthn- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 11 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 ES OF 
 
 PED 
 
 EL. 
 
 S' 
 
 longitude. 
 
 10' 
 
 longitude. 
 
 15' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 
 longitude. 
 
 30' 
 longitude. 
 
 DEVELO 
 PARALL 
 
 21°00' 
 
 lO 
 20 
 
 30 
 40 
 
 SO 
 
 22 00 
 
 10 
 20 
 30 
 40 
 50 
 
 23 00 
 
 10 
 20 
 
 30 
 40 
 
 so 
 
 2400 
 
 10 
 20 
 30 
 40 
 50 
 
 2500 
 
 10 
 20 
 30 
 40 
 SO 
 
 2600 
 
 10 
 20 
 
 30 
 40 
 
 50 
 27 00 
 
 10 
 
 20 
 30 
 40 
 SO 
 
 2800 
 
 Inches. 
 68.787 
 
 Inches. 
 
 5-384 
 5-378 
 5-366 
 
 5-359 
 5-353 
 
 5-347 
 
 5-341 
 5-334 
 5-328 
 5.322 
 
 5-315 
 S-309 
 5.302 
 
 5.282 
 5.276 
 
 5.269 
 
 S-263 
 5.256 
 5.249 
 5.242 
 5-235 
 
 5.227 
 
 5.220 
 
 5-213 
 5.206 
 
 5-199 
 5-191 
 
 5.184 
 
 5-177 
 S-169 
 5.162 
 
 S-154 
 S-147 
 
 5.140 
 
 5-132 
 5.124 
 5.116 
 
 5-109 
 5.101 
 
 5-093 
 
 Inches. 
 10.768 
 
 10.755 
 
 10.743 
 10.731 
 10.719 
 10.707 
 
 10.694 
 
 10.682 
 10.669 
 10.656 
 10.643 
 10.631 
 
 10.618 
 
 10.604 
 10.591 
 10.578 
 10.565 
 10.551 
 
 10.538 
 
 10.526 
 10.512 
 10.498 
 10.483 
 10.469 
 
 10.455 
 
 10.441 
 10.426 
 10.412 
 10.397 
 10.383 
 
 10.369 
 
 10.354 
 
 10-339 
 10.324 
 10.309 
 10.294 
 
 10.279 
 
 10.264 
 10.248 
 10.233 
 
 10.2l8 
 10.202 
 
 10.187 
 
 Inches. 
 16.151 
 
 16-133 
 16.115 
 16.097 
 16.078 
 16.060 
 
 16.042 
 
 16.022 
 16.003 
 15.984 
 
 15-965 
 15.946 
 
 15-927 
 
 i5'.887 
 15.867 
 15.847 
 15.827 
 
 15.807 
 
 15.789 
 15.767 
 15.746 
 
 15-725 
 15.704 
 
 15.682 
 
 15.661 
 
 15-639 
 15.618 
 15.596 
 15-575 
 
 15-553 
 
 15-531 
 15-508 
 15.486 
 15-463 
 15-441 
 
 15.419 
 
 15-396 
 15-373 
 15-349 
 15.326 
 
 15-303 
 15.280 
 
 Inches. 
 
 21-535 
 
 21.511 
 21.486 
 21.462 
 21.438 
 21.413 
 
 21.389 
 
 21.363 
 21.338 
 21.312 
 21.287 
 21.261 
 
 21.236 
 
 21.209 
 21.182 
 21.156 
 21.129 
 21.102 
 
 21.076 
 
 21.052 
 21.023 
 20.995 
 20.967 
 20.938 
 
 20.910 
 
 20.881 
 20.852 
 20.824 
 
 20.795 
 20.766 
 
 20.737 
 
 20.708 
 20.678 
 20.648 
 20.618 
 20.588 
 
 20.558 
 
 20.528 
 
 20.497 
 20.466 
 
 20.435 
 20.404 
 
 20.374 
 
 Inches. 
 26.919 
 
 26.889 
 26.858 
 26.828 
 26.797 
 26.767 
 
 26.736 
 
 26.704 
 26.672 
 26.641 
 26.609 
 26.577 
 
 26.545 
 
 26.511 
 26.478 
 26.445 
 26.412 
 26.378 
 
 26.345 
 
 26.315 
 26.279 
 26.244 
 26.209 
 26.173 
 
 26.137 
 
 26.101 
 26.065 
 26.029 
 
 25-993 
 25.958 
 
 25-922 
 
 25.884 
 25.847 
 25.810 
 25.772 
 25-735 
 
 25.698 
 
 25.659 
 25.621 
 25-582 
 25-544 
 25-505 
 
 25.467 
 
 Inches. 
 
 32-303 
 
 32.266 
 32.230 
 
 32-193 
 32.156 
 32.120 
 
 32.083 
 
 32.045 
 32.006 
 31.969 
 31-930 
 31-892 
 
 31-853 
 
 31-813 
 31-774 
 31-733 
 31.694 
 
 31-654 
 31.614 
 
 31-577 
 
 31-535 
 31-493 
 31-450 
 31.408 
 
 31-365 
 
 31-322 
 31-279 
 31-235 
 31.192 
 
 31-149 
 31.106 
 
 31.061 
 31.017 
 30.972 
 30.927 
 30.882 
 
 30.838 
 
 30.791 
 
 30-745 
 30.699 
 
 30-653 
 30.607 
 
 30.560 
 
 ■B13 
 
 !i 
 
 5' 
 10 
 
 15 
 20 
 
 25 
 3° 
 
 21° 
 
 22° 
 
 11.466 
 22.932 
 
 34-,397 
 45.863 
 
 57-329 
 68.795 
 
 Inches. 
 
 0.001 
 .006 
 ■013 
 .022 
 
 ■035 
 .051 
 
 Inches. 
 
 0.001 
 .006 
 •013 
 .023 
 .036 
 .052 
 
 11. 467 
 22.934 
 34.401 
 45.868 
 57-336 
 
 68.803 
 
 
 23° 
 
 24" 
 
 11.469 
 22.937 
 34.406 
 45-874 
 57-343 
 
 68.812 
 
 S 
 10 
 
 15 
 20 
 
 25 
 30 
 
 O.OOI 
 
 .006 
 
 .014 
 .024 
 
 •038 
 .054 
 
 0.002 
 .006 
 .014 
 .025 
 
 •039 
 .056 
 
 11.470 
 22.940 
 34.410 
 45.880 
 57-350 
 
 68.821 
 
 25° 
 
 26° 
 
 5 
 10 
 
 IS 
 20 
 
 25 
 30 
 
 0.002 
 .006 
 
 .026 
 .040 
 
 .058 
 
 0.002 
 .007 
 
 .026 
 .041 
 .059 
 
 11.472 
 22.943 
 
 45.886 
 57-358 
 
 68.830 
 
 11-473 
 22.946 
 
 34-419 
 45.892 
 
 57-365 
 68.838 
 
 5 
 10 
 
 15 
 
 20 
 
 25 
 30 
 
 27" 
 
 28° 
 
 0.002 
 
 .007 
 -015 
 
 .027 
 .042 
 .061 
 
 0.002 
 .007 
 .016 
 .028 
 
 •043 
 .063 
 
 11-475 
 22.950 
 34.424 
 45.899 
 57-374 
 
 68.849 
 
 Smithsonian Tables. 
 
 "3 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE fTsinS' 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 *J3 
 
 ional dis> 
 sfrom 
 degree 
 lels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 ORDINATES OF 
 DEVELOPED 
 
 
 o 
 
 ^1 
 
 
 
 
 
 
 
 
 
 •liil 
 
 5' 
 
 10' 
 
 IS' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 'Js. 
 
 l^s^a 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 Inc^s. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 I'^ 
 
 
 
 
 28°00' 
 10 
 
 68.849 
 
 s-093 
 5.085 
 
 10.187 
 10.171 
 
 15.280 
 15.256 
 
 20.374 
 20.342 
 
 25-467 
 25-427 
 
 30.560 
 30-513 
 
 § B 
 
 28° 
 
 29" 
 
 
 U.476 
 
 
 
 
 
 
 20 
 
 22.953 
 
 s-077 
 
 10-155 
 
 15-232 
 
 20.310 
 
 25-387 
 
 30.465 
 
 
 Inches. 
 
 Inches. 
 
 
 3° 
 
 34430 
 
 5-069 
 
 10.139 
 
 15.208 
 
 20.278 
 
 25-347 
 
 30-417 
 
 5' 
 
 0.002 
 
 0.002 
 
 
 40 
 
 45.906 
 
 5.061 
 
 10.123 
 
 15.185 
 
 20.246 
 
 25.308 
 
 30-369 
 
 10 
 
 .007 
 
 .007 
 .016 
 
 
 so 
 
 S7-383 
 
 5.054 
 
 10.107 
 
 15-161 
 
 20.214 
 
 25.268 
 
 30.321 
 
 IS 
 
 > 
 .016 
 
 
 
 
 
 
 
 
 
 
 20 
 
 .028 
 
 .028 
 
 
 2900 
 10 
 
 68.859 
 
 s-046 
 5-037 
 
 10.091 
 10.075 
 
 15-137 
 15.II2 
 
 20.182 
 20.150 
 
 25.228 
 25.187 
 
 30.274 
 30.224 
 
 25 
 
 30 
 
 -043 
 .063 
 
 .044 
 .064 
 
 
 11.478 
 
 
 20 
 
 22.957 
 
 5-029 
 
 10.058 
 
 15.087 
 
 20.117 
 
 25.146 
 
 30-175 
 30.126 
 
 
 
 
 
 30 
 
 34-435 
 
 5.021 
 
 10.042 
 
 15-063 
 15.038 
 
 20.084 
 
 25.105 
 
 
 
 
 
 40 
 
 45-913 
 
 S-013 
 
 10.025 
 
 20.051 
 
 25.064 
 
 30.076 
 
 
 
 
 
 SO 
 
 3000 
 
 10 
 
 57-391 
 68.870 
 
 5.004 
 4.996 
 4.988 
 
 10.009 
 
 9-993 
 9.976 
 
 15.013 
 14.989 
 14.963 
 
 20.018 
 19.985 
 19.951 
 
 25.022 
 24.981 
 24.939 
 
 30.027 
 29.978 
 29.927 
 
 
 
 
 
 
 30° 
 
 '31° 
 
 
 11.480 
 
 
 
 
 
 
 20 
 
 22.960 
 
 4-979 
 
 9-959 
 
 14.938 
 
 '9-§i7 
 
 24.896 
 
 29.876 
 
 S 
 
 0.002 
 
 0.002 
 
 
 30 
 
 34.440 
 
 4.971 
 
 9.942 
 
 14.912 
 
 19.883 
 
 24.854 
 
 29.825 
 
 10 
 
 .007 
 
 .007 
 
 
 40 
 
 45.920 
 
 4.962 
 
 9.925 
 
 14.887 
 
 19.849 
 
 24.812 
 
 29.774 
 
 15 
 
 .016 
 
 .017 
 
 
 SO 
 
 57.400 
 
 4-954 
 
 9.908 
 
 14.862 
 
 19-815 
 
 24.769 
 
 29.723 
 
 20 
 25 
 
 .029 
 .045 
 
 .030 
 .046 
 
 
 31 00 
 10 
 
 68.880 
 
 4-945 
 4-937 
 
 9.891 
 9-873 
 
 14.836 
 14.810 
 
 19.782 
 19-747 
 
 24.727 
 24-683 
 
 29.672 
 29.620 
 
 30 
 
 .065 
 
 .067 
 
 
 11.482 
 
 
 20 
 
 22.964 
 
 4.928 
 
 9?56 
 
 14.784 
 
 19.712 
 
 24.640 
 
 29.568 
 
 
 
 
 
 30 
 
 34.446 
 
 4.919 
 
 9-838 
 
 14-758 
 
 19.677 
 
 24.596 
 
 29-515 
 
 
 
 
 
 40 
 SO 
 
 45.927 
 57.409 
 
 4.910 
 4.902 
 
 9.821 
 9.804 
 
 14-731 
 14-705 
 
 19.642 
 19.607 
 
 24.552 
 24-509 
 
 29.463 
 29.411 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 32° 
 
 33° 
 
 
 32 00 
 10 
 
 68.891 
 
 4-893 
 4.884 
 
 9.786 
 9.768 
 
 14.679 
 14.652 
 
 19-572 
 19-536 
 
 24.465 
 24.420 
 
 29-358 
 29.305 
 
 
 
 
 
 s 
 
 0.002 
 
 0.002 
 
 
 11.484 
 
 
 20 
 
 22.967 
 
 '^l^l 
 
 9.750 
 
 14.625 
 
 19.500 
 
 24-376 
 
 29.251 
 
 10 
 
 .007 
 
 .008 
 
 
 30 
 
 34-451 
 
 4.866 
 
 9-732 
 
 14-598 
 
 19.465 
 
 24-331 
 
 29.197 
 
 IS 
 
 .017 
 
 .017 
 
 
 40 
 
 45-934 
 
 4.857 
 
 9-714 
 
 14-572 
 
 19.429 
 
 24.286 
 
 29.143 
 
 20 
 
 •030 
 
 .031 
 
 
 SO 
 
 57-418 
 
 4.848 
 
 9.696 
 
 14-545 
 
 19-393 
 
 24.241 
 
 29.089 
 
 25 
 30 
 
 ■1% 
 
 .048 
 .069 
 
 
 3300 
 
 10 
 
 68.902 
 
 4-839 
 4.830 
 
 9.679 
 9.660 
 
 14.518 
 14.490 
 
 19-357 
 19.320 
 
 24.196 
 24.150 
 
 29.036 
 28.980 
 
 
 
 
 
 11.485 
 
 
 20 
 
 22.971 
 
 4.821 
 
 9.642 
 
 14.462 
 
 19.283 
 
 24.104 
 
 28.925 
 
 
 
 
 
 30 
 40 
 
 34-456 
 45.942 
 
 4.812 
 4.802 
 
 9-623 
 9.605 
 
 14-435 
 14.407 
 
 19.246 
 19.210 
 
 24.058 
 24.012 
 
 28.870 
 28.814 
 
 
 
 
 
 
 -1.0 
 
 -IfO 
 
 
 SO 
 
 3400 
 
 10 
 
 57-427 
 68.913 
 
 4-793 
 4.784 
 
 4-774 
 
 9.586 
 9-568 
 9-549 
 
 14-379 
 14-352 
 14-323 
 
 19-173 
 19.136 
 19.098 
 
 23.966 
 23.920 
 23.872 
 
 28.759 
 28.704 
 28.647 
 
 
 34 
 
 35 
 
 
 5 
 10 
 
 IS 
 20 
 
 25 
 30 
 
 0.002 
 .008 
 .017 
 .031 
 
 -049 
 .070 
 
 0.002 
 .008 
 .018 
 
 
 11.487 
 
 
 20 
 30 
 
 22.975 
 34.462 
 
 4-765 
 4-755 
 
 9-530 
 9-5" 
 
 14.295 
 14.267 
 
 19.060 
 19.022 
 
 23.825 
 23-778 
 
 28.590 
 28-533 
 
 .031 
 .049 
 .071 
 
 
 40 
 
 5° 
 
 45-949 
 57-437 
 
 4-746 
 4-737 
 
 9.492 
 9-473 
 
 14.238 
 14.210 
 
 18.946 
 
 23-730 
 23.683 
 
 28.476 
 28.420 
 
 
 3SOO 
 
 68.924 
 
 4.727 
 
 9-454 
 
 14.181 
 
 18.908 
 
 23-636 
 
 28.363 
 
 
 
 
 
 
 
 Smithsonian Tables. 
 
 114 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF IVIAPS. SCALE t^^T)- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 5' 
 
 longitude. 
 
 10' 
 
 longitude. 
 
 IS' 
 
 longitude. 
 
 20' 
 lon^tude. 
 
 25' 
 longitude. 
 
 30' 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 10 
 
 20 
 
 30 
 40 
 
 50 
 3600 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 3700 
 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 3800 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 3900 
 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 4000 
 
 10 
 20 
 
 30 
 40 
 
 so 
 
 41 00 
 
 10 
 20 
 30 
 40 
 
 SO 
 
 Inches. 
 68.924 
 
 H.489 
 22.978 
 34.468 
 4S'9S7 
 S7r446 
 
 68.93s 
 
 1 1. 491 
 22.983 
 34-474 
 4S-96S 
 S74S7 
 
 68.94S 
 
 "•493 
 22.986 
 34.480 
 
 57.466 
 
 68.959 
 
 11.495 
 22.990 
 34-485 
 45-980 
 57-475 
 
 68.970 
 
 11.497 
 22.994 
 
 34-491 
 45.988 
 
 57-485 
 68.982 
 
 11.499 
 22.998 
 34-497 
 45-996 
 57-495 
 
 68.994 
 
 II. 501 
 23.002 
 
 34-503 
 46.004 
 57-506 
 
 42 00 69.007 
 
 Inches* 
 4.727 
 
 4-717 
 4.708 
 
 4-6g8 
 4.688 
 4.679 
 
 4.669 
 
 4.659 
 4.649 
 
 4-639 
 4.629 
 4.619 
 
 4.609 
 
 4-599 
 4-589 
 4-579 
 4-568 
 4-558 
 
 4.548 
 
 4-538 
 4-527 
 4-517 
 4.506 
 
 4-496 
 4-486 
 
 4-475 
 4.464 
 
 4-454 
 4-443 
 4-433 
 
 4.422 
 
 4-4" 
 4.400 
 
 4-389 
 4-378 
 4-368 
 
 4-357 
 
 4-346 
 4-335 
 4-324 
 4-312 
 4.301 
 
 4.290 
 
 Inches. 
 
 9-454 
 
 9-435 
 9.415 
 
 9-396 
 9-377 
 9-357 
 
 9-338 
 
 9.318 
 9.298 
 9.278 
 9.258 
 9.238 
 
 9.219 
 
 9-198 
 9.178 
 9-157 
 9-137 
 9.1 17 
 
 9.096 
 
 9.076 
 9.055 
 
 9-034 
 9.013 
 8.992 
 
 8-971 
 
 8.950 
 8.929 
 8.908 
 8.886 
 8.865 
 
 8.844 
 
 8.822 
 8.800 
 8.779 
 8-757 
 8-735 
 
 8-713 
 
 8.691 
 8.669 
 8.647 
 8.625 
 8.603 
 
 Inches. 
 I4.181 
 
 14.152 
 14.123 
 14.094 
 14.065 
 14.036 
 
 14.007 
 
 13-977 
 13-947 
 13-917 
 13-887 
 13.858 
 
 13.828 
 
 13-797 
 13-767 
 13-736 
 13.706 
 
 13-675 
 
 13-645 
 
 13-613 
 13.582 
 
 13-551 
 13.520 
 13.488 
 
 13-457 
 
 13-425 
 13-393 
 13-361 
 13-330 
 13.298 
 
 13.266 
 
 13-233 
 13.201 
 13.168 
 
 13-135 
 13-103 
 
 13.070 
 
 13-037 
 13.004 
 1 2.97 1 
 12.937 
 12.904 
 
 12.871 
 
 Inches. 
 18.908 
 
 18.870 
 18.831 
 18.792 
 
 18.753 
 18.714 
 
 18.676 
 
 18.636 
 18.596 
 18.556 
 18.517 
 18.477 
 
 18-437 
 
 18.396 
 18.356 
 
 18.315 
 18.274 
 18.234 
 
 18.193 
 
 18.151 
 18.109 
 18.068 
 18.026 
 17.984 
 
 17-943 
 
 17.900 
 17.858 
 17.815 
 17-773 
 17-730 
 
 17.688 
 
 17.644 
 17.601 
 
 17-557 
 17.514 
 17.470 
 
 17.427 
 
 17-383 
 17-338 
 17.294 
 17.250 
 17.205 
 
 17. 161 
 
 23-636 
 
 23-587 
 23-539 
 23.490 
 23.442 
 23-393 
 
 23-345 
 
 23-295 
 23-245 
 23-195 
 23.146 
 23.096 
 
 23.046 
 
 22.995 
 22.944 
 22.894 
 22.843 
 22.792 
 
 22.741 
 
 22.689 
 22.637 
 22.585 
 
 22.533 
 22.481 
 
 22.429 
 
 22.375 
 22.322 
 22.269 
 22.216 
 22.163 
 
 22.110 
 
 22.055 
 22.001 
 21.947 
 21.892 
 21.838 
 
 21.784 
 
 21.728 
 21.673 
 
 21.6:8 
 21.562 
 21.507 
 
 21.451 
 
 Inches. 
 28.363 
 
 28.305 
 28.246 
 28.188 
 28.130 
 28.072 
 
 28.014 
 
 27.954 
 27.894 
 
 27-835 
 27.775 
 27.715 
 
 27.656 
 
 27.594 
 
 27-533 
 27.472 
 27.411 
 27-350 
 
 27.289 
 
 27.227 
 27.164 
 27.102 
 27.039 
 26.977 
 
 26.914 
 
 26.8151 
 26.787 
 26.723 
 26.659 
 26.595 
 
 26.532 
 
 26.466 
 26.401 
 26.336 
 26.271 
 26.206 
 
 26.140 
 
 26.074 
 26.007 
 25-941 
 25-875 
 25.808 
 
 25.742 
 
 •afe 
 
 35° 
 
 Inches. 
 
 0.002 
 .008 
 .018 
 .031 
 .049 
 .071 
 
 37° 
 
 0.002 
 .008 
 .018 
 .032 
 .050 
 -073 
 
 39° 
 
 0.002 
 .008 
 .018 
 
 -033 
 .051 
 
 -074 
 
 36° 
 
 Inches. 
 
 0.002 
 .008 
 .018 
 -032 
 •050 
 .072 
 
 41" 
 
 0-002 
 .008 
 .019 
 
 -033 
 .052 
 .075 
 
 0.002 
 .008 
 .018 
 -033 
 •051 
 -073 
 
 40° 
 
 0.002 
 .008 
 .019 
 
 ■033 
 .052 
 
 -074 
 
 42° 
 
 0-002 
 .008 
 .019 
 
 -033 
 .052 
 .075 
 
 Smithsonian Tables. 
 
 "S 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 SCALE ^Tsirs- 
 
 '■OSS 
 .2 WO " 
 
 ."i rt > H 
 
 42°00' 
 lO 
 
 20 
 
 3° 
 40 
 
 5° 
 4300 
 
 10 
 20 
 
 3° 
 40 
 
 5° 
 
 4400 
 
 10 
 
 20 
 
 3° 
 40 
 
 SO 
 
 45 00 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 4600 
 
 10 
 20 
 
 30 
 40 
 
 so 
 4700 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 48 00 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 49 00 
 
 Inches. 
 69.007 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 S' 
 longitude. 
 
 11.503 
 23.000 
 
 34-Sio 
 46.013 
 S7-SI6 
 
 69.019 
 
 11.505 
 23.010 
 
 34-5'S 
 46.020 
 
 57-525 
 69.030 
 
 11.507 
 23.014 
 34.522 
 46.029 
 S7'S36 
 
 69.043 
 
 U.509 
 23.018 
 34.528 
 46.037 
 57-546 
 
 69-055 
 
 :i.5ii 
 23.023 
 
 34-534 
 46.045 
 
 57-557 
 69.068 
 
 11-513 
 23.027 
 
 34-540 
 46.053 
 57-567 
 
 69.080 
 
 II. 516 
 23-031 
 34-546 
 46.062 
 
 57-577 
 69.093 
 
 4.290 
 
 4.279 
 4.268 
 4.256 
 4-245 
 4-234 
 
 4.222 
 
 4.2H 
 4.199 
 4.188 
 4.176 
 4.165 
 
 4-153 
 
 4.142 
 4.130 
 4. 1 18 
 4.106 
 4.095 
 
 4.083 
 
 4.071 
 4-059 
 4.047 
 
 4-035 
 4.023 
 
 4.01 1 
 
 3-999 
 3-987 
 3-975 
 3-963 
 3-951 
 
 3-938 
 
 3.926 
 3914 
 
 3.889 
 3-877 
 
 3-864 
 
 3.852 
 
 3-839 
 3.827 
 3.814 
 3.802 
 
 3-789 
 
 10' 
 
 longitude. 
 
 Inches. 
 8.581 
 
 8-558 
 
 8-535 
 8-513 
 8.490 
 8.467 
 
 8.445 
 
 8.422 
 
 8-399 
 8.376 
 
 8-353 
 8-330 
 
 8-307 
 
 8.283 
 8.260 
 8.236 
 8.213 
 8.189 
 
 8.166 
 
 8.142 
 8.1 18 
 8.094 
 8.070 
 8.046 
 
 8.023 
 
 7.998 
 7-974 
 7.950 
 
 7-925 
 7.901 
 
 7-877 
 
 7.852 
 7.827 
 7.803 
 7.778 
 7-753 
 
 7.729 
 
 7.704 
 7.679 
 
 7-653 
 7.628 
 7.603 
 
 7.578 
 
 IS' 
 
 longitude 
 
 Inches. 
 12.871 
 
 2.837 
 2.803 
 2.769 
 
 2-735 
 2.701 
 
 2.667 
 
 2-633 
 2-598 
 2.564 
 2.529 
 2.494 
 
 2.460 
 
 2.425 
 2.390 
 2-354 
 2.319 
 2.284 
 
 2.249 
 
 2.213 
 2.177 
 2.141 
 2.105 
 2.070 
 
 2.034 
 
 1-997 
 1.961 
 1.925 
 1.888 
 1.852 
 
 1.815 
 
 1.778 
 1.741 
 1.704 
 1.667 
 1.630 
 
 1-593 
 
 1-555 
 1. 518 
 1.480 
 1.442 
 1.405 
 
 11.367 
 
 20' 
 
 longitude. 
 
 Inches. 
 I7.161 
 
 I7.H6 
 17.071 
 17.025 
 16.980 
 16.935 
 
 16.890 
 
 16.844 
 16.798 
 16.751 
 16.705 
 16.659 
 
 16.613 
 
 16.566 
 16.519 
 
 16.473 
 16.426 
 
 16.379 
 16.332 
 
 16.284 
 16.236 
 
 i6.i88 
 16.141 
 16.093 
 
 16.045 
 
 15-997 
 15.948 
 15.899 
 15.851 
 15.802 
 
 15-754 
 15.704 
 
 15.606 
 15-556 
 15-507 
 
 15.457 
 
 15.407 
 15.357 
 15-307 
 15-257 
 15.206 
 
 15.156 
 
 25' 
 
 longitude. 
 
 Inches. 
 21.451 
 
 21.395 
 21.338 
 21.282 
 21.225 
 21.169 
 
 21.054 
 20.997 
 20.939 
 20.882 
 20.824 
 
 20.767 
 
 20.708 
 20.649 
 20.591 
 20.532 
 20.473 
 
 20.415 
 
 20.355 
 20.295 
 20.236 
 20.176 
 20.116 
 
 20.056 
 
 19.996 
 
 '9-935 
 19-974 
 19.813 
 
 19-753 
 19.692 
 
 19.630 
 19.569 
 
 19-507 
 19.445 
 
 19.383 
 19.322 
 
 19.259 
 19.196 
 
 19-134 
 19.071 
 19.008 
 
 18.945 
 
 30' 
 
 longitude. 
 
 Inches. 
 25.742 
 
 25.674 
 25.606 
 
 25.538 
 25.470 
 25.402 
 
 25-334 
 
 25.265 
 25.196 
 25.127 
 25.058 
 24.989 
 
 24.920 
 
 24.849 
 24.779 
 24.709 
 24.638 
 24-568 
 
 24.498 
 
 24.426 
 
 24-354 
 24.283 
 24.211 
 24.139 
 
 24.068 
 
 23-995 
 23.922 
 23.849 
 23.776 
 23-703 
 
 23.630 
 
 23.556 
 23.482 
 23.408 
 
 23-334 
 23.260 
 
 23.186 
 
 23.111 
 
 23-03S 
 22.960 
 
 22.885 
 22.810 
 
 22.734 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 42" 
 
 Inches. 
 ,002 
 ,008 
 ,019 
 
 033 
 
 ,052 
 ,075 
 
 44° 
 
 0.002 
 .008 
 .019 
 .034 
 .052 
 
 -075 
 
 46° 
 
 0.002 
 .008 
 .019 
 •034 
 -053 
 .076 
 
 48° 
 
 0.002 
 .008 
 .019 
 
 ■033 
 
 .052 
 .075 
 
 43° 
 
 Inches. 
 
 0.002 
 .008 
 .019 
 
 •033 
 .052 
 
 .075 
 
 45° 
 
 0.002 
 .008 
 .019 
 .034 
 
 47° 
 
 0.002 
 .008 
 .019 
 
 .034 
 .052 
 .075 
 
 49° 
 
 0.002 
 .008 
 .019 
 
 ■033 
 .052 
 
 •07 S 
 
 Smithsonian Tables. 
 
 116 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE 75^7. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 . 
 
 inal dis- 
 
 from 
 
 egree 
 
 Is. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 'ES OF 
 
 
 
 
 
 
 
 OKDINAl 
 
 ■S5 
 
 .2 »^J^ 
 
 Vni 
 
 
 
 
 
 
 
 DEVELOPED 
 
 ' p 
 
 S 
 
 10 
 
 15' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 : •3°' 
 
 g2 S a 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 Inches. 
 
 hiches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 
 
 ,49°oo' 
 10 
 
 69.093 
 
 3-789 
 3-776 
 
 7-578 
 7.553 
 
 11.367 
 
 11.329 
 11.291 
 
 15.156 
 
 15.105 
 15.054 
 
 18.945 
 18.882 
 
 22.734 
 
 22.658 
 22.581 
 
 
 49° 
 
 50° 
 
 II. 517 
 
 20 
 
 23-035 
 
 3-764 
 
 7.527 
 
 18.818 
 
 
 Inches. 
 
 Inches. 
 
 30 
 
 34-SS2 
 
 3-751 
 
 7.502 
 
 11.253 
 
 15.003 
 
 18.754 
 
 22.505 
 
 . 
 
 
 40 
 
 46.070 
 
 3-738 
 
 7.476 
 
 11.214 
 
 14-952 
 
 18.690 
 
 22.429 
 
 s 
 
 0.002 
 .008 
 .019 
 
 0.002 
 .008 
 .019 
 
 SO 
 
 S7-S87 
 
 3-725 
 
 7-451 
 
 11.176 
 
 14.901 
 
 18.627 
 
 22.352 
 
 10 
 
 15 
 
 5000 
 
 69.105 
 
 3-713 
 
 7-425 
 
 11.138 
 
 14.850 
 
 18.563 
 
 22.276 
 
 20 
 
 25 
 
 •033 
 -052 
 
 •033 
 .052 
 
 10 
 
 11.520 
 
 3.700 
 
 7-399 
 
 11.099 
 
 14.799 
 
 18.499 
 
 22.198 
 
 30 
 
 .075 
 
 -07S 
 
 20 
 
 23.039 
 
 3.687 
 
 7-374 
 
 11.060 
 
 14-747 
 
 18.434 
 
 22.121 
 
 
 
 
 30 
 
 3i-S5« 
 
 3-674 
 
 7-348 
 
 1 1. 021 
 
 14.695 
 
 18.369 
 
 22.043 
 
 
 
 
 40 
 
 46.078 
 
 3.661 
 
 7.322 
 
 10.983 
 
 14.644 
 
 18.305 
 
 2i!888 
 21.811 
 21.732 
 
 
 
 
 SO 
 
 51 00 
 
 10 
 
 57.598 
 69.117 
 
 3-648 
 
 3-63S 
 3.622 
 
 7.296 
 7.270 
 7.244 
 
 10.944 
 10.905 
 
 10.866 
 
 14-592 
 14.540 
 14.488 
 
 18.240 
 18.176 
 18.110 
 
 
 
 
 
 51° 
 
 52° 
 
 II. 521 
 
 
 
 
 20 
 
 23-043 
 
 3.609 
 
 7.218 
 
 10.827 
 
 14.436 
 
 18.045 
 
 21.653 
 
 s 
 
 10 
 
 0.002 
 
 0.002 
 
 30 
 
 34-564 
 
 3-596 
 
 7.191 
 
 10.787 
 
 14-383 
 
 17-979 
 
 21.574 
 
 .008 
 
 .008 
 
 40 
 
 46.086 
 
 3-583 
 
 7.165 
 
 10.748 
 
 14-330 
 
 17-913 
 
 21.496 
 
 15 
 
 20 
 
 .019 
 
 .018 
 
 50 
 
 57.607 
 
 3-S70 
 
 7-139 
 
 10.709 
 
 14.278 
 
 17.848 
 
 21.417 
 
 -033 
 
 -033 
 
 5200 
 10 
 
 69.128 
 
 3-556 
 3-543 
 
 7-113 
 7.086 
 
 10.669 
 10.629 
 
 14.226 
 14.172 
 
 17.782 
 17.716 
 
 21.338 
 21.259 
 
 25 
 30 
 
 .051 
 .074 
 
 .051 
 -073 
 
 11.523 
 
 20 
 
 23-047 
 
 3-530 
 
 7.060 
 
 10.589 
 
 14.119 
 
 17.649 
 
 21.179 
 
 
 
 
 30 
 
 34-S70 
 
 3-516 
 
 7-033 
 
 10.550 
 
 14.066 
 
 17-583 
 
 21.099 
 
 
 
 
 •40 
 
 46.094 
 57-617 
 
 3-503 
 
 7.006 
 6.980 
 
 10.510 
 
 14.013 
 13.960 
 
 17.516 
 
 21.019 
 
 
 
 
 SO 
 
 3-490 
 
 10.470 
 
 17.450 
 
 20.939 
 
 
 
 
 5300 
 10 
 
 69.140 
 
 . 3-477 
 3-463 
 
 6-953 
 6.926 
 
 10.430 
 10.389 
 
 13.906 
 13.852 
 
 17-383 
 17.316 
 
 20.860 
 20.779 
 
 
 53° 
 
 S4° 
 
 "-525 
 
 5 
 
 0.002 
 
 0.002 
 
 20 
 
 23.051 
 
 3-450 
 
 6.899 
 
 10.349 
 
 13-798 
 
 17.248 
 
 20.698 
 
 10 
 
 .008 
 
 .008 
 
 30 
 
 34-576 
 
 3-436 
 
 6.872 
 
 10.309 
 
 13-745 
 
 17.181 
 
 20.617 
 
 15 
 
 .018 
 
 .018 
 
 40 
 
 46.102 
 
 3-423 
 
 6.845 
 
 10.268 
 
 13.691 
 
 17.114 
 
 20.536 
 
 20 
 
 .032 
 
 .032 
 
 SO 
 
 57-627 
 
 3-409 
 
 6.8i8 
 
 10.228 
 
 13-637 
 
 17.046 
 
 20.455 
 
 25 
 30 
 
 .050 
 -073 
 
 -050 
 .072 
 
 S4 00 
 10 
 
 69.152 
 
 3-396 
 3-382 
 
 6.791 
 6.764 
 
 10.187 
 10.146 
 
 13-583 
 13.528 
 
 16.979 
 16.910 
 
 20.374 
 20.292 
 
 
 
 
 11.527 
 
 20 
 
 23-055 
 
 3-368 
 
 6-737 
 
 10.105 
 
 13-474 
 
 16.842 
 
 20.210 
 
 
 
 
 30 
 
 34- 582 
 
 3-3SS 
 3-341 
 
 6.709 
 6.682 
 
 10.064 
 
 13-419 
 13-364 
 
 16.774 
 
 20.128 
 
 
 
 
 40 
 
 46.109 
 
 10.023 
 
 16.706 
 
 20.047 
 
 
 - -0 
 
 56° 
 
 -SO 
 55 00 
 
 57-636 
 69.164 
 
 3-327 
 3-314 
 
 6.655 
 6.628 
 
 9.982 
 9-941 ' 
 
 13-310 
 13-255 
 
 16.637 
 16.569 
 
 19.964 
 19-883 
 
 
 SS 
 
 5 
 10 
 
 IS 
 
 20 
 
 0.002 
 
 0.002 
 
 10 
 
 11.529 
 
 3-300 
 
 6.600 
 
 ■9.900 
 
 13.200 
 
 16.500 
 
 19.800 
 
 .008 
 .018 
 
 .008 
 .018 
 
 20 
 
 23.059 
 
 3.286 
 
 6.572 
 
 9.859 
 
 13-145 
 
 16.431 
 
 19.717 
 
 .032 
 
 -049 
 .071 
 
 .031 
 
 -049 
 .070 
 
 30 
 
 34-5«8 
 
 3.272 
 
 6.545 
 
 9.817 
 
 13.089 
 
 16.362 
 
 19.634 
 
 25 
 30 
 
 40 
 
 46.117 
 
 3.258 
 
 6-517 
 
 9-776 
 
 13-034 
 
 16.293 
 
 19-551 
 
 50 
 
 57.646 
 
 3-24S 
 
 6.489 
 
 9-734 
 
 12.979 
 
 16.224 
 
 19.468 
 
 5600 
 
 69.176 
 
 3-231 
 
 6.462 
 
 9-693 
 
 12.924 
 
 16.155 
 
 19-385 
 
 
 
 
 
 Smithsonian Tables. 
 
 117 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 SCALE ^TsinTS- 
 
 o 
 
 II 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 ES OF 
 
 PED 
 
 EL. 
 
 S' 
 longitude. 
 
 10' 
 
 longitude. 
 
 '5' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 
 longitude. 
 
 DEVELC 
 PARALL 
 
 56°oo' 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 S7 00 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 5800 
 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 5900 
 
 10 
 20 
 30 
 40 
 SO 
 
 6000 
 
 10 
 20 
 30 
 40 
 SO 
 
 61 00 
 
 10 
 20 
 30 
 40 
 SO 
 
 62 00 
 
 10 
 
 20 
 30 
 40 
 SO 
 
 6300 
 
 Inches. 
 69.176 
 
 Inches. 
 3-231 
 
 3.217 
 
 3-203 
 3.189 
 
 3-175 
 3.161 
 
 3-147 
 
 3^133 
 3-"9 
 3.104 
 
 3^090 
 3.076 
 
 3.062 
 
 3.048 
 
 3^034 
 3.019 
 
 3^005 
 2.991 
 
 2.976 
 
 2.962 
 2.947 
 2-933 
 2.918 
 2.904 
 
 2.890 
 
 2.875 
 2.860 
 2.846 
 2.831 
 2.816 
 
 2.802 
 
 2.787 
 2.772 
 2.758 
 
 2.743 
 2.728 
 
 2.713 
 
 2.699 
 2.684 
 2.669 
 2.654 
 2.639 
 
 2.624 
 
 Inches. 
 6.462 
 
 %^ 
 6.378 
 6.350 
 6.322 
 
 6.294 
 
 6.266 
 
 6-237 
 6.209 
 
 6.i8i 
 6.152 
 
 6.124 
 
 6.096 
 6.067 
 6.038 
 6.010 
 5.981 
 
 5-953 
 5^924 
 
 S-779 
 5750 
 
 S72I 
 
 5.662 
 5^633 
 
 5.604 
 
 S^S74 
 5^545 
 
 "^\ 
 5.456 
 
 5.427 
 
 5-397 
 5^367 
 5-337 
 5-308 
 5.278 
 
 5.248 
 
 Inches, 
 
 9-693 
 
 9.651 
 9.609 
 9-567 
 9-525 
 9-483 
 
 9.441 
 
 9-398 
 9-356 
 9-314 
 9.271 
 9.229 
 
 9.186 
 
 9-'43 
 9.101 
 9.058 
 9.015 
 8.972 
 
 8.929 
 
 8.885 
 8.842 
 8.799 
 8755 
 8.712 
 
 8.669 
 
 8.625 
 8.581 
 8-537 
 8.493 
 8.450 
 
 8.406 
 
 8.361 
 
 8.3'7 
 8.273 
 8.229 
 8.184 
 
 8.140 
 
 8.096 
 8.051 
 8.006 
 7.961 
 7.917 
 
 7.872 
 
 Inches. 
 12.924 
 
 12.868 
 12.812 
 12.756 
 12.700 
 12.644 
 
 12.588 
 
 12.531 
 12.475 
 12.418 
 12.362 
 12.305 
 
 12.248 
 
 12.191 
 12.134 
 12.077 
 12.020 
 11.962 
 
 11.905 
 
 11.847 
 11.790 
 11.732 
 11.674 
 11.616 
 
 11.558 
 
 11.500 
 11.441 
 
 "-383 
 11.324 
 11.266 
 
 11.208 
 
 11.148 
 11.090 
 11.030 
 10.972 
 10.912 
 
 10.854 
 
 10.794 
 '0734 
 10.675 
 10.615 
 10.556 
 
 10.496 
 
 Inches. 
 
 16.155 
 
 16.085 
 16.015 
 
 '5-945 
 15.875 
 15.805 
 
 '5-735 
 
 15.664 
 '5-594 
 '5-523 
 15.452 
 15.381 
 
 '5-3" 
 
 '5-239 
 15.168 
 15.096 
 15.025 
 '4^953 
 
 14.882 
 
 14.809 
 
 14-737 
 14.665 
 14.592 
 14.520 
 
 14.448 
 
 '4-375 
 14.302 
 14.229 
 14.156 
 14.083 
 
 14.010 
 
 13.862 
 13788 
 
 '3715 
 13.641 
 
 13-567 
 
 13-493 
 13.418 
 
 13.269 
 i3^i95 
 
 13.120 
 
 Inches. 
 
 i9^385 
 
 19.301 
 19.217 
 
 I9^i34 
 19.050 
 18.966 
 
 18.882 
 
 18.797 
 18.712 
 18.627 
 18.542 
 18.457 
 
 18-373 
 
 18.287 
 18.201 
 18.115 
 18.029 
 17.944 
 
 '^-858 
 
 17:684 
 
 17.597 
 17.510 
 17.424 
 
 17-337 
 
 17.249 
 17.162 
 17-074 
 
 16.899 
 
 16.81 1 
 
 16.723 
 16.634 
 16.546 
 16.457 
 16.369 
 
 16.280 
 
 16.191 
 16.102 
 16.012 
 15-923 
 '5-833 
 
 '5-744 
 
 ■S • 
 
 56° 
 
 57° 
 
 "•S3' 
 23-063 
 
 34-594 
 46.125 
 57-656 
 
 69.188 
 
 5' 
 10 
 
 '5 
 20 
 
 25 
 30 
 
 Inches. 
 
 0.002 
 .008 
 .018 
 .031 
 
 •049 
 .070 
 
 Inches. 
 
 0.002 
 .008 
 .017 
 .031 
 .048 
 .069 
 
 "•533 
 23.066 
 
 34-599 
 
 Si 
 
 69.199 
 
 
 58° 
 
 S9° 
 
 "-S3S 
 23-070 
 34-6Q5 
 46.140 
 57-675 
 
 69.210 
 
 S 
 10 
 
 IS 
 20 
 
 25 
 
 30 
 
 0.002 
 .008 
 .017 
 •030 
 
 •047 
 .068 
 
 0.002 
 .007 
 .017 
 •030 
 .046 
 .067 
 
 "•537 
 23.074 
 34.610 
 46.147 
 57.684 
 
 69.221 
 
 
 60° 
 
 61° 
 
 "•539 
 23.077 
 34-616 
 46.154 
 57-693 
 
 69.232 
 
 5 
 10 
 
 '5 
 
 20 
 
 25 
 30 
 
 0.002 
 .007 
 .016 
 .029 
 
 .045 
 .065 
 
 0.002 
 .007 
 .016 
 .029 
 
 .064 
 
 11.540 
 23.081 
 
 46.162 
 57.702 
 
 69.242 
 
 
 62° 
 
 63° 
 
 S 
 10 
 
 15 
 20 
 
 25 
 
 3° 
 
 0.002 
 .007 
 .016 
 .028 
 
 -044 
 .063 
 
 0.002 
 .007 
 .015 
 .027 
 
 11.542 
 23.084 
 34.626 
 46.168 
 
 577 10 
 
 69253 
 
 
 Smithsonian Tables. 
 
 1x8 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^i^- 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •S3 
 
 513 
 
 h> c V H 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 5' 
 
 longitude. 
 
 lO' 
 longitude. 
 
 IS' 
 
 longitude. 
 
 20 
 longitude. 
 
 2S' 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 63°00' 
 
 lO 
 
 20 
 
 3° 
 40 
 
 5° 
 6400 
 
 10 
 20 
 
 3° 
 40 
 
 5° 
 6500 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 66 00 
 
 10 
 20 
 
 30 
 40 
 
 50 
 67 00 
 
 10 
 
 20 
 
 30 
 40 
 
 5° 
 6800 
 
 10 
 
 20 
 
 30 
 40 
 
 so 
 6900 
 
 10 
 
 20 
 30 
 40 
 SO 
 
 7000 
 
 Inches. 
 
 69-2S3 
 
 11.544 
 23.087 
 34-631 
 
 46.17s 
 57.718 
 
 69.262 
 
 11.545 
 23.091 
 34-636 
 46.182 
 57.727 
 
 69.272 
 
 11.547 
 23.094 
 34.641 
 46.188 
 S7-73S 
 
 69.282 
 
 11.548 
 23.097 
 34-646 
 46.194 
 57-742 
 
 69.291 
 
 11.550 
 23.100 
 34.650 
 46.200 
 57-750 
 
 69.300 
 
 11.552 
 23.103 
 
 34-654 
 46.206 
 57-758 
 
 69.309 
 
 "■SS3 
 23.106 
 
 34-659 
 46.212 
 
 57-764 
 
 69.317 1.977 
 
 Inches. 
 2.624 
 
 2.609 
 
 2-594 
 2.579 
 2.564 
 2.549 
 
 2-534 
 
 2.519 
 2.504 
 2.488 
 
 2-473 
 
 2.458 
 
 2-443 
 
 2.428 
 2.412 
 
 2-397 
 2.382 
 2.366 
 
 2-3SI 
 
 2-336 
 2.320 
 2.305 
 2.290 
 2.274 
 
 2.259 
 
 2.243 
 2.228 
 2.212 
 2.197 
 2.181 
 
 2.166 
 
 2.150 
 
 2.134 
 2.119 
 2.103 
 2.088 
 
 2.072 
 
 2.056 
 2.040 
 2.025 
 2.009 
 1-993 
 
 Inches. 
 
 S.248 
 
 5.218 
 5.188 
 
 s-iss 
 5.128 
 5.098 
 
 s-068 
 
 s-037 
 5.007 
 
 4-977 
 4-947 
 4.916 
 
 4.886 
 
 4-855 
 4.825 
 
 4-794 
 4.764 
 
 4-733 
 4.702 
 
 4.672 
 4.641 
 4.610 
 
 4-579 
 4.548 
 
 4.518 
 
 4.487 
 
 4-4SS 
 4.424 
 
 4-393 
 4.362 
 
 4-331 
 
 4.300 
 4.269 
 
 4-237 
 4.206 
 
 4-175 
 4.144 
 
 4.112 
 
 4.081 
 4.049 
 4.018 
 3.986 
 
 3-9SS 
 
 Inches. 
 
 7.872 
 
 7.827 
 7.782 
 
 7-737 
 7.692 
 7.647 
 
 7.602 
 
 7-556 
 7.511 
 7-465 
 7.420 
 7-374 
 
 7-329 
 
 7.283 
 7-237 
 7-191 
 7-145 
 7.100 
 
 7.054 
 
 7.007 
 6.961 
 6.915 
 6.869 
 6.823 
 
 6.776 
 
 6.730 
 6.683 
 6.637 
 6.590 
 6-543 
 
 6-497 
 
 6.450 
 6.403 
 6-356 
 
 ^■^^^ 
 6.263 
 
 6.2 1 6 
 
 6.169 
 6.121 
 6.074 
 6.027 
 5.980 
 
 S-932 
 
 Inches. 
 10.496 
 
 10.436 
 10.376 
 10.316 
 10.256 
 10.196 
 
 10.136 
 
 10.075 
 10.014 
 
 9-954 
 9-893 
 9.832 
 
 9.772 
 
 9.711 
 9.650 
 9.588 
 
 9527 
 g.466 
 
 9.405 
 
 9-343 
 9.282 
 9.220 
 9.158 
 9.097 
 
 9-035 
 
 8-973 
 8.911 
 8.849 
 8.787 
 8.724 
 
 8.662 
 
 8.600 
 8.538 
 
 8.475 
 8.412 
 8.350 
 
 8.288 
 
 8.225 
 8.162 
 8.099 
 8.036 
 7-973 
 
 7.910 
 
 Inches. 
 
 13.120 
 
 13-045 
 12.970 
 12.895 
 12.820 
 12.745 
 
 12.670 
 
 12.594 
 12.518 
 12.442 
 12.367 
 12.291 
 
 12.215 
 
 12.139 
 12.062 
 11.986 
 11.909 
 "-833 
 
 11.756 
 
 11.679 
 11.602 
 11.525 
 11.448 
 "■371 
 
 11.294 
 
 11.217 
 11.139 
 11.061 
 10.984 
 10.906 
 
 10.828 
 
 10.750 
 10.672 
 10.594 
 10.516 
 10.438 
 
 10.360 
 
 10.281 
 10.202 
 10.124 
 10.045 
 9.966 
 
 9.888 
 
 Inches. 
 
 15-744 
 
 15-654 
 15.564 
 15-473 
 15-383 
 15-293 
 
 15.203 
 
 15.112 
 15.022 
 14.930 
 14.840 
 14.749 
 
 14.658 
 
 14.566 
 14.474 
 
 14-383 
 14.291 
 14.199 
 
 14.107 
 
 14.015 
 13.922 
 13-830 
 13-738 
 13-645 
 
 '3-553 
 
 13.460 
 13.366 
 
 13-273 
 13.180 
 13.087 
 
 12.994 
 
 i2.goo 
 12.806 
 12.712 
 12.619 
 12.525 
 
 12.431 
 
 12.337 
 12.242 
 12.148 
 12.054 
 11. 959 
 
 11.865 
 
 
 
 Inches 
 
 s' 
 
 0.002 
 
 10 
 
 .007 
 
 15 
 
 .015 
 
 20 
 
 .027 
 
 25 
 
 -043 
 
 30 
 
 .061 
 
 63° 
 
 65° 
 
 5 
 10 
 
 0.002 
 .006 
 
 15 
 20 
 
 .014 
 .026 
 
 25 
 30 
 
 .040 
 .058 
 
 67° 
 
 O.OOI 
 
 .006 
 .014 
 .024 
 .038 
 -054 
 
 69° 
 
 O.OOI 
 
 .006 
 
 .013 
 
 .022 
 
 -03s 
 .051 
 
 64° 
 
 0.002 
 .007 
 .015 
 .026 
 .041 
 .060 
 
 66° 
 
 0.002 
 .006 
 .014 
 .025 
 
 •039 
 .056 
 
 68° 
 
 O.OOI 
 
 .006 
 -013 
 .023 
 .036 
 
 •053 
 
 70" 
 
 0.001 
 
 .005 
 
 .012 
 .022 
 ■034 
 ■049 
 
 Smithsonian Tables. 
 
 TI9 
 
Table 22. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vrhrr- 
 
 [Derivation of table explained on p. liii-lvi.] 
 
 .82 
 
 Meridionel dis- 
 tances from 
 even degree 
 parallels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 ES OF 
 
 PED 
 
 EL. 
 
 
 S' 
 longitude. 
 
 10' 
 
 longitude. 
 
 'S' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 longitude. 
 
 DEVELO 
 PARALL 
 
 
 7o°oo' 
 
 10 
 20 
 
 30 
 40 
 50 
 
 71 00 
 
 lO 
 20 
 
 3° 
 40 
 
 50 
 
 72 00 
 
 10 
 20 
 30 
 40 
 
 50 
 7300 
 
 10 
 20 
 
 30 
 40 
 
 50 
 
 7400 
 
 10 
 20 
 30 
 40 
 SO 
 
 7500 
 
 10 
 20 
 
 30 
 40 
 50 
 
 7600 
 
 10 
 20 
 30 
 40 
 
 50 
 7700 
 
 Inches. 
 69-317 
 
 Inches. 
 1.977 
 
 1.962 
 1.946 
 1.930 
 I.914 
 1.898 
 
 1.882 
 
 1.866 
 1.850 
 
 1.835 
 1.819 
 •1.803 
 
 1.787 
 
 1.771 
 1755 
 1.739 
 1723 
 1.707 
 
 1.691 
 
 1.674 
 1.658 
 1.642 
 1.626 
 1.610 
 
 1.594 
 
 1.578 
 1.562 
 
 1-545 
 1.529 
 
 "-513 
 1.497 
 
 1.480 
 1.464 
 1.448 
 1.432 
 1.415 
 
 1.399 
 
 1.350 
 1-334 
 1.317 
 
 1.301 
 
 Inches. 
 
 3-955 
 
 3923 
 3.892 
 3.860 
 3.828 
 3796 
 
 ■3;765 
 
 3-733 
 
 3-^5' 
 3.669 
 
 3-637 
 3.605 
 
 3-574 
 
 3542 
 3-509 
 3-477 
 3-445 
 3-413 
 
 3.381 
 
 3.349 
 3.317 
 3.284 
 3.252 
 3.220 
 
 3-188 
 
 3-155 
 3-123 
 3.091 
 
 3-058 
 3.026 
 
 2-993 
 
 2.961 
 2.928 
 2.896 
 2.863 
 2.831 
 
 2.798 
 
 2.765 
 2733 
 2.700 
 2.667 
 2.634 
 
 2.602 
 
 Inches. 
 
 5-932 
 
 5.885 
 5-837 
 5.790 
 5.742 
 5-695 
 
 5.647 
 
 5.600 
 5-552 
 5-504 
 5.456 
 5.408 
 
 5.360 
 
 5-312 
 5-264 
 5.216 
 5.168 
 5.120 
 
 5.072 
 
 5.024 
 
 4-975 
 4.927 
 
 4.830 
 
 4.782 
 
 4-636 
 
 4-587 
 4.539 
 
 4.490 
 
 4.441 
 4-392 
 4.344 
 4-295 
 4.246 
 
 4.197 
 
 4.148 
 4.099 
 4.050 
 
 4.001 
 
 3952 
 3.903 
 
 Inches. 
 7.910 
 
 7.846 
 
 7783 
 7.720 
 7.656 
 7.593 
 
 7.530 
 
 7.466 
 7.402 
 
 7-338 
 7.275 
 7.2H 
 
 7-147 
 
 7-083 
 7.019 
 
 6-955 
 6.891 
 6.826 
 
 6.762 
 
 6.698 
 6.634 
 6.569 
 6.504 
 6.440 
 
 6.376 
 
 6.311 
 6.246 
 6.181 
 6.116 
 6.052 
 
 5-987 
 
 5-922 
 5-856 
 5.792 
 
 5.661 
 
 5-596 
 
 5.530 
 5-465 
 5.400 
 
 5.204 
 
 Inches. 
 9.888 
 
 9.808 
 
 9-729 
 9.650 
 
 9-571 
 9.491 
 
 9.412 
 
 9-333 
 9253 
 9-173 
 9.094 
 9.014 
 
 8.934 
 
 8.854 
 8.774 
 S.694 
 8.614 
 8.533 
 
 8.453 
 
 8-373 
 8.292 
 8.211 
 8-131 
 8.050 
 
 7.970 
 
 7.727 
 7-645 
 7.565 
 
 7-484 
 
 7.402 
 7.321 
 7.240 
 
 7-158 
 7.077 
 
 6.995 
 
 6.913 
 6.832 
 6.750 
 6.668 
 6.586 
 
 6.505 
 
 Inches. 
 11.865 
 
 11.770 
 11.675 
 
 "■579 
 11.485 
 
 11.389 
 11.294 
 
 11.199 
 II. 103 
 11.008 
 10.912 
 10.816 
 
 10.721 
 
 10.625 
 10.528 
 10.432 
 10.336 
 10.240 
 
 10.144 
 
 10.047 
 9.950 
 9-853 
 
 9.660 
 
 9-563 
 
 9.466 
 
 9-369 
 9.272 
 
 9-175 
 9-077 
 
 8.980 
 
 8.882 
 8.785 
 8.687 
 8.590 
 8.492 
 
 8.394 
 
 8.296 
 8.198 
 8.099 
 8.002 
 7.903 
 
 7.805 
 
 11 
 ■|| 
 
 5' 
 10 
 
 15 
 20 
 
 25 
 30 
 
 5 
 10 
 
 15 
 20 
 
 25 
 30 
 
 70= 
 
 71» 
 
 
 ".554 
 23.109 
 34-663 
 46.217 
 57.772 
 
 69.326 
 
 
 Inches. 
 
 0.001 
 .005 
 .012 
 .022 
 -034 
 •049 
 
 Inches. 
 
 O.OOI 
 .005 
 .012 
 .021 
 .032 
 .047 . 
 
 
 11.556 
 23.ni 
 34.667 
 46.222 
 57.778 
 
 69.334 
 
 
 72° 
 
 73° 
 
 
 "•557 
 23.114 
 34-670 
 46.227 
 57784 
 
 69.341 
 
 
 O.OOI 
 
 .005 
 .oil 
 
 .020 
 
 .031 
 •044 
 
 O.OOI 
 
 .005 
 
 .oil 
 .019 
 .029 
 .042 
 
 
 11.558 
 23.116 
 
 34-674 
 46.232 
 57.790 
 
 69.348 
 
 
 5 
 10 
 
 15 
 20 
 
 25 
 30 
 
 5 
 10 
 
 15 
 20 
 
 25 
 30 
 
 74° 
 
 75° 
 
 
 "-559 
 23.118 
 
 34-677 
 46.236 
 
 57796 
 69-355 
 
 0.001 
 .004 
 .010 
 .018 
 .028 
 .040 
 
 0.001 
 .004 
 .009 
 .017 
 .026 
 .038 
 
 
 11.560 
 23.120 
 34-681 
 46.241 
 57.801 
 
 69.361 
 
 
 760 
 
 77° 
 
 
 O.OOI 
 
 .004 
 .009 
 .016 
 .025 
 
 .036 
 
 0.001 
 .004 
 .008 
 .015 
 .023 
 .033 
 
 
 11.561 
 23.122 
 34.683 
 46.244 
 57.806 
 
 69.367 
 
 
 
 
 Smithsonian Tables. 
 
 120 
 
Table 22. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE sTsitT!- 
 [Derivation of table explained on p. liii-lvi.J 
 
 "S 
 
 .a 
 •°B8 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALEL. 
 
 
 
 
 «-s 
 
 
 
 
 
 
 
 
 
 •1" 
 
 •is-!= 
 
 
 
 
 
 
 
 DEVELOPED 
 
 
 P 
 
 |iil 
 
 S 
 longitude. 
 
 10' 
 longitude. 
 
 15' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 longitude. 
 
 PARALLEL. 
 
 
 
 Inches* 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 i-^ 
 
 
 
 
 77°oo' 
 
 10 
 
 69.367 
 
 1.301 
 
 1.284 
 1.268 
 1.252 
 
 2.602 
 
 2.569 
 2-536 
 2-503 
 
 3-903 
 
 3-854 
 3.804 
 
 3-755 
 
 5.204 
 
 5-138 
 5.072 
 5.006 
 
 6.505 
 
 6.423 
 6.341 
 6.258 
 
 7.805 
 
 7.707 
 7.609 
 7-510 
 
 *5c Jj 
 
 77" 
 
 78° 
 
 
 11.562 
 23.124 
 34.686 
 
 
 20 
 
 30 
 
 5' 
 10 
 
 IS 
 
 Inches. 
 
 Inches. 
 
 
 40 
 
 46.248 
 
 I-23S 
 
 2.470 
 
 3.706 
 
 4.941 
 
 6.176 
 
 7.41 1 
 
 0.00 1 
 
 O.OOI 
 
 
 5° 
 
 57.810 
 
 1.219 
 
 2.438 
 
 3-656 
 
 4-875 
 
 6.094 
 
 7-313 
 
 .008 
 
 .003 
 .008 
 
 
 7800 
 10 
 
 69-373 
 
 1.202 
 1.186 
 
 2.40s 
 2.372 
 
 3-607 
 3-558 
 
 4.810 
 4-744 
 
 6.012 
 
 s-930 
 
 7.214 
 7.I15 
 
 20 
 25 
 30 
 
 -015 
 .023 
 
 •033 
 
 .014 
 .021 
 ■031 
 
 
 11.563 
 
 
 20 
 
 23.126 
 
 1. 169 
 
 2-339 
 
 3.508 
 
 4.678 
 
 5847 
 
 7.016 
 
 
 
 
 
 30 
 
 34689 
 
 I-IS3 
 
 2.306 
 
 3-459 
 
 4.612 
 
 5-765 
 
 6.918 
 
 
 
 
 
 40 
 5° 
 
 46.252 
 57-814 
 
 1-136 
 
 2.273 
 
 3.410 
 3-360 
 
 4.546 
 4.480 
 
 5-683 
 
 5.600 
 
 6.819 
 6.720 
 
 
 
 
 
 
 
 
 
 
 
 7900 
 10 
 
 69-377 
 
 1.104 
 1.087 
 
 2.207 
 2.174 
 
 3-3" 
 3.261 
 
 4.414 
 4-348 
 
 5-518 
 
 5-435 
 
 6.621 
 6.522 
 
 
 79° 
 
 80° 
 
 
 11.564 
 
 s 
 
 O.OOI 
 
 O.OOI 
 
 
 20 
 
 23.127 
 
 1.070 
 
 2.141 
 
 3.211 
 
 4.282 
 
 5-352 
 
 6.422 
 
 10 
 
 -003 
 
 .003 
 
 
 30 
 
 34.691 
 
 1.054 
 
 2.108 
 
 3.162 
 
 4.216 
 
 5.270 
 
 6-323 
 
 •5 
 
 .007 
 
 .006 
 
 
 40 
 
 4^-?SS 
 
 1-037 
 
 2.075 
 
 3.112 
 
 4.150 
 4.083 
 
 5.187 
 
 6.224 
 
 20 
 
 .013 
 
 .on 
 
 
 SO 
 
 57.818 
 
 1. 02 1 
 
 2.042 
 
 3.062 
 
 5-104 
 
 6.125 
 
 25 
 
 .020 
 
 .018 
 
 
 
 
 
 
 
 
 
 
 30 
 
 .028 
 
 .026 
 
 
 8000 
 
 69.382 
 
 1.004 
 
 2.009 
 
 3-013 
 
 4.017 
 
 5.022 
 
 6.026 
 
 
 
 
 
 
 
 
 
 Smithsonian Tables. 
 
 121 
 
Table 23. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE aooStft - 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •s . 
 
 ■=sg 
 
 
 CO-ORDINATES Of 
 
 DEVELOPED PARALLEL FOR— 
 
 iq/ loDgitude. 
 
 20' longitude. 
 
 ao' longitude 
 
 40^ longitude. 
 
 50' longitude. 
 
 1° longitude. 
 
 3- 
 
 Meridio 
 tances 
 even d 
 paralle 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 tnm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 o°oo' 
 
 
 92.8 
 
 .0 
 
 185.5 
 
 .0 
 
 278.3 
 
 .0 
 
 371.I 
 
 .0 
 
 463.8 
 
 .0 
 
 556.6 
 
 .0 
 
 lO 
 
 92.1 
 
 92.8 
 
 .0 
 
 i8S-S 
 
 .0 
 
 278.3 
 
 .0 
 
 371.I 
 
 .0 
 
 463.8 
 
 .0 
 
 556-6 
 
 .0 
 
 20 
 
 184.3 
 
 92.8 
 
 .0 
 
 185.5 
 
 .0 
 
 278.3 
 
 .0 
 
 37I.I 
 
 .0 
 
 463.8 
 
 .0 
 
 556.6 
 
 .0 
 
 3° 
 
 276.4 
 
 92.8 
 
 .0 
 
 185.5 
 
 .0 
 
 278.3 
 
 .0 
 
 371.0 
 
 .0 
 
 463.8 
 
 .0 
 
 556.6 
 
 .0 
 
 40 
 
 368.6 
 
 92.8 
 
 .0 
 
 185.S 
 
 .0 
 
 278.3 
 
 .0 
 
 371.0 
 
 .0 
 
 463.8 
 
 .0 
 
 5S6.6 
 
 .0 
 
 SO 
 
 460.7 
 
 92.8 
 
 .0 
 
 185.5 
 
 .0 
 
 278.3 
 
 .0' 
 
 371.0 
 
 .0 
 
 463-7 
 
 .0 
 
 556.5 
 
 .1 
 
 I oo 
 
 
 92.8 
 
 .0 
 
 185.5 
 
 .0 
 
 278.3 
 
 .0 
 
 371.0 
 
 .0 
 
 463-7 
 
 .1 
 
 556.5 
 
 .1 
 
 10 
 
 92.1 
 
 92.7 
 
 .0 
 
 185.5 
 
 .0 
 
 278.2 
 
 .0 
 
 371.0 
 
 .0 
 
 463-7 
 
 .1 
 
 556-4 
 
 .1 
 
 20 
 
 184.3 
 
 92.7 
 
 .0 
 
 185.5 
 
 .0 
 
 278.2 
 
 .0 
 
 37 1 -o 
 
 .0 
 
 463-7 
 
 .1 
 
 556-4 
 
 .1 
 
 3° 
 
 ^Ifi 
 
 92.7 
 
 .0 
 
 185.5 
 
 .0 
 
 278.2 
 
 .0 
 
 370.9 
 
 .0 
 
 463-7 
 
 .1 
 
 556-4 
 
 .1 
 
 40 
 
 368.6 
 
 92.7 
 
 .0 
 
 185.4 
 
 .0 
 
 278.2 
 
 .0 
 
 370.9 
 
 .0 
 
 463.6 
 
 .1 
 
 556-3 
 
 .1 
 
 SO 
 
 460.7 
 
 92.7 
 
 .0 
 
 185.4 
 
 .0 
 
 278.2 
 
 .0 
 
 370.9 
 
 .1 
 
 463.6 
 
 .1 
 
 556-3 
 
 .2 
 
 2 00 
 
 
 92.7 
 
 .0 
 
 185.4 
 
 .0 
 
 278.1 
 
 .0 
 
 370.8 
 
 .1 
 
 463.6 
 
 .1 
 
 556-3 
 
 .2 
 
 10 
 
 92.1 
 
 92.7 
 
 .0 
 
 185.4 
 
 .0 
 
 278.1 
 
 .0 
 
 370.8 
 
 .1 
 
 463-5 
 
 .1 
 
 556-2 
 
 .2 
 
 20 
 
 184.3 
 
 92.7 
 
 .0 
 
 185.4 
 
 .0 
 
 278.1 
 
 .0 
 
 370.8 
 
 .1 
 
 463-4 
 
 .1 
 
 S56-1 
 
 .2 
 
 30 
 
 ^IH 
 
 92.7 
 
 .0 
 
 185.3 
 
 .0 
 
 278.0 
 
 .0 
 
 370.7 
 
 .1 
 
 463-4 
 
 .1 
 
 556-0 
 
 .2 
 
 40 
 
 368.6 
 
 92.7 
 
 .0 
 
 185.3 
 
 .0 
 
 278.0 
 
 .0 
 
 370.6 
 
 .1 
 
 463-3 
 
 .2 
 
 SS6-o 
 
 .2 
 
 SO 
 
 460.7 
 
 92.7 
 
 .0 
 
 185-3 
 
 .0 
 
 278.0 
 
 .1 
 
 370.6 
 
 .1 
 
 463.2 
 
 .2 
 
 555-9 
 
 .2 
 
 300 
 
 
 92.6 
 
 .0 
 
 185.3 
 
 .0 
 
 277.9 
 
 .1 
 
 370.6 
 
 .1 
 
 463-2 
 
 .2 
 
 555-8 
 
 .2 
 
 10 
 
 92. 1 
 
 92.6 
 
 .0 
 
 185.2 
 
 .0 
 
 277.9 
 
 .1 
 
 370.5 
 
 .1 
 
 463.1 
 
 .2 
 
 555-7 
 
 -3 
 
 20 
 
 184.3 
 
 92.6 
 
 .0 
 
 185.2 
 
 .0 
 
 277.8 
 
 .1 
 
 370.4 
 
 .1 
 
 463.0 
 
 .2 
 
 555-7 
 
 -3 
 
 30 
 
 276.4 
 
 92.6 
 
 .0 
 
 185.2 
 
 .0 
 
 277.8 
 
 .1 
 
 370.4 
 
 .1 
 
 463-0 
 
 .2 
 
 SSS-S 
 
 •3 
 
 40 
 
 368.6 
 
 92.6 
 
 .0 
 
 185.1 
 
 .0 
 
 277.7 
 
 .1 
 
 370.3 
 
 .1 
 
 462.8 
 
 .2 
 
 5S5-4 
 
 -3 
 
 SO 
 
 460.7 
 
 92.6 
 
 .0 
 
 185.1 
 
 .0 
 
 277.7 
 
 .1 
 
 370.2 
 
 .1 
 
 462.8 
 
 .2 
 
 SSS-4 
 
 •3 
 
 400 
 
 
 92.5 
 
 .0 
 
 185.1 
 
 .0 
 
 277.6 
 
 .1 
 
 370.2 
 
 .2 
 
 462.7 
 
 .2 
 
 555-2 
 
 •3 
 
 10 
 
 1^4-3 
 
 92.S 
 
 .0 
 
 185.0 
 
 .0 
 
 277.6 
 
 .1 
 
 370.1 
 
 .2 
 
 462.6 
 
 .2 
 
 SSS-' 
 
 -3 
 
 20 
 
 92.S 
 
 .0 
 
 185.0 
 
 .0 
 
 277.5 
 
 .1 
 
 370.0 
 
 .2 
 
 462.5 
 
 .2 
 
 555-0 
 
 .3 
 
 30 
 
 'IH 
 
 92.5 
 
 .0 
 
 185.0 
 
 .0 
 
 277.4 
 
 .1 
 
 309.9 
 
 .2 
 
 462.4 
 
 .2 
 
 554-9 
 
 554-8 
 
 -3 
 
 40 
 
 368.6 
 
 92.5 
 
 .0 
 
 184.9 
 
 .0 
 
 277.4 
 
 .1 
 
 369.8 
 
 .2 
 
 462.3 
 
 ■3 
 
 -4 
 
 SO 
 
 460.7 
 
 92.4 
 
 .0 
 
 184.9 
 
 .0 
 
 277.3 
 
 .1 
 
 369.8 
 
 .2 
 
 462.2 
 
 -3 
 
 S54.6 
 
 -4 
 
 500 
 
 
 92.4 
 
 .0 
 
 184.8 
 
 .0 
 
 277-3 
 
 .1 
 
 369.7 
 
 .2 
 
 462.1 
 
 -3 
 
 554.5 
 
 •4 
 
 10 
 
 92.2 
 
 92.4 
 
 .0 
 
 184.8 
 
 .1 
 
 277.2 
 
 .1 
 
 369.6 
 
 .2 
 
 462.0 
 
 -3 
 
 554.3 
 
 -4 
 
 20 
 
 184.3 
 
 92.4 
 
 .0 
 
 184.7 
 
 .1 
 
 277.1 
 
 .1 
 
 369.5 
 
 .2 
 
 461.8 
 
 ■3 
 
 554.2 
 
 -4 
 
 30 
 
 276.4 
 
 92-3 
 
 .0 
 
 184.7 
 
 .1 
 
 277.0 
 
 .1 
 
 369.4 
 
 .2 
 
 461.7 
 
 .3 
 
 554.0 
 
 ■4 
 
 40 
 
 368.6 
 
 92-3 
 
 .0 
 
 184.6 
 
 .1 
 
 276.9 
 
 .1 
 
 369.2 
 
 .2 
 
 461.6 
 
 .3 
 
 SS3-9 
 
 .5 
 
 so 
 
 460.7 
 
 92-3 
 
 .0 
 
 184.6 
 
 .1 
 
 276.9 
 
 .1 
 
 369.2 
 
 .2 
 
 461.4 
 
 -3 
 
 553-7 
 
 -S 
 
 600 
 
 
 92-3 
 
 .0 
 
 184.5 
 
 .1 
 
 276.8 
 
 .1 
 
 369.0 
 
 .2 
 
 461.3 
 
 -4 
 
 553-6 
 
 .5 
 
 10 
 
 ■ 92.2 
 
 92.2 
 
 .0 
 
 184.5 
 
 .1 
 
 276.7 
 
 .1 
 
 368.9 
 368.8 
 
 .2 
 
 461.2 
 
 -4 
 
 553-4 
 
 -S 
 
 20 
 
 184.3 
 
 92.2 
 
 .0 
 
 184.4 
 
 .1 
 
 276.6 
 
 .1 
 
 .2 
 
 461.0 
 
 -4 
 
 553-2 
 
 .5 
 
 30 
 
 276.4 
 
 92.2 
 
 .0 
 
 184.3 
 
 .1 
 
 276.5 
 
 .1 
 
 36S.7 
 
 .2 
 
 460.8 
 
 -4 
 
 553-0 
 
 .5 
 
 40 
 
 368.6 
 
 92.1 
 
 .0 
 
 184.3 
 
 .1 
 
 276.4 
 
 .1 
 
 368.6 
 
 .2 
 
 460.7 
 
 •4 
 
 552-8 
 
 .§ 
 
 SO 
 
 460.7 
 
 92.1 
 
 .0 
 
 184.2 
 
 .1 
 
 276.3 
 
 .1 
 
 368.4 
 
 .2 
 
 460.6 
 
 -4 
 
 552-7 
 
 .6 
 
 700 
 
 
 92.1 
 
 .0 
 
 184.2 
 
 .1 
 
 276.2 
 
 .1 
 
 368.3 
 
 •3 
 
 460.4 
 
 -4 
 
 552-5 
 
 .6 
 
 10 
 
 92.2 
 
 92.0 
 
 .0 
 
 184.1 
 
 .1 
 
 276.1 
 
 .1 
 
 368.2 
 
 •3 
 
 460.2 
 
 -4 
 
 552.2 
 
 .6 
 
 20 
 
 184.3 
 
 92.0 
 
 .0 
 
 184.0 
 
 .1 
 
 276.0 
 
 .1 
 
 368.0 
 
 •3 
 
 460.0 
 
 ■4 
 
 SS2-I 
 
 .6 
 
 30 
 
 276.4 
 
 92.0 
 
 .0 
 
 184.0 
 
 .1 
 
 275.9 
 27S.8 
 
 .1 
 
 367-9 
 
 .3 
 
 459-9 
 
 -4 
 
 55'-9 
 
 .6 
 
 40 
 
 368.6 
 
 91.9 
 
 .0 
 
 ■f3-9 
 
 .1 
 
 .1 
 
 367.8 
 
 .3 
 
 459-7 
 
 -4 
 
 > 
 551-6 
 
 .6 
 
 SO 
 
 460.7 
 
 91.9 
 
 .0 
 
 183.8 
 
 .1 
 
 27S.7 
 
 .1 
 
 367.6 
 
 .3 
 
 459-5 
 
 •5 
 
 551-4 
 
 •7 
 
 800 
 
 
 91.9 
 
 .0 
 
 183.7 
 
 .1 
 
 275.6 
 
 .2 
 
 367-5 
 
 -3 
 
 4S9-4 
 
 •S 
 
 551.2 
 
 ■7 
 
 Smithsonian 
 
 Tables. 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table 23. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE udlKf 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Latitude of 
 parallel. 
 
 Meridional dis- 
 tances from 
 even degree 
 parallels. 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 
 10/ longitude. 
 
 2.tJ longitude. 
 
 30/ longitude. 
 
 40' longitude. 
 
 Sof longitude. 
 
 1° longitude. 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 ?nm. 
 
 mm. 
 
 mm. 
 
 mvu 
 
 mm. 
 
 m»t. 
 
 mm. 
 
 
 8°oo' 
 
 
 91.9 
 
 .0 
 
 1837 
 
 .1 
 
 275.6 
 
 .2 
 
 367.5 
 
 •3 
 
 459-4 
 
 •s 
 
 551-2 
 
 •7 
 
 
 10 
 
 92.2 
 
 91.8 
 
 .0 
 
 183-7 
 
 .1 
 
 27S.5 
 
 .2 
 
 367.3 
 
 •3 
 
 459-2 
 
 ■5 
 
 551.0 
 
 •7 
 
 
 20 
 
 184.3 
 
 91.8 
 
 .0 
 
 183.6 
 
 .1 
 
 275.4 
 
 .2 
 
 367.2 
 
 •3 
 
 459.0 
 
 -S 
 
 550-7 
 
 -7 
 
 
 30 
 
 276.5 
 
 91.8 
 
 .0 
 
 183.5 
 
 .1 
 
 275.2 
 
 .2 
 
 357-2 
 
 ■3 
 
 458.8 
 
 •5 
 
 S50-5 
 
 •7 
 
 
 40 
 
 368.6 
 
 91.7 
 
 .0 
 
 183.4 
 
 .1 
 
 275.1 
 
 .2 
 
 366.8 
 
 -3 
 
 458.6 
 
 ■S 
 
 550-3 
 
 -7 
 
 
 so 
 
 460.8 
 
 91.7 
 
 .0 
 
 183.3 
 
 .1 
 
 275.0 
 
 .z 
 
 366.7 
 
 •3 
 
 458.4 
 
 •5 
 
 550.0 
 
 -7 
 
 
 900 
 
 
 91.6 
 
 .0 
 
 183.3 
 
 .1 
 
 274.9 
 
 .2 
 
 366.5 
 
 •3 
 
 458.2 
 
 •S 
 
 549.8 
 
 .8 
 
 
 10 
 
 92.2 
 
 91.6 
 
 .0 
 
 183.2 
 
 .1 
 
 274.8 
 
 .2 
 
 366.4 
 
 •3 
 
 458.0 
 
 •5 
 
 549-5 
 
 .8 
 
 
 20 
 
 184.3 
 
 91-5 
 
 .0 
 
 183.1 
 
 .1 
 
 274.6 
 
 .2 
 
 3^^-" 
 
 •3 
 
 457-7 
 
 •S 
 
 549.2 
 
 .8 
 
 
 30 
 
 276.5 
 
 91.5 
 
 .0 
 
 183.0 
 
 .1 
 
 274.5 
 
 .2 
 
 366.0 
 
 •3 
 
 457.5 
 
 .5 
 
 549-2 
 
 .8 
 
 
 40 
 
 368.6 
 
 91-5 
 
 .0 
 
 182.9 
 
 .1 
 
 274.4 
 
 .2 
 
 365.8 
 
 -4 
 
 457-3 
 
 .6 
 
 548.8 
 
 .8 
 
 
 5° 
 
 460.8 
 
 91.4 
 
 .0 
 
 182.8 
 
 .1 
 
 274.2 
 
 .2 
 
 365.6 
 
 -4 
 
 457-0 
 
 .6 
 
 548.5 
 
 .8 
 
 
 1000 
 
 
 91.4 
 
 .0 
 
 182.7 
 
 .1 
 
 274.1 
 
 .2 
 
 365.5 
 
 •4 
 
 456.8 
 
 .6 
 
 548.2 
 
 .8 
 
 
 10 
 
 92.2 
 
 91-3 
 
 .0 
 
 182.6 
 
 .1 
 
 274.0 
 
 .2 
 
 365.3 
 
 •4 
 
 456.6 
 
 .6 
 
 547-9 
 
 .8 
 
 
 20 
 
 184.3 
 
 9'-3 
 
 .0 
 
 182.5 
 
 .1 
 
 273.8 
 
 .2 
 
 365.1 
 
 -4 
 
 456.4 
 
 .6 
 
 547-6 
 
 -9 
 
 
 30 
 
 276.5 
 
 91.2 
 
 .0 
 
 182.4 
 
 .1 
 
 273.7 
 
 .2 
 
 364.9 
 
 -4 
 
 456.1 
 
 .6 
 
 547-3 
 
 -9 
 
 
 40 
 
 368.7 
 
 91.2 
 
 .0 
 
 182.3 
 
 .1 
 
 273.5 
 
 .2 
 
 364.7 
 
 .4 
 
 455-9 
 
 .6 
 
 547.0 
 
 -9 
 
 
 5° 
 
 460.8 
 
 91.1 
 
 .0 
 
 182.2 
 
 .1 
 
 273.4 
 
 .2 
 
 364.5 
 
 .4 
 
 455-6 
 
 .6 
 
 546-7 
 
 -9 
 
 
 II 00 
 
 
 91. 1 
 
 .0 
 
 182.I 
 
 .1 
 
 273.2 
 
 .2 
 
 364.3 
 
 •4 
 
 455-4 
 
 .6 
 
 546.4 
 
 -9 
 
 
 10 
 
 92.2 
 
 91.0 
 
 .0 
 
 182.0 
 
 .1 
 
 273.1 
 
 .2 
 
 364.1 
 
 .4 
 
 4S5-' 
 
 .6 
 
 546.1 
 
 -9 
 
 
 20 
 
 184.3 
 
 91.0 
 
 .0 
 
 181.9 
 
 .1 
 
 272.9 
 
 .2 
 
 363.8 
 
 .4 
 
 454-8 
 
 .6 
 
 545.8 
 
 -9 
 
 
 30 
 
 276.5 
 
 90.9 
 
 .0 
 
 181.8 
 
 .1 
 
 272.7 
 
 .2 
 
 363-6 
 
 .4 
 
 454-6 
 
 -7 
 
 545-5 
 
 -9 
 
 
 40 
 
 368.7 
 
 90.9 
 
 .0 
 
 181.7 
 
 .1 
 
 272.6 
 
 .2 
 
 363-4 
 
 .4 
 
 454-3 
 
 -7 
 
 545.2 
 
 I.O 
 
 
 5° 
 
 460.8 
 
 90.8 
 
 .0 
 
 181.6 
 
 .1 
 
 272.4 
 
 .2 
 
 363-2 
 
 .4 
 
 454-0 
 
 -7 
 
 544-8 
 
 I.O 
 
 
 12 00 
 
 
 90.8 
 
 .0 
 
 181. 5 
 
 .1 
 
 272.2 
 
 .2 
 
 363-0 
 
 .4 
 
 453-8 
 
 -7 
 
 544-5 
 
 1.0 
 
 
 10 
 
 92.2 
 
 90.7 
 
 .0 
 
 181.4 
 
 .1 
 
 272.1 
 
 _2 
 
 362.8 
 
 •4 
 
 453-4 
 
 -7 
 
 544.1 
 
 I.O 
 
 
 20 
 
 184.4 
 
 90.6 
 
 .0 
 
 181.3 
 
 .1 
 
 271.9 
 
 .2 
 
 362.5 
 
 .4 
 
 453-2 
 
 -7 
 
 543.8 
 
 I.O 
 
 
 30 
 
 276.5 
 
 90.6 
 
 .0 
 
 181. 1 
 
 .1 
 
 271.7 
 
 ■3 
 
 362.3 
 
 .4 
 
 452.8 
 
 -7 
 
 543.4 
 
 I.O 
 
 
 40 
 
 368.7 
 
 90.5 
 
 .0 
 
 181.O 
 
 .1 
 
 271.6 
 
 •3 
 
 362.1 
 
 .4 
 
 452.6 
 
 -7 
 
 543-1 
 
 I.O 
 
 
 5° 
 
 460.9 
 
 90.5 
 
 .0 
 
 180.9 
 
 .1 
 
 271.4 
 
 .3 
 
 361.8 
 
 -S 
 
 452-3 
 
 •7 
 
 542.8 
 
 I.I 
 
 
 1300 
 
 
 90.4 
 
 .0 
 
 180.8 
 
 .1 
 
 271.2 
 
 ■3 
 
 361.6 
 
 -5 
 
 452.0 
 
 -7 
 
 542-4 
 
 I.I 
 
 
 10 
 
 92.2 
 
 903 
 
 .0 
 
 180.7 
 
 .1 
 
 271.0 
 
 •3 
 
 361.4 
 
 -5 
 
 451-7 
 
 -7 
 
 542.0 
 
 I.I 
 
 
 20 
 
 184.4 
 
 90-3 
 
 .0 
 
 180.6 
 
 .1 
 
 270.8 
 
 .3 
 
 361. 1 
 
 -5 
 
 451.4 
 
 .8 
 
 541-7 
 
 I.I 
 
 
 30 
 
 276.6 
 
 90.2 
 
 .0 
 
 180.4 
 
 .1 
 
 270.6 
 
 •3 
 
 360.8 
 
 -5 
 
 451.0 
 
 .8 
 
 541-3 
 
 I.I 
 
 
 40 
 
 368.8 
 
 90.2 
 
 .0 
 
 180.3 
 
 .1 
 
 270.4 
 
 •3 
 
 360.6 
 
 -5 
 
 450.8 
 
 .8 
 
 540.9 
 
 I.I 
 
 
 50 
 
 461.0 
 
 90.1 
 
 .0 
 
 180.2 
 
 .1 
 
 270.3 
 
 •3 
 
 360.4 
 
 -S 
 
 450-4 
 
 .8 
 
 540.5 
 
 I.I 
 
 
 1400 
 
 
 90.0 
 
 .0 
 
 180.I 
 
 .1 
 
 270.1 
 
 •3 
 
 360.1 
 
 •5 
 
 450.2 
 
 .8 
 
 540.2 
 
 I.I 
 
 
 10 
 
 92.2 
 
 90.0 
 
 .0 
 
 179.9 
 
 .1 
 
 269.9 
 
 •3 
 
 359.8 
 
 -5 
 
 449.8 
 
 .8 
 
 539-8 
 
 1.2 
 
 
 20 
 
 184.4 
 
 89.9 
 
 .0 
 
 179.8 
 
 .1 
 
 269.7 
 
 .3 
 
 359.6 
 
 •5 
 
 449-5 
 
 .8 
 
 539-4 
 
 1.2 
 
 
 3° 
 
 276.6 
 
 89.8 
 
 .0 
 
 179.7 
 
 .1 
 
 269.5 
 
 .3 
 
 3S9-3 
 
 •S 
 
 449.2 
 
 .8 
 
 539-0 
 
 1.2 
 
 
 40 
 
 368.8 
 
 89.8 
 
 .0 
 
 179.S 
 
 .1 
 
 269.3 
 
 ■3 
 
 359-9 
 
 -5 
 
 448.8 
 
 .8 
 
 538.6 
 
 1.2 
 
 
 SO 
 
 461.0 
 
 89.7 
 
 .0 
 
 179.4 
 
 .1 
 
 269.1 
 
 .3 
 
 358.8 
 
 •5 
 
 448.5 
 
 .8 
 
 538.2 
 
 1.2 
 
 
 1500 
 
 
 89.6 
 
 .0 
 
 179.3 
 
 .1 
 
 268.9 
 
 .3 
 
 358-S 
 
 •5 
 
 448.2 
 
 .8 
 
 537-8 
 
 1.2 
 
 
 10 
 
 92.2 
 
 89.6 
 
 .0 
 
 I79.I 
 
 .1 
 
 268.7 
 
 •3 
 
 358-2 
 
 •S 
 
 447-8 
 
 .8 
 
 537-4 
 
 1.2 
 
 
 20 
 
 184.4 
 
 89.5 
 
 .0 
 
 179.0 
 
 .1 
 
 268.5 
 
 .3 
 
 358-0 
 
 .6 
 
 447-4 
 
 .8 
 
 536-9 
 
 1.2 
 
 
 30 
 
 276.6 
 
 89.4 
 
 .0 
 
 178.8 
 
 .1 
 
 268.3 
 
 •3 
 
 357-7 
 
 .6 
 
 447.1 
 
 -9 
 
 536-5 
 
 1.2 
 
 
 40 
 
 368.8 
 
 89-3 
 
 .0 
 
 178.7 
 
 .1 
 
 268.0 
 
 .3 
 
 357-4 
 
 .6 
 
 446-7 
 
 -9 
 
 536.0 
 
 1-3 
 
 
 SO 
 
 461.0 
 
 89-3 
 
 .0 
 
 178.5 
 
 .1 
 
 267.8 
 
 ■3 
 
 357-1 
 
 .6 
 
 446-4 
 
 -9 
 
 535-6 
 
 1-3 
 
 
 1600 
 
 
 89.2 
 
 .0 
 
 178.4 
 
 .1 
 
 267.6 
 
 •3 356.8 
 
 .6 
 
 446.0 
 
 -9 
 
 535-2 1.3 
 
 
 Smithsonian Tables. 
 
 123 
 
Table 23. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^Wtmr- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 i6°oo' 
 
 10 
 
 92.2 
 
 20 
 
 184.4 
 
 3° 
 
 276.6 
 
 40 
 
 368.8 
 
 SO 
 
 461.0 
 
 17 00 
 
 
 10 
 
 92.2 
 
 20 
 
 184.4 
 
 3° 
 
 276.7 
 
 40 
 
 368.9 
 
 SO 
 
 461. 1 
 
 i8oo 
 
 
 10 
 
 92.2 
 
 20 
 
 184.5 
 
 30 
 
 276.7 
 
 40 
 
 368.9 
 
 so 
 
 461.2 
 
 1900 
 
 10 
 20 
 30 
 40 
 SO 
 
 zo 00 
 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 21 00 
 10 
 20 
 30 
 40 
 
 SO 
 
 22 00 
 10 
 20 
 
 30 
 40 
 SO 
 
 23 00 
 10 
 20 
 30 
 40 
 SO 
 
 2400 
 
 ■5 a s 
 
 .S ufOM 
 
 lass. 
 
 92.2 
 184.S 
 276.7 
 369.0 
 461.2 
 
 92.2 
 184.5 
 276.8 
 369.0 
 461.2 
 
 92-3 
 184.5 
 276.8 
 369.0 
 461.3 
 
 92-3 
 184.S 
 276.8 
 369.1 
 461.4 
 
 92.3 
 184.6 
 276.8 
 369.1 
 461.4 
 
 CO-ORDINATKS OF DEVELOPED PARALLEL FOR — 
 
 10^ longitude. 
 
 89.2 
 '89.1 
 89.0 
 89.0 
 88,9 
 
 S8.7 
 88.7 
 88.6 
 88.5 
 88.4 
 88.3 
 
 88.3 
 88.2 
 88.1 
 88.0 
 
 111 
 
 87.7 
 87.6 
 87.6 
 87.5 
 87.4 
 
 87.3 
 
 87.2 
 87.1 
 87.0 
 86.9 
 86.8 
 86.7 
 
 86.6 
 86.5 
 86.4 
 86.3 
 86.2 
 86.1 
 
 86.0 
 
 in 
 85.7 
 85.6 
 
 8S.S 
 
 85.4 
 
 85.3 
 85.2 
 85.1 
 85.0 
 84.9 
 
 84.8 
 
 20^ longitude. 
 
 178.4 
 178.2 
 178.1 
 177.9 
 177.8 
 177.6 
 
 177-S 
 177-3 
 177.2 
 177.0 
 176.8 
 176.7 
 
 176.5 
 
 176-3 
 176.2 
 176.0 
 175.8 
 175.6 
 
 I7S-S 
 I7S-3 
 175.1 
 174.9 
 174.8 
 174.6 
 
 174.4 
 174.2 
 174.0 
 173-8 
 173-7 
 I73-S 
 
 '73-3 
 1 73- 1 
 172.9 
 172.7 
 172.5 
 172-3 
 
 172.1 
 171.9 
 171.7 
 171.5 
 
 171-3 
 171. 1 
 
 170.9 
 170.7 
 170.4 
 170.Z 
 170.0 
 169.8 
 
 169.6 
 
 30/ longitude. 
 
 267.6 
 267.4 
 267.2 
 266.9 
 266.7 
 266.5 
 
 266.2 
 266.0 
 265.7 
 265.5 
 265.2 
 265.0 
 
 264.8 
 264.5 
 264.2 
 264.0 
 263.7 
 263.5 
 
 263.2 
 263.0 
 2627 
 262.4 
 262.1 
 261.9 
 
 261.6 
 261.3 
 261.0 
 260.8 
 260.5 
 260.2 
 
 2S9-9 
 259.6 
 
 259-3 
 
 258.8 
 258.4 
 
 258.2 
 257.8 
 257.6 
 
 257.2 
 
 256.6 
 
 256.3 
 256.0 
 255-7 
 255-3 
 255-0 
 254-7 
 
 254-4 
 
 40^ longitude. 
 
 356.8 
 
 356-5 
 356.2 
 
 355-9 
 355-6 
 3SS-3 
 
 355-0 
 354-6 
 354-3 
 354-0 
 3S3-6 
 353-3 
 
 3S3-0 
 352.6 
 
 352-3 
 352.0 
 
 351-6 
 
 351-3 
 
 351-0 
 350.6 
 350.2 
 349-9 
 349-5 
 349-2 
 
 348-8 
 
 348.4 
 348.0 
 
 347-7 
 347-3 
 346.9 
 
 346-6 
 346.2 
 345-8 
 345-4 
 345-0 
 344-6 
 
 344-2 
 343-8 
 343-4 
 343-0 
 342-6 
 342.2 
 
 341.8 
 341-3 
 340-9 
 340-4 
 340-0 
 339-6 
 
 339-2 
 
 so' longitude. 
 
 446.0 
 
 445-6 
 445-2 
 444-8 
 444-4 
 444-1 
 
 443-7 
 443-3 
 442.9 
 442.5 
 442.0 
 441.6 
 
 441.2 
 440.8 
 
 440.4 
 440.0 
 439-6 
 439-1 
 
 438-7 
 438-2 
 437-8 
 437-4 
 436-9 
 436-4 
 
 436.0 
 435-6 
 435-0 
 434-6 
 434-2 
 433-6 
 
 433-2 
 432-7 
 432.2 
 
 431-7 
 431.2 
 
 430-8 
 
 430-2 
 429.8 
 429.2 
 428.8 
 428.2 
 427.7 
 
 427.2 
 426.6 
 426.1 
 425-6 
 425.0 
 424.5 
 
 424.0 1.3 
 
 9 
 9 
 •9 
 9 
 9 
 ■9 
 
 ■9 
 •9 
 1.0 
 1.0 
 i.o 
 i.o 
 
 1.2 
 1.2 
 1.2 
 1.2 
 1.2 
 1.2 
 
 1° longitude. 
 
 535-2 
 S34-7 
 534-3 
 533-8 
 533-3 
 532-9 
 
 532-4 
 532.0 
 
 531-S 
 531-0 
 530-5 
 530.0 
 
 529-5 
 529.0 
 528.5 
 528.0 
 
 527-5 
 526.9 
 
 526.4 
 525-9 
 525-4 
 524.8 
 
 524-3 
 523-7 
 
 523.2 
 522.7 
 522.1 
 
 521.5 
 521.0 
 520.4 
 
 519-8 
 519.2 
 518.6 
 518.0 
 
 517-5 
 516.9 
 
 516-3 
 51 5-7 
 515.1 
 
 SI4-S 
 S13-8 
 513-2 
 
 512.6 
 512.0 
 511-3 
 510.7 
 5101 
 509.4 
 
 508.7 
 
 Smithsonian Tables. 
 
 124 
 
CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Table 23. 
 SCALE Winnr- 
 
 •s 
 
 11 
 
 dional dis- ' 
 :es from 
 1 degree 
 lUels. 
 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 lo^ longitude. 
 
 iof longitude. 
 
 30^ longitude. 
 
 40/ longitude. 
 
 so' longitude. 
 
 1° longitude. 
 
 II 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 24°0O' 
 
 
 84.8 
 
 .0 
 
 169.6 
 
 .2 
 
 254-4 
 
 
 339-2 
 
 .8 
 
 424.0 
 
 1-3 
 
 508.7 
 
 1.8 
 
 10 
 
 184.6 
 
 IH 
 
 .0 
 
 169.4 
 
 .2 
 
 254.0 
 
 
 338-7 
 
 .8 
 
 423-4 
 
 '•3 
 
 508.1 
 
 ,1.8 
 
 20 
 
 84.6 
 
 .0 
 
 169.I 
 
 .2 
 
 253-7 
 
 
 338-3 
 337-8 
 
 .8 
 
 422.8 
 
 «-3 
 
 507.4 
 
 1.8 
 
 3° 
 
 276.9 
 
 84.S 
 
 .0 
 
 168.9 
 
 .2 
 
 253-4 
 
 
 .8 
 
 422.3 
 
 1-3 
 
 506.8 
 
 1.8 
 
 40 
 
 369-2 
 
 84.4 
 
 .0 
 
 168.7 
 
 .2 
 
 253.0 
 
 
 337-4 
 
 .8 
 
 421.8 
 
 1-3 
 
 506.1 
 
 1.8 
 
 SO 
 
 461.S 
 
 84.2 
 
 .0 
 
 168.5 
 
 .2 
 
 252.7 
 
 
 337-0 
 
 .8 
 
 421.2 
 
 1-3 
 
 505.4 
 
 1.9 
 
 2500 
 
 
 84.1 
 
 .1 
 
 168.3 
 
 .2 
 
 252.4 
 
 
 336-s 
 
 .8 
 
 420.6 
 
 >-3 
 
 504-8 
 
 1.9 
 
 10 
 
 2^-? 
 
 84.0 
 
 .1 
 
 168.0 
 
 .2 
 
 252.0 
 
 
 336-0 
 
 .8 
 
 420.0 
 
 1-3 
 
 504.1 
 
 1-9 
 
 20 
 
 184.6 
 
 In 
 
 .1 
 
 167.8 
 
 .2 
 
 251.7 
 
 
 335-6 
 
 .8 
 
 419-5 
 
 1-3 
 
 503-4 
 
 1.9 
 
 30 
 
 276.9 
 
 83-8 
 
 .1 
 
 167.6 
 
 .2 
 
 251-3 
 
 
 335-1 
 
 .8 
 
 418.9 
 
 1-3 
 
 502.7 
 
 1.9 
 
 40 
 
 369.2 
 
 83-7 
 
 .1- 
 
 167.3 
 
 .2 
 
 251.0 
 
 
 334-6 
 
 .8 
 
 418.3 
 417-8 
 
 1-3 
 
 502.0 
 
 1-9 
 
 SO 
 
 461.6 
 
 83.6 
 
 .1 
 
 167.1 
 
 .z 
 
 250.6 
 
 
 334-2 
 
 .8 
 
 1-3 
 
 501.3 
 
 1-9 
 
 2600 
 
 
 83-4 
 
 .1 
 
 166.9 
 
 .2 
 
 250.3 
 
 
 333-7 
 
 -9 
 
 417.2 
 
 1-3 
 
 500.6 
 
 1-9 
 
 10 
 
 §^•3 
 
 83-3 
 
 .1 
 
 166.6 
 
 .2 
 
 249.9 
 
 
 333-2 
 
 -9 
 
 416.6 
 
 1-3 
 
 499-9 
 
 1.9 
 
 20 
 
 184.6 
 
 83.2 
 
 .1 
 
 166.4 
 
 .2 
 
 249.6 
 
 
 332-8 
 
 -9 
 
 416.0 
 
 '■3 
 
 499-1 
 
 1.9 
 
 30 
 
 277.0 
 
 83-1 
 
 .1 
 
 166.1 
 
 .2 
 
 249.2 
 
 
 332-3 
 
 -9 
 
 415-4 
 
 1-3 
 
 498.4 
 
 1.9 
 
 40 
 
 369-3 
 
 82.9 
 
 .1 
 
 165.9 
 
 .2 
 
 248.8 
 
 
 331-8 
 
 -9 
 
 414-8 
 
 1-4 
 
 497-7 
 
 2.0 
 
 SO 
 
 461.6 
 
 82.8 
 
 .1 
 
 165.7 
 
 .z 
 
 248.5 
 
 
 331-3 
 
 •9 
 
 414.2 
 
 ■1.4 
 
 497-0 
 
 2.0 
 
 27 00 
 
 
 IH 
 
 .1 
 
 165.4 
 
 .2 
 
 248.1 
 
 
 330-8 
 
 -9 
 
 413.6 
 
 1-4 
 
 496-3 
 
 2.0 
 
 10 
 
 ■■■92.3" 
 
 82.6 
 
 .1 
 
 165.2 
 
 .2 
 
 247.8 
 
 
 330-4 
 
 •9 
 
 413.0 
 
 1-4 
 
 49S-S 
 494.8 
 
 2.0 
 
 20 
 
 184-7 
 
 82.S 
 
 .1 
 
 164.9 
 
 .2 
 
 247.4 
 
 r 
 
 329-8 
 
 -9 
 
 412.3 
 
 1-4 
 
 2.0 
 
 30 
 
 277.0 
 
 82.3 
 
 .1 
 
 164.7 
 
 .2 
 
 247.0 
 
 
 329-4 
 
 -9 
 
 411.7 
 
 1-4 
 
 494.0 
 
 2.0 
 
 40 
 
 369-3 
 
 82.2 
 
 .1 
 
 164.4 
 
 .2 
 
 246.7 
 
 
 328.9 
 
 -9 
 
 411. 1 
 
 1-4 
 
 493-3 
 
 2.0 
 
 SO 
 
 461.6 
 
 82.1 
 
 .1 
 
 164.2 
 
 .2 
 
 246.3 
 
 
 328.4 
 
 •9 
 
 410.4 
 
 1.4 
 
 492.5 
 
 2.0 
 
 2800 
 
 
 82.0 
 
 .1 
 
 163.9 
 
 .2 
 
 245-9 
 
 
 327-9 
 
 •9 
 
 409.8 
 
 1.4 
 
 491.8 
 
 2.0 
 
 10 
 
 92.4 
 
 81.8 
 
 .1 
 
 163-7 
 
 .2 
 
 245.5 
 
 
 327-4 
 
 •9 
 
 409.2 
 
 1.4 
 
 491.0 
 
 2.0 
 
 20 
 
 184.7 
 
 81.7 
 
 .1 
 
 163.4 
 
 .2 
 
 24S-.I 
 
 
 326.8 
 
 .9 
 
 408.6 
 
 1.4 
 
 490-3 
 
 2.0 
 
 30 
 
 277.0 
 
 81.6 
 
 .1 
 
 163.2 
 
 .2 
 
 244.7 
 
 
 326.3 
 
 -9 
 
 407-9 
 
 1.4 
 
 481:1 
 
 2.0 
 
 40 
 
 369-4 
 
 81.5 
 
 .1 
 
 162.9 
 
 .2 
 
 244-4 
 
 
 325-8 
 
 -9 
 
 407-3 
 
 1.4 
 
 2.0 
 
 SO 
 
 461.8 
 
 81.3 
 
 .1 
 
 162.7 
 
 .2 
 
 244.0 
 
 
 325-3 
 
 ■9 
 
 406.6 
 
 1.4 
 
 488.0 
 
 2.1 
 
 2900 
 
 
 81.2 
 
 .1 
 
 162.4 
 
 .2 
 
 243.6 
 
 
 324.8 
 
 •9 
 
 406.0 
 
 1.4 
 
 487.2 
 
 2.1 
 
 10 
 
 92.4 
 
 81.1 
 
 .1 
 
 i6z.i 
 
 .2 
 
 243-2 
 
 
 324-3 
 
 -9 
 
 405-4 
 
 1.4 
 
 486.4 
 
 2.1 
 
 20 
 
 184-7 
 
 80.9 
 
 .1 
 
 1 61. 9 
 
 .2 
 
 242.8 
 
 
 323-8 
 
 •9 
 
 404.7 
 
 1.4 
 
 485-6 
 
 2.1 
 
 30 
 
 277.1 
 
 80.8 
 
 .1 
 
 161.6 
 
 .2 
 
 242.4 
 
 
 323-2 
 
 -9 
 
 404.0 
 
 1.4 
 
 484.8 
 
 2.1 
 
 40 
 
 369-4 
 
 80.7 
 
 .1 
 
 161.3 
 
 .2 
 
 242.0 
 
 
 322-7 
 
 •9 
 
 403-4 
 
 1.4 
 
 484.0 
 
 2.1 
 
 SO 
 
 461.8 
 
 80.5 
 
 .1 
 
 161. 1 
 
 .2 
 
 241.6 
 
 
 322.2 
 
 -9 
 
 402.7 
 
 i-S 
 
 483.2 
 
 2.1 
 
 3000 
 
 
 80.4 
 
 .1 
 
 160.8 
 
 .2 
 
 241.2 
 
 
 321.6 
 
 -9 
 
 402.0 
 
 1-5 
 
 482.5 
 
 2.1 
 
 10 
 
 92.4 
 
 80.3 
 
 .1 
 
 160.5 
 
 .2 
 
 240.8 
 
 
 321.1 
 
 -9 
 
 401.4 
 
 J-S 
 
 481.6 
 
 2.1 
 
 20 
 
 184.8 
 
 80.1 
 
 .1 
 
 160.3 
 
 .2 
 
 240.4 
 
 
 320.6 
 
 -9 
 
 400.7 
 
 i-S 
 
 480.8 
 
 2.1 
 
 30 
 
 277.1 
 
 80.0 
 
 .1 
 
 160.0 
 
 .2 
 
 240.0 
 
 
 320.0 
 
 -9 
 
 400.0 
 
 1-5 
 
 480.0 
 
 2.1 
 
 40 
 
 369- s 
 
 79-9 
 
 .1 
 
 IS9-7 
 
 .2 
 
 239.6 
 
 
 319-4 
 
 -9 
 
 399-3 
 
 1-5 
 
 479-2 
 
 2.1 
 
 SO 
 
 461.9 
 
 79-7 
 
 .1 
 
 IS9-S 
 
 .2 
 
 239.2 
 
 
 318.9 
 
 -9 
 
 398.6 
 
 1-5 
 
 478.4 
 
 2.1 
 
 3100 
 
 
 79.6 
 
 .1 
 
 159.2 
 
 .2 
 
 238.8 
 
 
 318.4 
 
 I.O 
 
 398.0 
 
 1-5 
 
 477.5 
 
 2.1 
 
 10 
 
 92.4 
 
 79-4 
 
 .1 
 
 158.9 
 
 .2 
 
 238.4 
 
 
 317-8 
 
 I.O 
 
 397-2 
 
 1-5 
 
 476.7 
 
 2.1 
 
 20 
 
 184.8 
 
 79-3 
 
 .1 
 
 158.6 
 
 .2 
 
 237-9 
 
 
 317-2 
 
 1.0 
 
 396-6 
 
 1-5 
 
 475.9 
 
 2.2 
 
 30 
 
 277.2 
 
 79.2 
 
 .1 
 
 158-3 
 
 .2 
 
 237-5 
 
 
 3.'6.7 
 
 I.O 
 
 395-8 
 
 1-5 
 
 475.0 
 
 2.2 
 
 40 
 
 369.6 
 
 79.0 
 
 .1 
 
 I58.r 
 
 .2 
 
 237-1 
 
 
 316.1 
 
 I.O 
 
 395-2 
 
 1-5 
 
 474.2 
 
 2.2 
 
 so 
 
 462.0 
 
 78.9 
 
 .1 
 
 157.8 
 
 .2 
 
 236.7 
 
 
 315-6 
 
 I.O 
 
 394-4 
 
 i-S 
 
 473-3 
 
 2.2 
 
 3200 
 
 
 78.8 
 
 .1 
 
 IS7-S 
 
 .2 
 
 236.2 
 
 •s 
 
 31S-0 
 
 I.O 
 
 393-8 
 
 1-5 
 
 472.5 
 
 2.2 
 
 Smithsonian Tables. 
 
 I2S 
 
Table 23. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^mW- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 ■s_^. 
 
 onal dis- 
 > from 
 iegree 
 
 els. 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 
 10' longitude. 
 
 jo' longitude. 
 
 30' longitude. 
 
 40' longitude. 
 
 50' longitude. 
 
 1° longitude. 
 
 
 3^ 
 
 Igsl 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 igs- 
 
 llsa 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 mm. 
 
 9Km. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 WW. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 
 aa'oo' 
 
 
 78.8 
 
 
 IS7-S 
 
 .2 
 
 236.2 
 
 
 3 1 5-0 
 
 I.O 
 
 393-8 
 
 i-S 
 
 472.5 
 
 2.2 
 
 
 lO 
 
 92.4 
 
 78.6 
 
 
 157.2 
 
 .2 
 
 235.8 
 
 
 314-4 
 
 I.O 
 
 393-0 
 
 i-S 
 
 471.6 
 
 2.2 
 
 
 20 
 
 184.8 
 
 78.5 
 
 
 156.9 
 
 .2 
 
 235-4 
 
 
 3'3-8 
 
 1.0 
 
 392-3 
 
 1-5 
 
 470.8 
 
 2.2 
 
 
 3° 
 
 277-2 
 
 78-3 
 
 
 156.6 
 
 .2 
 
 235.0 
 
 
 313-3 
 
 I.O 
 
 391.6 
 
 i-S 
 
 469.9 
 
 2.2 
 
 
 40 
 
 3696 
 
 78.2 
 
 
 156-3 
 
 .2 
 
 234-5 
 
 
 312.7 
 
 I.O 
 
 390.8 
 
 i-S 
 
 469.0 
 
 2.2 
 
 
 50 
 
 462.0 
 
 78.0 
 
 
 156.0 
 
 .2 
 
 234.1 
 
 
 312.1 
 
 I.O 
 
 390.1 
 
 I -5 
 
 468.1 
 
 2.2 
 
 
 3300 
 
 
 77-9 
 
 
 155.8 
 
 .2 
 
 233-6 
 
 .6 
 
 3"-5 
 
 I.O 
 
 3ig-5 
 
 i-S 
 
 467-3 
 
 2.2 
 
 
 10 
 
 92.4 
 
 77-7 
 
 
 155-5 
 
 .2 
 
 233-2 
 
 .6 
 
 310.9 
 
 I.O 
 
 388.6 
 
 1-5 
 
 466.4 
 
 2.2 
 
 
 20 
 
 184.8 
 
 77.6 
 
 .1 
 
 iSS-2 
 
 .2 
 
 232-7 
 
 .6 
 
 3i°-3 
 
 I.O 
 
 387-9 
 
 1.5 
 
 465.5 
 
 2.2 
 
 
 3° 
 
 277-3 
 
 77-4 
 
 
 154.9 
 
 .2 
 
 232.3 
 
 .6 
 
 3°9-7 
 
 I.O 
 
 387.2 
 
 1.6 
 
 464.6 
 
 2.2 
 
 
 40 
 
 369-7 
 
 77-3 
 
 
 154.6 
 
 .2 
 
 231.9 
 
 .6 
 
 309.2 
 
 I.O 
 
 386.4 
 
 1.6 
 
 463.7 
 
 2.2 
 
 
 5° 
 
 462.1 
 
 77-1 
 
 
 154-3 
 
 .£. 
 
 231.4 
 
 .6 
 
 308.6 
 
 I.O 
 
 385-7 
 
 1.6 
 
 462.8 
 
 2.2 
 
 
 3400 
 
 
 77-0 
 
 ,1 
 
 154.0 
 
 •3 
 
 231.0 
 
 .6 
 
 308.0 
 
 I.O 
 
 384-9 
 
 1.6 
 
 461.9 
 
 2-3 
 
 
 10 
 
 92.4 
 
 76.8 
 
 
 153-7 
 
 -3 
 
 230.5 
 
 .6 
 
 307-4 
 
 I.O 
 
 384-2 
 
 1.6 
 
 461.0 
 
 2-3 
 
 
 20 
 
 184.9 
 
 76.7 
 
 
 153-4 
 
 -3 
 
 230.0 
 
 .6 
 
 306.7 
 
 I.O 
 
 383-4 
 
 1.6 
 
 460.1 
 
 2.3 
 
 
 3° 
 
 277-3 
 
 76.5 
 
 
 I S3-! 
 
 •3 
 
 229.6 
 
 .6 
 
 306.1 
 
 I.O 
 
 382.6 
 
 1.6 
 
 459.2 
 
 2-3 
 
 
 40 
 
 369-7 
 
 76.4 
 
 
 152.8 
 
 -3 
 
 229.1 
 
 .6 
 
 305-5 
 
 I.O 
 
 381.9 
 
 1.6 
 
 458-3 
 
 2-3 
 
 
 SO 
 
 462.1 
 
 76.2 
 
 
 152.4 
 
 -3 
 
 228.7 
 
 .6 
 
 304-9 
 
 I.O 
 
 381. 1 
 
 1.6 
 
 457-3 
 
 2-3 
 
 
 3S00 
 
 
 76.1 
 
 
 152. 1 
 
 •3 
 
 228.2 
 
 .6 
 
 304-3 
 
 I.O 
 
 380.4 
 
 1.6 
 
 456-4 
 
 2-3 
 
 
 10 
 
 92.4 
 
 7S-9 
 
 
 1 51. 8 
 
 ■3 
 
 227.8 
 
 .6 
 
 303-7 
 
 I.O 
 
 379-6 
 
 1.6 
 
 455-5 
 
 2-3 
 
 
 20 
 
 184.9 
 
 75.8 
 
 
 151-S 
 
 -3 
 
 227.3 
 
 .6 
 
 303-0 
 
 I.O 
 
 378-8 
 
 1.6 
 
 454-6 
 
 2-3 
 
 
 30 
 
 277.4 
 
 75-6 
 
 
 151.2 
 
 -3 
 
 226.8 
 
 .6 
 
 302.4 
 
 I.O 
 
 378-0 
 
 1.6 
 
 453-6 
 
 2-3 
 
 
 40 
 
 369-8 
 
 7S-4 
 
 
 150.9 
 
 -3 
 
 226.4 
 
 .6 
 
 301.8 
 
 I.O 
 
 377-2 
 
 1.6 
 
 452-7 
 
 2-3 
 
 
 SO 
 
 462.Z 
 
 7S-3 
 
 
 150.6 
 
 -3 
 
 225.9 
 
 .6 
 
 301.2 
 
 I.O 
 
 376-S 
 
 1.6 
 
 451.8 
 
 2-3 
 
 
 3600 
 
 
 7S-I 
 
 
 150.3 
 
 -3 
 
 225.4 
 
 .6 
 
 300.6 
 
 I.O 
 
 375-7 
 
 1.6 
 
 450.8 
 
 2-3 
 
 
 10 
 
 92-S 
 
 7S-0 
 
 .1 
 
 150.0 
 
 -3 
 
 224.9 
 
 .6 
 
 299.9 
 
 I.O 
 
 374-9 
 
 1.6 
 
 449-9 
 
 2-3 
 
 
 20 
 
 184.9 
 
 74-8 
 
 
 149.6 
 
 -3 
 
 224.5 
 
 .6 
 
 299-3 
 
 I.O 
 
 374-1 
 
 1.6 
 
 448.9 
 
 2-3 
 
 
 30 
 
 277.4 
 
 74-7 
 
 
 149-3 
 
 •3 
 
 224.0 
 
 .6 
 
 298.6 
 
 I.O 
 
 373-3 
 
 1.6 
 
 448.0 
 
 2-3 
 
 
 40 
 
 369-8 
 
 74- S 
 
 
 149.0 
 
 -3 
 
 223.5 
 
 .6 
 
 298.0 
 
 I.O 
 
 372-5 
 
 1.6 
 
 447.0 
 
 2-3 
 
 
 SO 
 
 462.3 
 
 74-3 
 
 
 148.7 
 
 •3 
 
 223.0 
 
 .6 
 
 297-4 
 
 I.O 
 
 371-7 
 
 1.6 
 
 446.0 
 
 2-3 
 
 
 3700 
 
 
 74.2 
 
 
 148.4 
 
 •3 
 
 222.5 
 
 .6 
 
 296.7 
 
 I.O 
 
 370.9 
 
 1.6 
 
 445.1 
 
 2-3 
 
 
 10 
 
 92-S 
 
 74.0 
 
 .1 
 
 148.0 
 
 -3 
 
 222.1 
 
 .6 
 
 296.1 
 
 I.O 
 
 370.1 
 
 1.6 
 
 444.1 
 
 2-3 
 
 
 20 
 
 185.0 
 
 73-8 
 
 
 147-7 
 
 -3 
 
 221.6 
 
 .6 
 
 295.4 
 
 I.O 
 
 369-2 
 
 1.6 
 
 443-1 
 
 2-3 
 
 
 30 
 
 277.4 
 
 73-7 
 
 
 147.4 
 
 -3 
 
 221. 1 
 
 .6 
 
 294.8 
 
 I.O 
 
 368-4 
 
 1.6 
 
 442.1 
 
 2-3 
 
 
 40 
 
 3699 
 
 73-S 
 
 -^ 
 
 147-1 
 
 -3 
 
 220.6 
 
 .6 
 
 294.1 
 
 I.O 
 
 367.6 
 
 1.6 
 
 441.2 
 
 2.4 
 
 
 SO 
 
 462.4 
 
 73-4 
 
 
 146.7 
 
 -3 
 
 220.1 
 
 .6 
 
 293-4 
 
 I.O 
 
 366.8 
 
 1.6 
 
 440.2 
 
 2.4 
 
 
 3800 
 
 
 73-2 
 
 
 146.4 
 
 ■3 
 
 219.6 
 
 .6 
 
 292.8 
 
 I.O 
 
 366.0 
 
 1.6 
 
 439-2 
 
 2.4 
 
 
 10 
 
 92. s 
 
 73-0 
 
 
 146. 1 
 
 ■3 
 
 219.1 
 
 .6 
 
 292.1 
 
 I.O 
 
 365-1 
 
 1.6 
 
 438.2 
 
 2.4 
 
 
 20 
 
 185.0 
 
 72.9 
 
 
 I4S-7 
 
 -3 
 
 218.6 
 
 .6 
 
 291.4 
 
 I.I 
 
 364-3 
 
 1.6 
 
 437-2 
 
 2.4 
 
 
 3° 
 
 277.5 
 
 72.7 
 
 
 145.4 
 
 ■3 
 
 218.1 
 
 .6 
 
 290.8 
 
 I.I 
 
 363-5 
 
 1.6 
 
 436.2 
 
 2.4 
 
 
 40 
 
 370.0 
 
 72.5 
 
 
 145.1 
 
 •3 
 
 217.6 
 
 .6 
 
 290.1 
 
 I.I 
 
 362.6 
 
 1.6 
 
 435-2 
 
 2.4 
 
 
 SO 
 
 462.5 
 
 72.4 
 
 
 144.7 
 
 -3 
 
 217.1 
 
 .6 
 
 289.4 
 
 I.I 
 
 361.8 
 
 1.6 
 
 434.2 
 
 2.4 
 
 
 3900 
 
 
 72.2 
 
 
 144.4 
 
 -3 
 
 216.6 
 
 .6 
 
 288.8 
 
 I.I 
 
 361.0 
 
 1-7 
 
 433-1 
 
 2.4 
 
 
 10 
 
 92-S 
 
 72.0 
 
 
 144.0 
 
 -3 
 
 216.1 
 
 .6 
 
 288.1 
 
 I.I 
 
 360.1 
 
 1-7 
 
 432-1 
 
 2.4 
 
 
 20 
 
 185.0 
 
 71.8 
 
 
 143-7 
 
 -3 
 
 215.6 
 
 .6 
 
 287.4 
 
 I.I 
 
 359-2 
 
 1-7 
 
 431-1 
 
 2.4 
 
 
 30 
 
 277.5 
 
 71.7 
 
 
 143-4 
 
 -3 
 
 215.0 
 
 .6 
 
 286.7 
 
 I.I 
 
 358-4 
 
 1-7 
 
 430.1 
 
 2.4 
 
 
 40 
 
 370.0 
 
 71-5 
 
 
 143.0 
 
 -3 
 
 214.5 
 
 .6 
 
 286.0 
 
 I.I 
 
 357-S 
 
 1-7 
 
 429.0 
 
 2.4 
 
 
 SO 
 
 462.6 
 
 71-3 
 
 
 142.7 
 
 ■3 
 
 214.0 
 
 .6 
 
 285-3 
 
 I.I 
 
 356-6 
 
 1-7 
 
 428.0 
 
 2.4 
 
 
 4000 
 
 
 71.2 
 
 .1 
 
 142.3 
 
 -3 
 
 213-5 
 
 .6 
 
 284.6 
 
 I.I 
 
 3S5-8 
 
 1-7 
 
 427.0 
 
 2.4 
 
 
 Smithsonian Tables. 
 
 126 
 
Table 23. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^mrr. 
 [Derivation of table explained on pp. liii-lvi.] 
 
 ■si 
 
 „ B 01 
 ^ 2 M * 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR— 
 
 lo' longitude. 
 
 X y 
 
 20^ longitude. 
 
 30/ longitude. 
 
 40^ longitude. 
 
 50^ longitude. 
 
 1° lonptude. 
 
 40"oo 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 41 00 
 10 
 
 20 
 
 30 
 40 
 
 50 
 
 42 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 4300 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 44 00 
 10 
 20 
 
 30 
 40 
 SO 
 
 4500 
 10 
 20 
 
 30 
 40 
 SO 
 
 4600 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 4700 
 10 
 20 
 
 30 
 40 
 
 50 
 4800 
 
 02.5 
 185.1 
 277.6 
 370-1 
 462.6 
 
 92.5 
 185. r 
 277.6 
 370.2 
 462.7 
 
 92.6 
 185.1 
 277.7 
 370.2 
 462.8 
 
 92.6 
 
 185.2 
 277.7 
 370.3 
 462.9 
 
 92.6 
 185.2 
 277.8 
 370.4 
 463.0 
 
 92.6 
 185.2 
 277.8 
 
 370.4 
 463.0 
 
 92.6 
 
 185-3 
 277.9 
 
 370-S 
 463.1 
 
 92.6 
 
 i8S-3 
 277.9 
 370.6 
 463.2 
 
 71.2 
 71.0 
 70.8 
 70.6 
 70-S 
 70-3 
 
 70.1 
 69.9 
 69.8 
 69.6 
 69.4 
 69.2 
 
 69.0 
 68.9 
 68.7 
 68.5 
 68.3 
 68.1 
 
 68.0 
 67.8 
 67.6 
 67.4 
 67.2 
 67.0 
 
 66.8 
 66.6 
 66.5 
 66.3 
 66.1 
 65.9 
 
 65-7 
 65.5 
 
 65-3 
 65.1 
 64.9 
 64.7 
 
 64.6 
 64.4 
 64.2 
 64.0 
 63.8 
 63.6 
 
 63-4 
 63-2 
 63.0 
 62.8 
 62.6 
 62.4 
 
 62.2 
 
 142.3 
 142.0 
 141.6 
 
 141-3 
 140.9 
 140.6 
 
 140.2 
 139-9 
 I39-S 
 139.2 
 138.8 
 138.4 
 
 1 38. 1 
 137-7 
 137-4 
 137-0 
 136.6 
 
 136-3 
 
 135-9 
 135-5 
 135-2 
 134.8 
 
 134-4 
 134.0 
 
 133-7 
 133-3 
 132.9 
 132.6 
 132.2 
 131.8 
 
 131-4 
 131.0 
 130.6 
 
 130-3 
 129.9 
 129.5 
 
 
 1 29. 1 
 
 128.7 
 128.3 
 
 
 127.9 
 
 127.6 
 127.2 
 
 
 126.8 
 
 126.4 
 126.0 
 125.6 
 
 
 125.2 
 
 
 124.8 
 
 .1 
 
 124.4 
 
 213-5 
 212.9 
 212.4 
 211.9 
 211.4 
 210.8 
 
 209.8 
 209.2 
 208.7 
 208.2 
 207.7 
 
 207.1 
 206.6 
 206.0 
 205.5 
 204-9 
 204.4 
 
 Z03.8 
 203-3 
 
 202. 7 
 202.2 
 201.6 
 201.1 
 
 200.5 
 200.0 
 
 99-4 
 
 98.8 
 
 98-3 
 
 97-7 
 
 97.1 
 96.6 
 96.0 
 95-4 
 94-8 
 94-2 
 
 93-6 
 93-1 
 92.5 
 91.9 
 
 91-3 
 90.7 
 
 90.1 
 
 89-5 
 88.9 
 88.3 
 87.8 
 87.2 
 
 186.6 
 
 284.6 
 283.9 
 283.2 
 282.6 
 281.8 
 281.1 
 
 280.4 
 279.7 
 279.0 
 
 278-3 
 277.6 
 276-9 
 
 276-2 
 275.4 
 
 274-7 
 274.0 
 273.2 
 272-5 
 
 271.8 
 271.0 
 270.3 
 269.0 
 268.8 
 268.1 
 
 267-4 
 266.6 
 265-8 
 265.1 
 264.4 
 263.6 
 
 262.8 
 262.1 
 261.3 
 260.5 
 259.8 
 259.0 
 
 258.2 
 
 257-4 
 256.6 
 
 255-9 
 
 255- 
 
 254-3 
 
 253-S 
 252.7 
 251.9 
 251.1 
 250.4 
 249.6 
 
 248.8 
 
 I.I 
 i.i 
 I.I 
 i.i 
 1.1 
 i.i 
 
 1.1 
 1.1 
 1.1 
 I.I 
 I.I 
 I.I 
 
 I.I 
 1.1 
 I.I 
 1.1 
 I.I 
 I.I 
 
 I.I 
 I.I 
 I.I 
 I.I 
 1.1 
 1.1 
 
 I.I 
 1.1 
 I.I 
 I.I 
 1.1 
 I.I 
 
 1.1 
 I.I 
 I.I 
 1.1 
 
 I.I 
 
 I.I 
 I.I 
 1.1 
 1.1 
 I.I 
 1.1 
 
 1.1 
 1.1 
 I.I 
 I.I 
 I.I 
 I.I 
 
 355-8 
 354-9 
 354-0 
 353-2 
 352-3 
 351-4 
 
 350.6 
 349-6 
 348.8 
 
 347-9 
 347-0 
 346-1 
 
 345-2 
 344-3 
 343-4 
 342-4 
 341-S 
 340.6 
 
 339-8 
 338-8 
 337-9 
 337-0 
 336-0 
 335-1 
 
 334-2 
 333-2 
 332-3 
 331-4 
 330-4 
 329-5 
 
 328-6 
 327.6 
 326.6 
 325.6 
 324-7 
 323-7 
 
 322.8 
 321-8 
 320.8 
 319.8 
 318.9 
 317-9 
 
 316.9 
 315-9 
 3 '4-9 
 313-9 
 313-0 
 312.0 
 
 311.0 
 
 427.0 
 
 425-9 
 424.9 
 
 423-8 
 422.8 
 421.7 
 
 420.7 
 419.6 
 418.5 
 
 417-5 
 416-4 
 
 415-3 
 
 414.2 
 413.2 
 412.1 
 410.9 
 409.9 
 408.8 
 
 407.7 
 406.6 
 
 405-5 
 404.4 
 
 403-3 
 402.1 
 
 401.0 
 
 399-9 
 398.8 
 
 397-7 
 396-5 
 395-4 
 
 394-3 
 393-1 
 391-9 
 390-8 
 389.6 
 388.4 
 
 386.2 
 
 3f5-9 
 383-8 
 382.7 
 
 381 -5 
 
 380-3 
 379-1 
 377-9 
 376-7 
 375-5 
 374-3 
 
 373-1 
 
 Smithsonian Tables. 
 
 127 
 
Table 23. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TtD^Tv* 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •3_. 
 
 nal dis- 
 from 
 egree 
 Is. 
 
 CO-ORDINATES OF DEVELOPED PARALLEL FOR- 
 
 
 lo' longitude. 
 
 20' longitude. 
 
 30' longitude. 
 
 40^ longitude. 
 
 so' longitude. 
 
 1° longitude. 
 
 
 |1 
 
 'tfd 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2^ 
 
 |iS£S, 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 tnm. 
 
 
 48°oo' 
 
 
 62.2 
 
 
 124.4 
 
 -3 
 
 186.6 
 
 .6 
 
 248.8 
 
 I.I 
 
 311.0 
 
 1.7 
 
 373-1 
 
 2.4 
 
 
 10 
 
 92.7 
 
 62.0 
 
 
 124.0 
 
 ■3 
 
 186.0 
 
 .6 
 
 248.0 
 
 I.l 
 
 310.0 
 
 1-7 
 
 371-9 
 
 2.4 
 
 
 20 
 
 185-3 
 278.0 
 
 61.8 
 
 
 123.6 
 
 •3 
 
 185.4 
 
 .6 
 
 247.2 
 
 l.I 
 
 309.0 
 
 1-7 
 
 370.7 
 
 2.4 
 
 
 3° 
 
 61.6 
 
 
 123.2 
 
 •3 
 
 184.7 
 
 .6 
 
 246.3 
 
 I.I 
 
 307.9 
 
 1-7 
 
 369.5 
 
 2.4 
 
 
 40 
 
 370.6 
 
 61.4 
 
 
 122.8 
 
 -3 
 
 184.I 
 
 .6 
 
 245.5 
 
 I.l 
 
 306.9 
 
 1-7 
 
 368-3 
 
 2.4 
 
 
 SO 
 
 463-3 
 
 61.2 
 
 
 122.4 
 
 -3 
 
 183-5 
 
 .6 
 
 244.7 
 
 I.l 
 
 305.9 
 
 1-7 
 
 367-1 
 
 2.4 
 
 
 49 00 
 
 
 61.0 
 
 
 122.0 
 
 -3 
 
 182.9 
 
 .6 
 
 243.9 
 
 I.I 
 
 304.9 
 
 1.7 
 
 365-9 
 
 2.4 
 
 
 10 
 
 '"92-7" 
 
 60.8 
 
 
 121.6 
 
 ■3 
 
 182.3 
 
 .6 
 
 243.1 
 
 1.1 
 
 303.9 
 
 1-7 
 
 364.7 
 
 2.4 
 
 
 20 
 
 185.4 
 278.0 
 
 60.6 
 
 
 121.1 
 
 -3 
 
 181.7 
 
 .6 
 
 242.3 
 
 I.I 
 
 302.8 
 
 1-7 
 
 363-4 
 
 2.4 
 
 
 3° 
 
 60.4 
 
 
 120.7 
 
 -3 
 
 181.1 
 
 .6 
 
 241.4 
 
 1.1 
 
 301.8 
 
 1.7 
 
 362.2 
 
 2.4 
 
 
 40 
 
 370.7 
 
 60.2 
 
 
 120.3 
 
 -3 
 
 180.5 
 
 .6 
 
 240.6 
 
 I.I 
 
 300.8 
 
 1.7 
 
 361.0 
 
 2.4 
 
 
 SO 
 
 463-4 
 
 60.0 
 
 
 1 19.9 
 
 •3 
 
 179.9 
 
 .6 
 
 239.8 
 
 I.l 
 
 299.8 
 
 1.7 
 
 359-8 
 
 2.4 
 
 
 5000 
 
 
 59.8 
 
 
 119-5 
 
 -3 
 
 179.2 
 
 .6 
 
 239.0 
 
 I.l 
 
 298.8 
 
 1.7 
 
 358-5 
 
 2.4 
 
 
 10 
 
 92.7 
 
 S9-S 
 
 
 1 19.1 
 
 -3 
 
 178.6 
 
 .6 
 
 238.2 
 
 I.I 
 
 297.7 
 
 1.7 
 
 357.2 
 
 2.4 
 
 
 20 
 
 185.4 
 278.1 
 
 S9-3 
 
 
 118.7 
 
 •3 
 
 178.0 
 
 .6 
 
 237.3 
 
 I.l 
 
 2966 
 
 1.7 
 
 356.0 
 
 2.4 
 
 
 30 
 
 59.1 
 
 
 118.2 
 
 •3 
 
 177-4 
 
 .6 
 
 236.5 
 
 I.l 
 
 295.6 
 
 1-7 
 
 354.7 
 
 2.4 
 
 
 40 
 
 370.8 
 
 S8-9 
 
 
 II 7.8 
 
 •3 
 
 176.8 
 
 .6 
 
 235-7 
 
 l.I 
 
 294.6 
 
 1.7 
 
 353-5 
 
 2.4 
 
 
 SO 
 
 463-4 
 
 58.7 
 
 
 117.4 
 
 -3 
 
 176.1 
 
 .6 
 
 234-8 
 
 I.I 
 
 293.6 
 
 1-7 
 
 352-3 
 
 2.4 
 
 
 SI 00 
 
 
 sf-s 
 
 
 1 17.0 
 
 •3 
 
 17S-S 
 
 .6 
 
 234.0 
 
 l.I 
 
 292.5 
 
 1-7 
 
 351.0 
 
 2.4 
 
 
 10 
 
 "f^'-j' 
 
 58-3 
 
 .1 
 
 116.6 
 
 •3 
 
 174-9 
 
 .6 
 
 233.2 
 
 1.1 
 
 291.4 
 
 1.6 
 
 349-7 
 
 2.4 
 
 
 20 
 
 185.4 
 278.1 
 
 58.1 
 
 • I 
 
 1 16.2 
 
 •3 
 
 174-2 
 
 .6 
 
 232.3 
 
 l.I 
 
 290.4 
 
 1.6 
 
 348-S 
 
 2.4 
 
 
 30 
 
 S7-9 
 
 •I 
 
 115-7 
 
 -3 
 
 173-6 
 
 .6 
 
 231.5 
 
 l.I 
 
 289.4 
 
 1.6 
 
 347-2 
 
 2.4 
 
 
 40 
 
 ^l°-l 
 
 57-6 
 
 .1 
 
 iiS-3 
 
 •3 
 
 173-0 
 
 .6 
 
 230.6 
 
 I.l 
 
 288.2 
 
 1.6 
 
 345-9 
 
 2.4 
 
 
 SO 
 
 463-6 
 
 57-4 
 
 
 "4-9 
 
 •3 
 
 172.3 
 
 .6 
 
 229.8 
 
 1.1 
 
 287.2 
 
 1.6 
 
 344.6 
 
 2.4 
 
 
 5200 
 
 
 57-2 
 
 .1 
 
 114.5 
 
 •3 
 
 171.7 
 
 .6 
 
 228.9 
 
 1.0 
 
 286.2 
 
 1.6 
 
 343.4 
 
 2.4 
 
 
 10 
 
 92.7 
 
 5^2 
 
 ■ I 
 
 114.0 
 
 •3 
 
 171.1 
 
 .6 
 
 228.1 
 
 1.0 
 
 285.1 
 
 1.6 
 
 342.1 
 
 2.4 
 
 
 20 
 
 185.4 
 
 56.8 
 
 .1 
 
 113.6 
 
 •3 
 
 170.4 
 
 .6 
 
 227.2 
 
 I.O 
 
 284.0 
 
 1.6 
 
 340.8 
 
 2.4 
 
 
 30 
 
 278.2 
 
 56.6 
 
 .1 
 
 113.2 
 
 •3 
 
 169.8 
 
 .6 
 
 226.4 
 
 1.0 
 
 283.0 
 
 1.6 
 
 
 2.3 
 
 
 40 
 
 370-9 
 
 56.4 
 
 
 112.8 
 
 -3 
 
 i6g.i 
 
 .6 
 
 225.5 
 
 1.0 
 
 281.9 
 
 1.6 
 
 338.3 
 
 2.3 
 
 
 SO 
 
 463-6 
 
 56.2 
 
 -I 
 
 112.3 
 
 •3 
 
 168.5 
 
 .6 
 
 224.6 
 
 1.0 
 
 280.8 
 
 1.6 
 
 337.0 
 
 2-3 
 
 
 S3 00 
 
 
 56.0 
 
 .1 
 
 111.9 
 
 •3 
 
 167.9 
 
 .6 
 
 223.8 
 
 1.0 
 
 279.8 
 
 1.6 
 
 335-7 
 
 2.3 
 
 
 10 
 
 92."7 
 
 SS-7 
 
 .1 
 
 111.5 
 
 -3 
 
 167.2 
 
 .6 
 
 222.9 
 
 1.0 
 
 278.6 
 
 1.6 
 
 334-4 
 
 2-3 
 
 
 20 
 
 185.5 
 278.2 
 
 SS-S 
 
 .1 
 
 III.O 
 
 -3 
 
 166.6 
 
 .6 
 
 222.1 
 
 1.0 
 
 277.6 
 
 1.6 
 
 333-1 
 
 2.3 
 
 
 30 
 
 SS-3 
 
 • 1 
 
 110.6 
 
 -3 
 
 165-9 
 
 .6 
 
 221.2 
 
 1.0 
 
 276.5 
 
 1.6 
 
 331-8 
 
 2.3 
 
 
 40 
 
 37I-0 
 
 SS-i 
 
 .1 
 
 110.2 
 
 •3 
 
 165.2 
 
 .6 
 
 220.3 
 
 1.0 
 
 275-4 
 
 1.6 
 
 330.5 
 
 2.3 
 
 
 SO 
 
 463-7 
 
 54-9 
 
 •1 
 
 109.7 
 
 •3 
 
 164.6 
 
 .6 
 
 219.5 
 
 1.0 
 
 274.4 
 
 1.6 
 
 329.2 
 
 2.3 
 
 
 54 00 
 
 
 54.6 
 
 .1 
 
 109.3 
 
 -3 
 
 164.0 
 
 .6 
 
 218.6 
 
 1.0 
 
 273.2 
 
 1.6 
 
 3279 
 
 2.3 
 
 
 10 
 
 92.8 
 
 54-4 
 
 
 108.9 
 
 •3 
 
 163.3 
 
 .6 
 
 217.7 
 
 1.0 
 
 272.1 
 
 1.6 
 
 326.6 
 
 2.3 
 
 
 20 
 
 185.5 
 
 54-2 
 
 
 108.4 
 
 •3 
 
 162.6 
 
 .6 
 
 216.8 
 
 1.0 
 
 271.0 
 
 1.6 
 
 325.3 
 
 2-3 
 
 
 30 
 
 278.3 
 
 54.0 
 
 ■ I 
 
 108.0 
 
 ■3 
 
 162.0 
 
 .6 
 
 216.0 
 
 1.0 
 
 269.9 
 
 1.6 
 
 323.9 
 
 2-3 
 
 
 40 
 
 3?'-2 
 
 S3-8 
 
 
 107.5 
 
 •3 
 
 161.3 
 
 .6 
 
 215.I 
 
 1.0 
 
 268.8 
 
 1.6 
 
 322.6 
 
 2-3 
 
 
 50 
 
 463-8 
 
 53-6 
 
 
 107.1 
 
 •3 
 
 160.6 
 
 .6 
 
 214.2 
 
 1.0 
 
 267.7 
 
 1.6 
 
 321.3 
 
 2.3 
 
 
 SSoo 
 
 
 S3-3 
 
 .1 
 
 106.7 
 
 -3 
 
 160.0 
 
 .6 
 
 213-3 
 
 1.0 
 
 266.6 
 
 1.6 
 
 320.0 
 
 2-3 
 
 
 10 
 
 92.8 
 
 S3-I 
 
 .1 
 
 106.2 
 
 -3 
 
 1S9-3 
 158.7 
 
 .6 
 
 212.4 
 
 1.0 
 
 265.6 
 
 1.6 
 
 318.7 
 
 2.3 
 
 
 20 
 
 185.5 
 278.3 
 
 S2-9 
 
 
 105.8 
 
 -3 
 
 .6 
 
 211.6 
 
 1.0 
 
 264.4 
 
 1.6 
 
 317-3 
 
 2-3 
 
 
 30 
 
 S2-7 
 
 • 1 
 
 ioS-3 
 
 •3 
 
 158.0 
 
 .6 
 
 210.7 
 
 1.0 
 
 263.4 
 
 1.6 
 
 316.0 
 
 2.3 
 
 
 40 
 
 ^v-i 
 
 52-4 
 
 
 104.9 
 
 •3 
 
 157.3 
 
 .6 
 
 209.8 
 
 1.0 
 
 262.2 
 
 1.6 
 
 314.6 
 
 2-3 
 
 
 SO 
 
 463-8 
 
 S2.2 
 
 •1 
 
 104.4 
 
 -3 
 
 156.7 
 
 .6 
 
 208.9 
 
 1.0 
 
 261.1 
 
 1.6 
 
 313.3 
 
 2.3 
 
 
 5600 
 
 
 52.0 
 
 .1 
 
 104.0 
 
 .2 
 
 156.0 
 
 .6 
 
 208.0 
 
 1.0 
 
 260.0 
 
 1.6 
 
 312.0 
 
 2-3 
 
 
 Smithsonian Tables. 
 
 128 
 
Table 23. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Winnr- 
 [Derivation of table explained on pp. liii-lvi.} 
 
 ■3 . 
 
 naldis- 
 from 
 egree 
 Is. 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 ia> longitude. 
 
 20^ longitude. 
 
 30' longitude. 
 
 40^ longitude. 
 
 so' longitude. 
 
 1° longitude. 
 
 •S3 
 
 .2«-e^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^lls. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 mm. 
 
 tnm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 fftm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 S6°oo' 
 
 
 52.0 
 
 
 104.0 
 
 .2 
 
 156.0 
 
 .6 
 
 208.0 
 
 1.0 
 
 260.0 
 
 1.6 
 
 312.0 
 
 2-3 
 
 lO 
 
 '"ga.'s" 
 
 51.8 
 
 .1 
 
 103.6 
 
 .2 
 
 iSS-3 
 
 .6 
 
 207.1 
 
 I.O 
 
 258.9 
 
 1.6 
 
 310.7 
 
 2-3 
 
 20 
 
 185.6 
 278.4 
 
 51.6 
 
 .1 
 
 1 03. 1 
 
 .2 
 
 154.6 
 
 .6 
 
 206.2 
 
 1.0 
 
 257.8 
 
 1.6 
 
 309-3 
 
 2.2 
 
 30 
 
 Si-3 
 
 
 102.6 
 
 .2 
 
 154.0 
 
 .6 
 
 205.3 
 
 1.0 
 
 256.6 
 
 1.6 
 
 307-9 
 
 2.2 
 
 40 
 
 371.2 
 
 Si.i 
 
 
 102.2 
 
 .2 
 
 1 53-3 
 
 .6 
 
 204.4 
 
 1.0 
 
 2SS-S 
 
 i-S 
 
 306.6 
 
 2.2 
 
 SO 
 
 464.0 
 
 50.9 
 
 
 101.8 
 
 .2 
 
 152.6 
 
 .6 
 
 203.5 
 
 1.0 
 
 254.4 
 
 i-S 
 
 3OS-3 
 
 2.2 
 
 57 00 
 
 
 50.6 
 
 
 IOI.3 
 
 .2 
 
 152.0 
 
 .6 
 
 202.6 
 
 1.0 
 
 253.2 
 
 1-5 
 
 303-9 
 
 2.2 
 
 10 
 
 '"92.8" 
 
 50-4 
 
 
 100.8 
 
 .2 
 
 iSi-3 
 
 .6 
 
 201.7 
 
 1.0 
 
 252.1 
 
 I-S 
 
 302.5 
 
 2.2 
 
 20 
 
 185.6 
 278.4 
 
 50.2 
 
 
 100.4 
 
 .2 
 
 150.6 
 
 .6 
 
 200.8 
 
 1.0 
 
 251.0 
 
 I-S 
 
 301.1 
 
 2.2 
 
 30 
 
 50.0 
 
 .1 
 
 99-9 
 
 .2 
 
 149.9 
 
 .6 
 
 199.8 
 
 1.0 
 
 249.8 
 
 1-5 
 
 299.8 
 
 2.2 
 
 40 
 
 371-2 
 
 49-7 
 
 .1 
 
 99-S 
 
 .2 
 
 149.2 
 
 .6 
 
 199.0 
 
 1.0 
 
 248.7 
 
 I-S 
 
 298.4 
 
 2.2 
 
 SO 
 
 464.0 
 
 49-S 
 
 •I 
 
 99-0 
 
 .2 
 
 148.5 
 
 •S 
 
 198.0 
 
 1.0 
 
 247.6 
 
 I-S 
 
 297.1 
 
 2.2 
 
 S8oo 
 
 
 49-3 
 
 ,1 
 
 98.6 
 
 .2 
 
 147.8 
 
 •s 
 
 197.1 
 
 1.0 
 
 246.4 
 
 I-S 
 
 295-7 
 
 2.2 
 
 io 
 
 "92.8 
 
 49.0 
 
 .1 
 
 98.1 
 
 .2 
 
 147.2 
 
 •s 
 
 196.2 
 
 1.0 
 
 245.2 
 
 i-S 
 
 294-3 
 
 2.2 
 
 20 
 
 185.6 
 
 48.8 
 
 .1 
 
 97-6 
 
 .2 
 
 146.5 
 145-8 
 
 •5 
 
 I9S-3 
 
 1.0 
 
 244.1 
 
 i-S 
 
 292.9 
 
 2.2 
 
 30 
 
 278.5 
 
 48.6 
 
 • I 
 
 97-2 
 
 .2 
 
 •S 
 
 194.4 
 
 1.0 
 
 243.0 
 
 I-S 
 
 291-5 
 
 2.2 
 
 40 
 
 371-3 
 
 48.4 
 
 .1 
 
 96.7 
 
 .2 
 
 I4S-I 
 
 •S 
 
 193-4 
 
 1.0 
 
 241.8 
 
 1-5 
 
 290.2 
 
 2.2 
 
 SO 
 
 464.1 
 
 48.1 
 
 -1 
 
 96-3 
 
 .2 
 
 144.4 
 
 •S 
 
 192.5 
 
 1.0 
 
 240.6 
 
 I-S 
 
 288.8 
 
 2.1 
 
 59 00 
 
 
 47-9 
 
 ,1 
 
 95.8 
 
 .2 
 
 143-7 
 
 •s 
 
 191.6 
 
 1.0 
 
 239-5 
 
 1-5 
 
 287.4 
 
 2.1 
 
 10 
 
 92.8 
 
 47-7 
 
 .1 
 
 9S-3 
 
 .2 
 
 143-0 
 
 •s 
 
 190.7 
 
 1.0 
 
 238.4 
 
 I-S 
 
 286.0 
 
 2.1 
 
 20 
 
 185.7 
 278.5 
 
 47-4 
 
 .1 
 
 94-9 
 
 .2 
 
 142.3 
 
 -s 
 
 189.7 
 
 1.0 
 
 237.2 
 
 I-S 
 
 284.6 
 
 2.1 
 
 30 
 
 47-2 
 
 
 94-4 
 
 .2 
 
 141.6 
 
 •s 
 
 188.8 
 
 1.0 
 
 236.0 
 
 1-5 
 
 ^§3-? 
 
 2.1 
 
 40 
 
 371-3 
 
 47.0 
 
 
 93-9 
 
 .2 
 
 140.9 
 
 •s 
 
 187.9 
 
 -9 
 
 234.8 
 
 1-5 
 
 281.8 
 
 2.1 
 
 SO 
 
 464.3 
 
 46.7 
 
 
 93-S 
 
 .2 
 
 140.2 
 
 •s 
 
 186.9 
 
 •9 
 
 233-6 
 
 I-S 
 
 280.4 
 
 2.1 
 
 6ooo 
 
 
 46.5 
 
 ,1 
 
 93-0 
 
 .2 
 
 m 
 
 •s 
 
 186.0 
 
 -9 
 
 232- s 
 
 I-S 
 
 279.0 
 
 2.1 
 
 10 
 
 92.8 
 
 46.3 
 
 
 92.5 
 
 .2 
 
 -s 
 
 185.0 
 
 -9 
 
 231-3 
 
 I-S 
 
 277.6 
 
 2.1 
 
 20 
 
 278.6 
 
 46.0 
 
 .1 
 
 92.1 
 
 .2 
 
 138-1 
 
 •s 
 
 184.1 
 
 9 
 
 230.2 
 
 1-4 
 
 276.2 
 
 2.1 
 
 30 
 
 45.8 
 
 .1 
 
 91.6 
 
 .2 
 
 137-4 
 
 -s 
 
 183.2 
 
 •9 
 
 229.0 
 
 1-4 
 
 274.8 
 
 2.1 
 
 40 
 
 371-4 
 
 45.6 
 
 
 91.1 
 
 .2 
 
 136-7 
 
 ■s 
 
 182.2 
 
 -9 
 
 227.8 
 
 1-4 
 
 273-4 
 
 2.1 
 
 SO 
 
 464.2 
 
 4S-3 
 
 •I 
 
 90.6 
 
 .2 
 
 136.0 
 
 •s 
 
 181.3 
 
 •9 
 
 226.6 
 
 1-4 
 
 271-9 
 
 2.1 
 
 61 00 
 
 
 45.1 
 
 
 90.2 
 
 .2 
 
 '35-3 
 
 -s 
 
 180.4 
 
 •9 
 
 225.4 
 
 1-4 
 
 270.5 
 
 2.1 
 
 10 
 
 92.9 
 
 44-8 
 
 ,1 
 
 89-7 
 
 .2 
 
 134.6 
 
 -s 
 
 179-4 
 
 -9 
 
 224.2 
 
 1-4 
 
 269.1 
 
 2.1 
 
 20 
 
 185.7 
 
 44-6 
 
 .1 
 
 89.2 
 
 .2 
 
 133-9 
 
 -s 
 
 178.5 
 
 •9 
 
 223.1 
 
 1-4 
 
 267.7 
 
 2.1 
 
 30 
 
 278,6 
 
 44-4 
 
 
 88.8 
 
 .2 
 
 133-1 
 
 -s 
 
 177-S 
 
 -9 
 
 221.9 
 
 1-4 
 
 266.3 
 
 2.0 
 
 40 
 
 371-4 
 
 44-1 
 
 
 88.3 
 
 .2 
 
 132.4 
 
 -s 
 
 176.6 
 
 -9 
 
 220.7 
 
 1-4 
 
 264.8 
 
 2.0 
 
 SO 
 
 464-3 
 
 43-9 
 
 
 87.8 
 
 .2 
 
 13 1 -7 
 
 -s 
 
 175.6 
 
 •9 
 
 219.6 
 
 1-4 
 
 263.5 
 
 2.0 
 
 6200 
 
 
 43-7 
 
 
 87-3 
 
 .2 
 
 131-0 
 
 •s 
 
 174-7 
 
 -9 
 
 218.4 
 
 1-4 
 
 262.0 
 
 2.0 
 
 10 
 
 92.9 
 
 43-4 
 
 
 86.9 
 
 .2 
 
 130-3 
 
 -s 
 
 173-7 
 
 -9 
 
 217.2 
 
 1-4 
 
 260.6 
 
 2.0 
 
 20 
 
 185.7 
 278.6 
 
 43-2 
 
 ,1 
 
 86.4 
 
 .2 
 
 129.6 
 
 •s 
 
 172.8 
 
 •9 
 
 216.0 
 
 1-4 
 
 259.1 
 
 2.0 
 
 30 
 
 43-0 
 
 ,1 
 
 85-9 
 
 .2 
 
 128.8 
 
 -s 
 
 171.8 
 
 •9 
 
 214.8 
 
 1-4 
 
 257-7 
 
 2.0 
 
 40 
 
 37I-S 
 
 42.7 
 
 
 85-4 
 
 .2 
 
 128.1 
 
 -s 
 
 170.8 
 
 -9 
 
 213-6 
 
 1-4 
 
 256.3 
 
 2.0 
 
 so 
 
 464-4 
 
 42.5 
 
 •I 
 
 84.9 
 
 .2 
 
 127.4 
 
 -s 
 
 169.9 
 
 -9 
 
 212.4 
 
 1-4 
 
 254.8 
 
 2.0 
 
 6300 
 
 
 42.2 
 
 
 84-S 
 
 .2 
 
 126.7 
 
 ■s 
 
 168.9 
 
 -9 
 
 211.2 
 
 1.4 
 
 253-4 
 
 2.0 
 
 10 
 
 92.9 
 
 42.0 
 
 ,1 
 
 84.0 
 
 .2 
 
 126.0 
 
 -s 
 
 168.0 
 
 -9 
 
 210.0 
 
 1.4 
 
 251.9 
 
 2.0 
 
 20 
 
 185.8 
 
 41.7 
 
 ,1 
 
 83-5 
 
 .2 
 
 125.2 
 
 -s 
 
 167.0 
 
 •9 
 
 208.8 
 
 1-4 
 
 250.5 
 
 2.0 
 
 30 
 
 278.7 
 
 41.5 
 
 
 83.0 
 
 .2 
 
 124.5 
 
 •5 
 
 166.0 
 
 -9 
 
 207.5 
 
 1-3 
 
 249.0 
 
 1.9 
 
 40 
 
 371-6 
 
 41-3 
 
 
 82.5 
 
 .2 
 
 123.8 
 
 ■5 
 
 165.0 
 
 -9 
 
 206.3 
 
 1-3 
 
 247.6 
 
 1-9 
 
 SO 
 
 464-4 
 
 41.0 
 
 
 82.0 
 
 .2 
 
 123.1 
 
 -s 
 
 1 64. 1 
 
 -9 
 
 205.1 
 
 1-3 
 
 246.1 
 
 1-9 
 
 6400 
 
 
 40.8 
 
 .1 
 
 81.6 
 
 .2 
 
 122.3 
 
 ■s 
 
 163.1 
 
 -9 
 
 203.9 
 
 1-3 
 
 244-7 
 
 1.9 
 
 Smithsonian Tables. 
 
 129 
 
Table 23. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Wtnnr- 
 
 [Derivation of table explained on pp. liii.-lviii.] 
 
 "8_. 
 
 [onal dis- 
 s from 
 degree 
 
 els. 
 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 lo' longitude. 
 
 20' longitude. 
 
 30' longitude. 
 
 40/ longitude. 
 
 so' longitude. 
 
 1° longitude. 
 
 Be 
 
 |sgl 
 
 
 
 
 
 
 
 
 
 
 
 3I 
 
 gssa 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 mm. 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 mm. 
 
 X 
 
 y 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 WW. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 64°oo' 
 
 
 40.8 
 
 .1 
 
 81.6 
 
 .2 
 
 122.3 
 
 •5 
 
 1 63. 1 
 
 ■% 
 
 203-9 
 
 1-3 
 
 244-7 
 
 1.9 
 
 10 
 
 92.9 
 
 40-5 
 
 .1 
 
 81. 1 
 
 .2 
 
 I2I.6 
 
 -5 
 
 162.2 
 
 .8 
 
 202.7 
 
 1-3 
 
 243.2 
 
 1.9 
 
 20 
 
 185.8 
 
 40-3 
 
 .1 
 
 80.6 
 
 .2 
 
 120.9 
 
 •5 
 
 161.2 
 
 .8 
 
 201.4 
 
 1-3 
 
 241.7 
 
 1.9 
 
 3° 
 
 278.7 
 
 40.0 
 
 .1 
 
 80.1 
 
 .2 
 
 1 20. 1 
 
 -S 
 
 160.2 
 
 .8 
 
 200.2 
 
 1-3 
 
 240.2 
 
 1.9 
 
 40 
 
 371.6 
 
 39-8 
 
 .1 
 
 79.6 
 
 .2 
 
 1 19.4 
 
 -S 
 
 159.2 
 
 .8 
 
 199.0 
 
 1-3 
 
 238.8 
 
 1.9 
 
 SO 
 
 464-5 
 
 39-6 
 
 .1 
 
 79.1 
 
 .2 
 
 1 18.7 
 
 •S 
 
 158.2 
 
 .8 
 
 197.8 
 
 '•3 
 
 237-4 
 
 1.9 
 
 6500 
 
 
 39-3 
 
 .1 
 
 78.6 
 
 .2 
 
 II7.9 
 
 -s 
 
 157.2 
 
 .8 
 
 196.6 
 
 1-3 
 
 235-9 
 
 1-9 
 
 10 
 
 92.9 
 
 39-1 
 
 .1 
 
 78.1 
 
 .2 
 
 II7.2 
 
 •5 
 
 156.2 
 
 .8 
 
 ■95-3 
 
 1-3 
 
 234-4 
 
 1.9 
 
 20 
 
 278.7 
 
 38-8 
 
 .1 
 
 77-6 
 
 .2 
 
 II6.5 
 
 -5 
 
 155-3 
 
 .8 
 
 194.1 
 
 1-3 
 
 232-9 
 
 1.8 
 
 30 
 
 38.6 
 
 .1 
 
 77-2 
 
 .2 
 
 1 1 5.7 
 
 -S 
 
 1 54-3 
 
 .8 
 
 192.9 
 
 1-3 
 
 231-5 
 
 1.8 
 
 40 
 
 37 1 -0 
 
 38-3 
 
 .1 
 
 76-7 
 
 .2 
 
 1 15.0 
 
 -5 
 
 153-3 
 
 .8 
 
 191.6 
 
 1-3 
 
 230.0 
 
 1.8 
 
 50 
 
 464.6 
 
 38.1 
 
 •" 
 
 76.2 
 
 .i 
 
 II4.2 
 
 ■5 
 
 152-3 
 
 .8 
 
 190.4 
 
 1-3 
 
 228.5 
 
 1.8 
 
 66 00 
 
 
 37-8 
 
 .1 
 
 75-7 
 
 .2 
 
 1 1 3- 5 
 
 •S 
 
 151-4 
 
 .8 
 
 '!?■" 
 
 1-3 
 
 227.0 
 
 1.8 
 
 10 
 
 92.9" 
 
 37-6 
 
 .0 
 
 75-2 
 
 .2 
 
 1 1 2.8 
 
 -4 
 
 150.4 
 
 .8 
 
 188.0 
 
 13 
 
 225-5 
 
 1.8 
 
 zo 
 
 
 37-3 
 
 .0 
 
 74-7 
 
 .2 
 
 H2.0 
 
 ■4 
 
 149.4 
 
 .8 
 
 186.7 
 
 1.2 
 
 224.0 
 
 1.8 
 
 30 
 
 278.8 
 
 37-1 
 
 .0 
 
 74.2 
 
 .2 
 
 III. 3 
 
 -4 
 
 148.4 
 
 .8 
 
 185.4 
 
 1.2 
 
 222.5 
 
 1.8 
 
 40 
 
 3717 
 
 36.8 
 
 .0 
 
 73-7 
 
 .2 
 
 1 10.6 
 
 •4 
 
 147.4 
 
 .8 
 
 184.2 
 
 1.2 
 
 221. 1 
 
 1.8 
 
 5° 
 
 464.6 
 
 36.6 
 
 .0 
 
 73-2 
 
 .2 
 
 109.8 
 
 •4 
 
 146.4 
 
 .8 
 
 183.0 
 
 1.2 
 
 219.6 
 
 1.8 
 
 67 00 
 
 
 36-4 
 
 .0 
 
 72.7 
 
 .2 
 
 109.0 
 
 -4 
 
 145.4 
 
 .8 
 
 181.8 
 
 1.2 
 
 218.1 
 
 1.8 
 
 10 
 
 278.8 
 
 361 
 
 .0 
 
 72.2 
 
 .2 
 
 108.3 
 
 -4 
 
 144.4 
 
 .8 
 
 180.5 
 
 1.2 
 
 216.6 
 
 1-7 
 
 20 
 
 35-8 
 
 .0 
 
 71.7 
 
 .2 
 
 107.6 
 
 -4 
 
 143-4 
 
 .8 
 
 179.2 
 
 1.2 
 
 215.1 
 
 1-7 
 
 3° 
 
 35-6 
 
 .0 
 
 71.2 
 
 .2 
 
 106.8 
 
 •4 
 
 142.4 
 
 .8 
 
 178.0 
 
 1.2 
 
 213.6 
 
 1-7 
 
 40 
 
 371.8 
 
 35-4 
 
 .0 
 
 70.7 
 
 .2 
 
 106.0 
 
 •4 
 
 141.4 
 
 .8 
 
 176.8 
 
 1.2 
 
 2I2.I 
 
 1-7 
 
 SO 
 
 464.7 
 
 35-1 
 
 .0 
 
 70.2 
 
 .2 
 
 105-3 
 
 •4 
 
 140.4 
 
 .8 
 
 175-5 
 
 1.2 
 
 210.6 
 
 1-7 
 
 6800 
 
 
 34-8 
 
 .0 
 
 69-7 
 
 .2 
 
 104.6 
 
 •4 
 
 139-4 
 
 .8 
 
 174.2 
 
 1.2 
 
 209.1 
 
 1-7 
 
 10 
 
 93-0 
 
 34-6 
 
 .0 
 
 
 .2 
 
 103.8 
 
 ■4 
 
 138.4 
 
 -7 
 
 173-0 
 
 1.2 
 
 207.6 
 
 1-7 
 
 20 
 
 %n 
 
 34-4 
 
 .0 
 
 68.7 
 
 .2 
 
 103.0 
 
 ■4 
 
 137-4 
 
 -7 
 
 171.8 
 
 1.2 
 
 206.1 
 
 1-7 
 
 30 
 
 34-1 
 
 .0 
 
 68.2 
 
 .2 
 
 102.3 
 
 •4 
 
 136.4 
 
 -7 
 
 170.4 
 
 i.i 
 
 204.5 
 
 1-7 
 
 40 
 
 37' 1 
 
 33-8 
 
 .0 
 
 67-7 
 
 .2 
 
 101.5 
 100.8 
 
 •4 
 
 135-4 
 
 -7 
 
 169.2 
 168.0 
 
 1.1 
 
 203.0 
 
 1-7 
 
 SO 
 
 464.8 
 
 33-6 
 
 .0 
 
 67.2 
 
 .2 
 
 •4 
 
 134-4 
 
 -7 
 
 I.I 
 
 201.5 
 
 1.6 
 
 6900 
 
 
 33-3 
 
 .0 
 
 66.7 
 
 .2 
 
 1 00.0 
 
 •4 
 
 133-4 
 
 -7 
 
 166.7 
 
 I.I 
 
 200.0 
 
 1.6 
 
 10 
 
 93-0 
 
 33-1 
 
 .0 
 
 66.2 
 
 .2 
 
 99.3 
 
 -4 
 
 132.4 
 
 •7 
 
 165.4 
 
 1.1 
 
 198.5 
 
 1.6 
 
 20 
 
 185.9 
 278.9 
 
 32.8 
 
 .0 
 
 65-7 
 
 .2 
 
 98.5 
 
 -4 
 
 131-3 
 
 •7 
 
 164.2 
 
 I.I 
 
 197.0 
 
 1.6 
 
 30 
 
 32-6 
 
 .0 
 
 65.2 
 
 .2 
 
 97-7 
 
 •4 
 
 130-3 
 
 -7 
 
 162.9 
 
 I.I 
 
 I9S-S 
 
 1.6 
 
 40 
 
 371-8 
 
 32-3 
 
 .0 
 
 64-7 
 
 .2 
 
 97.0 
 
 -4 
 
 129-3 
 
 -7 
 
 161.6 
 
 i.i 
 
 194.0 
 
 1.6 
 
 SO 
 
 464.8 
 
 32.1 
 
 .0 
 
 64.1 
 
 .2 
 
 96.2 
 
 -4 
 
 128.3 
 
 •7 
 
 160.4 
 
 I.I 
 
 192.4 
 
 1.6 
 
 70 00 
 
 
 31.8 
 
 .0 
 
 63.6 
 
 .2 
 
 9S-5 
 
 •4 
 
 127-3 
 
 -7 
 
 1 59- 1 
 
 i.i 
 
 190.9 
 
 1.6 
 
 10 
 
 930 
 
 31-6 
 
 .0 
 
 63.1 
 
 .2 
 
 94-7 
 
 •4 
 
 126.2 
 
 ■7 
 
 157-8 
 
 I.I 
 
 189.4 
 
 1.6 
 
 20 
 
 1 8 5.9 
 278.9 
 
 31-3 
 
 .0 
 
 62.6 
 
 .2 
 
 93-9 
 
 •4 
 
 125.2 
 
 -7 
 
 156.6 
 
 1.1 
 
 187.9 
 
 1.6 
 
 30 
 
 3'-i 
 
 .0 
 
 62.1 
 
 .2 
 
 93-2 
 
 -4 
 
 124.2 
 
 •7 
 
 155-3 
 
 I.I 
 
 186.4 
 
 '•5 
 
 40 
 
 371-9 
 
 30.8 
 
 .0 
 
 61.6 
 
 .2 
 
 92-4 
 
 •4 
 
 123.2 
 
 ■7 
 
 154-0 
 
 1.1 
 
 184.8 
 
 i-S 
 
 SO 
 
 464.9 
 
 30-S 
 
 .0 
 
 61. 1 
 
 .2 
 
 91.6 
 
 -4 
 
 122.2 
 
 ■7 
 
 152-7 
 
 1.0 
 
 183.2 
 
 ^■\ 
 
 71 00 
 
 
 30-3 
 
 .0 
 
 60.6 
 
 .2 
 
 90.9 
 
 •4 
 
 121.2 
 
 -7 
 
 151-4 
 
 I.O 
 
 181.7 
 
 i-S 
 
 10 
 
 186.0 
 
 30-0 
 
 .0 
 
 60.1 
 
 .2 
 
 90.1 
 
 ■4 
 
 120.2 
 
 -7 
 
 150.2 
 
 I.O 
 
 180.2 
 
 i-S 
 
 20 
 
 29.8 
 
 .0 
 
 59.6 
 
 .2 
 
 89-3 
 
 -4 
 
 119.1 
 
 •7 
 
 148.9 
 
 1.0 
 
 178.7 
 
 i-S 
 
 30 
 
 278.9 
 
 29.5 
 
 .0 
 
 S9-0 
 
 .2 
 
 88.0 
 
 •4 
 
 1 18.1 
 
 ■7 
 
 147.6 
 
 1.0 
 
 177.1 
 
 1-5 
 
 40 
 
 371-9 
 
 29-3 
 
 .0 
 
 58-5 
 
 .2 
 
 87.8 
 
 •4 
 
 117.1 
 
 .6 
 
 146.4 
 
 1.0 
 
 175.6 
 
 I-S 
 
 SO 
 
 464.9 
 
 29.0 
 
 .0 
 
 58.0 
 
 .2 
 
 87.1 
 
 •4 
 
 116.1 
 
 .6 
 
 145.1 
 
 I.O 
 
 174.1 
 
 1.4 
 
 72 00 
 
 
 28.8 
 
 .0 
 
 S7-S 
 
 .2 
 
 86.3 
 
 •4 
 
 115.0 
 
 .6 
 
 143.8 
 
 I.O 
 
 172.6 
 
 1.4 
 
 Smithsonian Tables. 
 
 130 
 
Table 23. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE aoAtO - 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •s 
 
 ■S3 
 i% 
 
 ridionaldis- 
 ices from 
 iT\ degree 
 rallels. 
 
 
 CO-ORDINATES OF 
 
 DEVELOPED PARALLEL FOR— 
 
 
 
 10' longitude. 
 
 3& longitude. 
 
 30^ longitude. 
 
 40' longitude. 
 
 50/ longitude. 
 
 1° longitude. 
 
 
 3^ 
 
 ^att 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 TKtn. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 
 72°00' 
 
 
 28.8 
 
 .0 
 
 S7-S 
 
 .2 
 
 86.3 
 
 ■4 
 
 1 1 5.0 
 
 .6 
 
 143-8 
 
 I.O 
 
 172.6 
 
 1-4 
 
 
 10 
 
 93.0 
 
 28.5 
 
 .0 
 
 S7.0 
 
 .2 
 
 8S-S 
 
 •4 
 
 1 14.0 
 
 .6 
 
 142.5 
 
 I.O 
 
 I7I.0 
 
 1.4 
 
 
 20 
 
 186.0 
 
 28.2 
 
 .0 
 
 56.5 
 
 
 84.7 
 
 •3 
 
 1 13.0 
 
 .6 
 
 141. 2 
 
 1.0 
 
 169.4 
 
 1-4 
 
 
 3° 
 
 279.0 
 
 28.0 
 
 .0 
 
 56.0 
 
 
 83-9 
 
 •3 
 
 III.9 
 
 .6 
 
 139-9 
 
 I.O 
 
 167.9 
 
 1.4 
 
 
 40 
 
 372.0 
 
 27.7 
 
 .0 
 
 SS-S 
 
 
 83.2 
 
 •3 
 
 1 10.9 
 
 .6 
 
 138.6 
 
 I.O 
 
 166.4 
 
 1-4 
 
 
 SO 
 
 465.0 
 
 27.5 
 
 .0 
 
 S4-9 
 
 
 82.4 
 
 •3 
 
 109.9 
 
 .6 
 
 137-4 
 
 I.O 
 
 164.8 
 
 1.4 
 
 
 7300 
 
 
 27.2 
 
 .0 
 
 54-4 
 
 
 81.6 
 
 •3 
 
 108.8 
 
 .6 
 
 136.0 
 
 -9 
 
 163-3 
 
 1.4 
 
 
 10 
 
 93-0 
 
 27.0 
 
 .0 
 
 S3-9 
 
 
 80.8 
 
 •3 
 
 107.8 
 
 .6 
 
 134.8 
 
 -9 
 
 161.7 
 
 1.4 
 
 
 20 
 
 186.0 
 
 26.7 
 
 .0 
 
 S34 
 
 
 80.1 
 
 •3 
 
 106.8 
 
 .6 
 
 133-4 
 
 -9 
 
 160.1 
 
 1-3 
 
 
 3° 
 
 279.0 
 
 26.4 
 
 .0 
 
 52.9 
 
 
 79-3 
 
 •3 
 
 105.7 
 
 .6 
 
 132.2 
 
 -9 
 
 158.6 
 
 1-3 
 
 
 40 
 
 372.0 
 
 26.2 
 
 .0 
 
 5^-=5 
 
 
 78.5 
 
 ■3 
 
 104.7 
 
 .6 
 
 130.8 
 
 -9 
 
 157.0 
 
 1-3 
 
 
 SO 
 
 465.0 
 
 25.9 
 
 .0 
 
 5i.g 
 
 
 77-7 
 
 •3 
 
 103.6 
 
 .6 
 
 129.6 
 
 •9 
 
 'SS-S 
 
 1-3 
 
 
 7400 
 
 
 25.6 
 
 .0 
 
 S'-3 
 
 
 77.0 
 
 ■3 
 
 102.6 
 
 .6 
 
 128.2 
 
 •9 
 
 IS3-9 
 
 1-3 
 
 
 10 
 
 93-0 
 
 25.4 
 
 .0 
 
 50.8 
 
 
 76.2 
 
 •3 
 
 I0I.6 
 
 .6 
 
 127.0 
 
 -9 
 
 152-3 
 
 1-3 
 
 
 20 
 
 186.0 
 
 25.1 
 
 .0 
 
 So-3 
 
 
 7S-4 
 
 •3 
 
 100.5 
 
 .6 
 
 125.6 
 
 ■9 
 
 150.8 
 
 1-3 
 
 
 30 
 
 279.0 
 
 24.9 
 
 .0 
 
 49-7 
 
 
 74.6 
 
 •3 
 
 99.5 
 
 .6 
 
 124.4 
 
 •9 
 
 149.2 
 
 1-3 
 
 
 40 
 
 372.0 
 
 24.6 
 
 .0 
 
 49.2 
 
 
 73-8 
 
 •3 
 
 98.4 
 
 .6 
 
 123.0 
 
 -9 
 
 147-7 
 
 1.2 
 
 
 SO 
 
 465.0 
 
 24.4 
 
 .0 
 
 48.7 
 
 
 73-0 
 
 •3 
 
 97-4 
 
 •S 
 
 121.8 
 
 ■9 
 
 146.1 
 
 1.2 
 
 
 7500 
 
 
 24.1 
 
 .0 
 
 48.2 
 
 
 72-3 
 
 •3 
 
 96.4 
 
 •S 
 
 120.4 
 
 .8 
 
 144.5 
 
 1.2 
 
 
 10 
 
 93.0 
 
 23.8 
 
 .0 
 
 47-7 
 
 
 7I-S 
 
 ■3 
 
 95-3 
 
 •s 
 
 1 19.2 
 
 .8 
 
 143.0 
 
 1.2 
 
 
 20 
 
 186.0 
 
 23.6 
 
 .0 
 
 47.1 
 
 
 70.7 
 
 ■3 
 
 94.2 
 
 ■s 
 
 1 1 7.8 
 
 .8 
 
 141.4 
 
 1.2 
 
 
 30 
 
 279.1 
 
 23-3 
 
 .0 
 
 46.6 
 
 
 69.9 
 
 •3 
 
 93-2 
 
 ■s 
 
 116.5 
 
 .8 
 
 139.8 
 
 1.2 
 
 
 40 
 
 372.1 
 
 23.0 
 
 .0 
 
 46.1 
 
 
 69.1 
 
 •3 
 
 92.2 
 
 •s 
 
 1 1 5.2 
 
 .8 
 
 138.2 
 
 1.2 
 
 
 5° 
 
 465.1 
 
 22.8 
 
 .0 
 
 4S-S 
 
 
 68.3 
 
 •3 
 
 91. 1 
 
 •s 
 
 1 1 3.8 
 
 .8 
 
 136.6 
 
 I.I 
 
 
 7600 
 
 
 22.S 
 
 .0 
 
 45.0 
 
 
 tu 
 
 •3 
 
 90.0 
 
 ■s 
 
 1 1 2.6 
 
 .8 
 
 13s- 1 
 
 i.i 
 
 
 10 
 
 93.0 
 
 22.2 
 
 .0 
 
 44-5 
 
 
 ■3 
 
 89.0 
 
 ■s 
 
 III. 2 
 
 .8 
 
 133-S 
 
 I.I 
 
 
 20 
 
 186.1 
 
 22.0 
 
 .0 
 
 44.0 
 
 
 65.9 
 
 •3 
 
 87.9 
 
 •s 
 
 109.9 
 
 .8 
 
 131-9 
 
 1. 1 
 
 
 30 
 
 279.1 
 
 21.7 
 
 .0 
 
 43-4 
 
 
 65.2 
 
 •3 
 
 f^-f 
 
 •s 
 
 108.6 
 
 .8 
 
 130-3 
 
 I.I 
 
 
 40 
 
 3721 
 
 21.5 
 
 .0 
 
 42.9 
 
 
 64.4 
 
 •3 
 
 85.8 
 
 •s 
 
 107.3 
 
 .8 
 
 128.8 
 
 I.I 
 
 
 SO 
 
 465.1 
 
 21.2 
 
 .0 
 
 42.4 
 
 
 63.6 
 
 •3 
 
 84.8 
 
 •s 
 
 106.0 
 
 ■7 
 
 127.1 
 
 I.I 
 
 
 7700 
 
 
 20.9 
 
 .0 
 
 41.9 
 
 
 62.8 
 
 •3 
 
 83.7 
 
 •s 
 
 104.6 
 
 •7 
 
 125.6 
 
 I.I 
 
 
 10 
 
 93.0 
 
 20.7 
 
 .0 
 
 41-3 
 
 
 62.0 
 
 •3 
 
 82.7 
 
 ■s 
 
 103.4 
 
 •7 
 
 124.0 
 
 I.I 
 
 
 20 
 
 i86.i 
 
 20.4 
 
 .0 
 
 40.8 
 
 
 61.2 
 
 •3 
 
 81.6 
 
 ■5 
 
 102.0 
 
 -7 
 
 122.4 
 
 1.0 
 
 
 30 
 
 279.1 
 
 20.1 
 
 .0 
 
 40-3 
 
 
 60.4 
 
 ■3 
 
 80.6 
 
 ■s 
 
 100.7 
 
 •7 
 
 120.8 
 
 1.0 
 
 
 40 
 
 372.2 
 
 19.9 
 
 .0 
 
 39-« 
 
 
 59.6 
 
 •3 
 
 79-S 
 
 •4 
 
 99-4 
 
 •7 
 
 "9-3 
 
 I.O 
 
 
 SO 
 
 465.2 
 
 19.6 
 
 .0 
 
 39-2 
 
 
 58.8 
 
 •3 
 
 78.4 
 
 ■4 
 
 98.0 
 
 •7 
 
 117.7 
 
 I.O 
 
 
 78 00 
 
 
 19.4 
 
 .0 
 
 387 
 
 
 58.0 
 
 .2 
 
 77-4 
 
 •4 
 
 96.8 
 
 •7 
 
 116.1 
 
 I.O 
 
 
 10 
 
 93.0 
 
 19.1 
 
 .0 
 
 38.2 
 
 
 57.2 
 
 .2 
 
 76.3 
 
 •4 
 
 9S-4 
 
 •7 
 
 "4-5 
 
 I.O 
 
 
 20 
 
 1 86. 1 
 
 18.8 
 
 .0 
 
 37-6 
 
 
 56.5 
 
 .2 
 
 7S-3 
 
 •4 
 
 94.1 
 
 ■7 
 
 112.9 
 
 1.0 
 
 
 30 
 
 279.1 
 
 18.6 
 
 .0 
 
 37-1 
 
 
 SS-7 
 
 .2 
 
 74.2 
 
 ■4 
 
 92.8 
 
 -7 
 
 111.4 
 
 I.O 
 
 
 40 
 
 372.2 
 
 18.3 
 
 .0 
 
 36.6 
 
 
 54.9 
 
 .2 
 
 73-2 
 
 •4 
 
 91.4 
 
 .6 
 
 109.7 
 
 -9 
 
 
 SO 
 
 465.2 
 
 18.0 
 
 .0 
 
 36.0 
 
 
 54.1 
 
 .2 
 
 72.1 
 
 ■4 
 
 90.1 
 
 .6 
 
 108.1 
 
 ■9 
 
 
 7900 
 
 
 17.8 
 
 .0 
 
 3S-S 
 
 
 S3-3 
 
 .2 
 
 71.0 
 
 ■4 
 
 88.8 
 
 .6 
 
 106.6 
 
 •9 
 
 
 10 
 
 93-° 
 
 I7-S 
 
 .0 
 
 3S-0 
 
 
 52.5 
 
 .2 
 
 70.0 
 
 ■4 
 
 ?2-4 
 
 .6 
 
 104.9 
 
 •9 
 
 
 20 
 
 186.1 
 
 17.2 
 
 .0 
 
 34-5 
 
 
 S»-7 
 
 .2 
 
 68.9 
 
 •4 
 
 86.2 
 
 .6 
 
 103.4 
 
 •9 
 
 
 30 
 
 279.2 
 
 17.0' 
 
 .0 
 
 33-9 
 
 
 So-9 
 
 .2 
 
 67.8 
 
 ■4 
 
 84.8 
 
 .6 
 
 101.8 
 
 ■§ 
 
 
 40 
 
 372.2 
 
 16.7 
 
 .0 
 
 33-4 
 
 
 50.1 
 
 .2 
 
 66.8 
 
 •4 
 
 83-4 
 
 .6 
 
 100.1 
 
 .8 
 
 
 SO 
 
 465.2 
 
 16.4 
 
 .0 
 
 32-9 
 
 
 49-3 
 
 .2 
 
 65.7 
 
 ■4 
 
 82.2 
 
 .6 
 
 98.6 
 
 .8 
 
 
 8000 
 
 
 16.2 
 
 .0 
 
 32-3 
 
 .1 
 
 48.5 
 
 .2 
 
 64.6 
 
 •4 
 
 80.8 
 
 .6 
 
 97.0 
 
 .8 
 
 
 Smithsonian Tables. 
 
 131 
 
Table 24. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jnhsv- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •S H 
 
 o"oo' 
 
 10 
 
 20 
 
 3° 
 40 
 
 5° 
 
 1 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 2 00 
 10 
 20 
 
 3° 
 40 
 
 5° 
 
 300 
 10 
 20 
 
 30 
 40 
 
 5° 
 
 400 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 500 
 10 
 20 
 
 30 
 40 
 
 so 
 
 6 00 
 10 
 20 
 
 30 
 40 
 
 so 
 
 7 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 800 
 
 2ES 
 
 230.4 
 460.7 
 691.0 
 921.4 
 
 iiSi-8 
 
 230.4 
 460.7 
 691.0 
 921.4 
 1151.8 
 
 230.4 
 460.7 
 691.0 
 921.4 
 1151.8 
 
 230.4 
 460.7 
 691. 1 
 921.4 
 1151.8 
 
 230.4 
 460.7 
 691. 1 
 921.4 
 1151.8 
 
 230.4 
 460.7 
 691. 1 
 921.5 
 1151.8 
 
 230.4 
 460.8 
 691. 1 
 921.5 
 1151.9 
 
 230.4 
 460.8 
 691.1 
 921.5 
 1151.9 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 longitude. 
 
 16.0 
 16.0 
 16.0 
 16.0 
 16.0 
 15.9 
 
 15.9 
 15.9 
 
 iS-9 
 15.9 
 
 iS-9 
 ■S-9 
 
 IS-9 
 15.9 
 
 IS-9 
 15.8 
 15.8 
 15.8 
 
 15.8 
 
 1 5-8 
 15.8 
 
 IS7 
 15-7 
 iS-7 
 
 iS-7 
 
 1 5-6 
 15.6 
 15.6 
 15.6 
 
 iS-S 
 iS-S 
 iS-S 
 15-4 
 15.4 
 15.4 
 
 iS-3 
 iS-3 
 15.2 
 15.2 
 15.2 
 iS-i 
 
 1 5-1 
 iS-i 
 15.0 
 15.0 
 
 14-9 
 14.9 
 
 114.8 
 
 10' 
 longitude. 
 
 231.9 
 231.9 
 231.9 
 231.9 
 231.9 
 231.9 
 
 231.9 
 231.9 
 231.8 
 231.8 
 231.8 
 231.8 
 
 231.8 
 231.8 
 2317 
 231-7 
 231-7 
 231.6 
 
 231.6 
 231.6 
 23I-S 
 
 23I-S 
 231-4 
 231-4 
 
 229.7 
 
 IS' 
 
 longitude. 
 
 347-9 
 347-9 
 347-8 
 347-8 
 347-8 
 347-8 
 
 347-8 
 347-8 
 347-8 
 347-7 
 347-7 
 347-7 
 
 347-7 
 347-6 
 347-6 
 347-S 
 347-S 
 347-S 
 
 347-4 
 347-3 
 347-3 
 347-2 
 347-2 
 347-1 
 
 231-4 
 
 347-0 
 
 231-3 
 
 347-0 
 
 231-3 
 
 346-9 
 
 231.2 
 
 346.8 
 
 231.1 
 
 346-7 
 
 231. 1 
 
 346.6 
 
 231.0 
 
 346.6 
 
 231.0 
 
 346.5 
 
 230.9 
 
 346-4 
 
 230.8 
 
 346-3 
 
 230.8 
 
 346.2 
 
 230.7 
 
 346.1 
 
 230.7 
 
 346.0 
 
 230.6 
 
 34S-9 
 
 230.5 
 
 345-8 
 
 230.4 
 
 34S-7 
 
 230.4 
 
 345-5 
 
 230-3 
 
 345-4 
 
 230.2 
 
 345-3 
 
 230.1 
 
 345-2 
 
 230.0 
 
 345-0 
 
 229.9 
 
 344-9 
 
 229.9 
 
 344-8 
 
 229.8 
 
 344-6 
 
 344-5 
 
 20' 
 
 longitude. 
 
 463-8 
 463.8 
 463-8 
 463.8 
 463.8 
 463-8 
 
 463.8 
 463-7 
 463-7 
 463-6 
 463.6 
 463.6 
 
 463-6 
 463-S 
 463-4 
 463-4 
 463-3 
 463-3 
 
 463.2 
 463.1 
 463.0 
 463.0 
 462-9 
 462.8 
 
 462.7 
 462.6 
 462.5 
 462.4 
 462.3 
 462.2 
 
 462.1 
 462.0 
 461.8 
 461.7 
 
 461 -6 
 461.4 
 
 461.3 
 461.2 
 461.0 
 460.9 
 460.7 
 460.6 
 
 460.4 
 460.2 
 460.0 
 459-9 
 459-7 
 4S9-5 
 
 459-4 
 
 25' 
 longitude. 
 
 579-8 
 579-8 
 
 579-8 
 579.8 
 579-8 
 579-7 
 
 579-7 
 579-6 
 579-6 
 5796 
 579-6 
 579-S 
 
 579-4 
 579-4 
 579-3 
 579-2 
 579-2 
 579-1 
 
 579.0 
 
 578.8 
 
 578.6 
 578-5 
 
 578.4 
 578.2 
 578.2 
 578.0 
 577.8 
 577-8 
 
 577-6 
 577-4 
 S77-3 
 577-1 
 577-0 
 576.8 
 
 576.6 
 576.4 
 576.2 
 576-1 
 575-9 
 S75-7 
 
 575-5 
 S7S-3 
 575-0 
 574-8 
 574.6 
 574.4 
 
 574-2 
 
 30' 
 
 longitude. 
 
 695.8 
 695.8 
 695.7 
 695-7 
 695-7 
 695.6 
 
 695.6 
 
 695-5 
 695.5 
 
 695-5 
 695.4 
 
 695-3 
 695-3 
 695-2 
 695.0 
 695.0 
 694-9 
 
 694.8 
 
 694-7 
 694.6 
 694.4 
 
 694-3 
 694.2 
 
 694.1 
 693-9 
 
 693.6 
 693-4 
 693-3 
 
 693.1 
 692-9 
 692.8 
 692.5 
 692.3 
 692.2 
 
 692.0 
 
 691-7 
 691.5 
 691.3 
 691. 1 
 690.8 
 
 690.6 
 690.4 
 690.1 
 689.8 
 689.6 
 689.3 
 
 689.0 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 I>S 
 
 mm, 
 0.0 
 0.0 
 0.0 
 0.0 
 0.0 
 0.0 
 
 0.0 
 0.0 
 0.0 
 0.0 
 0.1 
 0.1 
 
 0.0 
 0.0 
 0.1 
 0.1 
 0.1 
 0.2 
 
 0.0 
 0.0 
 O.I 
 O.l 
 0.2 
 
 0-3 
 
 8° 
 
 0.0 
 0.0 
 
 0.1 
 0.2 
 
 0-3 
 0.4 
 
 mm. 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 O.I 
 
 0.0 
 0.0 
 0.0 
 0.1 
 0.1 
 0.2 
 
 0.0 
 0.0 
 O.I 
 O.I 
 0.2 
 
 0-3 
 
 0.0 
 0.0 
 O.I 
 0.2 
 
 0.3 
 0.4 
 
 Smithsonian Tables. 
 
 1,^2 
 
CO-ORDINATES FOR PROJECTION OF MAPS. 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 Table 24. 
 
 SCALE 8 a tail - 
 
 21 
 ■9S 
 
 i 
 =308 
 
 .3 o-oiJ 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 S' 
 longitude. 
 
 10' 
 longitude. 
 
 IS' 
 
 longitude. 
 
 20' 
 longitude. 
 
 ZS' 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 8°00' 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 900 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 
 10 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 11 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 12 00 
 10 
 20 
 
 30 
 40 
 SO 
 
 1300 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 1400 
 10 
 
 20 
 
 30 
 40 
 
 SO 
 iSpo 
 
 10 
 20 
 
 30 
 40 
 
 so 
 i6oo 
 
 230.4 
 460.8 
 
 691.2 
 
 921.6 
 
 II 52.0 
 
 230.4 
 460.8 
 691.2 
 921.6 
 
 1152.0 
 
 230.4 
 460.8 
 691.3 
 921.7 
 
 II52.I 
 
 230.4 
 
 460.9 
 
 691.3 
 
 921.8 
 1152.2 
 
 230.4 
 
 460.9 
 
 691.2 
 
 921.8 
 
 1152.2 
 
 230-5 
 460.9 
 691.4 
 921.9 
 1 1 52.4 
 
 230.5 
 
 461.0 
 
 691.5 
 
 922.0 
 
 1 1 52.4 
 
 230.5 
 
 461.0 
 
 691.5 
 
 922.0 
 
 1 1 52.6 
 
 1 14.8 
 
 1 14.8 
 114.7 
 114.7 
 
 1 14.6 
 114.6 
 
 1 14.5 
 H4.5 
 114.4 
 114.4 
 
 1 14-3 
 1 14-3 
 
 114.2 
 1 1 4.2 
 114.1 
 1 1 4.0 
 114.0 
 1 1 3-9 
 
 113-8 
 113.8 
 
 113-7 
 113.6 
 1 1 3-6 
 "3-S 
 
 113-4 
 1 1 3-4 
 113-3 
 113.2 
 1 13.2 
 113.1 
 
 11 3-0 
 
 1 1 2.9 
 1 1 2.8 
 1 1 2.8 
 112.7 
 
 1 1 2.6 
 
 112.5 
 112.5 
 1 1 2.4 
 112.3 
 112.2 
 112.1 
 
 112.0 
 111.9 
 1 1 1.8 
 111.8 
 111.7 
 111.6 
 
 229.7 
 
 344- S 
 
 229.6 
 
 344-4 
 
 229.5 
 
 344-2 
 
 229.4 
 
 344-1 
 
 229.3 
 
 343-9 
 
 229.2 
 
 343-ii 
 
 229.1 
 229.0 
 228.9 
 228.7 
 228.6 
 228.5 
 
 228.4 
 228.3 
 228.2 
 228.0 
 227.9 
 227.8 
 
 227.7 
 227.5 
 227.4 
 227.3 
 
 227.1 
 227.0 
 
 226.9 
 226.7 
 226.6 
 226.4 
 
 226.3 
 
 226.2 
 226.0 
 
 225.9 
 225.7 
 
 225.6 
 
 225.4 
 
 225.2 
 
 225.1 
 224.9 
 
 224.7 
 
 224.6 
 224.4 
 224.2 
 
 224.1 
 
 223.9 
 223.7 
 
 223-S 
 223-3 
 
 223.2 
 
 I I 1.5 223.0 
 
 343-6 
 343-4 
 343-3 
 343-1 
 343-0 
 342.8 
 
 342.6 
 342-4 
 342-3 
 342.1 
 
 341-9 
 341-7 
 
 341-S 
 341-3 
 341-1 
 340-9 
 340-7 
 340-5 
 
 340-3 
 340.1 
 
 339-9 
 339-7 
 339-4 
 339-2 
 
 339-0 
 338-8 
 338-6 
 338-3 
 338-1 
 337-9 
 
 337-6 
 337-4 
 
 337-! 
 336-8 
 336-6 
 336-4 
 
 336-1 
 335-8 
 33S-6 
 335-3 
 33S-0 
 334-7 
 
 459-4 
 459.2 
 
 459-0 
 458.8 
 458-6 
 458-4 
 
 458.2 
 457-9 
 457-7 
 457-5 
 457-3 
 457-0 
 
 456-8 
 456-6 
 456.4 
 456.1 
 455-8 
 455.6 
 
 455-4 
 4SS-? 
 454-8 
 454-6 
 454-3 
 454-0 
 
 453-8 
 453-5 
 453-2 
 452.9 
 452.6 
 452-3 
 
 452.0 
 
 451-7 
 451.4 
 451.1 
 450.8 
 450.5 
 
 450-2 
 449-8 
 449-5 
 449.1 
 448.8 
 448.5 
 
 448.1 
 447.8 
 
 447-4 
 447.0 
 446.7 
 446-3 
 
 334-S 446-0 
 
 574.2 
 574.0 
 573-7 
 573-4 
 S73-2 
 S73-0 
 
 572.7 
 572.4 
 572.2 
 
 S7I-8 
 571.6 
 
 571-3 
 
 571.0 
 570-8 
 570-4 
 570.1 
 569.8 
 569.5 
 
 569.2 
 568.8 
 568.6 
 568.2 
 567.8 
 567-6 
 
 567-2 
 566.8 
 
 566.5 
 566.1 
 565.8 
 565.4 
 
 564.6 
 564.2 
 563-9 
 563-5 
 563-1 
 
 562-7 
 562.3 
 561.8 
 561-4 
 561.0 
 560.6 
 
 560.2 
 
 559-7 
 559.2 
 558.8 
 558-4 
 557-9 
 
 SS7-4 
 
 689.0 
 688.7 
 688.4 
 688.1 
 687.8 
 687.5 
 
 687.2 
 686.9 
 686.6 
 686.2 
 685.9 
 685.6 
 
 685.3 
 684.9 
 684.5 
 684.1 
 683.8 
 683.4 
 
 683.0 
 682.6 
 682.3 
 681.8 
 681.4 
 681.1 
 
 680.6 
 680.2 
 679-8 
 
 679-3 
 678.9 
 
 678.5 
 
 678.1 
 677.6 
 677.1 
 676.7 
 676.2 
 675-7 
 
 675.2 
 674.8 
 674.2 
 
 673-7 
 673.2 
 672.7 
 
 672.2 
 671.6 
 671.1 
 670.6 
 670.0 
 669.5 
 
 668.9 
 
 •as 
 
 go 
 
 0.0 
 0.0 
 0.1 
 0.2 
 
 03 
 0.4 
 
 0.0 
 
 0.1 
 0.1 
 0.2 
 0.4 
 0.5 
 
 0.0 
 O.I 
 
 0.2 
 
 0-3 
 0.4 
 0.6 
 
 14" 
 
 s 
 
 0.0 
 
 10 
 
 O.I 
 
 M 
 
 0.2 
 
 20 
 
 0-3 
 
 2S 
 
 0.5 
 
 30 
 
 0.7 
 
 16° 
 
 0.0 
 
 0.1 
 0.2 
 0.4 
 
 0.6 
 0.8 
 
 mvi. 
 0.0 
 O.I 
 0.1 
 0.2 
 
 0-3 
 
 0.5 
 
 0.0 
 0.1 
 O.I 
 0.2 
 0.4 
 
 0.6 
 
 13° 
 
 0.0 
 
 O.I 
 
 0.2 
 0-3 
 
 0.5 
 
 0-7 
 
 15° 
 
 0.0 
 0.1 
 0.2 
 
 0-3 
 0.5 
 0.8 
 
 Smithsonian Tables. 
 
 133 
 
Table 24. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE -gnhsV' 
 [Derivation of table explained on pp. liii-lvi.] 
 
 II 
 
 .2 »-DJi 
 
 M S P, 
 
 s 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 5' 
 
 longitude. 
 
 lO' 
 
 longitude. 
 
 longitude. 
 
 20' 
 longitude. 
 
 25 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 l6°oo' 
 lo 
 zo 
 
 3° 
 
 40 
 
 5° 
 
 17 00 
 10 
 20 
 
 3° 
 40 
 
 5° 
 
 18 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 1900 
 10 
 20 
 
 30 
 40 
 
 50 
 
 2000 
 10 
 20 
 
 30 
 40 
 
 50 
 
 21 00 
 10 
 20 
 
 3° 
 40 
 
 SO 
 
 22 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 23 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 2400 
 
 230.5 
 461. 1 
 691.6 
 922.1 
 1 1 52.6 
 
 230.6 
 461. 1 
 691.6 
 922.2 
 1152.8 
 
 230.6 
 46r.i 
 691.7 
 922.3 
 1 1 52.8 
 
 230.6 
 461.2 
 691.8 
 922.4 
 1153.0 
 
 230.6 
 461.2 
 691.9 
 922.5 
 IIS3-I 
 
 230.6 
 461.3 
 692.0 
 922.6 
 "53-2 
 
 230.7 
 461.4 
 692.0 
 922.7 
 "534 
 
 230.7 
 461.4 
 692.1 
 922.8 
 "53-6 
 
 1H.5 
 111.4 
 111.3 
 III. 2 
 III. I 
 
 III.O 
 
 1 10.9 
 1 10.8 
 1 10.7 
 1 10.6 
 1 10.5 
 1 10.4 
 
 110.3 
 110.2 
 
 IIO.I 
 
 1 1 0.0 
 
 109.9 
 
 109.8 
 
 109.7 
 
 109.6 
 
 109.5 
 109.4 
 
 109.2 
 109.1 
 
 109.0 
 108.9 
 108.8 
 108.7 
 108.5 
 108.4 
 
 108.3 
 
 108.2 
 
 108. 1 
 107.9 
 107.8 
 107.7 
 
 107.6 
 
 107.4 
 
 107-3 
 107.2 
 107.1 
 106.9 
 
 106.8 
 106.7 
 106.5 
 106.4 
 106.3 
 1 06. 1 
 
 106.0 
 
 223.0 
 
 334-5 
 
 222.8 
 
 334-2 
 
 222.6 
 
 333-9 
 
 222.4 
 
 333-6 
 
 222.2 
 
 333-3 
 
 222.0 
 
 333-1 
 
 221.8 
 
 221.6 
 221.4 
 221.2 
 221.0 
 220.8 
 
 220.6 
 220.4 
 220.2 
 220.0 
 219.8 
 219.6 
 
 219.4 
 219.1 
 218.9 
 218.7 
 218.5 
 218.2 
 
 218.0 
 217.8 
 ZI7.5 
 
 21 7-3 
 217.1 
 216.8 
 
 216.6 
 216.4 
 216.1 
 215.9 
 215.6 
 215.4 
 
 215.1 
 214.9 
 214.6 
 214.4 
 214.1 
 213.9 
 
 213.6 
 
 213-3 
 213.1 
 212.8 
 212.5 
 212.3 
 
 212.0 
 
 332-8 
 332-5 
 332-2 
 331-9 
 331-6 
 331-3 
 
 331-0 
 330-6 
 330-3 
 330-0 
 329-7 
 329-4 
 
 329.0 
 328.7 
 328.4 
 328.0 
 327-7 
 327-4 
 
 327.0 
 326-7 
 326-3 
 326.0 
 325.6 
 325-3 
 
 324-9 
 324-5 
 324.2 
 
 323-8 
 323-4 
 323-1 
 
 322.7 
 322-3 
 321.9 
 321.6 
 321.2 
 320.8 
 
 320.4 
 320.0 
 319.6 
 319.2 
 318.8 
 318.4 
 
 318.0 
 
 446.0 
 445-6 
 445.2 
 444.8 
 444.4 
 444.1 
 
 443-7 
 443-3 
 442-9 
 442.5 
 442.1 
 441.7 
 
 441-3 
 440.8 
 440.4 
 440.0 
 439-6 
 439-2 
 
 438-7 
 438-3 
 437-8 
 437-4 
 436-9 
 436-5 
 
 436.0 
 435-6 
 435-1 
 434-6 
 434-2 
 433-7 
 
 433-2 
 432-7 
 432-2 
 
 431-7 
 431.2 
 430.8 
 
 430-3 
 429.8 
 429.2 
 428.8 
 428.2 
 427.7 
 
 427.2 
 426.6 
 426.1 
 425.6 
 425.0 
 424-5 
 
 424.0 
 
 557-4 
 557-0 
 
 556-5 
 556.0 
 
 555-6 
 
 55S-I 
 
 554-6 
 S54-I 
 553-6 
 
 553-1 
 552.6 
 
 552-1 
 
 551-6 
 
 551.0 , 
 
 550.6 
 
 550.0 
 
 549-4 
 
 549-0 
 
 548-4 
 547-8 
 
 547-3 
 546.8 
 546.1 
 545-6 
 
 545-0 
 544-4 
 543-8 
 543-3 
 542.7 
 542.1 
 
 S4I-5 
 540-9 
 540-3 
 S39-6 
 539-0 
 538-4 
 
 537-8 
 
 53I-? 
 536-6 
 
 536.0 
 
 535-3 
 534-6 
 
 534-0 
 
 533-3 
 532-6 
 532.0 
 
 531-3 
 530-6 
 
 530.0 
 
 668.9 
 668.3 
 667.8 
 667.2 
 666.7 
 666.1 
 
 664.9 
 664.3 
 663.7 
 663.1 
 662.5 
 
 661.9 
 661.3 
 660.7 
 660.0 
 
 659-3 
 658.7 
 
 658.1 
 
 657-4 
 656.8 
 656.1 
 
 655-4 
 654.7 
 
 654.1 
 
 652.6 
 652.0 
 651.2 
 650-5 
 
 649.8 
 649.1 
 648.4 
 647.6 
 646.9 
 646.1 
 
 645.4 
 644.6 
 
 643-9 
 643.1 
 642.4 
 641.6 
 
 640.8 
 640.0 
 639.2 
 638.4 
 637.6 
 636.8 
 
 636.0 
 
 = 13 
 
 
 mm. 
 
 THm. 
 
 .5' 
 
 0.0 
 
 0.0 
 
 TO 
 
 O.I 
 
 O.I 
 
 15 
 
 0.2 
 
 0.2 
 
 20 
 
 0.4 
 
 0.4 
 
 2S 
 
 0.6 
 
 0.6 
 
 30 
 
 0.8 
 
 0.8 
 
 16° 
 
 18° 
 
 0.0 
 0.1 
 
 0.2 
 
 0.4 
 
 0.6 
 0.9 
 
 0.0 
 0.1 
 0.2 
 0.4 
 0.7 
 
 I.O 
 
 s 
 
 0.0 
 
 10 
 
 0.1 
 
 15 
 
 0-3 
 
 20 
 
 o-S 
 
 25 
 
 0-7 
 
 30 
 
 1.1 
 
 24° 
 
 0.0 
 
 0.1 
 
 0-3 
 0.5 
 0.8 
 I.I 
 
 17" 
 
 19° 
 
 0.0 
 0.1 
 0.2 
 0.4 
 0.6 
 0.9 
 
 0.0 
 
 O.I 
 
 0.3- 
 
 0-5 
 0-7 
 1.0 
 
 23° 
 
 0.0 
 0.1 
 
 0-3 
 0.5 
 0.8 
 i.i 
 
 Smithsonian Tables. 
 
 134 
 
Table 24. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE tish^- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 •S3 
 
 24"00' 
 10 
 20 
 
 3° 
 40 
 
 SO 
 
 2500 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 2600 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 27 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 28 00 
 10 
 20 
 
 30 
 40 
 
 so 
 
 2900 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 3000 
 10 
 20 
 30 
 40 
 SO 
 
 31 00 
 10 
 20 
 30 
 40 
 SO 
 
 3200 
 
 I 
 
 s S > « 
 ^S S p. 
 
 230.7 
 461.5 
 
 692.2 
 923.0 
 
 "S37 
 
 230.8 
 461.5 
 692.3 
 923.1 
 1153.8 
 
 230.8 
 461.6 
 692.4 
 923.2 
 1 1 54.0 
 
 230.8 
 461.7 
 692.5 
 
 923-3 
 1 1 54.2 
 
 230.9 
 461.7 
 692.6 
 
 923-5 
 1 1 54.4 
 
 230.9 
 461.8 
 692.7 
 923.6 
 1154.5 
 
 230.9 
 461.9 
 692.8 
 923.8 
 1 1 54.7 
 
 231.0 
 461.9 
 692.9 
 923-9 
 1 154-8 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 longitude. 
 
 106.0 
 105.9 
 105.7 
 105.6 
 105.4 
 "05-3 
 
 105.2 
 105.0 
 104.9 
 
 104-7 
 104.6 
 104.4 
 
 104.3 
 1 04. 1 
 104.0 
 103.8 
 103-7 
 "03-5 
 
 103.4 
 103.2 
 103.1 
 102.9 
 102.8 
 102.6 
 
 102.5 
 102.3 
 102. 1 
 102.0 
 101.8 
 101.7 
 
 101.5 
 101.3 
 101.2 
 
 lOI.O 
 
 100.8 
 100.7 
 
 100.5 
 100.3 
 100.2 
 1 00.0 
 99-8 
 99-6 
 
 99-S 
 99-3 
 99.1 
 99.0 
 98.8 
 98.6 
 
 98.4 
 
 10' 
 longitude. 
 
 212.0 
 
 21 1.7 
 
 2II.4 
 
 2II.2 
 210.9 
 210.6 
 
 210.3 
 210.0 
 209.7 
 209.4 
 209.2 
 208.9 
 
 208.6 
 208.3 
 208.0 
 207.7 
 207.4 
 207.1 
 
 206.8 
 206.5 
 206.2 
 205.8 
 
 205-S 
 205.2 
 
 204.9 
 204.6 
 204.3 
 204.0 
 203.6 
 203-3 
 
 203.0 
 202.7 
 202.3 
 202.0 
 201.7 
 201.4 
 
 201.0 
 200.7 
 200.3 
 200.0 
 199.6 
 
 "99-3 
 
 199.0 
 198.6 
 198.3 
 197.9 
 197.6 
 197.2 
 
 196.9 
 
 "S' 
 
 longitude. 
 
 318.0 
 3"7-6 
 317-2 
 
 3"6.7 
 3'6.3 
 3" 5-9 
 
 3"S-5 
 3'S-o 
 314.6 
 314.2 
 3"3-7 
 313-3 
 
 312.9 
 312.4 
 312.0 
 
 3"-S 
 311. 1 
 310.6 
 
 310.2 
 
 309-7 
 309.2 
 308.8 
 308.3 
 307-9 
 
 307-4 
 306.9 
 306.4 
 305-9 
 30S-S 
 305.0 
 
 304-5 
 304.0 
 
 303-5 
 303-0 
 302.5 
 302.0 
 
 301-S 
 301.0 
 
 300.5 
 300.0 
 299.5 
 299.0 
 
 298.4 
 297.9 
 297.4 
 296.9 
 296.3 
 295.8 
 
 295-3 
 
 20' 
 longitude. 
 
 424.0 
 
 423-4 
 422.9 
 422.3 
 421.8 
 421.2 
 
 420.6 
 420.0 
 419.5 
 418.9 
 418.3 
 4"7-7 
 
 417.2 
 416.6 
 416.0 
 
 415-4 
 414.8 
 414.2 
 
 413.6 
 413.0 
 412.3 
 41 1.7 
 411. 1 
 410.5 
 
 409.8 
 409.2 
 408.6 
 407-9 
 407-3 
 406.6 
 
 406.0 
 
 405-4 
 404.7 
 404.0 
 
 403-4 
 402.7 
 
 402.0 
 401.4 
 400.7 
 400.0 
 
 399-3 
 398.6 
 
 397-9 
 397.2 
 
 396-5 
 395-8 
 395-" 
 394-4 
 
 393-7 
 
 25' 
 
 longitude. 
 
 530.0 
 
 529-3 
 528.6 
 527.9 
 527.2 
 526.5 
 
 525.8 
 525.0 
 
 524-4 
 523.6 
 522.9 
 522.2 
 
 521.4 
 520.7 
 520.0 
 519.2 
 518.4 
 S17-7 
 
 S"7-o 
 516.2 
 
 S"S-4 
 514.6 
 
 5"3-8 
 5" 3-" 
 
 5" 2.3 
 511.5 
 510.7 
 509.9 
 509.1 
 S08.3 
 
 S07-S 
 506.7 
 505.8 
 
 505-0 
 504.2 
 
 503-4 
 
 502.6 
 501.7 
 500.8 
 500.0 
 
 499-1 
 498.2 
 
 497-4 
 496.5 
 495-6 
 494.8 
 
 493-9 
 493-0 
 
 492.2 
 
 30' 
 longitude. 
 
 636.0 
 635.2 
 634-3 
 
 632.6 
 631.8 
 
 631.0 
 630.1 
 629.2 
 628.3 
 627.5 
 626.6 
 
 625.7 
 624.8 
 
 623-9 
 623.0 
 622.1 
 621.2 
 
 620.3 
 619.4 
 618.5 
 617.5 
 616.6 
 615.7 
 
 614.8 
 613.8 
 612.8 
 611.9 
 610.9 
 610.0 
 
 609.0 
 608.0 
 607.0 
 606.0 
 605.0 
 604.1 
 
 603.1 
 602.0 
 601.0 
 
 599-9 
 598.9 
 
 597-9 
 
 596-9 
 595.8 
 594-8 
 593-8 
 592-7 
 591.6 
 
 590.6 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 go 
 1-1 •■" 
 
 24- 
 
 mtn. 
 0.0 
 O.I 
 
 0-3 
 0.5 
 
 0.8 
 I.I 
 
 26° 
 
 0.0 
 0.1 
 
 0-3 
 0.5 
 0.8 
 1.2 
 
 28° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 1-3 
 
 30° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 "•3 
 
 32° 
 
 0.0 
 0.2 
 
 0-3 
 0.6 
 0.9 
 1.4 
 
 25" 
 
 0.0 
 
 O.I 
 
 0-3 
 
 1.2 
 
 27- 
 
 0.0 
 
 0.1 
 
 0-3 
 0.5 
 0.8 
 1.2 
 
 29° 
 
 0.0 
 0.1 
 
 o. 
 o. 
 
 0.9 
 
 "•3 
 
 ;.i 
 
 3"° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 "•3 
 
 Smithsonian Tables. 
 
 "35 
 
Table 24. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE g^,^. 
 [Derivation of table explained on pp. liii-lvi.] 
 
 "S 
 
 ional dis- 
 :5 from 
 degree 
 
 lels. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 ORDINATES OF 
 DEVELOPED 
 
 ll 
 
 
 
 
 
 
 
 
 rm 
 
 S' 
 
 10' 
 
 >S' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 Hs. 
 
 gjsSS. 
 
 longitude. 
 
 longitude. 
 
 longitude . 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 fnm. 
 
 mm. 
 
 mm. 
 
 ■II 
 
 
 
 32000' 
 
 
 98.4 
 
 196.9 
 
 295-3 
 
 393-7 
 
 492.2 
 
 si2-3 
 
 32° 
 
 33° 
 
 10 
 
 20 
 30 
 
 231.0 
 462.0 
 693.0 
 
 98.2 
 98.1 
 
 97-9 
 
 196.5 
 
 ig6.i 
 195.8 
 
 294.8 
 294.2 
 293-7 
 
 393-0 
 392-3 
 391.6 
 
 491.2 
 490-4 
 
 ij 
 
 
 
 
 mm. 
 
 mm. 
 
 40 
 
 924.0 
 
 977 
 
 195.4 
 
 293.1 
 
 390.8 
 
 488.6 
 
 586.3 
 
 S' 
 10 
 
 0.0 
 
 0.0 
 
 SO 
 
 "SS-o 
 
 97-S 
 
 195.1 
 
 292.6 
 
 390.1 
 
 487.6 
 
 585.2 
 
 0.2 
 
 0.2 
 
 3300 
 
 
 974 
 
 194.7 
 
 292.1 
 
 389.4 
 
 486.8 
 
 584.1 
 
 IS 
 
 20 
 
 ai 
 
 oi 
 
 10 
 
 231.0 
 
 97.2 
 
 194-3 
 
 291.5 
 
 388.6 
 
 485.8 
 
 583-0 
 
 25 
 30 
 
 0.9 
 1-4 
 
 1.0 
 
 20 
 
 462.1 
 
 97-° 
 
 194.0 
 
 290.9 
 
 387-9 
 
 484.9 
 
 580.8 
 
 1.4 
 
 30 
 
 693.2 
 
 96.8 
 
 193.6 
 
 290.4 
 
 387.2 
 
 484.0 
 
 
 40 
 
 924.2 
 
 96.6 
 96.4 
 
 193.2 
 192.8 
 
 289.S 
 289.3 
 
 386.4 
 38S-7 
 
 483-0 
 482.1 
 
 S79-7 
 578-5 
 
 
 
 
 5° 
 
 USS-2 
 
 
 
 
 3400 
 10 
 
 23I.I 
 
 96.2 
 96.0 
 
 192.5 
 192.1 
 
 288.7 
 288.Z 
 
 385-0 
 384-2 
 
 481.2 
 480.2 
 
 S774 
 576.3 
 
 
 34' 
 
 35° 
 
 
 
 
 20 
 
 462.2 
 
 9S-9 
 
 191.7 
 
 287.6 
 
 383-4 
 
 479-3 
 
 S7S-2 
 
 s 
 
 0.0 
 
 0.0 
 
 30 
 
 693.2 
 
 9S7 
 
 191 -3 
 
 287.0 
 
 382.6 
 
 478.3 
 
 574-0 
 
 10 
 
 0.2 
 
 0.2 
 
 40 
 
 9243 
 
 9SS 
 
 190.9 
 
 286.4 
 
 381.9 
 
 4774 
 
 572.8 
 
 «5 
 
 0.4 
 
 0.4 
 
 SO 
 
 "SS4 
 
 9S-3 
 
 190.6 
 
 285.8 
 
 381.1 
 
 476.4 
 
 5717 
 
 20 
 25 
 
 0.6 
 
 I.O 
 
 0.6 
 1.0 
 
 3500 
 
 
 95.1 
 
 '§°-S 
 
 285.3 
 
 380.4 
 
 4754 
 
 570.5 
 
 30 
 
 1.4 
 
 1.4 
 
 10 
 
 Z31.1 
 
 94.9 
 
 189.8 
 
 284.7 
 
 379-6 
 
 474-S 
 
 5694 
 
 
 
 
 20 
 30 
 
 462.Z 
 6934 
 
 947 
 945 
 
 189.4 
 189.0 
 
 284.1 
 283.5 
 
 378.8 
 378-0 
 
 473-S 
 472-S 
 471.6 
 
 568.2 
 567.0 
 
 
 
 
 
 
 
 40 
 
 924-S 
 
 94-3 
 
 188.6 
 
 282.9 
 
 377-2 
 
 565-9 
 
 
 36° 
 
 37° 
 
 so 
 3600 
 
 1155.6 
 
 94.1 
 93-9 
 
 188.2 
 187.8 
 
 282.4 
 281.8 
 
 376-S 
 37S7 
 
 470.6 
 469.6 
 
 564.7 
 S63-S 
 
 
 
 
 5 
 
 0.0 
 
 0.0 
 
 10 
 
 231.2 
 
 937 
 
 187.4 
 
 281.2 
 
 374-9 
 
 468.6 
 
 562-3 
 
 10 
 
 0.2 
 
 0.2 
 
 20 
 
 462.3 
 
 93-S 
 
 187.0 
 
 280.6 
 
 374-1 
 
 467.6 . 
 
 561.1 
 
 IS 
 
 0.4 
 
 0.4 
 
 3° 
 
 693-5 
 924.6 
 
 93-3 
 
 186.6 
 
 280.0 
 
 373-3 
 
 466.6 
 
 559-9 
 5587 
 
 20 
 
 0.6 
 
 0.6 
 
 40 
 
 93' I 
 
 186.2 
 
 279.4 
 
 372-S 
 
 465-6 
 
 25 
 
 1.0 
 
 1.0 
 
 so 
 
 1 1 55.8 
 
 92.9 
 
 185.8 
 
 278.8 
 
 3717 
 
 464.6 
 
 S57-S 
 
 30 
 
 1.4 
 
 I-S 
 
 3700 
 10 
 
 
 92.7 
 92-S 
 
 185.4 
 185.0 
 
 278.2 
 277.6 
 
 370-9 
 370.1 
 
 462.6 
 
 556-3 
 SS5-I 
 
 
 
 
 231.2 
 
 
 38° 
 
 irfi 
 
 20 
 
 693.6 
 
 92-3 
 
 184.6 
 184.2 
 
 276.9 
 276.3 
 
 369.2 
 368.4 
 
 461.6 
 460.5 
 
 S53-9 
 552.6 
 
 
 39 
 
 30 
 
 92.1 
 
 
 
 
 40 
 
 924.8 
 
 91.9 
 
 183.8 
 
 2757 
 
 3f7-6 
 
 459-5 
 
 5514 
 
 5 
 
 0.0 
 
 0.0 
 
 SO 
 
 1 1 56.0 
 
 91.7 
 
 183.4 
 
 27S-I 
 
 366.8 
 
 458-S 
 
 550-2 
 
 10 
 
 0.2 
 
 0.2 
 
 3800 
 
 
 9I-S 
 
 183.0 
 
 274.5 
 
 366.0 
 
 457-4 
 
 548.9 
 
 IS 
 20 
 
 0.4 
 0.7 
 1.0 
 
 0.4 
 0.7 
 
 10 
 
 231.2 
 
 9' -3 
 
 182.6 
 
 273.8 
 
 365-1 
 
 456.4 
 
 5477 
 
 25 
 
 1.0 
 
 20 
 
 462.5 
 
 91.1 
 
 182.1 
 
 273.2 
 
 364-3 
 
 4554 
 
 546-4 
 
 30 
 
 i-S 
 
 1-5 
 
 30 
 
 6937 
 
 90.9 
 
 181.7 
 
 272.6 
 
 363-5 
 
 4544 
 
 S45-2 
 
 
 40 
 
 925.0 
 1156.2 
 
 90.7 
 
 181.3 
 180.9 
 
 272.0 
 
 362.6 
 361.8 
 
 453-3 
 
 544-0 
 
 
 
 ^ 
 
 5° 
 
 90.4 
 
 271.4 
 
 452.2 
 
 542-7 
 
 
 ._,o 
 
 
 3900 
 10 
 
 231.3 
 
 90.2 
 
 180.5 
 180.1 
 
 270.7 
 270.1 
 
 361.0 
 360.1 
 
 451.2 
 450.2 
 
 541.4 
 540.2 
 
 
 40° 
 
 
 
 
 
 20 
 
 462.6 
 
 89.8 
 
 179.6 
 
 269.4 
 
 359-2 
 
 449.0 
 
 538.9 
 
 5 
 
 0.0 
 
 
 30 
 
 693.8 
 
 89.6 
 
 179.2 
 
 268.8 
 
 358-4 
 
 448.0 
 
 537-6 
 
 10 
 
 0.2 
 
 
 40 
 
 925.1 
 
 89.4 
 
 178.8 
 
 268.2 
 
 357-6 
 
 447.0 
 
 536.3 
 
 15 
 
 0.4 
 
 
 SO 
 
 1156.4 
 
 89.2 
 
 178-3 
 
 267.5 
 
 356-7 
 
 445-8 
 
 535-0 
 
 20 
 
 25 
 
 0.7 
 1.0 
 
 
 4000 
 
 
 89.0 
 
 177.9 
 
 266.9 
 
 355-8 
 
 -444-8 
 
 533-8 
 
 30 
 
 i-S 
 
 
 Smithsonian Tables. 
 
 136 
 
Table 24i 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE srshns- 
 [Derivation of table explained on pp. liii-lvij 
 
 S'rs 
 
 40"oo 
 
 10 
 20 
 
 30 
 40 
 
 so 
 
 41 CO 
 10 
 20 
 
 30 
 40 
 
 so 
 
 42 00 
 
 10 
 
 20 
 
 30 
 40 
 
 so 
 
 4300 
 
 10 
 20 
 
 30 
 40 
 
 so 
 
 4400 
 10 
 
 20 
 
 30 
 40 
 
 so 
 4S00 
 
 10 
 20 
 
 30 
 40 
 
 so 
 46 00 
 
 10 
 20 
 
 30 
 40 
 
 so 
 
 4700 
 
 10 
 20 
 
 30 
 40 
 
 so 
 4800 
 
 'as 
 
 462.6 
 694.0 
 
 1 1 56.0 
 
 2314 
 462.7 
 694.1 
 
 1 1 56.8 
 
 231 -4 
 
 462.8 
 
 694.2 
 
 925.6 
 
 1157.0 
 
 231.4 
 462.9 
 
 694-3 
 
 925.8 
 
 II57.2 
 
 23I-S 
 
 463.0 
 694.4 
 92S-9 
 
 1157.4 
 
 231-5 
 463.1 
 
 694.6 
 926.1 
 
 1157.6 
 
 231.6 
 463.1 
 694.7 
 
 926.3 
 
 II57.8 
 
 231.6 
 
 463.2 
 694.8 
 
 926.4 
 1 1 58.0 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 s' 
 
 longitude. 
 
 89.0 
 88.7 
 88.5 
 88.3 
 
 88.1 
 
 87.9 
 
 87.6 
 87.4 
 
 87.2 
 87.0 
 
 86.8 
 
 86.5 
 86.3 
 
 86.1 
 
 85.8 
 85.6 
 85.4 
 
 85.2 
 
 84.9 
 84.7 
 84-5 
 
 84.2 
 84.0 
 
 83.8 
 
 83.6 
 
 83-3 
 83.1 
 
 82.8 
 82.6 
 82.4 
 
 82.1 
 81.9 
 81.6 
 81.4 
 81.2 
 80.9 
 
 80.7 
 80.4 
 80.2 
 80.0 
 
 79-7 
 79-S 
 
 79-2 
 79.0 
 78.7 
 78.5 
 78.2 
 78.0 
 
 77-7 
 
 10' 
 longitude. 
 
 177-9 
 I77-S 
 177-0 
 176.6 
 176.2 
 I7S-7 
 
 I7S-3 
 174-8 
 174.4 
 
 173-9 
 I73-S 
 173-0 
 
 172.6 
 172.1 
 17 1. 7 
 171.2 
 170.8 
 170-3 
 
 169.9 
 169.4 
 169.0 
 168.5 
 168.0 
 167.6 
 
 167.1 
 166.6 
 166.2 
 165.7 
 165.2 
 164.7 
 
 164.3 
 163.8 
 163-3 
 162.8 
 162.3 
 1 61 .9 
 
 1 61. 4 
 160.9 
 160.4 
 159.9 
 
 •594 
 158.9 
 
 1585 
 158.0 
 
 157-5 
 157-0 
 156.5 
 156.0 
 
 IS5-5 
 
 «S' 
 
 longitude. 
 
 266.9 
 266.2 
 265.6 
 264.9 
 264.2 
 263.6 
 
 262.9 
 262.3 
 261.6 
 260.9 
 260.2 
 259.6 
 
 258.9 
 258.2 
 257.6 
 256.9 
 256.2 
 
 2SS-S 
 
 254.8 
 254.1 
 253-4 
 
 252.8 
 252.0 
 
 251-3 
 
 250.6 
 249.9 
 249.2 
 248.5 
 247.8 
 247.1 
 
 246.4 
 
 245.7 
 245.0 
 244.2 
 
 243-5 
 242.8 
 
 242.1 
 241.4 
 240.6 
 
 239-9 
 239.2 
 
 238.4 
 
 237-7 
 236.9 
 236.2 
 235-S 
 234-7 
 234.0 
 
 233.2 
 
 20' 
 longitude. 
 
 355-8 
 355-0 
 354-1 
 353-2 
 352-3 
 3514 
 
 350.6 
 
 349-7 
 348-8 
 347-9 
 347-0 
 346.1 
 
 345-2 
 344-3 
 343-4 
 342- 5 
 341.6 
 
 340.7 
 
 339? 
 338-8 
 337-9 
 337-0 
 336-0 
 335-1 
 
 334-2 
 333-2 
 332-3 
 3314 
 330-4 
 329-5 
 
 328-S 
 327.6 
 326.6 
 325.6 
 324-7 
 323-7 
 
 322.8 
 321.8 
 320.8 
 319.8 
 318.9 
 317.9 
 
 316.9 
 315.9 
 314-9 
 314-0 
 313-0 
 312.0 
 
 311.0 
 
 25' 
 
 longitude. 
 
 444.8 
 
 443-7 
 442.6 
 
 441-5 
 440.4 
 
 439-3 
 
 438.2 
 
 437-1 
 436.0 
 434.8 
 
 433-8 
 432.6 
 
 431-5 
 430.4 
 429.2 
 428.1 
 427.0 
 425.8 
 
 424.7 
 423.6 
 422.4 
 421.2 
 420.0 
 418.9 
 
 417.8 
 416.6 
 415.4 
 414.2 
 413.0 
 411.8 
 
 410.6 
 
 409-4 
 408.2 
 407.0 
 405.8 
 404.6 
 
 4034 
 402.2 
 401.0 
 39q.8 
 398.6 
 397.4 
 
 396.2 
 
 394.9 
 3936 
 392.4 
 391.2 
 
 390-0 
 
 388.7 
 
 533^8 
 
 532-4 
 
 S3'-' 
 529.8 
 
 528.5 
 527.2 
 
 525.8 
 524.5 
 
 523-1 
 521.8 
 520.5 
 519.1 
 
 517.8 
 516.4 
 S'S-i 
 S13-7 
 512.3 
 511.0 
 
 509.6 
 508.3 
 506.9 
 
 S05-S 
 504.1 
 502.7 
 
 501-3 
 499-9 
 498-5 
 497.0 
 
 495-6 
 494.2 
 
 492.8 
 
 491-3 
 489.9 
 488.5 
 487.0 
 485.6 
 
 484.1 
 482.7 
 481.2 
 479.8 
 
 478-3 
 476.8 
 
 475-4 
 473.9 
 472.4 
 470.9 
 469.4 
 467-9 
 
 466.4 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 'aii 
 
 ^■s 
 
 
 
 mm. 
 
 5' 
 
 0.0 
 
 10 
 
 0.2 
 
 IS 
 
 0.4 
 
 20 
 
 0.7 
 
 25 
 
 I.O 
 
 30 
 
 1-S 
 
 40° 
 
 42" 
 
 46° 
 
 5 
 
 0.0 
 
 10 
 
 0.2 
 
 IS 
 
 0.4 
 
 20 
 
 0.7 
 
 25 
 
 i.i 
 
 30 
 
 1-5 
 
 41" 
 
 43° 
 
 47" 
 
 Smithsonian Tables. 
 
 137 
 
Table 24. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE snhnS' 
 
 [Derivation of table explained on pp. Uii-lvi.] 
 
 
 A 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 *« 
 
 lis 
 
 
 
 
 
 
 
 ORDINATES OF 
 DEVELOPED 
 
 o 
 
 01 — 
 
 •diJ 
 
 2 c 61) J 
 
 
 
 
 
 
 
 11 
 
 SsSs 
 
 S' 
 
 10' 
 
 "5' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 S^ 
 
 gasa 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 m?n. 
 
 mm. 
 
 tnm. 
 
 mm. 
 
 mm. 
 
 tntn. 
 
 mm. 
 
 1-^ 
 
 
 
 48°oo' 
 
 
 77-7 
 
 iSS-S 
 
 233-2 
 
 311.0 
 
 388.7 
 
 466.4 
 
 |.i 
 
 48° 
 
 49° 
 
 10 
 
 231.6 
 
 77-S 
 
 155.0 
 
 232.5 
 
 310.0 
 
 387-4 
 
 464-9 
 
 
 
 20 
 
 463-3 
 
 77-2 
 
 IS4-S 
 
 231.7 
 
 308.9 
 
 386.2 
 
 463-4 
 
 
 
 
 
 
 
 3° 
 
 695.0 
 
 77.0 
 
 154.0 
 
 230.9 
 
 307-9 
 
 384-9 
 
 461.9 
 
 
 mm 
 
 mm. 
 
 40 
 
 926.6 
 
 76.7 
 
 IS3-S 
 
 230.2 
 
 306.9 
 
 383-6 
 
 460.4 
 
 s' 
 
 10 
 
 0.0 
 
 0.0 
 
 SO 
 
 1158.2 
 
 76.5 
 
 IS2-9 
 
 229.4 
 
 305-9 
 
 382.4 
 
 458.8 
 
 0.2 
 
 0.2 
 
 49 00 
 
 
 76.2 
 
 152.4 
 
 228.7 
 
 304-9 
 
 381.1 
 
 457-3 
 455.8 
 
 15 
 
 20 
 
 0.4 
 0.7 
 
 0.4 
 0.7 
 
 10 
 
 2317 
 
 76.0 
 
 I SI -9 
 
 227.9 
 
 303-8 
 
 379-8 
 
 25 
 30 
 
 1.0 
 
 1.0 
 
 20 
 
 463.4 
 
 75-7 
 
 1 51. 4 
 
 227.1 
 
 302.8 
 
 378.6 
 
 454-3 
 
 i.S 
 
 1-S 
 
 3° 
 
 695.1 
 
 7S-4 
 
 150.9 
 
 226.4 
 
 301.8 
 
 377-2 
 
 452-7 
 
 40 
 SO 
 
 926.8 
 1 1 58.4 
 
 75.2 
 
 150.4 
 
 225.6 
 224.8 
 
 300.8 
 299.8 
 
 376.0 
 374-7 
 
 451.1 
 449.6 
 
 
 
 
 74-9 
 
 149.9 
 
 
 
 
 5000 
 10 
 
 231-7 
 
 74-7 
 74-4 
 
 w^ 
 
 224.0 
 223.3 
 
 298.7 
 297.7 
 
 373-4 
 372-1 
 
 448.1 
 446.5 
 
 
 50° 
 
 51° 
 
 
 
 
 20 
 
 463-S 
 
 74.2 
 
 148.3 
 
 147-8 
 
 222.5 
 
 296.6 
 
 370.8 
 
 445-0 
 
 5 
 
 0.0 
 
 0.0 
 
 30 
 
 695.2 
 
 73-9 
 
 221.7 
 
 295.6 
 
 S.i 
 
 443-4 
 
 10 
 
 0.2 
 
 0.2 
 
 40 
 
 926.9 
 
 73-6 
 
 '47-3 
 
 220.9 
 
 294.6 
 
 441.8 
 
 15 
 
 0.4 
 
 0.4 
 
 SO 
 
 1158.6 
 
 73-4 
 
 146.8 
 
 220.1 
 
 293-S 
 
 366.9 
 
 440-3 
 
 20 
 
 25 
 
 0.7 
 1.0 
 
 0-7 
 t.o 
 
 51 00 
 
 
 73-1 
 
 146.2 
 
 219.4 
 218.6 
 
 292.5 
 
 365.6 
 
 438-7 
 
 30 
 
 i-S 
 
 i-S 
 
 10 
 
 231.8 
 
 72.9 
 
 I4S-7 
 
 291.4 
 
 364-3 
 
 437-2 
 
 
 
 
 20 
 
 t^-^ 
 
 72.6 
 
 145.2 
 
 217.8 
 
 290.4 
 
 363-0 
 
 43S-S 
 
 
 
 
 30 
 
 69S-3 
 
 72-3 
 
 144.7 
 
 217.0 
 
 289.3 
 
 361.6 
 
 434-0 
 
 
 
 
 40 
 
 9^Z-J 
 
 72.1 
 
 144.1 
 
 216.2 
 
 288.3 
 
 360.4 
 
 432-4 
 
 
 52° 
 
 53° 
 
 SO 
 52 00 
 
 1158.8 
 
 71.8 
 71-S 
 
 143-6 
 I43-" 
 
 215.4 
 214.6 
 
 287.2 
 286.2 
 
 359.0 
 357.7 
 
 430.8 
 429.2 
 
 
 
 
 S 
 
 0.0 
 
 0.0 
 
 10 
 
 231.8" 
 
 71-3 
 
 142- s 
 
 213.8 
 
 285.1 
 
 356.4 
 
 427.6 
 
 10 
 
 0.2 
 
 0.2 
 
 20 
 
 463.6 
 
 71.0 
 
 142.0 
 
 213.0 
 
 284.0 
 
 355.0 
 
 426.1 
 
 15 
 
 0.4 
 
 0.4 
 
 30 
 
 695.4 
 
 70.7 
 
 141.5 
 
 212.2 
 
 283.0 
 
 353-7 
 
 424.4 
 
 20 
 
 0.7 
 
 0.6 
 
 40 
 
 927.2 
 
 70-S 
 
 140.9 
 
 2II.4 
 
 ^^'•f 
 
 352-4 
 
 422.8 
 
 25 
 
 1.0 
 
 1.0 
 
 SO 
 
 1 1 59.0 
 
 70.2 
 
 140.4 
 
 210.6 
 
 280.8 
 
 ■ 
 
 351-0 
 
 421.3 
 
 30 
 
 1-5 
 
 i-S 
 
 S3 00 
 
 
 69.9 
 
 139-9 
 
 209.8 
 
 279.8 
 278.7 
 
 349-7 
 348.4 
 
 419.6 
 418.0 
 
 
 
 
 10 
 
 ""V3V.8' 
 
 69.7 
 
 '39-3 
 
 209.0 
 
 
 
 xkO 
 
 20 
 
 463-7 
 
 69.4 
 
 i3|.g 
 
 208.2 
 
 277.6 
 
 347-0 
 
 416.4 
 
 
 54° 
 
 S5 
 
 30 
 40 
 
 695.6 
 927.4 
 
 
 138-3 
 137-7 
 
 206.6 
 
 276.5 
 275.4 
 
 345-6 
 344-2 
 
 414.8 
 413-1 
 
 
 
 
 5 
 10 
 
 0.0 
 
 0.0 
 
 SO 
 
 1159.2 
 
 68.6 
 
 137-2 
 
 205-7 
 
 274-3 
 
 342-9 
 
 411.5 
 
 0.2 
 
 0.2 
 
 S4 00 
 10 
 
 231.9 
 
 68.3 
 68.0 
 
 136.6 
 136.1 
 
 204.9 
 204.1 
 
 273.2 
 
 272.2 
 
 341-6 
 340.2 
 
 409.9 
 408.2 
 
 15 
 20 
 
 0.6 
 1 
 
 0.4 
 0.6 
 
 1.0 
 
 20 
 
 463.8 
 
 67.8 
 
 I3S-S 
 
 203-3 
 
 271.0 
 
 338-8 
 
 406.6 
 
 25 
 
 1-4 
 
 1.4 
 
 30 
 
 69S-7 
 
 67.5 
 
 135-0 
 
 202.4 
 
 269.9 
 
 3374 
 
 404.9 
 
 30 
 
 40 
 SO 
 
 927.6 
 1159.4 
 
 67.2 
 66.9 
 
 134-4 
 
 201.6 
 200.8 
 
 267.8 
 
 336-0 
 
 403-3 
 401.6 
 
 
 
 
 '33-9 
 
 334-7 
 
 
 
 
 SSoo 
 10 
 
 231-9 
 
 66.7 
 66.4 
 
 133-3 
 132.8 
 
 200.0 
 1 99. 1 
 
 266.6 
 265-5 
 
 333-3 
 33 1-9 
 
 460.0 
 398-3 
 
 
 56° 
 
 
 
 
 20 
 
 463-9 
 
 66.1 
 
 132.2 
 
 198.3 
 
 264.4 
 
 330-5 
 
 396.6 
 
 5 
 
 0.0 
 
 
 30 
 
 695-8 
 
 ^5-8 
 
 131-7 
 
 197-5 
 
 263.3 
 
 329.2 
 
 395-0 
 
 10 
 
 0.2 
 
 . 
 
 40 
 
 927.7 
 
 65.6 
 
 131. 1 
 
 196.6 
 
 262.2 
 
 327.8 
 
 393-3 
 
 IS 
 
 0.4 
 
 
 SO 
 
 1159.6 
 
 6S-3 
 
 '30-5 
 
 195.8 
 
 261. 1 
 
 326.4 
 
 391.6 
 
 20 
 
 25 
 
 0.6 
 1.0 
 
 
 5600 
 
 
 65.0 
 
 130.0 
 
 195.0 
 
 260.0 
 
 325-0 
 
 389-9 
 
 30 
 
 1-4 
 
 
 Smithsonian Tables. 
 
 138 
 
Table 24. 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^W 
 
 [Derivation of table explained on pp. liii-lvi.] 
 
 11 
 
 2is 
 
 .2 m-a^ 
 
 ABSCISSAS OF DEVELOPED PARALLEL. 
 
 longitude, 
 
 10' 
 longitude. 
 
 IS' 
 
 longitude. 
 
 2& 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 
 longitude. 
 
 ORDINATES OF 
 DEVELOPED 
 PARALLEL. 
 
 seooo' 
 
 10 
 
 20 
 
 30 
 
 40 
 
 5° 
 
 S7 00 
 10 
 20 
 
 30 
 
 40 
 
 SO 
 
 5800 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 S9 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 6000 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 61 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 62 00 
 10 
 20 
 
 30 
 40 
 
 SO 
 
 6300 
 10 
 20 
 
 30 
 40 
 
 SO 
 6400 
 
 232.0 
 
 463-9 
 
 69S-9 
 927.9 
 
 IIS9-8 
 
 232.0 
 464.0 
 696.0 
 928.0 
 1 1 60.0 
 
 232.0 
 464.1 
 696.1 
 928.2 
 
 Il60.2 
 
 232.1 
 464.2 
 
 696.2 
 
 928.3 
 
 1 160.4 
 
 232.1 
 464.2 
 696.4 
 
 1 1 60.6 
 
 232.2 
 
 464-3 
 
 696.4 
 928.6 
 1 160.8 
 
 232.2 
 
 464.4 
 
 696.6 
 
 928.8 
 
 II6I.0 
 
 232.2 
 
 464-4 
 
 696.7 
 
 928.9 
 
 II6I.I 
 
 65.0 
 
 64.7 
 64.4 
 
 64.2 
 
 63.6 
 
 63-3 
 63.0 
 62.7 
 62.5 
 62.2 
 61.9 
 
 61.6 
 61.3 
 61.0 
 60.7 
 60.4 
 60.2 
 
 S9-9 
 59-6 
 59-3 
 59-0 
 58.7 
 58-4 
 
 58.. 
 57-8 
 57-5 
 S7-Z 
 S7.0 
 56-7 
 
 56.4 
 56.1 
 SS-8 
 55-5 
 55-2 
 54-9 
 
 54-6 
 54-3 
 S4-0 
 S3-7 
 53-4 
 S3- 1 
 
 52.8 
 52.S 
 52.2 
 SI.9 
 51.6 
 
 S'-3 
 Si-o 
 
 30.0 
 29.4 
 28.9 
 28.3 
 27.7 
 27.2 
 
 26.6 
 26.0 
 
 25-5 
 24.9 
 
 24.3 
 23.8 
 
 23.2 
 22.6 
 
 22.0 
 
 2I-S 
 20.9 
 20.3 
 
 19.7 
 19.2 
 18.6 
 18.0 
 17.4 
 16.8 
 
 16.3 
 
 '5-7 
 15.1 
 
 14-5 
 13-9 
 13-3 
 
 12.7 
 12.1 
 ii.S 
 10.9 
 10.3 
 09.8 
 
 09.2 
 08.6 
 08.0 
 07.4 
 06.8 
 06.2 
 
 05.6 
 65.0 
 04.4 
 03.8 
 03.1 
 02.5 
 
 101.9 
 
 195.0 
 194.1 
 
 193-3 
 192.4 
 191.6 
 190.8 
 
 189.9 
 189.1 
 188.2 
 187.4 
 186.5 
 185.6 
 
 184.8 
 183.9 
 183.1 
 182.2 
 181.4 
 180.5 
 
 179.6 
 178.7 
 177.9 
 177.0 
 176.1 
 I7S-3 
 
 174.4 
 
 173-5 
 172.6 
 171.7 
 170.8 
 170.0 
 
 169.1 
 168.2 
 
 167-3 
 166.4 
 
 '^^■| 
 164.6 
 
 163.7 
 162.8 
 161.9 
 161.0 
 160.1 
 159.2 
 
 158-3 
 IS7-4 
 156-S 
 iSS-6 
 154-7 
 153-8 
 
 152.9 
 
 260.0 
 258.8 
 
 257-7 
 256.6 
 
 255-S 
 254.4 
 
 253.2 
 252.1 
 251.0 
 249.8 
 248.7 
 247-5 
 
 246.4 
 245.2 
 244.1 
 242.9 
 241.8 
 240.6 
 
 239-5 
 238-3 
 237-2 
 236.0 
 234-8 
 233-7 
 
 232-S 
 231.4 
 230.2 
 229.0 
 227.8 
 226.6 
 
 225.4 
 224.2 
 223.1 
 221.9 
 220.7 
 219-5 
 
 218.3 
 217.1 
 215.9 
 214.7 
 
 213-5 
 212.3 
 
 2X1. 1 
 209.9 
 208.7 
 
 207-5 
 206.3 
 205.1 
 
 203.9 
 
 325-0 
 323-6 
 322.2 
 320.8 
 
 3'9-4 
 318.0 
 
 316.6 
 31 5-1 
 3>3-7 
 312.3 
 310-8 
 309-4 
 
 308.0 
 306.6 
 305-" 
 303-6 
 302.2 
 300.8 
 
 299-4 
 297.9 
 296.4 
 295.0 
 293.6 
 292.1 
 
 290.6 
 289.2 
 287.7 
 286.2 
 284.8 
 283-3 
 
 281.8 
 280.3 
 278.8 
 277-4 
 275-8 
 274.4 
 
 272.9 
 271.4 
 269.9 
 268.4 
 266.9 
 265.4 
 
 263.9 
 262.4 
 260.9 
 259.4 
 257-8 
 256.4 
 
 254.8 
 
 388-3 
 386.6 
 
 384-9 
 383-2 
 381-S 
 
 379-9 
 378.1 
 376-4 
 374-8 
 373-0 
 371-3 
 
 369.6 
 367-9 
 365-1 
 364-4 
 362.7 
 361.0 
 
 359-2 
 357-5 
 355-7 
 354-0 
 352.3 
 3S0-S 
 
 348-8 
 347-0 
 345-2 
 343-4 
 341.7 
 340.0 
 
 338.2 
 336.4 
 334.6 
 332.8 
 331.0 
 329-3 
 
 327-5 
 325-7 
 323-9 
 322.1 
 
 320-3 
 318-S 
 
 316.7 
 3 '4.9 
 313-1 
 3"-3 
 309-4 
 307.6 
 
 305.8 
 
 2S 
 
 ■an 
 
 S6° 
 
 tntn. 
 0.0 
 0.2 
 0.4 
 0.6 
 I.O 
 
 1-4 
 
 58° 
 
 0.0 
 0.2 
 
 0.6 
 1.0 
 1.4 
 
 60° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 1-3 
 
 62° 
 
 0.0 
 0.1 
 
 0.3 
 0.6 
 0.9 
 1-3 
 
 64° 
 
 0.0 
 0.1 
 0.3 
 
 Ik 
 1.2 
 
 57" 
 
 mm. 
 0.0 
 0.2 
 
 0.6 
 1.0 
 
 1-4 
 
 59° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 1-3 
 
 61° 
 
 0.0 
 0.1 
 
 0.6 
 0.9 
 ».3 
 
 63° 
 
 0.0 
 0.1 
 
 0-3 
 0.5 
 0.9 
 1.2 
 
 Smithsonian Tables. 
 
 139 
 
Table 24. 
 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Tshv' 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 
 "S 
 
 
 
 
 
 
 
 ORDINATES OF 
 DEVELOPED 
 
 
 h 
 
 
 
 
 
 
 
 
 
 It 
 
 Merid 
 tance 
 even 
 paral 
 
 S' 
 longitude. 
 
 10' 
 
 longitude. 
 
 15' 
 
 longitude. 
 
 20' 
 
 longitude. 
 
 25' 
 longitude. 
 
 30' 
 
 longitude. 
 
 PARALLEL. 
 
 
 
 mm. 
 
 mm. 
 
 mm. 
 
 vtm. 
 
 9flfH. 
 
 ftlTH. 
 
 fnm. 
 
 tl 
 
 
 
 
 64°oo' 
 
 
 51.0 
 
 IOI.9 
 
 152.9 
 
 203.9 
 
 254.8 
 
 305.8 
 
 64° 
 
 6S° 
 
 
 10 
 
 464-5 
 
 S0.7 
 
 IOI.3 
 
 152.0 
 
 202.6 
 
 253-3 
 
 304.0 
 
 3-- 
 
 
 
 
 20 
 
 S0.4 
 
 100.7 
 
 151.1 
 
 201.4 
 
 251.8 
 
 302.2 
 
 
 
 
 
 3° 
 
 696.8 
 
 S0.1 
 
 1 00. 1 
 
 150.2 
 
 200.2 
 
 ^5°-3 
 
 300.4 
 
 
 mm. 
 
 mm. 
 
 
 4° 
 
 929.0 
 
 49-8 
 
 99- S 
 
 149.2 
 
 199.0 
 
 248.8 
 
 298.5 
 
 5' 
 
 0.0 
 
 0.0 
 
 
 SO 
 
 I161.2 
 
 49.4 
 
 98.9 
 
 148.3 
 
 197.8 
 
 247.2 
 
 296.6 
 
 10 
 
 0.1 
 
 0.1 
 
 
 6500 
 
 
 49-1 
 
 98-3 
 
 147.4 
 
 196.6 
 
 245-7 
 
 294.8 
 
 15 
 
 20 
 
 0-3 
 0.5 
 0.8 
 
 03 
 
 
 10 
 
 232.3 
 
 48.8 
 
 97.7 
 
 146.5 
 
 195-3 
 
 244.2 
 
 293.0 
 
 25 
 
 
 20 
 
 464.6 
 
 48.5 
 
 97-1 
 
 145.6 
 
 194-1 
 
 242.6 
 
 291.2 
 
 30 
 
 1.2 
 
 1.2 
 
 
 30 
 
 696.9 
 
 48.2 
 
 96.4 
 
 144.7 
 
 192.9 
 
 241.1 
 
 289.3 
 
 
 
 
 
 40 
 
 929.1 
 I161.4 
 
 47-9 
 47-6 
 
 95-8 
 
 143-7 
 142.8 
 
 191.6 
 
 239.6 
 238.0 
 
 287.5 
 285.7 
 
 
 
 
 
 5° 
 
 95.2 
 
 190.4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 66° 
 
 67° 
 
 
 6600 
 10 
 
 232.3 
 
 47-3 
 47.0 
 
 94.6 
 94-0 
 
 141.9 
 141 .0 
 
 189.2 
 188.0 
 
 236-5 
 235.0 
 
 283.8 
 281.9 
 
 
 
 
 
 
 
 
 20 
 
 464.6 
 
 46.7 
 
 93-4 
 
 140.0 
 
 186.7 
 
 233-4 
 
 280.1 
 
 S 
 
 0.0 
 
 0.0 
 
 
 3° 
 
 697.0 
 
 46.4 
 
 92.7 
 
 I39-I 
 
 185-5 
 
 231-8 
 
 278.2 
 
 10 
 
 0.1 
 
 0.1 
 
 
 40 
 
 929-3 
 
 46.1 
 
 92.1 
 
 138.2 
 
 184.2 
 
 230.3 
 
 276.4 
 
 15 
 
 0.3 
 
 0.3 
 
 
 5° 
 
 I161.6 
 
 45.8 
 
 91.S 
 
 137-2 
 
 183.0 
 
 228.8 
 
 274.5 
 
 20 
 25 
 
 °ok 
 
 C:^ 
 
 
 67 00 
 
 
 4S-4 
 
 90.9 
 
 136-3 
 
 181.8 
 
 227.2 
 
 272.6 
 
 30 
 
 i.i 
 
 I.I 
 
 
 10 
 
 232.4 
 
 4S.I 
 
 §°-? 
 
 I3S-4 
 
 180.5 
 
 225.6 
 
 270.8 
 
 
 
 
 
 20 
 30 
 
 464.7 
 697.0 
 
 44-8 
 44-S 
 
 89.6 
 
 134-4 
 133-S 
 
 179.2 
 178.0 
 
 224.0 
 222.5 
 
 268.9 
 267.0 
 
 
 
 
 
 
 
 
 
 40 
 
 929.4 
 
 44.2 
 
 88.4 
 
 132.6 
 
 176.8 
 
 221.0 
 
 265.1 
 
 
 68° 
 
 69° 
 
 
 SO 
 6800 
 
 ii6i.8 
 
 43-9 
 43-6 
 
 87.7 
 87.1 
 
 131.6 
 
 130-7 
 
 175-S 
 174.2 
 
 219.4 
 217.8 
 
 263.2 
 261.4 
 
 
 
 
 
 5 
 
 0.0 
 
 0.0 
 
 
 10 
 
 232.4 
 
 43-2 
 
 86.5 
 
 129.8 
 
 173-0 
 
 216.2 
 
 259.5 
 
 10 
 
 0.1 
 
 0.1 
 
 
 20 
 
 464.8 
 
 42-9 
 
 85.9 
 
 128.8 
 
 171.7 
 
 214.6 
 
 257-6 
 
 IS 
 
 0.3 
 
 0.3 
 
 
 30 
 
 697.1 
 
 42.6 
 
 85.2 
 
 127-9 
 
 170-5 
 
 213.1 
 
 25S-7 
 
 20 
 
 0.5 
 
 OS 
 
 
 40 
 
 929.5 
 
 42.3 
 
 84.6 
 
 126.9 
 
 169.2 
 
 211.6 
 
 253.9 
 
 25 
 
 0.7 
 
 0.7 
 
 
 so 
 
 1 161.9 
 
 42.0 
 
 84.0 
 
 126.0 
 
 168.0 
 
 210.0 
 
 251.9 
 
 30 
 
 1.1 
 
 1.0 
 
 
 6900 
 10 
 
 
 41.7 
 41.4 
 
 83.4 
 82.7 
 
 125.0 
 124.1 
 
 166.7 
 165.4 
 
 208.4" ' 
 206.8 
 
 250.1 
 248.2 
 
 
 
 
 
 232.4 
 
 
 wrtO 
 
 110 
 
 
 20 
 
 464.8 
 697.2 
 
 41.0 
 40.7 
 
 82.1 
 81.S 
 80.8 
 
 123.2 
 122.2 ' 
 
 164.2 
 162.9 
 
 205.2 
 203.6 
 
 246.3 
 
 
 70 
 
 71 
 
 
 3° 
 
 244.4 
 
 
 
 
 
 40 
 
 929.6 
 
 40-4 
 
 121.2 
 
 161.6 
 
 202.0 
 
 242.5 
 
 5 
 10 
 
 0.0 
 
 0.0 
 
 
 SO 
 
 1 162.0 
 
 40.1 
 
 80.2 
 
 120.3 
 
 160.4 
 
 200.5 
 
 240.6 
 
 O.I 
 
 0.1 
 
 
 70 00 
 
 
 39-8 
 
 79-6 
 
 "9-3 
 
 159.1 
 
 198-9 
 
 238.7 
 
 IS 
 
 20 
 
 0.2 
 
 0.4 
 0.7 
 
 I.O 
 
 0.2 
 0.4 
 
 
 10 
 
 232.4 
 
 39-S 
 
 78.9 
 
 1 18.4 
 
 157.8 
 
 197.3 
 
 236.8 
 
 25 
 30 
 
 0.7 
 0.9 
 
 
 20 
 
 464.9 
 
 
 78.3 
 
 117.4 
 
 156.6 
 
 195-7 
 
 234.8 
 
 
 30 
 
 697-3 
 
 38.8 
 
 77.6 
 
 116.5 
 
 155-3 
 
 194.1 
 
 232.9 
 
 
 
 40 
 
 929-7 
 1 162.2 
 
 38-S 
 38.2 
 
 77-0 
 76.4 
 
 "5-5 
 1 14.6 
 
 154.0 
 152.8 
 
 192.6 
 
 231.1 
 
 
 
 
 
 SO 
 
 191.0 
 
 229.1 
 
 
 
 
 
 71 00 
 10 
 
 "'232.s' 
 
 37-9 
 37-6 
 
 75-7 
 75-1 
 
 113.6 
 112.6 
 
 151-5 
 150-2 
 
 187.8 
 
 227.2 
 225.3 
 
 
 72° 
 
 
 
 
 
 
 
 20 
 
 464.9 
 
 37-2 
 
 74-S 
 73-8 
 
 111.7 
 
 148.9 
 
 186.2 
 
 223.4 
 
 s 
 
 0.0 
 
 
 
 30 
 
 697-4 
 
 36-9 
 
 110.7 
 
 147-6 
 
 184-5 
 
 221.4 
 
 10 
 
 O.I 
 
 
 
 40 
 
 929.8 
 
 36-6 
 
 73-2 
 
 ;s^ 
 
 146.3 
 
 182.9 
 
 219.5 
 
 15 
 
 0.2 
 
 
 
 SO 
 
 1 162.3 
 
 36.3 
 
 72.5 
 
 145.0 
 
 181.3 
 
 217.6 
 
 20 
 
 0.4 
 
 
 
 
 
 
 
 
 
 
 
 25 
 
 0.6 
 
 
 
 7200 
 
 
 359 
 
 71.9 
 
 107.8 
 
 143.8 
 
 179-7 
 
 215.6 
 
 30 
 
 0.9 
 
 
 
 -Smithsonian Tables. 
 
 140 
 
Table 24 
 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE -wnhm- 
 [Derivation of table explained on pp. liii-lvi.] 
 
 
 ■K. 
 
 ABSCISSAS 
 
 OF DEVELOPED 
 
 PARALLEL. 
 
 
 
 
 
 UH 
 
 .2 g-aJ! 
 
 
 
 
 
 
 
 ORDINATES OF 
 DEVELOPED 
 
 
 |i 
 
 
 
 
 
 
 
 
 13 
 
 ■egg's 
 
 s' 
 
 10' 
 
 IS' 
 
 20' 
 
 25' 
 
 30' 
 
 PARALLEL. 
 
 
 ^3°- 
 
 gS S S, 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 longitude. 
 
 
 
 
 
 
 mm. 
 
 fntn. 
 
 fitm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 u 
 
 
 
 
 72°oo' 
 
 
 3S-9 
 
 71.9 
 
 107.8 
 
 143.8 
 
 179.7 
 
 215.6 
 
 11 
 
 72° 
 
 73° 
 
 
 10 
 20 
 
 2*32.5 
 465.0 
 
 3S-6 
 3S.3 
 
 71.2 
 70.6 
 
 106.9 
 105.9 
 
 142.5 
 I41.2 
 
 178.1 
 176.5 
 
 213.7 
 2II.8 
 
 ^.a 
 
 
 
 
 
 
 
 
 
 30 
 
 697.4 
 
 35-0 
 
 70.0 
 
 104.9 
 
 \n 
 
 174.9 
 
 209.9 
 
 
 mm. 
 
 ntfH. 
 
 
 40 
 
 929.9 
 
 34.6 
 
 69.3 
 
 104.0 
 
 173-2 
 
 207.9 
 
 s' 
 
 10 
 
 0.0 
 
 0.0 
 
 
 SO 
 
 1 162.4 
 
 34-3 
 
 68.7 
 
 103.0 
 
 137.3 
 
 171.6 
 
 206.0 
 
 O.I 
 
 0.1 
 
 
 7300 
 
 
 34.0 
 
 68.0 
 
 102.0 
 
 136.0 
 
 170.0 
 
 204.1 
 
 IS 
 
 20 
 
 0.2 
 
 0.2 
 0.4 
 
 0.6 
 
 
 10 
 
 232.5 
 465.0 
 
 33.7 
 
 67.4 
 
 lOI.O 
 
 134-7 
 
 168.4 
 
 202.1 
 
 2S 
 
 30 
 
 
 20 
 
 33-4 
 
 66.7 
 
 lOO.I 
 
 133.4 
 
 166.8 
 
 200.2 
 
 0.9 
 
 0.9 
 
 
 30 
 
 697.5 
 
 33.0 
 
 66.1 
 
 99.1 
 
 132.2 
 
 165.2 
 
 198.2 
 
 
 40 
 
 1162.6 
 
 32.7 
 
 ^S-4 
 64.8 
 
 98.1 
 
 130.8 
 
 163.6 
 161.9 
 
 196.3 
 
 
 
 
 
 SO 
 
 32.4 
 
 97.1 
 
 129.5 
 
 194-3 
 
 
 
 
 
 7400 
 10 
 
 232.5 
 
 32.1 
 317 
 
 64.1 
 f3-S 
 
 96.2 
 
 95.2 
 
 128.2 
 127.0 
 
 160.3 
 158.7 
 
 192.4 
 190-4 
 
 
 74° 
 
 7S° 
 
 
 
 
 
 
 20 
 
 465.1 
 
 31-4 
 
 62.8 
 
 94.2 
 
 125.6 
 
 157.0 
 
 188.5 
 
 s 
 
 0.0 
 
 0.0 
 
 
 30 
 
 697.6 
 
 3'i 
 
 62.2 
 
 93.2 
 
 124.3 
 
 ISS.4 
 
 186.5 
 
 10 
 
 0.1 
 
 0.1 
 
 
 40 
 
 930.1 
 
 30.8 
 
 61.S 
 
 92.3 
 
 123.0 
 
 IS3-8 
 
 184.6 
 
 IS 
 
 0.2 
 
 0.2 
 
 
 SO 
 
 1 162.6 
 
 30-4 
 
 60.9 
 
 91.3 
 
 121.8 
 
 152.2 
 
 182.6 
 
 20 
 
 25 
 
 l;l 
 
 0-3 
 0.5 
 
 
 7500 
 
 
 30.1 
 
 60.2 
 
 90.3 ' 
 
 120.4 
 
 150.6 
 
 180.7 
 
 30 
 
 0.8 
 
 0.8 
 
 
 10 
 
 232.6' 
 
 29.8 
 
 59.6 
 
 89.3 
 
 119.1 
 
 148.9 
 
 178.7 
 
 
 
 
 
 20 
 
 465.1 
 
 29.4 
 
 58.9 
 S8-3 
 
 8S.4 
 
 1 1 7.8 
 
 147.2 
 
 176.7 
 
 
 
 
 
 30 
 
 697.6 
 
 29.1 
 
 874 
 
 1 16.5 
 
 145.6 
 
 174.8 
 
 
 76° 
 
 
 
 40 
 
 930.2 
 
 28.8 
 
 57.6 
 
 86.4 
 
 1 15.2 
 
 144.0 
 
 172.8 
 
 
 77° 
 
 
 SO 
 
 1 1 62.8 
 
 28.5 
 
 S6.9 
 
 85.4 
 
 "39 
 
 142.4 
 
 170.8 
 
 
 
 
 
 
 
 
 
 7600 
 
 
 28.1 
 
 S6-3 
 
 84.4 
 
 112.6 
 
 140.7 
 
 168.8 
 
 s 
 
 10 
 
 0.0 
 0.1 
 
 0.0 
 0.1 
 
 
 10 
 
 232.6 
 
 27.8 
 
 55.6 
 
 83.4 
 
 111.2 
 
 139.0 
 
 . 166.9 
 
 15 
 
 20 
 
 0.2 
 
 0.2 
 
 
 20 
 
 465.1 
 
 27.5 
 
 SS.O 
 
 82.4 
 
 109.9 
 
 137.4 
 
 164.9 
 
 0-3 
 0.5 
 0.7 
 
 0.3 
 0.5 
 0.7 
 
 
 30 
 
 697.7 
 
 ^I'l 
 
 S4.3 
 
 81.4 
 
 108.6 
 
 135.8 
 
 162.9 
 
 2S 
 
 30 
 
 
 40 
 
 930.3 
 
 26.8 
 
 S3-7 
 
 80.5 
 
 107.3 
 
 134.2 
 
 161.0 
 
 
 SO 
 7700 
 
 H62.8 
 
 26.5 
 26.2 
 
 S3-0 
 
 79-S 
 78.5 
 
 106.0 
 
 132.S 
 130.8 
 
 159.0 
 
 IS7.0 
 
 
 
 52.3 
 
 104.7 
 
 
 
 
 
 10 
 
 232.6 
 
 25.8 
 
 Si-7 
 
 77.S 
 
 103.4 
 
 129.2 
 
 155.0 
 
 
 78° 
 
 79° 
 
 
 20 
 30 
 
 465.2 
 697.8 
 
 2S.S 
 25.2 
 
 51.0 
 50.4 
 
 76.S 
 7S.S 
 
 102.0 
 100.7 
 
 127.6 
 125.9 
 
 153.1 
 151.1 
 
 
 
 
 
 
 
 
 40 
 
 930.4 
 
 24.8 
 
 49-7 
 
 74.6 
 
 99.4 
 
 124.2 
 
 149.1 
 
 S 
 
 0.0 
 
 0.0 
 
 
 SO 
 
 1 1 63.0 
 
 24.5 
 
 49.0 
 
 73-6 
 
 98.1 
 
 122.6 
 
 147.1 
 
 10 
 15 
 
 0.1 
 0.2 
 
 0.1 
 0.1 
 
 
 7800 
 
 
 24.2 
 
 48.4 
 
 72.6 
 
 96.8 
 
 121.0 
 
 145.1 
 
 20 
 
 0.3 
 
 0-3 
 
 
 10 
 
 232.6 
 
 23.9 
 
 47-7 
 
 71.6 
 
 9S.4 
 
 "9-3 
 
 143-2 
 
 25 
 
 0.4 
 
 04 
 
 
 20 
 
 465.2 
 
 23. s 
 
 47.1 
 
 70.6 
 
 94.1 
 
 H7.6 
 
 141.2 
 
 30 
 
 0.6 
 
 0.6 
 
 
 30 
 
 697.8 
 
 23.2 
 
 46.4 
 
 69.6 
 
 92.8 
 
 1 1 6.0 
 
 139.2 
 
 
 
 
 
 40 
 
 930.4 
 1 163.0 
 
 22.9 
 
 45.7 
 
 67;6 
 
 91.4 
 
 114.3 
 112.6 
 
 137-2 
 
 
 
 
 
 SO 
 
 22.5 
 
 45.1 
 
 90.1 
 
 135.2 
 
 
 
 
 
 7900 
 10 
 
 232.6' 
 
 22.2 
 21.9 
 
 44.4 
 43-7 
 
 66.6 
 65.6 
 
 88.8 
 87.S 
 
 III.O 
 
 109.4 
 
 133.2 
 131.2 
 
 
 80° 
 
 
 
 
 
 
 20 
 
 465.2 
 
 21.5 
 
 43-1 
 
 64.6 
 
 86.1 
 
 107.6 
 
 129.2 
 
 5 
 
 0.0 
 
 
 
 30 
 
 697.9 
 
 21.2 
 
 42.4 
 
 63.6 
 
 84.8 
 
 106.0 
 
 127.2 
 
 10 
 
 0.1 
 
 
 
 40 
 
 930-5 
 
 20.9 
 
 41.7 
 
 62.6 
 
 83.S 
 
 104.4 
 
 125.2 
 
 15 
 
 0.1 
 
 
 
 SO 
 
 1163.1 
 
 20.5 
 
 41. 1 
 
 61.6 
 
 82.1 
 
 102.6 
 
 123.2 
 
 20 
 
 2S 
 
 C.2 
 0.4 
 
 
 
 8000 
 
 
 20.2 
 
 40.4 
 
 60.6 
 
 80.8 
 
 lOI.O 
 
 121.2 
 
 30 
 
 0.5 
 
 
 
 Smithsonian Tables. 
 
 141 
 
Table 25. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10° EXTENT 
 IN LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. l-lii.] 
 
 Middle 
 
 Latitude of 
 
 Quadrilateral. 
 
 Area in 
 Square Miles. 
 
 0° 
 
 474653 
 
 S 
 
 47289s 
 
 10 
 
 467631 
 
 IS 
 
 458891 
 
 20 
 
 446728 
 
 2S 
 
 43'2i3 
 
 30 
 
 412442 
 
 35 
 
 390533 
 
 40 
 
 365627 
 
 45 
 
 337890 
 
 50 
 
 307514 
 
 55 
 
 274714 
 
 6o 
 
 239730 
 
 65 
 
 202S23 
 
 70 
 
 164279 
 
 75 
 
 124400 
 
 So 
 
 83504 
 
 85 
 
 41924 
 
 Smithsonian Tables. 
 
 142 
 
Table 26. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1° EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 o£ quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 o°oo' 
 
 30 
 
 1 00 
 I 30 
 
 4752-33 
 4752.16 
 
 4751-63 
 4750-75 
 
 26° 00' 
 
 26 30 
 
 27 00 
 27 30 
 
 4282.50 
 4264.51 
 4246.20 
 4227.56 
 
 52° 00' 
 
 52 30 
 
 53 00 
 53 30 
 
 295o.q8 
 2851.68 
 
 2 00 
 
 2 30 
 
 3 00 
 3 30 
 
 4749-52 
 4747-93 
 4746.00 
 
 4743-71 
 
 28 00 
 
 28 30 
 
 29 00 
 29 30 
 
 4208.61 
 
 4189-33 
 4169.74 
 4149.83 
 
 54 00 
 
 54 30 
 
 55 00 
 55 30 
 
 2818.27 
 2784.62 
 2750.76 
 2716.67 
 
 4 00 
 
 4 30 
 
 5 00 
 
 s 30 
 
 4741.07 
 4738.08 
 4734-74 
 4731.04 
 
 30 00 
 
 30 30 
 
 31 00 
 
 31 30 
 
 4129.60 
 4109.06 
 4088.21 
 4067.05 
 
 56 00 
 
 56 30 
 
 57 00 
 57 30 
 
 2682.37 
 2647.85 
 2613.13 
 2578.19 
 
 6 00 
 
 6 30 
 
 7 00 
 7 30 
 
 4727.00 
 4722.61 
 4717.86 
 4712.76 
 
 32 00 
 
 32 30 
 
 33 00 
 33 30 
 
 4045-57 
 4023.79 
 4001.69 
 3979-30 
 
 58 00 
 
 58 30 
 
 59 00 
 59 30 
 
 2543.05 
 2507.70 
 2472.16 
 2436.42 
 
 8 00 
 
 8 30 
 
 9 00 
 9 30 
 
 4707-32 
 4701.52 
 
 34 00 
 
 34 30 
 
 35 00 
 35 30 
 
 3956-59 
 3933-59 
 3910.28 
 3886.67 
 
 60 00 
 
 60 30 
 
 61 00 
 61 30 
 
 2400.48 
 
 2364.34 
 2338.02 
 2291.51 
 
 10 00 
 
 10 30 
 
 11 00 
 n 30 
 
 4682.05 
 4674.86 
 4667.32 
 4659-43 
 
 36 00 
 
 36 30 
 
 37 00 
 37 30 
 
 3862.76 
 3838.56 
 3814.06 
 3789.26 
 
 62 00 
 
 62 30 
 
 63 00 
 63 30 
 
 2254.82 
 2217.94 
 2180.89 
 2143.66 
 
 12 00 
 
 12 30 
 
 13 00 
 13 30 
 
 4651.20 
 4642.63 
 
 4633-71 
 4624.44 
 
 38 00 
 
 38 30 
 
 39 00 
 39 30 
 
 3764-18 
 3738-80 
 
 37i3-i4 
 3687.18 
 
 64 00 
 
 64 30 
 
 65 00 
 65 30 
 
 2106.26 
 2068.68 
 2030.94 
 1993.04 
 
 14 00 
 
 14 30 
 
 15 00 
 
 IS 30 
 
 4614.82 
 4604.87 
 
 4594-57 
 4583.92 
 
 40 00 
 
 40 30 
 
 41 00 
 41 30 
 
 3660.95 
 3634-42 
 3607.62 
 3580.54 
 
 66 00 
 
 66 30 
 
 67 00 
 67 30 
 
 1954.97 
 1916.75 
 
 i839;i4 
 
 16 00 
 
 16 30 
 
 17 00 
 17 30 
 
 4572.94 
 4561.61 
 4549.94 
 4537-93 
 
 42 00 
 
 42 30 
 
 43 00 
 43 30 
 
 3553-17 
 3525-54 
 3497-62 
 3469-44 
 
 68 00 
 
 68 30 
 
 69 00 
 69 30 
 
 1801.16 
 1762.33 
 1723.36 
 1684.24 
 
 18 00 
 i8 30 
 
 19 00 
 19 30 
 
 4525-59 
 4512.90 
 4499.87 
 4486.51 
 
 44 00 
 
 44 30 
 
 45 00 
 45 30 
 
 3440-98 
 3412.26 
 3383-27 
 3354-01 
 
 70 00 
 
 70 30 
 
 71 00 
 71 30 
 
 1645.00 
 1605.62 
 1566.10 
 1526.46 
 
 20 00 
 
 20 30 
 
 21 00 
 21 30 
 
 4472-81 
 
 4458-78 
 4444.41 
 
 4429.71 
 
 46 00 
 
 46 30 
 
 47 00 
 47 30 
 
 3324-49 
 3294-71 
 3264.68 
 
 3234-39 
 
 72 00 
 
 72 30 
 
 73 00 
 73 30 
 
 1486.70 
 1446.81 
 1406.81 
 1366.69 
 
 22 00 
 
 22 30 
 
 23 00 
 23 30 
 
 4414.67 
 
 4399-30 
 4383.60 
 
 4367-57 
 
 48 00 
 
 48 30 
 
 49 00 
 49 30 
 
 3203.84 
 3173-04 
 3141-99 
 3110.69 
 
 74 00 
 
 74 30 
 
 75 00 
 75 30 
 
 1326.46 
 1286.12 
 1245.68 
 1205.13 
 
 24 00 
 
 24 30 
 
 25 00 
 25 30 
 
 4351.21 
 4334-52 
 4317-51 
 4300.17 
 
 50 00 
 
 50 30 
 
 51 00 
 
 51 30 
 
 3079-15 
 3047-37 
 3015-34 
 2983.08 
 
 76 00 
 
 76 30 
 
 77 00 
 77 3° 
 
 1164.49 
 "23.75 
 1082.91 
 1041.99 
 
 Smithsonian Tables. 
 
 144 
 

 
 
 
 
 Table 26. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1° 
 LATITUDE AND LONGITUDE. 
 
 EXTENT IN 
 
 [Derivation of table explained on pp. l-lii.] 
 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 78° oo' 
 
 78 30 
 
 79 00 
 79 30 
 
 1000.99 
 959-90 
 918.73 
 877.49 
 
 82° 00' 
 
 82 30 
 
 83 00 
 83 30 
 
 670.27 
 628.64 
 586.97 
 545-24 
 
 86° 00' 
 
 86 30 
 
 87 00 
 87 30 
 
 336.02 
 294.08 
 252.11 
 
 210.12 
 
 80 00 
 
 80 30 , 
 8i 00 
 
 81 30 
 
 836.18 
 794-79 
 753-34 
 711-83 
 
 84 00 
 
 84 30 
 
 85 00 
 
 85 30 
 
 461.66 
 419.81 
 377-93 
 
 88 00 
 
 88 30 
 
 89 00 
 89 30 
 
 168.12 
 
 126.10 
 
 84.07 
 
 42.04 
 
 Smithsonian Tables. 
 
 '45 
 
Table 27. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 30' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 o°oo' 
 IS 
 o 30 
 4S 
 
 1 188.10 
 1188.08 
 1188.05 
 1188.00 
 
 13° 00' 
 13 15 
 13 30 
 13 45 
 
 1158.44 
 1157.29 
 1156.12 
 1154-93 
 
 26° 00' 
 26 IS 
 26 30 
 26 45 
 
 1070.64 
 1068.40 
 1066.14 
 1063.86 
 
 I 00 
 
 ' IS 
 I 30 
 
 I 4S 
 
 1187.92 
 1187.82 
 U87.70 
 1187.56 
 
 14 00 
 14 IS 
 14 30 
 14 45 
 
 1153-72 
 1 1 52.48 
 1151.23 
 1149.95 
 
 27 00 
 27 15 
 27 30 
 27 45 
 
 1061.56 
 1059.24 
 1056.90 
 1054-54 
 
 2 00 
 2 IS 
 2 30 
 2 45 
 
 1187.39 
 1187.20 
 1186.99 
 H86.76 
 
 15 00 
 15 IS 
 15 30 
 15 45 
 
 1148.65 
 
 1147-33 
 1145.99 
 1144.63 
 
 28 00 
 28 IS 
 28 30 
 28 45 
 
 1052.16 
 1049.76 
 
 1047-34 
 1044.90 
 
 3 00 
 3 15 
 3 30 
 3 45 
 
 1186.51 
 1186.24 
 1185.95 
 1185.62 
 
 16 00 
 16 15 
 16 30 
 16 45 
 
 1143.25 
 1 141.84 
 1 1 40.41 
 1138.96 
 
 29 00 
 29 15 
 29 30 
 29 45 
 
 1042.44 
 1039.97 
 1037-47 
 1034-95 
 
 4 00 
 4 IS 
 4 30 
 4 45 
 
 1185.28 
 1184.92 
 1184.53 
 1 184.13 
 
 17 00 
 17 IS 
 17 30 
 17 45 
 
 "37-5° 
 1136.00 
 1134.49 
 1132.96 
 
 30 00 
 30 IS 
 30 30 
 30 45 
 
 1032.41 
 1029.85 
 
 1024.68 
 
 5 00 
 5 IS 
 5 30 
 S 45 
 
 1183.70 
 1183.24 
 1182.77 
 1182.28 
 
 18 00 
 18 15 
 18 30 
 18 45 
 
 1131-41 
 1129.83 
 1128.24 
 1126.62 
 
 31 00 
 31 IS 
 31 30 
 31 45 
 
 1022.06 
 1019.43 
 1016.77 
 1014.10 
 
 6 00 
 
 6 30 
 6 45 
 
 1181.76 
 1181.22 
 1180.66 
 1180.08 
 
 19 00 
 19 IS 
 19 30 
 19 45 
 
 1124.98 
 1123.32 
 1121.64 
 1119.93 
 
 32 00 
 32 IS 
 32 30 
 32 45 
 
 1011.40 
 1008.69 
 1005.96 
 1003.20 
 
 7 00 
 7 15 
 7 3° 
 7 45 
 
 1179.48 
 1178.85 
 1178.20 
 1177-53 
 
 20 00 
 20 IS 
 20 30 
 20 45 
 
 1 1 18.21 
 1116.47 
 1114.71 
 1112.92 
 
 33 00 
 33 15 
 33 30 
 33 45 
 
 1000.43 
 997.64 
 994.83 
 992.00 
 
 8 00 
 
 8 30 
 8 45 
 
 1176.84 
 1 176.13 
 
 "75-39 
 1174.63 
 
 21 00 
 21 15 
 21 30 
 21 45 
 
 iiii.ii 
 1109.28 
 1107.44 
 1105.57 
 
 34 00 
 34 IS 
 34 30 
 34 45 
 
 989.16 
 986.29 
 
 983-41 
 980.50 
 
 9 00 
 9. 15 
 9 3° 
 9 45 
 
 1173.86 
 1173.06 
 1172.23 
 1171-39 
 
 22 00 
 22 IS 
 22 30 
 22 45 
 
 1103.68 
 1101.77 
 1099.84 
 1097.88 
 
 35 00 
 35 15 
 35 30 
 3S 45 
 
 977-58 
 974.64 
 971.68 
 968.70 
 
 10 00 
 10 IS 
 10 30 
 10 45 
 
 1170.52 
 1169.63 
 1168.73 
 1167.80 
 
 23 00 
 23 15 
 23 30 
 23 45 
 
 1095.91 
 1093.92 
 1091.90 
 1089.87 
 
 36 00 
 36 IS 
 36 30 
 36 45 
 
 965.70 
 962.68 
 
 959-65 
 956.60 
 
 II 00 
 II IS 
 II 30 
 II 45 
 
 1166.84 
 1165.86 
 1164.86 
 1163.85 
 
 24 00 
 
 24 IS 
 24 30 
 
 24 45 
 
 1087.81 
 1085.74 
 1083.64 
 1081.52 
 
 37 00 
 37 IS 
 37 30 
 37 45 
 
 953- 52 
 950-43 
 947-32 
 944.21 
 
 12 00 
 12 IS 
 
 12 30 
 
 12 4S 
 
 1162.81 
 
 1160.67 
 1159.56 
 
 25 00 
 25 IS 
 25 30 
 25 45 
 
 1079.39 
 1077.23 
 1075.05 
 1072.85 
 
 38 00 
 38 15 
 38 30 
 38 45 
 
 941.05 
 937-88 
 934-71 
 931-SI 
 
 Smithsonian Tables. 
 
 146 
 
Table 27. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 30' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-Iii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 39" 
 
 00 
 
 39 
 
 15 
 
 39 
 
 30 
 
 39 
 
 45 
 
 40 
 
 oo 
 
 40 
 
 15 
 
 40 
 
 3° 
 
 4° 
 
 45 
 
 41 
 
 00 
 
 41 
 
 IS 
 
 41 
 
 3° 
 
 41 
 
 45 
 
 42 
 
 oo 
 
 42 
 
 '5 
 
 42 
 
 .3° 
 
 42 
 
 45 
 
 43 
 
 oo 
 
 43 
 
 "5 
 
 43 
 
 30 
 
 43 
 
 45 
 
 44 00 
 
 44 15 
 
 44 30 
 
 44 45 
 
 45 00 
 45 15 
 45 30 
 
 45 45 
 
 46 00 
 46 IS 
 46 30 
 
 46 4S 
 
 47 00 
 47 15 
 47 30 
 
 47 45 
 
 48 00 
 48 15 
 48 30 
 
 48 45 
 
 49 00 
 49 15 
 49 30 
 
 49 45 
 
 50 00 
 
 50 IS 
 
 SO 30 
 
 50 45 
 
 51 00 
 51 15 
 51 30 
 SI 45 
 
 928.29 
 925.06 
 921.80 
 918.53 
 
 915.25 
 
 911-94 
 908.61 
 905.27 
 
 901.91 
 898.54 
 895.14 
 891.73 
 
 888.30 
 884.85 
 881.39 
 877.91 
 
 874.41 
 870.90 
 
 867.37 
 863.82 
 
 860.25 
 856.67 
 
 853.07 
 849.46 
 
 845.82 
 842.18 
 138-51 
 834.83 
 
 831.13 
 827.42 
 823.68 
 819.94 
 
 816.18 
 812.40 
 808.60 
 804.79 
 
 800.97 
 797.13 
 793-27 
 789-39 
 
 785.50 
 781.60 
 777.68 
 773-74 
 
 769.79 
 
 765-83 
 761.85 
 
 757.85 
 
 753-84 
 749.82 
 
 745-78 
 741.72 
 
 52° 00' 
 52 15 
 52 30 
 
 52 45 
 
 53 00 
 S3 IS 
 
 S3 30 
 
 53 45 
 
 54 00 
 54 15 
 54 30 
 
 54 45 
 
 55 00 
 55 15 
 55 30 
 
 55 45 
 
 56 00 
 
 56 IS 
 56 30 
 
 56 45 
 
 57 00 
 57 IS 
 57 30 
 57 45 
 
 58 IS 
 
 S8 30 
 
 58 45 
 
 59 00 
 59 15 
 59 30 
 
 59 45 
 
 60 00 
 60 15 
 60 30 
 
 60 45 
 
 61 00 
 61 15 
 61 30 
 
 61 45 
 
 62 00 
 62 15 
 62 30 
 
 62 45 
 
 63 00 
 63 15 
 63 30 
 
 63 4S 
 
 64 00 
 64 15 
 64 30 
 64 45 
 
 737.65 
 733-57 
 729-47 
 725-36 
 
 721.23 
 717.08 
 
 712-93 
 708.76 
 
 704-57 
 700.38 
 696.16 
 691.94 
 
 687.70 
 683.44 
 679.17 
 674.89 
 
 670.60 
 666.29 
 661.97 
 657-64 
 
 653.29 
 648.93 
 644.55 
 640.17 
 
 635-77 
 631-36 
 626.93 
 622.49 
 
 618.05 
 
 613.59 
 609.11 
 604.62 
 
 600.13 
 595.62 
 591.09 
 586.56 
 
 582.01 
 
 S77-4S 
 572.88 
 568.30 
 
 563-71 
 559-11 
 SS4-49 
 549-86 
 
 545-23 
 540.58 
 
 535-92 
 
 531-25 
 
 526.57 
 521.88 
 
 517.17 
 512.46 
 
 65° 00' 
 65 15 
 65 30 
 
 65 45 
 
 66 00 
 66 5 
 66 30 
 
 66 45 
 
 67 00 
 67 15 
 67 30 
 
 67 45 
 
 68 00 
 68 15 
 68 30 
 
 68 45 
 
 69 00 
 69 15 
 69 30 
 
 69 45 
 
 70 00 
 70 15 
 70 30 
 
 70 45 
 
 71 00 
 71 15 
 71 30 
 
 71 45 
 
 72 00 
 72 15 
 72 30 
 
 72 45 
 
 73 00 
 73 IS 
 73 30 
 
 73 45 
 
 74 00 
 74 IS 
 74 30 
 
 74 45 
 
 75 00 
 75 IS 
 75 30 
 
 75 45 
 
 76 00 
 76 15 
 76 30 
 
 76 45 
 
 77 00 
 77 IS 
 77 30 
 77 45 
 
 507.74 
 503.01 
 498.26 
 493-Si 
 
 488.75 
 483-97 
 479-19 
 474.40 
 
 469.60 
 464.78 
 459-96 
 455-13 
 
 450-29 
 
 445-45 
 440.59 
 
 435-72 
 
 430.84 
 425.96 
 421.06 
 416.16 
 
 411.25 
 406.34 
 401.41 
 396.47 
 
 3§'-53 
 
 386.58 
 
 381.62 
 376.65 
 
 371.68 
 
 366.70 
 
 361.71 
 356-71 
 
 346.69 
 341.68 
 336.65 
 
 331-62 
 326.58 
 
 321-53 
 316.48 
 
 311.42 
 306.36 
 
 301.28 
 296.21 
 
 291.12 
 286.04 
 280.94 
 
 275.84 
 270-73 
 
 265.62 
 260.50 
 
 255-38 
 
 Smithsonian Tables. 
 
 147 
 
Table 27. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 30' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 78° 00' 
 
 78 IS 
 78 30 
 
 78 4S 
 
 79 00 
 79 IS 
 79 30 
 
 79 4S 
 
 80 00 
 80 IS 
 80 30 
 
 80 4S 
 
 81 00 
 8i IS 
 81 30 
 8i 4S 
 
 Area in 
 square miles. 
 
 Z50.2S 
 245.12 
 239.98 
 234-83 
 
 229.68 
 224.53 
 
 219-37 
 214.21 
 
 209.05 
 203.88 
 198.70 
 I93-SZ 
 
 188.34 
 
 183-15 
 177.96 
 172.77 
 
 Middle latitude 
 of quadrilateral, 
 
 82° 00' 
 
 82 IS 
 
 82 30 
 
 82 45 
 
 83 00 
 83 IS 
 83 30 
 
 83 4S 
 
 84 00 
 
 84 IS 
 
 84 30 
 
 84 4S 
 
 8s 00 
 
 fS IS 
 
 85 30 
 85 45 
 
 Area in 
 square miles. 
 
 167.57 
 162.37 
 157.16 
 151-95 
 
 146.74 
 141-53 
 136-31 
 131-09 
 
 125.87 
 120.64 
 115.42 
 iio.iS 
 
 104.95 
 99.72 
 94.48 
 89.25 
 
 Middle latitude 
 of quadrilateral. 
 
 86° 00' 
 86 15 
 86 30 
 
 86 45 
 
 87 00 
 87 IS 
 87 30 
 
 87 4S 
 
 88 00 
 88 15 
 88 30 
 
 88 45 
 
 89 00 
 
 89 IS 
 89 30 
 89 45 
 
 Area in 
 square miles. 
 
 84.01 
 78.76 
 
 68!27 
 
 63.03 
 
 57-78 
 
 52-53 
 47.28 
 
 42.03 
 36-78 
 31-53 
 26.27 
 
 21.02 
 
 1576 
 
 10.51 
 
 5.26 
 
 Smithsonian Tables. 
 
 148 
 
Table 28. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 15' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. I-IU.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 o°07'3o" 
 o 15 oo 
 
 22 30 
 30 00 
 
 297,02 
 297.02 
 297.02 
 297.01 
 
 6° 37' 30" 
 6 45 00 
 
 6 52 30 
 
 7 00 00 
 
 295.09 
 295.02 
 294.95 
 294.87 
 
 13° 07' 30" 
 13 15 00 
 13 22 30 
 13 30 00 
 
 289.47 
 289.33 
 289.18 
 289.03 
 
 37 30 
 45 00 
 
 52 30 
 
 1 00 00 
 
 297.01 
 297.00 
 
 7 07 30 
 7 15 00 
 7 22 30 
 7 30 00 
 
 294.79 
 294.71 
 294.63 
 294-55 
 
 13 37 30 
 13 45 00 
 
 13 52 30 
 
 14 00 00 
 
 288.88 
 288.7-5 
 288.58 
 288.43 
 
 I 07 30 
 I 15 00 
 I 22 30 
 I 30 00 
 
 296.97 
 
 296.96 
 
 296.94 
 296.93 
 
 7 37 30 
 7 45 00 
 
 7 52 30 
 
 8 00 00 
 
 294.47 
 
 294-39 
 294.30 
 294.21 
 
 14 07 30 
 14 15 00 
 14 22 30 
 14 30 00 
 
 288.28 
 288.12 
 287.96 
 287.81 
 
 I 37 30 
 I 45 00 
 
 1 52 30 
 
 2 00 00 
 
 296.89 
 296.87 
 296.85 
 
 8 07 30 
 8 15 00 
 8 22 30 
 8 30 00 
 
 294.12 
 294.03 
 293.94 
 293-85 
 
 14 37 30 
 14 45 00 
 
 14 52 30 
 
 15 00 00 
 
 287.6s 
 287.49 
 
 287-33 
 287.17 
 
 2 07 30 
 2 15 00 
 2 22 30 
 2 30 00 
 
 296.82 
 296.80 
 296.77 
 296.75 
 
 8 37 30 
 8 45 00 
 
 8 52 30 
 
 9 00 00 
 
 293.66 
 293-56 
 293-47 
 
 IS 07 30 
 15 15 00 
 15 22 30 
 15 30 00 
 
 287.00 
 286.83 
 286.67 
 286.50 
 
 2 37 30 
 2 45 00 
 
 2 52 30 
 
 3 00 00 
 
 296.72 
 
 296.69 
 296.66 
 
 296.63 
 
 9 07 30 
 9 15 00 
 9 22 30 
 9 30 00 
 
 293-37 
 293.27 
 293.16 
 293.06 
 
 IS 37 30 
 15 45 00 
 
 15 52 30 
 
 16 00 00 
 
 286.33 
 286.16 
 285.99 
 285.82 
 
 3 07 30 
 3 15 00 
 3 22 30 
 3 30 00 
 
 296.60 
 
 296.56 
 296.53 
 296.49 
 
 9 37 30 
 9 45 00 
 
 9 52 30 
 10 00 00 
 
 292.9s 
 292.85 
 292.74 
 292.63 
 
 16 07 30 
 16 IS 00 
 16 22 30 
 16 30 00 
 
 28546 
 285.28 
 2S5.10 
 
 3 37 30. 
 3 45 00 
 
 3 52 30 
 
 4 00 00 
 
 296.45 
 296.41 
 296.36 
 296.32 
 
 10 07 30 
 10 15 00 
 10 22 30 
 10 30 00 
 
 292.52 
 292.41 
 292.30 
 292.19 
 
 16 37 30 
 16 45 00 
 
 16 52 30 
 
 17 00 00 
 
 284.92 
 284.74 
 284.56 
 284.38 
 
 4 07 30 
 4 15 00 
 4 22 30 
 4 30 00 
 
 296.28 
 296.23 
 296.18 
 
 296.13 
 
 10 37 30 
 10 45 00 
 
 10 52 30 
 
 11 00 00 
 
 292.07 
 291.95 
 291.83 
 291.71 
 
 17 07 30 
 17 15 00 
 17 22 30 
 17 30 00 
 
 284.19 
 284.00 
 283.81 
 283.62 
 
 4 37 30 
 4 45 00 
 
 4 52 30 
 
 5 00 00 
 
 296.08 
 296.03 
 295.98 
 
 295-93 
 
 II 07 30 
 11 15 00 
 II 22 30 
 II 30 00 
 
 291.59 
 291.47 
 
 291-34 
 291.22 
 
 17 37 30 
 17 45 00 
 
 17 52 30 
 
 18 00 00 
 
 283-43 
 283.24 
 
 S 07 30 
 5 15 00 
 5 22 30 
 5 30 00 
 
 295.87 
 295.81 
 
 295-75 
 295.69 
 
 II 37 30 
 II 45 00 
 
 11 52 30 
 
 12 00 00 
 
 291.09 
 290.96 
 290.83 
 290.70 
 
 18 07 30 
 18 15 00 
 18 22 30 
 18 30 00 
 
 282.66 
 282.46 
 282.26 
 282.06 
 
 S 37 30 
 
 5 45 00 
 s 52 30 
 
 6 00 00 
 
 295.63 
 
 295-57 
 295.51 
 
 295-44 
 
 12 07 30 
 12 15 00 
 12 22 30 
 12 30 00 
 
 290-57 
 290.44 
 290.30 
 290.17 
 
 18 37 30 
 18 45 00 
 
 18 52 30 
 
 19 00 00 
 
 281.86 
 281.66 
 281.45 
 281.25 
 
 6 07 30 
 6 15 00 
 6 22 30 
 6 30 00 
 
 295-37 
 295-31 
 295.24 
 
 295-17 
 
 12 37 30 
 12 45 00 
 
 12 52 30 
 
 13 00 00 
 
 290.03 
 289.89 
 
 289.75 
 289.61 
 
 19 07 30 
 19 15 00 
 19 22 30 
 19 30 00 
 
 281.04 
 280.83 
 280.62 
 280.41 
 
 Smithsonian Tables. 
 
 150 
 
Table 28. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 15' EXTENT IN 
 AKCAs ur V" LATITUDE AND LONGITUDE. 
 
 [Oenvation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 19° 37' 30" 
 19 45 00 
 
 19 52 30 
 
 20 CO 00 
 
 280.20 
 279.99 
 279.77 
 279-55 
 
 26° 07' 30" 
 26 15 00 
 26 22 30 
 26 30 00 
 
 267.38 
 267.10 
 266.82 
 266.54 
 
 32° 37' 30" 
 32 45 00 
 
 32 52 30 
 
 33 00 00 
 
 251-15 
 250.80 
 
 250.45 
 250.11 
 
 20 07 30 
 20 IS 00 
 20 22 30 
 20 30 00 
 
 279'34 
 279.12 
 278.90 
 278.68 
 
 26 37 30 
 26 45 00 
 
 26 52 30 
 
 27 00 00 
 
 266.25 
 
 265.97 
 265.68 
 
 265-39 
 
 33 07 30 
 33 15 00 
 33 22 30 
 33 30 00 
 
 249.76 
 
 249.41 
 249.06 
 248.71 
 
 20 37 30 
 20 45 00 
 
 20 52 30 
 
 21 00 00 
 
 278.46 
 278.23 
 278.00 
 277.78 
 
 27 07 30 
 27 15 00 
 27 22 30 
 27 30 00 
 
 265.10 
 264.81 
 264.52 
 264.23 
 
 33 37 30 
 33 45 00 
 
 33 52 30 
 
 34 00 00 
 
 248.36 
 248.00 
 
 247-65 
 247.29 
 
 21 07 30 
 21 IS 00 
 21 22 30 
 21 30 00 
 
 277-55 
 277-32 
 277.09 
 276.86 
 
 27 37 30 
 27 45 00 
 
 27 52 30 
 
 28 00 00 
 
 263.93 
 263.64 
 263.34 
 263.04 
 
 34 07 30 
 34 15 00 
 34 22 30 
 34 30 00 
 
 246.93 
 246.57 
 246.21 
 245.85 
 
 21 37 30 
 
 21 45 00 
 
 21 52 30 
 
 22 00 00 
 
 276.63 
 276.39 
 276.16 
 275.92 
 
 28 07 30 
 28 15 00 
 28 22 30 
 28 30 00 
 
 262.74 
 262.44 
 262.14 
 261.84 
 
 34 37 30 
 34 45 00 
 
 34 52 30 
 
 35 00 00 
 
 245.49 
 245-13 
 244.76 
 244.40 
 
 22 07 30 
 22 15 00 
 22 22 30 
 22 30 00 
 
 275.68 
 
 275-44 
 275.20 
 274.96 
 
 28 37 30 
 28 45 00 
 
 28 52 30 
 
 29 00 00 
 
 261.53 
 261.23 
 260.92 
 260.61 
 
 35 07 30 
 35 15 00 
 35 22 30 
 35 30 00 
 
 244.03 
 243-66 
 243.29 
 242.92 
 
 22 37 30 
 
 22 45 00 
 
 22 52 30 
 
 23 00 00 
 
 274.72 
 274.47 
 274.22 
 273.98 
 
 29 07 30 
 29 IS 00 
 29 22 30 
 29 30 00 
 
 260.30 
 259.99 
 259.68 
 259-37 
 
 35 37 30 
 35 45 00 
 
 35 52 30 
 
 36 00 00 
 
 242.55 
 242.18 
 241.80 
 241.43 
 
 23 07 30 
 23 15 00 
 23 22 30 
 23 30 00 
 
 273-73 
 273.48 
 
 273-23 
 272.9S 
 
 29 37 30 
 29 45 00 
 
 29 52 30 
 
 30 00 00 
 
 259-05 
 258.74 
 258.42 
 258.10 
 
 36 07 30 
 36 15 00 
 36 22 30 
 36 30 00 
 
 241.05 
 240.67 
 240.29 
 239.91 
 
 23 37 30 
 23 45 00 
 
 23 52 30 
 
 24 00 00 
 
 272.72 
 272.47 
 272.21 
 271.95 
 
 30 07 30 
 30 15 00 
 30 22 30 
 30 30 00 
 
 257.78 
 257-46 
 257.14 
 256.82 
 
 36 37 30 
 36 45 00 
 
 36 52 30 
 
 37 00 00 
 
 239-53 
 239-15 
 238.77 
 238.38 
 
 24 07 30 
 24 15 00 
 24 22 30 
 24 30 00 
 
 271.69 
 271-44 
 271.17 
 270.91 
 
 30 37 30 
 30 45 00 
 
 30 52 30 
 
 31 00 00 
 
 256.49 
 256.17 
 255.84 
 255-52 
 
 37 07 30 
 37 15 00 
 37 22 30 
 37 30 00 
 
 237-99 
 237.61 
 237,22 
 236.83 
 
 24 37 30 
 24 45 00 
 
 24 52 30 
 
 25 00 GO 
 
 270.65 
 270.38 
 270.1 1 
 269.85 
 
 31 07 30 
 31 15 00 
 31 22 30 
 31 30 00 
 
 255-19 
 254.86 
 
 254-53 
 254.19 
 
 37 37 30 
 37 45 00 
 
 37 52 30 
 
 38 00 00 
 
 236.44 
 236.05 
 235.66 
 235.26 
 
 25 07 30 
 
 25 IS 00 
 25 22 30 
 25 30 00 
 
 269.58 
 269.31 
 
 268.76 
 
 31 37 30 
 31 45 00 
 
 31 52 30 
 
 32 00 00 
 
 253-86 
 253-53 
 253-19 
 252.85 
 
 38 07 30 
 38 IS 00 
 38 22 30 
 38 30 00 
 
 234.87 
 234-47 
 
 233-68 
 
 ZS 37 30 
 25 45 00 
 
 25 52 30 
 
 26 00 00 
 
 268.49 
 268.21 
 267.94 
 267.66 
 
 32 07 30 
 32 15 00 
 32 22 30 
 32 30 00 
 
 252-51 
 252.17 
 251.83 
 251.49 
 
 38 37 30 
 38 45 00 
 
 38 52 30 
 
 39 00 00 
 
 233-28 
 232.88 
 232.48 
 232.07 
 
 Smithsonian Tables. 
 
 151 
 
Table 28. 
 
 AREAS OF QUADRILATERALS OP E&RTH'S SURFACE OF 15' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. l-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 39° 07' 30" 
 39 15 00 
 39 22 30 
 39 30 00 
 
 231.67 
 231.27 
 230.86 
 230.45 
 
 45°37'3°" 
 45 45 00 
 
 45 52 30 
 
 46 00 00 
 
 209.17 
 208.71 
 208.25 
 207.78 
 
 52° 07' 30" 
 52 IS 00 
 52 22 30 
 52 30 00 
 
 183.90 
 182.37 
 
 39 37 30 
 39 45 00 
 
 39 52 30 
 
 40 00 00 
 
 230.04 
 229.63 
 229.22 
 228.81 
 
 46 07 30 
 46 15 00 
 46 22 30 
 46 30 00 
 
 207.32 
 206.86 
 206.39 
 205.92 
 
 52 37 30 
 52 45 00 
 
 52 52 30 
 
 53 00 00 
 
 181.85 
 
 i8o!82 
 180.31 
 
 40 07 30 
 40 15 00 
 40 22 30 
 40 30 00 
 
 228.40 
 227.99 
 227.57 
 227.15 
 
 46 37 30 
 46 45 00 
 
 46 52 30 
 
 47 00 00 
 
 205.45 
 204.99 
 204.52 
 204.05 
 
 S3 07 30 
 S3 15 00 
 53 22 30 
 S3 30 00 
 
 179-79 
 179-27 
 178-75 
 178.23 
 
 40 37 30 
 40 45 00 
 
 40 52 3° 
 
 41 00 00 
 
 226.73 
 226.32 
 225.90 
 225.48 
 
 47 07 30 
 47 15 00 
 47 22 30 
 47 30 00 
 
 203.57 
 203.10 
 202.63 
 202.15 
 
 53 37 30 
 53 45 00 
 
 53 52 30 
 
 54 00 00 
 
 177.71 
 
 176.67 
 176.14 
 
 41 07 30 
 41 15 00 
 41 22 30 
 41 30 00 
 
 225.06 
 224.64 
 224.21 
 223.79 
 
 47 37 30 
 47 45 00 
 
 47 52 30 
 
 48 00 00 
 
 201.67 
 201.20 
 200.72 
 200.24 
 
 54 07 30 
 54 15 00 
 54 22 30 
 54 30 00 
 
 175.62 
 175.10 
 
 174-57 
 174.04 
 
 41 37 30 
 41 45 00 
 
 41 52 30 
 
 42 00 00 
 
 223.36 
 222.93 
 222.50 
 222.08 
 
 48 07 30 
 48 15 00 
 48 22 30 
 48 30 00 
 
 199.76 
 199.28 
 198.80 
 198.32 
 
 54 37 30 
 54 45 00 
 
 54 52 30 
 
 55 00 00 
 
 173.51 
 172.99 
 172.46 
 
 171.93 
 
 42 07 30 
 42 15 00 
 42 22 30 
 42 30 00 
 
 221.65 
 221.21 
 220.78 
 220.3s 
 
 48 37 30 
 48 45 00 
 
 48 52 30 
 
 49 00 00 
 
 19783 
 
 196.86 
 196.38 
 
 55 07 30 
 55 15 00 
 55 22 30 
 55 30 00 
 
 171-39 
 170-86 
 
 170-33 
 169-79 
 
 42 37 30 
 42 45 00 
 
 42 52 30 
 
 43 00 00 
 
 219.91 
 219.48 
 219.04 
 218.60 
 
 49 07 30 
 49 15 00 
 49 22 30 
 49 30 00 
 
 195-89 
 195.40 
 194.91 
 194.42 
 
 55 37 30 
 55 45 00 
 
 55 52 30 
 
 56 00 00 
 
 169.26 
 168.72 
 168.19 
 167.65 
 
 43 07 30 
 43 15 00 
 43 22 30 
 43 30 00 
 
 218.16 
 
 217-73 
 217.28 
 216.84 
 
 49 37 30 
 49 45 00 
 
 49 52 30 
 
 50 00 00 
 
 193-93 
 193-44 
 192.94 
 192.45 
 
 56 07 30 
 56 15 00 
 56 22 30 
 56 30 00 
 
 167. II 
 
 166.03 
 165-49 
 
 43 37 30 
 43 45 00 
 
 43 52 30 
 
 44 00 00 
 
 216.40 
 215.96 
 
 2'5-5i 
 215.06 
 
 50 07 30 
 50 15 00 
 50 22 30 
 50 30 00 
 
 191-95 
 191.46 
 190.96 
 190.46 
 
 56 37 30 
 56 45 00 
 
 56 52 30 
 
 57 00 00 
 
 164-95 
 163.32 
 
 44 07 30 
 44 15 00 
 44 22 30 
 44 30 00 
 
 214.61 
 214.17 
 213.72 
 213.27 
 
 SO 37 30 
 50 45 00 
 
 50 52 30 
 
 51 00 00 
 
 189.96 
 189.46 
 188.96 
 188.46 
 
 57 07 30 
 57 15 00 
 57 22 30 
 57 30 00 
 
 162.78 
 162.23 
 161.68 
 161. 14 
 
 44 37 30 
 44 45 00 
 
 44 52 30 
 
 45 00 00 
 
 212.82 
 212.37 
 211.91 
 211.46 
 
 51 07 30 
 SI IS 00 
 51 22 30 
 
 51 30 00 
 
 187.96 
 187.46 
 186.95 
 186.45 
 
 57 37 30 
 57 45 00 
 
 57 52 30 
 
 58 00 00 
 
 160.59 
 160.04 
 
 159-49 
 158-94 
 
 45 07 30 
 45 15 00 
 45 22 30 
 45 30 00 
 
 211.00 
 210.55 
 210.09 
 209.63 
 
 51 37 30 
 5' 45 00 
 
 51 52 30 
 
 52 00 00 
 
 185.94 
 
 185-43 
 184.92 
 184.41 
 
 58 07 30 
 58 15 00 
 58 22 30 
 
 58 30 CO 
 
 158-39 
 157.84 
 157-29 
 156.73 
 
 Smithsonian Tables. 
 
 152 
 
Table 28. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 15' EXTENT IN 
 ^ LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 58° 37' 30" 
 58 45 °o 
 
 58 52 30 
 
 59 00 00 
 
 156.18 
 155.62 
 155-07 
 154-51 
 
 65° 07' 30" 
 65 15 00 
 65 22 30 
 65 30 00 
 
 126.34 
 
 125-75 
 125.16 
 124.57 
 
 71° 37' 30" 
 71 45 00 
 
 71 52 30 
 
 72 00 00 
 
 94.78 
 94.16 
 
 93-54 ■ 
 92.92 
 
 59 07 30 
 59 IS 00 
 59 22 30 
 59 30 00 
 
 153-96 
 153-40 
 152.84 
 152.28 
 
 65 37 30 
 
 65 45 00 
 6s 52 30 
 
 66 00 00 
 
 123.97 
 123-38 
 122.78 
 122.19 
 
 72 07 30 
 72 15 00 
 72 22 30 
 72 30 00 
 
 92.30 
 91.68 
 91.05 
 90-43 
 
 59 37 30 
 59 45 00 
 
 59 52 30 
 
 60 00 00 
 
 151.72 
 151. 16 
 150.60 
 150.03 
 
 66 07 30 
 66 15 00 
 66 22 30 
 66 30 00 
 
 121.59 
 120.99 
 120.40 
 119.80 
 
 72 37 30 
 72 45 00 
 
 72 52 30 
 
 73 00 00 
 
 89.80 
 89.18 
 88-55 
 87-93 
 
 60 07 30 
 60 15 00 
 60 22 30 
 60 30 00 
 
 149-47 
 148.91 
 148.34 
 147-77 
 
 66 37 30 
 66 45 00 
 
 66 52 30 
 
 67 00 00 
 
 119.20 
 118.60 
 118.00 
 117.40 
 
 73 07 30 
 73 15 00 
 73 22 30 
 73 30 00 
 
 87.30 
 86.67 
 86.05 
 85.42 
 
 60 37 30 
 60 45 00 
 
 60 52 30 
 
 61 00 00 
 
 147.21 
 146.64 
 146.07 
 145.50 
 
 67 07 30 
 67 15 00 
 67 22 30 
 67 30 00 
 
 116.80 
 1 16.20 
 
 "5-59 
 114.99 
 
 73 37 30 
 73 45 00 
 
 73 52 30 
 
 74 00 00 
 
 84.79 
 84.16 
 
 8353 
 82.91 
 
 61 07 30 
 61 15 00 
 61 22 30 
 61 30 00 
 
 144-93 
 144.36 
 
 143-79 
 143.22 
 
 67 37 30 
 67 45 00 
 
 67 52 30 
 
 68 00 00 
 
 114-39 
 113.78 
 113.18 
 112.57 
 
 74 07 30 
 74 15 00 
 74 22 30 
 74 30 00 
 
 82.28 
 81.65 
 8r.oi 
 80.38 
 
 6i 37 30 
 61 45 00 
 
 61 52 30 
 
 62 00 00 
 
 142.65 
 142.08 
 141.50 
 140.93 
 
 68 07 30 
 68 15 00 
 68 22 30 
 68 30 00 
 
 1 1 1.97 
 1 1 1.36 
 110.76 
 110.15 
 
 74 37 30 
 74 45 00 
 
 74 52 30 
 
 75 00 00 
 
 79-75 
 79.12 
 
 78.49 
 77.86 
 
 62 07 30 
 62 15 00 
 62 22 30 
 62 30 00 
 
 140-35 
 139-78 
 139.20 
 138.62 
 
 68 37 30 
 68 45 00 
 
 68 52 30 
 
 69 00 00 
 
 109.54 
 108.93 
 108.32 
 107.71 
 
 75 07 30 
 75 IS 00 
 75 22 30 
 75 30 00 
 
 77-22 
 
 76-59 
 75-95 
 75-32 
 
 62 37 30 
 62 45 00 
 
 62 52 30 
 
 63 00 00 
 
 138.04 
 137-47 
 136.89 
 136-31 
 
 69 07 30 
 69 IS 00 
 69 22 30 
 69 30 00 
 
 107.10 
 106.49 
 105.88 
 105.27 
 
 75 37 30 
 75 45 00 
 
 75 52 30 
 
 76 00 00 
 
 74-69 
 74-05 
 73-42 
 72-78 
 
 63 07 30 
 63 IS 00 
 63 22 30 
 63 30 00 
 
 I3S-73 
 135-1S 
 134-56 
 133-98 
 
 69 37 30 
 69 4S 00 
 
 69 52 30 
 
 70 00 00 
 
 104.65 
 104.04 
 
 103.43 
 102.81 
 
 76 07 30 
 76 15 00 
 76 22 30 
 76 30 00 
 
 72.14 
 
 71-51 
 70.87 
 70.24 
 
 63 37 30 
 63 45 00 
 
 63 52 30 
 
 64 00 00 
 
 133-40 
 132.81 
 
 132-23 
 131.64 
 
 70 07 30 
 70 15 00 
 70 22 30 
 70 30 00 
 
 102.20 
 101.59 
 100.97 
 100.35 
 
 76 37 30 
 76 45 00 
 
 76 52 30 
 
 77 00 00 
 
 69.60 
 68.96 
 (8.32 
 67.6S 
 
 64 07 30 
 64 15 00 
 64 22 30 
 64 30 00 
 
 131.06 
 
 130-47 
 129.88 
 129.29 
 
 70 37 30 
 70 45 00 
 
 70 52 30 
 
 71 00 00 
 
 99-74 
 99.12 
 
 97:^8 
 
 77 07 30 
 77 15 00 
 77 22 30 
 77 30 00 
 
 66.41 
 65.77 
 65-13 
 
 64 37 30 
 64 45 00 
 
 64 52 30 
 
 65 00 00 
 
 128.70 
 128.12 
 
 127-53 
 126.94 
 
 71 07 30 
 71 IS 00 
 71 22 30 
 71 30 00 
 
 97.26 
 96.65 
 96-03 
 95-41 
 
 77 37 30 
 77 45 00 
 
 77 52 30 
 
 78 00 00 
 
 64.49 
 63.85 
 63.20 
 62.56 
 
 Smithsonian Tables. 
 
 153 
 
Table 28. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 15' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 78° 07' 30" 
 
 78 IS 00 
 
 78 22 30 
 
 78 30 00 
 
 78 37 30 
 
 78 4S 00 
 
 78 52 30 
 
 79 00 00 
 
 79 07 30 
 
 79 15 00 
 
 79 22 30 
 
 79 3° 00 
 
 79 37 30 
 
 79 45 0° 
 
 79 S2 30 
 
 80 00 00 
 
 80 07 30 
 
 80 15 00 
 
 80 22 30 
 
 80 30 00 
 
 80 37 30 
 
 80 45 00 
 
 80 52 30 
 
 81 00 00 
 
 81 07 30 
 
 81 
 81 
 
 15 00 
 22 30 
 
 01 30 00 
 
 81 37 30 
 
 81 45 00 
 
 81 52 30 
 
 82 00 00 
 
 Area in 
 square miles. 
 
 61.92 
 61.28 
 60.64 
 60.00 
 
 59-35 
 
 58.06 
 57.42 
 
 56.78 
 56-13 
 55-49 
 54-84 
 
 54-20 
 
 53-55 
 52.91 
 
 52.26 
 
 51.62 
 50.97 
 50.32 
 49.68 
 
 49-03 
 48.38 
 
 47-73 
 47.08 
 
 46.44 
 45-79 
 45-14 
 44.49 
 
 43-84 
 43-19 
 42.54 
 41. 89 
 
 Middle latitude 
 of quadrilateral. 
 
 15 00 
 22 30 
 
 82° 07' 30" 
 
 82 15 00 
 
 82 22 30 
 
 82 30 00 
 
 82 37 30 
 
 82 45 00 
 
 82 52 30 
 
 83 00 00 
 
 83 07 30 
 
 §3 - 
 
 83 30 00 
 
 83 37 30 
 
 83 45 00 
 
 83 52 30 
 
 84 00 00 
 
 84 07 30 
 
 84 15 00 
 
 84 22 30 
 
 84 30 00 
 
 84 37 30 
 
 84 45 00 
 
 84 52 30 
 
 85 00 00 
 
 85 07 30 
 
 85 15 00 
 
 85 22 30 
 
 85 30 00 
 
 85 37 30 
 
 85 45 00 
 
 85 52 30 
 
 86 00 00 
 
 Area in 
 square miles. 
 
 41.24 
 40.59 
 39-94 
 39-29 
 
 38.64 
 37-99 
 
 36-69 
 
 36-03 
 35-38 
 34-73 
 34.08 
 
 33-42 
 32-77 
 32.12 
 
 31-47 
 
 30-81 
 30.16 
 29.51 
 28.86 
 
 28.20 
 
 26.89 
 26.24 
 
 25-58 
 24-93 
 24.27 
 23.62 
 
 22.97 
 22.31 
 21.66 
 21.00 
 
 Middle latitude 
 of quadrilateral. 
 
 86° 07' 30" 
 
 86 15 00 
 
 86 22 30 
 
 86 30 00 
 
 86 37 30 
 
 86 45 00 
 
 86 52 30 
 
 87 00 00 
 
 87 07 30 
 
 87 15 00 
 
 87 22 30 
 
 87 30 00 
 
 87 37 30 
 
 87 45 00 
 
 87 52 30 
 
 88 00 00 
 
 88 07 30 
 
 88 15 00 
 
 88 22 30 
 
 88 30 00 
 
 88 37 30 
 
 88 45 00 
 
 88 52 30 
 
 89 00 00 
 
 89 07 30 
 15 00 
 22 30 
 
 89 
 
 89 30 00 
 
 89 37 30 
 
 89 45 00 
 
 89 52 30 
 
 Area in 
 square miles. 
 
 20.3s 
 19.69 
 19.04 
 18.38 
 
 17.72 
 17.07 
 16.41 
 15-76 
 
 15.10 
 14.44 
 13-79 
 13.13 
 
 12.48 
 11.82 
 11.16 
 10.51 
 
 9.85 
 9.20 
 
 8.54 
 7.88 
 
 7.22 
 
 6.57 
 5.91 
 5.26 
 
 4.60 
 3-94 
 3.28 
 2.63 
 
 1.97 
 
 '-31 
 0.66 
 
 Smithsonian Tables. 
 
 154 
 
Table 29. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 o°os' 
 IS 
 
 25 
 
 3S 
 
 132.01 
 132.01 
 132.01 
 132.00 
 
 8°4S' 
 
 8 55 
 
 9 OS 
 9 15 
 
 130-51 
 130-46 
 130.40 
 130-34 
 
 17° 25' 
 17 35 
 17 45 
 17 55 
 
 126.11 
 126.00 
 125.88 
 125-77 
 
 o 45 
 
 55 
 
 1 OS 
 
 I IS 
 
 132.00 
 i3>-99 
 131-99 
 131.98 
 
 9 25 
 9 35 
 9 45 
 9 55 
 
 130.28 
 130.22 
 130.15 
 130.09 
 
 18 05 
 18 15 
 18 25 
 18 35 
 
 125.65 
 125.54 
 125.42 
 125.30 
 
 I 25 
 I 35 
 
 I 45 
 I 55 
 
 131-97 
 131.96 
 
 131-95 
 131-94 
 
 10 05 
 10 IS 
 10 25 
 10 35 
 
 130.02 
 129.96 
 129.89 
 129.82 
 
 18 45 
 
 18 55 
 
 19 05 
 19 IS 
 
 125.18 
 125.06 
 124.94 
 124.81 
 
 2 OS 
 
 2 15 
 2 25 
 
 2 35 
 
 131-93 
 131.91 
 
 10 45 
 
 10 55 
 
 11 05 
 
 11 15 
 
 129.76 
 129.68 
 129.61 
 129.54 
 
 19 25 
 19 35 
 19 45 
 19 55 
 
 124.69 
 124-56 
 124.44 
 124.31 
 
 2 45 
 
 2 55 
 
 3 °S 
 3 15 
 
 131.86 
 131.84 
 131.82 
 131.80 
 
 II 2S 
 
 II 35 
 II 45 
 
 11 55 
 
 129-47 
 129-39 
 129.32 
 129.24 
 
 20 05 
 20 15 
 20 25 
 20 35 
 
 124.18 
 124.05 
 123.92 
 123-79 
 
 3 25 
 
 3 35 
 3 45 
 3 55 
 
 131.78 
 131.76 
 131-74 
 131-71 
 
 12 05 
 12 IS 
 12 25 
 12 35 
 
 129.16 
 129.08 
 129.00 
 128.92 
 
 20 45 
 
 20 55 
 
 21 OS 
 21 IS 
 
 123.66 
 
 123.52 
 
 123-39 
 123.25 
 
 4 OS 
 4 15 
 4 25 
 4 35 
 
 131.68 
 131.66 
 
 131-63 
 131.60 
 
 12 45 
 
 12 55 
 
 13 OS 
 13 15 
 
 128.84 
 128.76 
 128.67 
 128.59 
 
 21 25 
 21 35 
 21 4S 
 21 SS 
 
 123.12 
 122.98 
 122.84 
 12270 
 
 4 45 
 
 4 55 
 
 5 OS 
 S IS 
 
 131-57 
 131-54 
 131-50 
 131-47 
 
 13 25 
 13 35 
 13 4S 
 13 SS 
 
 128.50 
 128.41 
 128.33 
 128.24 
 
 22 05 
 22 15 
 22 25 
 22 35 
 
 122.56 
 122.42 
 122.28 
 122.13 
 
 S 25 
 5 35 
 5 45 
 S SS 
 
 131-44 
 131.40 
 
 131-36 
 131-33 
 
 14 OS 
 14 IS 
 14 25 
 14 35 
 
 128.14 
 128.05 
 127.96 
 127.87 
 
 22 45 
 
 22 55 
 
 23 OS 
 
 23 IS 
 
 121.99 
 121.84 
 121.69 
 121-55 
 
 6 05 
 
 ^ '5 
 6 25 
 
 6 35 
 
 131-29 
 131-25 
 131.21 
 
 131-16 
 
 14 45 
 
 14 55 
 
 15 05 
 IS IS 
 
 127-77 
 127.67 
 127.58 
 127.48 
 
 23 25 
 23 35 
 23 45 
 23 SS 
 
 121.40 
 121.25 
 121. 10 
 120.94 
 
 6 45 
 
 6 55 
 
 7 OS 
 7 IS 
 
 131-12 
 131-07 
 131-03 
 130.98 
 
 15 25 
 15 35 
 IS 45 
 IS SS 
 
 127.38 
 127.28 
 127.18 
 127.08 
 
 24 OS 
 24 15 
 24 25 
 
 24 35 
 
 120.79 
 120.64 
 120.48 
 120.33 
 
 7 25 
 7 35 
 7 45 
 7 SS 
 
 130-93 
 130.88 
 130.84 
 130-79 
 
 16 05 
 16 IS 
 16 25 
 16 35 
 
 126.98 
 126.87 
 126.77 
 126.66 
 
 24 45 
 
 24 SS 
 
 25 05 
 25 15 
 
 120.17 
 120.01 
 
 119-8S 
 119.69 
 
 8 OS 
 
 f 'S 
 8 25 
 
 8 3S 
 
 130.68 
 130-63 
 130-57 
 
 164s 
 
 16 55 
 
 17 05 
 17 IS 
 
 126.55 
 126.44 
 126.33 
 126.22 
 
 2S 25 
 25 35 
 25 45 
 25 55 
 
 119-53 
 119-37 
 119.21 
 119.04 
 
 Smithsonian Tabi 
 
 .ES. 
 
 
 
 
 
 156 
 
Table 29. 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 
 26° OS' 
 
 26 15 
 
 • 26 25 
 
 26 35 
 
 118.87 
 118.71 
 118.54 
 118.37 
 
 34=45' 
 
 34 55 
 
 35 05 
 35 15 
 
 108.94 
 108.73 
 108.51 
 108.29 
 
 43° 25' 
 43 35 
 43 45 
 43 55 
 
 96.50 
 96.24 
 95-98 
 95-71 
 
 
 26 45 
 
 26 55 
 
 27 05 
 27 15 
 
 1 18.21 
 118.04 
 117.87 
 117.69 
 
 35 25 
 
 35 35 
 35 45 
 35 55 
 
 108.07 
 107.85 
 107.63 
 107.41 
 
 44 05 
 44 IS 
 44 25 
 44 35 
 
 95-45 
 95-19 
 94-92 
 94.65 
 
 
 27 25 
 27 35 
 27 45 
 27 55 
 
 117.52 
 
 "7-35 
 117.17 
 116.99 
 
 36 05 
 
 36 IS 
 36 25 
 
 36 35 
 
 107.19 
 106.96 
 106.74 
 106.51 
 
 44 45 
 
 44 55 
 
 45 05 
 45 IS 
 
 94-38 
 94-11 
 93-84 
 93-58 
 
 
 28 05 
 28 15 
 28 25 
 28 35 
 
 116.82 
 1 16.64 
 116.46 
 r 16.28 
 
 36 45 
 
 36 55 
 
 37 OS 
 37 15 
 
 106.29 
 106.06 
 105.83 
 105.60 
 
 45 2S 
 45 35 
 45 45 
 45 55 
 
 93-30 
 93-03 
 92.76 
 92.48 
 
 
 28 45 
 
 28 55 
 
 29 05 
 29 IS 
 
 n6.io 
 115.92 
 "5-73 
 "5-55 
 
 37 25 
 37 35 
 37 45 
 37 55 
 
 105-37 
 105.14 
 104.91 
 104.68 
 
 46 OS 
 46 15 
 46 25 
 46 35 
 
 92.21 • 
 91.94 
 91.66 
 91-38 
 
 
 29 25 
 
 29 35 
 29 45 
 
 29 55 
 
 "5-37 
 115.18 
 114.99 
 1 14.81 
 
 38 05 
 38 15 ' 
 38 25 
 38 35 
 
 104.44 
 104.21 
 103-97 
 103-74 
 
 46 45 
 
 46 55 
 
 47 OS 
 47 IS 
 
 91.10 
 90.82 
 
 90-55 
 90.27 
 
 
 30 05 
 
 30 15 
 
 30 25 
 
 30 35 
 
 114.62 
 
 "4-43 
 114.24 
 114.04 
 
 38 45 
 
 38 55 
 
 39 OS 
 39 15 
 
 103.50 
 103.26 
 103.02 
 102.78 
 
 47 25 
 47 35 
 47 45 
 47 55 
 
 89.99 
 89.70 
 89.42 
 89.14 
 
 
 30 45 
 3° 55 
 3.1 05 
 
 31 '5 
 
 "Hi 
 113.66 
 
 ii3'47 
 113.27 
 
 39 25 
 39 35 
 39 45 
 39 55 
 
 102.54 
 102.30 
 102.06 
 101.82 
 
 48 OS 
 
 48 15 
 48 25 
 
 48 35 
 
 88.85 
 88.57 
 88.28 
 88.00 
 
 
 31 25 
 31 35 
 31 45 
 31 55 
 
 "3-07 
 112.88 
 112.68 
 112.48 
 
 40 05 
 40 15 
 40 25 
 40 35 
 
 101.57 
 
 101.33 
 101.08 
 100.83 
 
 48 45 
 
 48 55 
 
 49 OS 
 49 IS 
 
 87-71 
 87.42 
 
 87-13 
 86-84 
 
 
 32 05 
 32 IS 
 32 25 
 32 35 
 
 112.28 
 112.08 
 1 1 1.87 
 111.67 
 
 40 45 
 
 40 55 
 
 41 05 
 41 15 
 
 100.59 
 
 100.34 
 
 100.09 
 
 99.84 
 
 49 25 
 49 35- 
 49 45 
 49 55 
 
 86-55 
 86.26 
 85.97 
 85.68 
 
 
 32 45 
 
 32 55 
 
 33 05 
 33 15 
 
 111.47 
 111.26 
 111.06 
 1 10.85 
 
 41 25 
 41 35 
 41 45 
 41 55 
 
 99-59 
 
 98.83 
 
 50 05 
 SO IS 
 SO 25 
 SO 35 
 
 85-39 
 85.09 
 84.80 
 84.50 
 
 
 33 25 
 33 35 
 33 45 
 33 55 
 
 110.64 
 
 110.43 
 110.22 
 110.01 
 
 42 05 
 42 15 
 42 2S 
 42 35 
 
 98-57 
 98.32 
 98.06 
 97.80 
 
 SO 45 
 
 50 55 
 
 51 05 
 51 15 
 
 84.21 
 83.91 
 83.61 
 
 83-31 
 
 
 34 OS 
 34 IS 
 '34 25 
 34 35 
 
 109.80 
 109.59 
 
 109-37 
 109.16 
 
 42 45 
 
 42 55 
 
 43 OS 
 43 15 
 
 97-55 
 97.29 
 
 97-03 
 96.77 
 
 51 25 
 SI 35 
 51 45 
 51 55 
 
 83.01 
 82.71 
 82.41 
 82.11 
 
 
 Smithsonian Tables. 
 
 157 
 
Table 29. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. 1-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 
 52° 05' 
 52 15 
 52 25 
 52 35 
 
 81.81 
 81.51 
 81.20 
 80.90 
 
 60° 45' 
 
 60 55 
 
 61 05 
 6i 15 
 
 65.17 
 64.84 
 64.50 
 64.16 
 
 69° 25' 
 69 35 
 69 45 
 69 55 
 
 46.97 
 46.60 
 46.24 
 45-88 
 
 
 52 45 
 
 52 55 
 
 53 OS 
 53 15 
 
 80.60 
 80.29 
 79.9S 
 79.68 
 
 61 25 
 61 35 
 6i 45 
 61 55 
 
 63.82 
 63.48 
 63.14 
 62.80 
 
 70 05 
 70 15 
 70 25 
 70 35 
 
 4S-5I 
 45-15 
 44-7» 
 44.42 
 
 
 53 25 
 53 35 
 53 45 
 53 55 
 
 79-37 
 79-06 
 78.75 
 78.44 
 
 62 05 
 62 15 
 62 25 
 62 35 
 
 62.46 
 62.12 
 61.78 
 61.44 
 
 70 45 
 
 70 55 
 
 71 OS 
 
 71 15 
 
 44-05 
 43-69 
 43-32 
 42.95 
 
 
 54 OS 
 54 IS 
 54 25 
 54 35 
 
 78.13 
 77.82 
 
 77-51 
 77-19 
 
 62 45 
 
 62 55 
 
 63 OS 
 63 15 
 
 61.10 
 60.75 
 60.41 
 60.06 
 
 71 25 
 71 35 
 71 45 
 71 55 
 
 42.58 
 42.22 
 41.8s 
 41.48 
 
 
 ■ 54 45 
 
 54 55 
 
 55 OS 
 55 15 
 
 76.88 
 
 76.57 
 76.25 
 
 75-94 
 
 63 25 
 63 35 
 63 45 
 63 55 
 
 59-72 
 59-37 
 
 72 OS 
 72 15 
 72 25 
 72 35 
 
 41.11 
 40.74 
 
 40.37 
 40.00 
 
 
 55 25 
 55 35 
 
 55 45 
 55 55 
 
 75-62 
 75-30 
 74-99 
 74-67 
 
 64 05 
 64 15 
 64 25 
 64 35 
 
 58-33 
 57-99 
 57-64 
 57-29 
 
 72 45 
 
 72 55 
 
 73 OS 
 73 15 
 
 39-63 
 
 38^89 
 38.52 
 
 
 56 OS 
 S6 IS 
 56 25 
 56 35 
 
 74-35 
 74-03 
 73-71 
 73-39 
 
 64 45 
 
 64 55 
 
 65 OS 
 65 15 
 
 56-94 
 56-59 
 56-24 
 55-89 
 
 73 25 
 73 3S 
 73 45 
 73 55 
 
 38.1S 
 37.78 
 37-41 
 37-03 
 
 
 56 45 
 
 56 55 
 
 57 05 
 57 15 
 
 73-07 
 72-75 
 72.43 
 72.10 
 
 65 25 
 65 35 
 65 45 
 65 55 
 
 SS-S4 
 55-19 
 54-83 
 54-48 
 
 74 OS 
 74 15 
 74 25 
 74 35 
 
 36.66 
 36.29 
 35-91 
 35-54 
 
 
 57 25 
 57 35 
 57 45 
 57 55 
 
 71-78 
 71.46 
 
 71-13 
 70.80 
 
 66 05 
 66 15 
 66 25 
 66 35 
 
 54.13 
 53-78 
 53-42 
 53.06 
 
 74 45 
 
 74 55 
 
 75 OS 
 75 IS 
 
 35-17 
 34-79 
 34-42 
 34-04 
 
 
 58 05 
 58 IS 
 58 25 
 
 S8 35 
 
 70.48 
 
 70-15 
 69.82 
 69.49 
 
 ^< 45 
 
 66 55 
 
 67 05 
 67 IS 
 
 52-71 
 52-35 
 52.00 
 51.64 
 
 75 25 
 75 35 
 75 45 
 75 55 
 
 33-66 
 
 3329 
 32.91 
 
 32-53 
 
 
 58 45 
 
 58 55 
 
 59 05 
 59 15 
 
 ^11 
 68.51 
 68.18 
 
 67 25 
 67 35 
 67 45 
 67 5S 
 
 51.28 
 So-93 
 
 50-57 
 50.21 
 
 76 05 
 76 IS 
 76 25 
 76 35 
 
 32.16 
 31-78 
 31.40 
 
 31-03 
 
 
 59 25 
 59 35 
 59 45 
 59 55 
 
 67.84 
 
 67.18 
 66.85 
 
 68 OS 
 68 15 
 68 25 
 68 35 
 
 49-85 
 49.49 
 
 49-13 
 48.77 
 
 76 45 
 
 76 55 
 
 77 OS 
 77 15 
 
 30.65 
 30.27 
 29.89 
 29.51 
 
 
 60 05 
 60 15 
 60 25 
 60 35 
 
 66.51 
 66.18 
 65.84 
 65-51 
 
 68 45 
 
 68 ss 
 
 69 05 
 69 15 
 
 48.41 
 48.05 
 47-69 
 47-33 
 
 77 25 
 77 35 
 77 45 
 77 55 
 
 29.13 
 28.76 
 
 28.37 
 27.99 
 
 
 Smithsonian Tables. 
 
 158 
 
Table 29. 
 
 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN 
 LATITUDE AND LONGITUDE. 
 
 [Derivation of table explained on pp. I-lii.] 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 Middle latitude 
 of quadrilateral. 
 
 Area in 
 square miles. 
 
 
 78° OS' 
 78 15 
 78 25 
 78 35 
 
 27.62 
 27.24 
 26.85 
 26.47 
 
 82° OS' 
 82 IS 
 82 25 
 82 35 
 
 18.43 
 18.04 
 
 17-65 
 17.27 
 
 86° OS' 
 86 15 
 86 25 
 86 35 
 
 tit 
 8.36 
 7-97 
 
 
 78 45 
 
 78 55 
 
 79 OS 
 79 IS 
 
 26.09 
 25.71 
 
 25-33 
 24.9s 
 
 82 45 
 
 82 55 
 
 83 OS 
 83 IS 
 
 16.88 
 16.50 
 16.11 
 
 15-73 
 
 86 45 
 
 86 55 
 
 87 05 
 87 15 
 
 7-59 
 7.20 
 6.81 
 6.42 
 
 
 79 25 
 79 35 
 79 45 
 79 55 
 
 24-57 
 24.18 
 23.80 
 23.42 
 
 83 25 
 83 35 
 83 45 
 83 55 
 
 15-34 
 14.95 
 
 14-57 
 14.18 
 
 87 25 
 87 35 
 87 45 
 87 55 
 
 6.03 
 S-64 
 5-25 
 4.86 
 
 
 80 05 
 80 IS 
 80 25 
 80 35 
 
 23.04 
 22.65 
 22.27 
 21.89 
 
 84 05 
 84 15 
 84 25 
 84 35 
 
 13-79 
 13.40 
 13.02 
 12.63 
 
 88 OS 
 88 IS 
 88 25 
 88 35 
 
 4-47 
 4.09 
 3-70 
 3-31 
 
 
 80 45 
 
 80 55 
 
 81 OS 
 81 IS 
 
 21.50 
 21.12 
 20.73 
 20.3s 
 
 84 45 
 
 84 55 
 
 85 05 
 8s IS 
 
 12.24 
 11.86 
 11.47 
 11.08 
 
 88 45 
 
 88 55 
 
 89 OS 
 89 15 
 
 2.92 
 2-53 
 
 2.14 
 
 1-75 
 
 
 81 25 
 8i 35 
 81 45 
 81 55 
 
 19.97 
 19.58 
 19.20 
 18.81 
 
 8s 25 
 85 35 
 85 45 
 85 55 
 
 10.69 
 
 10.30 
 
 992 
 
 9-S3 
 
 89 25 
 89 35 
 89 45 
 89 55 
 
 1.36 
 0.97 
 0.58 
 0!i9 
 
 
 Smithsonian Tables. 
 
 159 
 
Table 30. 
 
 DETERMINATION OF HEIGHTS BY THE BAROMETER. 
 
 Formula of Babinet, 
 
 C(m feet) = 32494 fi + °"^ ^ | — English Measures. 
 L 900 J 
 
 t2 (to + t)-\ 
 I + — rrrr — I — Metric Measures. 
 
 In which Z= Difference of height of two stations in feet or metres. 
 
 .ffoi -5^ Barometric readings at the lower and upper stations respectively, corrected for 
 all sources of instrumental error. 
 /■(J, ^= Air temperatures at the lower and upper stations respectively. 
 
 Values of C. 
 
 ENGLISH MEASURES. 
 
 METRIC MEASURES. 
 
 ife+0- 
 
 logC. 
 
 C. 
 
 F. 
 
 
 Feet. 
 
 10° 
 
 4.69834 
 
 49928 
 
 IS 
 
 ■70339 
 
 505" 
 
 20 
 
 •70837 
 
 51094 
 
 2S 
 
 •71330 
 
 5'677 
 
 30 
 
 .71818 
 
 52261 
 
 35 
 
 4.72300 
 
 52844 
 
 40 
 
 .72777 
 
 53428 
 
 45 
 
 .73248 
 
 5401 1 
 
 SO 
 
 •7371 S 
 
 S4S9S 
 
 SS 
 
 •74177 
 
 55178 
 
 60 
 
 4-74633 
 
 55761 
 
 6S 
 
 •75085 
 
 56344 
 
 70 
 
 ■75532 
 
 56927 
 
 7S 
 
 •75975 
 
 575" 
 
 80 
 
 •76413 
 
 58094 
 
 85 
 
 4.76847 
 
 58677 
 
 90 
 
 .77276 
 
 59260 
 
 95 
 
 .77702 
 
 59844 
 
 100 
 
 .78123 
 
 60427 
 
 Smithsonian Tables. 
 
 i(to+i)- 
 
 logC. 
 
 C. 
 
 C. 
 
 
 Metres. 
 
 —10° 
 
 4.18639 
 
 15360 
 
 — 8 
 
 .19000 
 
 15488 
 
 — 6 
 
 ■19357 
 
 15616 
 
 — 4 
 
 .I97J2 
 
 15744 
 
 — 2 
 
 .20063 
 
 15872 
 
 
 
 4.20412 
 
 16000 
 
 +2 
 
 .20758 
 
 1 61 28 
 
 4 
 
 .21101 
 
 16256 
 16384 
 
 6 
 
 .21442 
 
 8 
 
 .21780 
 
 16512 
 
 10 
 
 4.22115 
 .22448 
 
 16640 
 
 12 
 
 16768 
 
 14 
 
 .22778 
 
 16896 
 
 16 
 
 .23106 
 
 17024 
 
 18 
 
 •23431 
 
 17152 
 
 20 
 
 4-23754 
 
 17280 
 
 22 
 
 .24075 
 
 17408 
 
 24 
 
 •24393 
 
 17536 
 
 26 
 
 .24709 
 
 17664 
 
 28 
 
 .25022 
 
 17792 
 
 30 
 
 4.25334 
 
 17920 
 
 32 
 
 •25643 
 
 18048 
 
 3i 
 
 .25950 
 
 18176 
 
 36 
 
 .26255 
 
 18304 
 
 160 
 
MEAN REFRACTION. 
 
 Table 31 , 
 
 a-3 
 
 o o 
 
 10 
 20 
 
 30 
 40 
 
 5° 
 I o 
 
 10 
 20 
 
 30 
 40 
 
 50 
 
 3 o 
 10 
 20 
 
 30 
 40 
 
 50 
 
 Refraction. 
 
 0.-B 
 
 34 54-1 
 
 32 49.2 
 30 S2-3 
 29 3-5 
 27 22.7 
 
 25 49-8 
 
 24 24.6 
 
 23 6.7 
 21 55-6 
 20 50.9 
 
 1951-9 
 18 58.0 
 
 18 8.6 
 
 17 23.0 
 1640.7 
 16 0.9 
 15234 
 14 47-8 
 
 14 14.6 
 
 4_o 
 10 
 
 20 
 
 30 
 40 
 
 50 
 
 5 o 
 
 60 
 
 7 o 
 
 13437 
 13 15.0 
 
 12 48.3 
 12 23.7 
 12 0.7 
 
 "38-9 
 
 II 18.3 
 
 10 58.6 
 10 39.6 
 
 10 21.2 
 
 10 3-3 
 
 946.5 
 
 930-9 
 916.0 
 
 9 1-9 
 
 848.4 
 
 835-6 
 
 823-3 
 
 811.6 
 8 0.3 
 7 49-S 
 7 39-2 
 7 29.2 
 
 719.7 
 
 124.9 
 116.9 
 
 ioS.8 
 100.8 
 92.9 
 85.2 
 77-9 
 71. 1 
 64.7 
 59.0 
 
 53-9 
 49.4 
 45.6 
 42-3 
 39-8 
 37-S 
 3S-6 
 33-2 
 3°-9 
 28.7 
 26.7 
 24.6 
 23.0 
 21.8 
 20.6 
 19.7 
 19.0 
 1S.4 
 17-9 
 16.8 
 15.6 
 14.9 
 14.1 
 '3-5 
 12.8 
 12.3 
 11.7 
 ".3 
 10.8 
 J0.3 
 
 lO.O 
 
 9-5 
 
 7 O 
 
 10 
 20 
 30 
 40 
 50 
 
 10 
 20 
 30 
 40 
 50 
 
 Refraction. 
 
 719-7 
 
 7 10.5 
 7 1.7 
 
 653-3 
 645.1 
 
 637-2 
 
 629.6 
 
 9 o 
 
 10 
 20 
 30 
 40 
 50 
 
 10 
 
 20 
 30 
 40 
 
 50 
 
 10 
 
 20 
 30 
 40 
 
 50 
 
 10 
 20 
 30 
 40 
 50 
 
 6 22.3 
 615.2 
 6 8.4 
 6 1.8 
 
 5 55-4 
 
 5 49-3 
 
 5 43-3 
 5 37-6 
 5320 
 526.5 
 521.3 
 
 5 16.2 
 
 5 II-2 
 
 5 6.4 
 5 1-7 
 457.2 
 
 452-8 
 
 4 44-3 
 440.2 
 
 436-3 
 432-4 
 428.7 
 
 425-0 
 
 421.4 
 4 18.0 
 414.6 
 
 4 "-3 
 4 8.1 
 
 13 o I 4 4-9 
 
 10 
 20 
 
 30 
 40 
 SO 
 
 14 o 
 
 4 1.8 
 358.8 
 3 55-9 
 3 53-0 
 350-2 
 
 347-4 
 
 8.4 
 8.2 
 
 7-9 
 7.6 
 
 7-3 
 7-1 
 6.8 
 6.6 
 6.4 
 6.1 
 6.0 
 5-7 
 5-6 
 S-5 
 5-2 
 5-1 
 50 
 4.8 
 4-7 
 4-5 
 4-4 
 4.3 
 4.2 
 4.1 
 3-9 
 3-9 
 3-7 
 3-7 
 3.6 
 3.4 
 3-4 
 3.3 
 3-2 
 3-2 
 
 3-1 
 3.0 
 2.9 
 2.9 
 2.8 
 2.8 
 
 a-43 
 
 14 O 
 
 20 
 40 
 
 Refraction. 
 
 3 47-4 
 
 3 42-1 
 ^37-o 
 
 IS o 
 
 20 
 40 
 
 17 o 
 
 20 
 40 
 
 20 
 40 
 
 19 o 
 
 20 
 40 
 
 20 
 40 
 
 20 
 40 
 
 20 
 40 
 
 23 o 
 
 20 
 40 
 
 24 o 
 
 332-1 
 
 327-4 
 322.9 
 
 318.6 
 
 314-5 
 310.5 
 
 6.6 
 
 3 2.9 
 259-3 
 
 255.8 
 
 252-5 
 2 49-3 
 
 246.1 
 
 243.L 
 2 40.2 
 
 237-3 
 
 234-5 
 231-9 
 
 2293 
 
 226.8 
 
 ^3.-3 
 2 21.9 
 
 2 19.6 
 217-4 
 
 15-2 
 
 213.0 
 2 10.9 
 
 2 8.9 
 
 20 
 40 
 
 25 o 
 
 20 
 40 
 
 26 o 
 
 20 
 40 
 
 27 o 
 
 20 
 40 
 
 28 o 
 
 7.0 
 
 S-i 
 
 3-2 
 
 2 1.4 
 '59-6 
 
 157.8 
 
 I 56.1 
 I 54-4 
 
 I 52.(! 
 
 I 51.2 
 '49-7 
 
 I 48.2 
 
 5-3 
 S-i 
 4.9 
 
 4-7 
 4-5 
 4-3 
 4-1 
 4.0 
 3-9 
 3-7 
 3-6 
 3 5 
 3-3 
 3-2 
 3-2 
 
 3.0 
 2.9 
 2.9 
 
 2.8 
 2.6 
 2.6 
 
 2-5 
 2-5 
 2-4 
 2-3 
 
 2.2 
 2.2 
 2.2 
 2.1 
 2.0 
 '■9 
 1.9 
 1-9 
 1.8 
 
 1.8 
 
 \.i 
 ■.7 
 1.7 
 
 1.6 
 1.6 
 1-5 
 
 l-S 
 
 
 28 o 
 
 20 
 40 
 
 29 o 
 
 20 
 40 
 
 30 o 
 
 20 
 40 
 
 31 o 
 
 20 
 40 
 
 32 O 
 
 20 
 40 
 
 33 o 
 
 20 
 40 
 
 34 o 
 
 20 
 
 40 
 
 35 
 
 20 
 40 
 
 36 o 
 
 20 
 40 
 
 37 o 
 
 20 
 40 
 
 38 
 
 20 
 39 o 
 
 20 
 
 40 
 
 40 o 
 
 20 
 40 
 
 41 o 
 
 20 
 40 
 
 42 o 
 
 Refraction 
 
 48.2 
 
 46.7 
 45-3 
 
 43-8 
 
 42-4 
 41.0 
 
 39-7 
 
 38.4 
 37-1 
 
 35-8 
 
 34-5 
 33-3 
 
 32.1 
 
 30-9 
 29.8 
 
 28.7 
 
 27.6 
 26.5 
 
 254 
 
 24-3 
 23-3 
 22.3 
 21-3 
 20.3 
 19-3 
 18.3 
 17.4 
 
 16.5 
 15.6 
 14-7 
 118 
 12.9 
 12.0 
 11.2 
 10.3 
 9-5 
 
 7-9 
 7-1 
 
 5-5 
 4-7 
 4.0 
 
 i3 3 
 
 i2. 
 
 60 
 
 JS. 
 
 Refraction. 
 
 64.0 
 61.8 
 
 59-7 
 57-7 
 5 5-7 
 53-8 
 
 51-9 
 
 i;o.2 
 
 46.7 
 45-1 
 43-5 
 
 41.9 
 40.4 
 38-9 
 
 37-5 
 36.1 
 
 26±. 
 
 33-3 
 
 32.0 
 
 30-7 
 
 194_ 
 
 28.2 
 
 26.9 
 
 25-7 
 
 24.5 
 
 23-3 
 
 22.2 
 
 21.0 
 
 18.8 
 17-7 
 
 16.6 
 15-5 
 14-5 
 
 13-4 
 12.3 
 1 1.2 
 
 10.2 
 
 9.1 
 8.1 
 4-1 
 
 2.2 
 2.1 
 2.0 
 2.0 
 1.9 
 1.9 
 1.7 
 1.8 
 1-7 
 1.6 
 1.6 
 1.6 
 l-S 
 1-5 
 1.4 
 1-4 
 1.4 
 1.4 
 
 1-3 
 1.3 
 1-3 
 1.2 
 1-3 
 1.2 
 
 E.O 
 4.0 
 4.1 
 
 Smithsonian Tables. 
 
 161 
 
Table 32. 
 
 FOR CONVERSION OF ARC INTO TIME. 
 
 o 
 
 
 h. m. 
 
 
 
 h. m. 
 
 
 
 h. m. 
 
 
 
 h. m. 
 
 
 
 h. in. 
 
 
 
 h. m. 
 
 / 
 
 m. s. 
 
 // 
 
 s. 
 
 
 
 
 60 
 
 4 
 
 120 
 
 8 
 
 180 
 
 12 
 
 240 
 
 16 
 
 300 
 
 20 
 
 
 
 
 
 
 
 0.000 
 
 
 I 
 
 4 
 
 61 
 
 4 4 
 
 121 
 
 8 4 
 
 181 
 
 12 4 
 
 241 
 
 16 4 
 
 301 
 
 20 4 
 
 I 
 
 4 
 
 I 
 
 0.067 
 
 
 2 
 
 8 
 
 62 
 
 4 8 
 
 122 
 
 8 8 
 
 182 
 
 12 8 
 
 242 
 
 16 8 
 
 302 
 
 20 8 
 
 2 
 
 8 
 
 2 
 
 OI33 
 
 
 1 
 
 12 
 
 61 
 
 412 
 
 123 
 
 812 
 
 183 
 
 12 12 
 
 243 
 
 16 12 
 
 303 
 
 2012 
 
 3 
 
 12 
 
 3 
 
 0.200 
 
 
 4 
 
 16 
 
 64 
 
 416 
 
 124 
 
 8i6 
 
 184 
 
 12 16 
 
 244 
 
 16 16 
 
 304 
 
 20 16 
 
 4 
 
 16 
 
 4 
 
 0.267 
 
 
 5 
 
 020 
 
 65 
 
 420 
 
 125 
 
 820 
 
 185 
 
 12 20 
 
 245 
 
 1620 
 
 305 
 
 20 20 
 
 5 
 
 020 
 
 5 
 
 °-333 
 
 
 6 
 
 024 
 
 66 
 
 424 
 
 126 
 
 824 
 
 186 
 
 12 24 
 
 246 
 
 1624 
 
 306 
 
 2024 
 
 6 
 
 024 
 
 6 
 
 0.400 
 
 
 7 
 
 028 
 
 67 
 
 428 
 
 127 
 
 828 
 
 187 
 
 1228 
 
 247 
 
 1628 
 
 307 
 
 2028 
 
 7 
 
 028 
 
 7 
 
 0.467 
 
 
 8 
 
 032 
 
 68 
 
 432 
 
 128 
 
 832 
 
 188 
 
 1232 
 
 248 
 
 1632 
 
 308 
 
 20 32 
 
 8 
 
 032 
 
 8 
 
 0-533 
 
 
 9 
 10 
 
 II 
 
 036 
 
 69 
 
 436 
 
 129 
 
 836 
 
 189 
 
 1236 
 
 249 
 
 1636 
 
 309 
 
 2036 
 
 9 
 10 
 
 036 
 
 9 
 
 0.600 
 
 
 040 
 
 70 
 
 440 
 
 130 
 
 840 
 
 190 
 
 1240 
 
 250 
 
 1640 
 
 310 
 
 2040 
 
 040 
 
 10 
 
 0.667 
 
 
 044 
 
 71 
 
 444 
 
 131 
 
 844 
 
 191 
 
 1244 
 
 2SI 
 
 1644 
 
 31^ 
 
 2044 
 
 II 
 
 044 
 
 II 
 
 °-733 
 
 
 12 
 
 048 
 
 72 
 
 448 
 
 132 
 
 848 
 
 192 
 
 12 48 
 
 2S2 
 
 1648 
 
 312 
 
 2048 
 
 12 
 
 048 
 
 12 
 
 0.800 
 
 
 n 
 
 OS2 
 
 73 
 
 452 
 
 133 
 
 «52 
 
 ■93 
 
 1252 
 
 253 
 
 16 52 
 
 313 
 
 20 52 
 
 13 
 
 OS2 
 
 13 
 
 0.867 
 
 
 14 
 
 056 
 
 74 
 
 4 56 
 
 134 
 
 8s6 
 
 ■94 
 
 12 56 
 
 2S4 
 
 16 
 
 314 
 
 2056 
 
 ■4 
 
 056 
 
 14 
 
 0-933 
 
 
 15 
 
 I 
 
 75 
 
 5 
 
 135 
 
 Q 
 
 19fa 
 
 ■3 
 
 255 
 
 17 
 
 315 
 
 21 
 
 15 
 
 I 
 
 15 
 
 1. 000 
 
 
 i6 
 
 I 4 
 
 76 
 
 5 4 
 
 136 
 
 9 4 
 
 196 
 
 ■3 4 
 
 256 
 
 17 4 
 
 316 
 
 21 4 
 
 16 
 
 I 4 
 
 16 
 
 1.067 
 
 
 17 
 
 I 8 
 
 77 
 
 5 » 
 
 137 
 
 9 » 
 
 197 
 
 ■3 « 
 
 257 
 
 17 « 
 
 3^7 
 
 21 8 
 
 17 
 
 I 8 
 
 17 
 
 ■•■33 
 
 
 i8 
 
 I 12 
 
 7« 
 
 512 
 
 i3« 
 
 912 
 
 198 
 
 13 12 
 
 2S8 
 
 17 12 
 
 3'« 
 
 21 12 
 
 18 
 
 I 12 
 
 18 
 
 1.200 
 
 
 iQ 
 
 I 16 
 
 79 
 
 SI5 
 
 139 
 
 9 16 
 
 199 
 
 13 16 
 
 259 
 
 17 16 
 
 319 
 
 21 16 
 
 ■9 
 
 116 
 
 ■9 
 20 
 
 1.267 
 
 
 20 
 
 21 
 
 I 20 
 
 80 
 
 520 
 
 140 
 
 920 
 
 200 
 
 1320 
 
 260 
 
 17 20 
 
 320 
 
 21 20 
 
 20 
 
 I 20 
 
 1-333 
 
 
 I 24 
 
 81 
 
 524 
 
 141 
 
 924 
 
 201 
 
 1324 
 
 261 
 
 ■724 
 
 321 
 
 21 24 
 
 21 
 
 124 
 
 21 
 
 1.400 
 
 
 22 
 
 I 28 
 
 82 
 
 528 
 
 142 
 
 928 
 
 202 
 
 1328 
 
 262 
 
 1728 
 
 322 
 
 2128 
 
 22 
 
 128 
 
 22 
 
 1.467 
 
 
 ZS 
 
 132 
 
 »3 
 
 532 
 
 143 
 
 932 
 
 203 
 
 1332 
 
 263 
 
 ■732 
 
 323 
 
 21 32 
 
 23 
 
 132 
 
 23 
 
 1-533 
 
 
 24 
 
 13b 
 
 84 
 
 536 
 
 144 
 
 936 
 
 204 
 
 ■3 36 
 
 264 
 
 1736 
 
 324 
 
 21 36 
 
 24 
 
 136 
 
 24 
 
 
 
 25 
 
 I 40 
 
 Ufa 
 
 540 
 
 14fa 
 
 940 
 
 20fa 
 
 1340 
 
 265 
 
 1740 
 
 325 
 
 21 40 
 
 25 
 
 I 40 
 
 25 
 
 1.667 
 
 
 26 
 
 144 
 
 86 
 
 544 
 
 146 
 
 9 44 
 
 206 
 
 1344 
 
 266 
 
 ■7 44 
 
 326 
 
 2144 
 
 26 
 
 144 
 
 26 
 
 ■-733 
 
 
 27 
 
 148 
 
 »7 
 
 5 4« 
 
 147 
 
 948 
 
 207 
 
 1348 
 
 267 
 
 ■748 
 
 327 
 
 2148 
 
 27 
 
 148 
 
 27 
 
 1.800 
 
 
 28 
 
 I S2 
 
 88 
 
 5 52 
 
 148 
 
 952 
 
 208 
 
 ■3 52 
 
 268 
 
 17 S2 
 
 32S 
 
 21 S2 
 
 28 
 
 I S2 
 
 28 
 
 1.867 
 
 
 29 
 30 
 
 156 
 
 89 
 
 5 5^' 
 
 149 
 
 9 5t> 
 
 209 
 
 ■3 5'' 
 
 269 
 
 175& 
 
 329 
 330 
 
 33^ 
 
 21 56 
 
 29 
 
 ■56 
 
 29 
 
 ■-933 
 
 
 2 
 
 90 
 
 6 
 
 150 
 
 10 
 
 210 
 
 14 
 
 270 
 
 18 
 
 22 
 22 4 
 
 30 
 
 31 
 
 2 
 
 30 
 
 2.000 
 
 
 ^i 
 
 2 4 
 
 91 
 
 6 4 
 
 151 
 
 10 4 
 
 211 
 
 14 4 
 
 271 
 
 18 4 
 
 2 4 
 
 31 
 
 2.067 
 
 
 7,2 
 
 2 8 
 
 92 
 
 6 8 
 
 152 
 
 10 8 
 
 212 
 
 14 8 
 
 272 
 
 18 8 
 
 332 
 
 22 8 
 
 32 
 
 2 8 
 
 32 
 
 2-133 
 
 
 ^^ 
 
 2 12 
 
 93 
 
 612 
 
 153 
 
 10 12 
 
 213 
 
 14 12 
 
 273 
 
 18 12 
 
 333 
 
 22 12 
 
 33 
 
 2 12 
 
 33 
 
 2.200 
 
 
 34 
 
 216 
 
 94 
 
 616 
 
 154 
 
 10 16 
 
 214 
 
 14 16 
 
 274 
 
 18 16 
 
 334 
 
 22 16 
 
 ik 
 
 216 
 
 34 
 
 2.267 
 
 
 35 
 
 2 20 
 
 95 
 
 620 
 
 155 
 
 1020 
 
 21fa 
 
 1420 
 
 275 
 
 1820 
 
 335 
 
 22 20 
 
 2 20 
 
 35 
 
 2-333 
 
 
 ^6 
 
 2 24 
 
 96 
 
 624 
 
 i5fe 
 
 1024 
 
 216 
 
 1424 
 
 276 
 
 1824 
 
 336 
 
 22 24 
 
 36 
 
 224 
 
 36 
 
 2.400 
 
 
 37 
 
 228 
 
 97 
 
 628 
 
 ■57 
 
 1028 
 
 217 
 
 1428 
 
 277 
 
 1828 
 
 337 
 
 2228 
 
 37 
 
 228 
 
 37 
 
 2.467 
 
 
 38 
 
 232 
 
 98 
 
 632 
 
 iS8 
 
 1032 
 
 218 
 
 1432 
 
 278 
 
 1832 
 
 33« 
 
 2232 
 
 3« 
 
 232 
 
 38 
 
 2-533 
 
 
 39 
 40 
 
 236 
 
 99 
 
 636 
 
 159 
 
 1036 
 
 219 
 
 1436 
 
 279 
 
 183b 
 
 339 
 340 
 
 341 
 
 2236 
 
 39 
 
 236 
 
 39 
 
 
 
 240 
 
 100 
 
 lOI 
 
 640 
 
 160 
 
 1040 
 
 220 
 
 1440 
 
 280 
 
 1840 
 
 2240 
 
 40 
 
 240 
 
 40 
 
 2.667 
 
 
 41 
 
 244 
 
 644 
 
 161 
 
 1044 
 
 221 
 
 1444 
 
 281 
 
 1844 
 
 2244 
 
 41 
 
 244 
 
 41 
 
 2-733 
 
 
 42 
 
 248 
 
 102 
 
 648 
 
 162 
 
 1048 
 
 222 
 
 1448 
 
 282 
 
 1848 
 
 342 
 
 2248 
 
 42 
 
 248 
 
 42 
 
 2.800 
 
 
 43 
 
 252 
 
 103 
 
 652 
 
 163 
 
 10 52 
 
 223 
 
 ■452 
 
 283 
 
 1852 
 
 343 
 
 22 S2 
 
 43 
 
 2 C,2 
 
 43 
 
 2.867 
 
 
 44 
 
 2 56 
 
 104 
 
 656 
 
 164 
 
 1056 
 
 224 
 
 1456 
 
 284 
 
 18 s6 
 
 344 
 
 22 56 
 
 44 
 
 2S6 
 
 44 
 
 2-933 
 
 
 45 
 
 3 
 
 105 
 
 7 
 
 165 
 
 II 
 
 22fa 
 
 ■5 
 
 285 
 
 19 
 
 345 
 
 23 
 
 45 
 
 3 ° 
 
 45 
 
 3.000 
 
 
 46 
 
 3 4 
 
 106 
 
 7 4 
 
 166 
 
 " ? 
 
 226 
 
 ■5 4 
 
 286 
 
 19 4 
 
 346 
 
 23 4 
 
 46 
 
 3 4 
 
 46 
 
 3.067 
 
 
 47 
 
 3 » 
 
 107 
 
 7 t> 
 
 167 
 
 II 8 
 
 227 
 
 ■5 a 
 
 287 
 
 ig 8 
 
 347 
 
 23 8 
 
 47 
 
 3 8 
 
 47 
 
 3133 
 
 
 48 
 
 312 
 
 108 
 
 712 
 
 168 
 
 II 12 
 
 228 
 
 ■5 12 
 
 288 
 
 19 12 
 
 348 
 
 2312 
 
 48 
 
 312 
 
 48 
 
 3.200 
 
 
 49 
 
 31b 
 
 109 
 
 7 16 
 
 169 
 
 n 16 
 
 229 
 
 15 16 
 
 289 
 
 19 16 
 
 349 
 
 2316 
 
 49 
 
 316 
 
 49 
 
 3.267 
 
 
 50 
 
 320 
 
 110 
 
 720 
 
 170 
 
 n 20 
 
 230 
 
 15 20 
 
 290 
 
 1920 
 
 350 
 
 351 
 
 23 20 
 2324 
 
 50 
 
 320 
 
 50 
 
 3-333 
 
 
 51 
 
 324 
 
 in 
 
 724 
 
 171 
 
 II 24 
 
 231 
 
 1524 
 
 291 
 
 1924 
 
 5^ 
 
 324 
 
 51 
 
 3.400 
 
 
 52 
 
 328 
 
 112 
 
 728 
 
 172 
 
 11 28 
 
 232 
 
 1528 
 
 292 
 
 1928 
 
 352 
 
 2328 
 
 52 
 
 328 
 
 52 
 
 3-467 
 
 
 53 
 
 iSf, 
 
 113 
 
 ^32 
 
 ■73 
 
 1132 
 
 233 
 
 ■532 
 
 293 
 
 1932 
 
 353 
 
 23 32 
 
 53 
 
 332 
 
 53 
 
 3-533 
 
 
 J';* 
 
 3 3t' 
 
 114 
 
 736 
 
 ■74 
 
 II 36 
 
 234 
 
 1536 
 
 294 
 
 1936 
 
 354 
 
 2336 
 
 54 
 
 336 
 
 ii 
 
 3.600 
 
 
 fab 
 
 340 
 
 Ufa 
 
 740 
 
 IVfa 
 
 II 40 
 
 23fa 
 
 1540 
 
 29fa 
 
 ■940 
 
 355 
 
 2340 
 
 55 
 
 340 
 
 3.667 
 
 
 5b 
 
 3 44 
 
 lib 
 
 744 
 
 17b 
 
 1144 
 
 236 
 
 1544 
 
 296 
 
 ■9 44 
 
 356 
 
 2344 
 
 56 
 
 344 
 
 56 
 
 3-733 
 
 
 H 
 
 348 
 
 ir/ 
 
 748 
 
 ■77 
 
 II 48 
 
 237 
 
 154a 
 
 297 
 
 1948 
 
 357 
 
 2348 
 
 57 
 
 348 
 
 ■57 
 
 3.800 
 
 
 5» 
 
 352 
 
 118 
 
 752 
 
 178 
 
 ■■52 
 
 238 
 
 1552 
 
 298 
 
 1952 
 
 35« 
 
 2352 
 
 58 
 
 352 
 
 58 
 
 3.867 
 
 
 59 
 
 3 5f 
 
 ny 
 
 7 5*' 
 
 ■79 
 180 
 
 II 56 
 
 239 
 
 ■5 50 
 
 299 
 
 19 Sb 
 
 359 
 360 
 
 23 s6 
 
 59 
 
 356 
 
 59 
 60 
 
 3-933 
 
 
 60 
 
 4 
 
 120 
 
 8 
 
 12 
 
 240 
 
 16 
 
 300 
 
 20 
 
 24 
 
 60 
 
 4 
 
 4.000 
 
 
 ^^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Smithsonian Tables. 
 
 162 
 
FOR CONVERSION OF TIME INTO ARC. 
 
 Table 33. 
 
 Hours of Time into Arc. 
 
 
 Time. 
 
 Arc. 
 
 Time. 
 
 Arc. 
 
 Time. 
 
 Arc. 
 
 Time. 
 
 Arc. 
 
 Time. 
 
 Arc. 
 
 Time. 
 
 Arc. 
 
 
 hrs. 
 
 
 
 hrs. 
 
 
 
 hrs. 
 
 
 
 hrs. 
 
 
 
 hrs. 
 
 
 
 hrs. 
 
 
 
 
 1 
 
 IS 
 
 5 
 
 75 
 
 9 
 
 135 
 
 13 
 
 195 
 
 17 
 
 255 
 
 21 
 
 315 
 
 
 2 
 
 3° 
 
 6 
 
 90 
 
 10 
 
 150 
 
 14 
 
 210 
 
 18 
 
 270 
 
 22 
 
 330 
 
 
 3 
 
 4S 
 
 7 
 
 105 
 
 11 
 
 165 
 
 15 
 
 225 
 
 19 
 
 285 
 
 23 
 
 34S 
 
 
 4 
 
 60 
 
 8 
 
 120 
 
 12 
 
 180 
 
 16 
 
 240 
 
 20 
 
 300 
 
 24 
 
 360 
 
 
 IVIinutes of Time into Arc. 
 
 Seconds of Time into Arc. 
 
 
 m. 
 
 / 
 
 m. 
 
 ' 
 
 m. 
 
 f 
 
 s. 
 
 / // 
 
 s. 
 
 f // 
 
 s. 
 
 / ft 
 
 
 1 
 
 015 
 
 21 
 
 S15 
 
 41 
 
 10 15 
 
 1 
 
 015 
 
 21 
 
 515 
 
 41 
 
 10 15 
 
 
 2 
 
 030 
 
 22 
 
 5 3° 
 
 42 
 
 10 30 
 
 2 
 
 030 
 
 22 
 
 530 
 
 42 
 
 1030 
 
 
 3 
 
 4S 
 
 23 
 
 5 45 
 
 43 
 
 1045 
 
 3 
 
 045 
 
 23 
 
 5 45 
 
 43 
 
 10 45 
 
 
 4 
 
 I 
 
 24 
 
 6 
 
 44 
 
 II 
 
 4 
 
 I 
 
 24 
 
 6 
 
 44 
 
 II 
 
 
 S 
 
 I IS 
 
 25 
 
 615 
 
 45 
 
 II IS 
 
 5 
 
 I IS 
 
 25 
 
 615 
 
 45 
 
 II 15 
 
 
 6 
 
 130 
 
 26 
 
 6 30 
 
 46 
 
 II 30 
 
 6 
 
 130 
 
 26 
 
 6 30 
 
 46 
 
 II 30 
 
 
 7 
 
 145 
 
 27 
 
 645 
 
 47 
 
 II 45 
 
 7 
 
 145 
 
 '^l 
 
 645 
 
 47 
 
 "45 
 
 
 8 
 
 2 
 
 28 
 
 7 
 
 48 
 
 12 
 
 8 
 
 2 
 
 28 
 
 7 
 
 48 
 
 12 
 
 
 9 
 
 215 
 
 29 
 
 7 IS 
 
 49 
 
 12 IS 
 
 9 
 
 215 
 
 29 
 
 7 IS 
 
 49 
 
 12 IS 
 
 
 10 
 
 230 
 
 30 
 
 730 
 
 50 
 
 12 30 
 
 10 
 
 230 
 
 30 
 
 7 3° 
 
 50 
 
 12 30 
 
 
 n 
 
 2 45 
 
 31 
 
 7 45 
 
 S' 
 
 1245 
 
 II 
 
 ^45 
 
 31 
 
 ^45 
 
 SI 
 
 12 45 
 
 
 12 
 
 3 
 
 32 
 
 8 
 
 S2 
 
 13 
 
 12 
 
 3 
 
 32 
 
 8 
 
 52 
 
 13 
 
 
 13 
 
 3 IS 
 
 33 
 
 815 
 
 53 
 
 13 15 
 
 13 
 
 3 15 
 
 33 
 
 **i5 
 
 S3 
 
 13 15 
 
 
 14 
 
 330 
 
 34 
 
 830 
 
 54 
 
 1330 
 
 14 
 
 3 3° 
 
 34 
 
 830 
 
 54 
 
 1330 
 
 
 IS 
 
 3 4S 
 
 35 
 
 845 
 
 55 
 
 1345 
 
 15 
 
 3 45 
 
 35 
 
 845 
 
 55 
 
 1345 
 
 
 i6 
 
 4 
 
 36 
 
 9 
 
 
 
 14 
 
 i6 
 
 4 ° 
 
 3" 
 
 9 ° 
 
 Sb 
 
 14 
 
 
 17 
 
 4 IS 
 
 37 
 
 915 
 
 S? 
 
 14 15 
 
 17 
 
 415 
 
 37 
 
 9 '5 
 
 K 
 
 1415 
 
 
 i8 
 
 4 3° 
 
 3« 
 
 930 
 
 S« 
 
 1430 
 
 18 
 
 430 
 
 3** 
 
 9 3° 
 
 s*» 
 
 1430 
 
 
 19 
 
 4 45 
 
 39 
 
 9 45 
 
 59 
 
 14 45 
 
 19 
 
 4 45 
 
 39 
 
 9 45 
 
 59 
 
 1445 
 
 
 20 
 
 S 
 
 40 
 
 10 
 
 60 
 
 15 ° 
 
 20 
 
 5 
 
 40 
 
 10 
 
 60 
 
 IS 0. 
 
 
 
 IHundr 
 
 edtlis of a Sec 
 
 ond of Time into Arc 
 
 " 
 
 
 
 Hundredths 
 
 
 
 
 
 
 
 
 
 
 
 
 of a Sec- 
 
 .00 
 
 .01 
 
 .02 
 
 .03 
 
 .04 
 
 .05 
 
 .06 
 
 .07 
 
 .08 
 
 .09 
 
 
 ond of Time. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 // 
 
 // 
 
 ff 
 
 // 
 
 // 
 
 „ 
 
 // 
 
 // 
 
 // 
 
 ft 
 
 
 0.00 
 
 0.00 
 
 0.15 
 
 0.30 
 
 0.4s 
 
 0.60 
 
 0.75 
 
 0.90 
 
 1.05 
 
 1.20 
 
 ^■l\ 
 
 
 .lO 
 
 1.50 
 
 i.6s 
 
 1.80 
 
 1-95 
 
 2.10 
 
 2.25 
 
 2.40 
 
 2-55 
 
 2.70 
 
 2.85 
 
 
 .20 
 
 3.00 
 
 3-15 
 
 3-3° 
 
 3-45 
 
 3-to 
 
 3-75 
 
 3-90 
 
 4.05 
 
 4.20 
 
 4-35 
 
 
 •30 
 
 4.50 
 
 4.65 
 
 4.80 
 
 4-95 
 
 5.10 
 
 S-25 
 
 5-4° 
 
 5-55 
 
 5.70 
 
 5.85 
 
 
 .40 
 
 6.00 
 
 6.1S 
 
 6.30 
 
 6.45 
 
 6.60 
 
 6.75 
 
 6.90 
 
 7-05 
 
 7.20 
 
 7-35 
 
 
 0.50 
 
 7-50 
 
 7.6^ 
 
 7.80 
 
 7-95 
 
 8.10 
 
 8.25 
 
 8.40 
 
 8.55 
 
 8.70 
 
 8.8s 
 
 
 .60 
 
 9.00 
 
 9.15 
 
 9-3° 
 
 9-45 
 
 9.60 
 
 9-75 
 
 9.90 
 
 10.05 
 
 10.20 
 
 ^^M 
 
 
 .70 
 
 10.50 
 
 10.65 
 
 10.80 
 
 10.95 
 
 II. 10 
 
 11.25 
 
 11.40 
 
 11-55 
 
 11.70 
 
 11.05 
 
 
 .80 
 
 12.00 
 
 12.15 
 
 12.30 
 
 12.45 
 
 12.60 
 
 12.75 
 
 12.90 
 
 I3-°S 
 
 13.20 
 
 '^■^5 
 
 
 .90 
 
 13-50 
 
 13-65 
 
 13.80 
 
 13-95 
 
 14.10 
 
 14.25 
 
 14.40 
 
 14-55 
 
 14.70 
 
 14.85 
 
 
 Smithsonian Tables. 
 
 163 
 
Table 34. 
 
 CONVERSION OF MEAN TIME INTO SIDEREAL TIME. 
 
 s 
 
 m 
 
 
 m 
 I 
 
 m 
 2 
 
 m 
 
 3 
 
 
 
 
 h m s 
 000 
 
 h m s 
 
 6 515 
 
 h m s 
 12 10 29 
 
 h m s 
 18 IS 44 
 
 s 
 0.00 
 
 in s 
 
 
 s 
 
 0.50 
 
 m s 
 3 3 
 
 I 
 
 2 
 
 3 
 4 
 
 1 
 9 
 
 065 
 12 10 
 18 16 
 24 21- 
 30 26 
 03631 
 04237 
 048 42 
 
 54 47 
 
 6 II 20 
 61725 
 62330 
 62936 
 
 641 46 
 64751 
 65356 
 702 
 
 12 1634 
 12 22 40 
 12 28 45 
 12 34 50 
 12 40 55 
 1247 1 
 12 S3 6 
 
 12 59 11 
 
 13 516 
 
 18 21 49 
 
 18 27 54 
 
 18 33 59 
 18 40 5 
 18 46 10 
 18 52 15 
 
 18 58 20 
 
 19 4 26 
 19 10 31 
 
 O.OI 
 
 0.02 
 0.03 
 0.04 
 0.0s 
 0.06 
 0.07 
 0.08 
 
 O.OQ 
 
 4 
 7 
 II 
 
 018 
 22 
 26 
 29 
 033 
 
 0.51 
 0.52 
 
 0-53 
 0.54 
 0.5s 
 0.56 
 
 0.58 
 0-59 
 
 3 6 
 
 310 
 314 
 317 
 321 
 
 ^2^ 
 332 
 
 3 35 
 
 lO 
 
 I 52 
 
 7 6 7 
 
 13 II 21 
 
 19 16 36 
 
 O.IO 
 
 037 
 
 0.60 
 
 3 39 
 
 II 
 
 12 
 
 13 
 U 
 IS 
 
 i 
 
 19 
 
 I 658 
 
 I 13 3 
 I 19 8 
 I 2513 
 131 19 
 I 37 24 
 14329 
 14934 
 I 55 40 
 
 7 12 12 
 
 7 18 17 
 72423 
 73028 
 73633 
 74238 
 74844 
 
 7 54 49 
 
 8 054 
 
 13 17 27 
 13 23 32 
 13 29 37 
 13 35 42 
 13 41 48 
 13 47 53 
 
 13 53 58 
 
 14 3 
 14 6 9 
 
 19 22 41 
 19 28 47 
 
 19 34 52 
 19 40 57 
 1947 2 
 19 53 7 
 
 19 59 13 
 zo 5 i8 
 
 20 I I 23 
 
 O.I I 
 0.12 
 0.13 
 0.14 
 0.15 
 0.16 
 0.17 
 0.18 
 0.19 
 
 40 
 044 
 047 
 051 
 
 058 
 
 1 2 
 I 6 
 1 9 
 
 o.6i 
 0.62 
 0.63 
 0.64 
 0.6s 
 0.66 
 0.67 
 0.68 
 0.69 
 
 3 50 
 3 54 
 
 3 57 
 
 4 I 
 
 412 
 
 20 
 
 2 I 45 
 
 8 659 
 
 14 12 14 
 
 20 17 28 
 
 0.20 
 
 1 13 
 
 0.70 
 
 4 16 
 
 21 
 22 
 
 23 
 24 
 
 2I 
 27 
 28 
 29 
 
 2 7 so 
 
 21355 
 2 20 I 
 2 26 6 
 232 n 
 238 16 
 24422 
 25027 
 2 56 32 
 
 813 5 
 8 19 10 
 82515 
 8 31 20 
 83726 
 
 84331 
 84936 
 
 85541 
 Q I 47 
 
 14 18 19 
 14 24 24 
 14 30 30 
 14 36 35 
 14 42 40 
 
 14 48 45 
 
 14 54 51 
 
 15 056 
 
 IS 7 I 
 
 20 23 34 
 20 29 39 
 20 35 44 
 20 41 49 
 20 47 5S 
 
 20 54 
 
 21 5 
 21 6 10 
 
 21 T2 16 
 
 0.21 
 0.22 
 0.23 
 0.24 
 0.25 
 0.26 
 
 0.28 
 0.29 
 
 I 17 
 I 20 
 I 24 
 1 28 
 131 
 13s 
 1 39 
 142 
 I 46 
 
 0.71 
 0.72 
 
 0-73 
 0.74 
 0.7s 
 0.76 
 
 0.77 
 0.78 
 0.79 
 
 419 
 423 
 427 
 430 
 4 34 
 438 
 441 
 4 45 
 4 49 
 
 30 
 
 3 237 
 
 9 7 52 
 
 11; 13 6 
 
 21 18 21 
 
 0.30 
 
 ISO 
 
 0.80 
 
 452 
 
 31 
 
 32 
 
 33 
 
 34 
 
 i 
 39 
 
 3 843 
 3 '448 
 32053 
 32658 
 3 33 3 
 3 39 9 
 34514 
 351 19 
 3 57 24 
 
 9 '3 57 
 9 20 2 
 926 8 
 
 93213 
 93818 
 94423 
 9 50 28 
 95634 
 10 239 
 
 15 1912 
 
 152517 
 1531 22 
 
 15 37 27 
 154333 
 154938 
 15.5543 
 
 16 I 48 
 
 16 7 54 
 
 21 24 26 
 21 30 31 
 21 36 37 
 21 42 42 
 21 48 47 
 
 21 54 52 
 
 22 58 
 22 7 3 
 22 13 8 
 
 0.31 
 0.32 
 
 0-33 
 0-34 
 
 0.36 
 
 0-37 
 0.38 
 0.39 
 
 1 53 
 
 1 57 
 
 2 I 
 
 2 4 
 2 8 
 2 II 
 215 
 2 19 
 2 22 
 
 0.8 1 
 0.82 
 0.83 
 0.84 
 0.85 
 0.86 
 0.87 
 0.88 
 0.89 
 
 456 
 
 4 59 
 
 5 3 
 5 7 
 5 10 
 514 
 5 '8 
 521 
 525 
 
 40 
 
 4 330 
 
 10 8 44 
 
 16 13 59 
 
 22 19 13 
 
 0.40 
 
 226 
 
 0.90 
 
 529 
 
 41 
 42 
 
 43 
 44 
 
 45 
 46 
 
 :^ 
 
 49 
 
 4 9 35 
 41540 
 42145 
 427 SI 
 43356 
 440 I 
 446 6 
 45212 
 45817 
 
 5 422 
 51027 
 51633 
 52238 
 
 52843 
 53448 
 54054 
 54659 
 5 53 4 
 5 59 9 
 
 10 14 49 
 10 20 55 
 10 27 
 
 1033 5 
 10 39 10 
 10 45 16 
 10 51 21 
 
 10 57 26 
 
 11 331 
 
 16 20 4 
 16 26 9 
 16 32 14 
 16 38 20 
 164425 
 16 so 30 
 
 16 56 35 
 
 17 241 
 17 846 
 
 22 25 19 
 22 31 24 
 22 37 29 
 22 43 34 
 22 49 39 
 
 22 55 45 
 
 23 I 50 
 23 7 55 
 2314 
 
 0.41 
 0.42 
 
 0-43 
 0.44 
 
 0.45 
 0.46 
 
 0.47 
 0.48 
 0.49 
 
 230 
 233 
 237 
 241 
 244 
 248 
 252 
 25s 
 259 
 
 0.91 
 0.92 
 
 0-93 
 0.94 
 0.9s 
 0.96 
 0.97 
 0.98 
 0.99 
 
 5 32 
 536 
 5 40 
 
 5 43 
 5 47 
 551 
 SS4 
 558 
 2 
 
 50 
 
 11 9 37 
 
 17 '4 51 
 
 23 20 6 
 
 0.50 
 
 3 3 
 
 1. 00 
 
 6 5 
 
 SI 
 
 52 
 
 53 
 54 
 
 i 
 59 
 
 II 1542 
 II 21 47 
 
 II 27 52 
 
 11 3358 
 II 40 3 
 II 46 8 
 
 115213 
 
 11 58 19 
 
 12 424 
 
 17 20 56 
 1727 2 
 1733 7 
 17 39 12 
 17 45 17 
 17 51 23 
 
 17 57 28 
 
 '§ 3 33 
 
 18 938 
 
 23 26 1 1 
 23 32 16 
 23 38 21 
 23 44 27 
 23 50 32 
 
 23 56 37 
 
 24 242 
 24 848 
 24 14 S3 
 
 Example : Let the given mean 
 time be 14' 57" 32'.s6. 
 
 The table gives 
 first for 14'' 54"" 51" 2" 27' 
 then for 2 41 0.44 
 2 27.44 
 
 The sum 
 
 14' W" 12".i:6 -1-2" ■il'.AA I c' o^rf 
 
 60 
 
 6 515 
 
 12 10 29 
 
 181544 
 
 24 20 58 
 
 is the 
 
 required s 
 
 idereal tii 
 
 .. 
 
 Smithsonian Tables. 
 
 164 
 
Table 35. 
 
 CONVERSION OF SIDEREAL TIME INTO MEAN TIME. 
 
 
 m 
 
 m 
 
 m 
 
 m 
 
 
 
 
 
 
 s 
 
 
 
 I 
 
 2 
 
 3 
 
 
 
 
 
 
 
 h m s 
 
 h m s 
 
 h m s 
 
 h m s 
 
 s 
 
 m s 
 
 s 
 
 m s 
 
 
 o 
 
 000 
 
 6 615 
 
 12 12 29 
 
 18 1844 
 
 0.00 
 
 
 
 0.50 
 
 3 3 
 
 
 I 
 
 066 
 
 6 12 21 
 
 12 18 35 
 
 18 24 50 
 
 0.0 1 
 
 4 
 
 0.51 
 
 3 7 
 
 
 z 
 
 12 12 
 
 618 27 
 
 12 24 42 
 
 18 30 56 
 
 0.02 
 
 7 
 
 0.52 
 
 310 
 
 
 3 
 
 18 19 
 
 62433 
 
 12 30 48 
 
 1837 2 
 
 0.03 
 
 II 
 
 0-53 
 
 3 '4 
 
 
 4 
 
 02425 
 
 6 30 40 
 
 12 36 54 
 
 1843 9 
 
 0.04 
 
 15 
 
 0.54 
 
 318 
 
 
 S 
 
 03031 
 
 6 36 46 
 
 1243 
 
 18 49 15 
 
 0.05 
 
 18 
 
 0.55 
 
 321 
 
 
 6 
 
 03637 
 
 ^'^oS? 
 
 1249 7 
 
 185521 
 
 0.06 
 
 22 
 
 0.56 
 
 32s 
 
 
 7 
 
 04244 
 
 648 58 
 
 12 55 13 
 
 19 I 27 
 
 0.07 
 
 26 
 
 0.57 
 
 329 
 
 
 8 
 
 48 SO 
 
 655 4 
 
 13 I 19 
 
 19 7 34 
 
 0.08 
 
 29 
 
 0.58 
 
 332 
 
 
 9 
 
 05456 
 
 7 I II 
 
 13 725 
 
 19 13 40 
 
 0.09 
 
 033 
 
 0.59 
 
 336 
 
 
 10 
 
 112 
 
 7 7 17 
 
 13 13 31 
 
 19 ig 46 
 
 O.IO 
 
 037 
 
 0.60 
 
 340 
 
 
 n 
 
 I 7 9 
 
 7 1323 
 
 13 19 38 
 
 19 25 52 
 
 O.I I 
 
 40 
 
 0.61 
 
 3 43 
 
 
 12 
 
 I 13 IS 
 
 7 1929 
 
 13 25 44 
 
 19 31 59 
 
 0.12 
 
 044 
 
 0.62 
 
 3 47 
 
 
 13 
 
 I 19 21 
 
 72536 
 
 13 31 50 
 
 1938 5 
 
 0.13 
 
 048 
 
 0.63 
 
 351 
 
 
 14 
 
 I 2527 
 
 73142 
 
 13 37 56 
 
 19 44 n 
 
 0.14 
 
 051 
 
 0.64 
 
 3 54 
 
 
 15 
 
 131 34 
 
 73748 
 
 1344 3 
 
 19 50 17 
 
 0.1 5 
 
 055 
 
 0.65 
 
 358 
 
 
 16 
 
 1 3740 
 
 7 43 54 
 
 1350 9 
 
 19 56 23 
 
 0.16 
 
 059 
 
 0.66 
 
 4 2 
 
 
 17 
 
 14346 
 
 7 50 I 
 
 13 56 IS 
 
 20 230 
 
 0.17 
 
 I 2 
 
 0.67 
 
 4 5 . 
 
 
 18 
 
 14952 
 
 756 7 
 
 14 221 
 
 20 836 
 
 0.18 
 
 I 6 
 
 0.68 
 
 4 9 
 
 
 19 
 
 I 55 55 
 
 8 2 13 
 
 14 828 
 
 20 14 42 
 
 0.19 
 
 I 10 
 
 0.69 
 
 413 
 
 
 20 
 
 2 2 5 
 
 8 8 19 
 
 14 14 34 
 
 20 20 48 
 
 0.20 
 
 I 13 
 
 0.70 
 
 4 16 
 
 
 21 
 
 2 8 II 
 
 8 1426 
 
 14 20 40 
 
 20 26 55 
 
 0.21 
 
 I 17 
 
 0.71 
 
 4 20 
 
 
 22 
 
 2 14 17 
 
 82032 
 
 14 26 46 
 
 2033 I 
 
 0.22 
 
 I 21 
 
 0.72 
 
 424 
 
 
 23 
 
 2 20 24 
 
 82638 
 
 14 32 53 
 
 2039 7 
 
 0.23 
 
 124 
 
 0.73 
 
 427 
 
 
 24 
 
 22630 
 
 83244 
 
 14 38 59 
 
 20 45 13 
 
 0.24 
 
 1 28 
 
 0.74 
 
 431 
 
 
 25 
 
 23236 
 
 83851 
 
 1445 5 
 
 20 51 20 
 
 0.25 
 
 I 32 
 
 0.75 
 
 ^^ 
 
 
 26 
 
 23842 
 
 84457 
 
 14 51 II 
 
 20 57 26 
 
 0.26 
 
 135 
 
 0.76 
 
 438 
 
 
 27 
 
 24449 
 
 851 3 
 
 14 57 18 
 
 21 332 
 
 0.27 
 
 I 39 
 
 0.77 
 
 442 
 
 
 28 
 
 2 5055 
 
 857 9 
 
 IS 324 
 
 21 938 
 
 0.28 
 
 143 
 
 0.78 
 
 446 
 
 
 29 
 
 2 S7 I 
 
 9 316 
 
 15 930 
 
 21 1545 
 
 0.29 
 
 I 46 
 
 0.79 
 
 4 49 
 
 
 .30 
 
 3 3 7 
 
 9 922 
 
 15 15 36 
 
 21 21 51 
 
 0.30 
 
 150 
 
 0.80 
 
 4 53 
 
 
 31 
 
 3 9 14 
 
 91528 
 
 15 21 43 
 
 21 27 57 
 
 0.31 
 
 154 
 
 0.81 
 
 4 57 
 
 
 32 
 
 31520 
 
 9zt 34 
 
 152749 
 
 2134 3 
 
 0.32 
 
 157 
 
 0.82 
 0.83 
 0.84 
 0.85 
 0.86 
 0.87 
 0.88 
 0.89 
 
 5 
 
 
 33 
 
 321 26 
 
 92741 
 
 IS 33 55 
 
 21 40 10 
 
 0.33 
 
 2 I 
 
 'st 
 
 
 34 
 
 32732 
 
 9 33 47 
 
 1540 I 
 
 21 46 16 
 
 0.34 
 
 2 5 
 2 8 
 
 
 35 
 
 33338 
 
 9 39 53 
 
 1546 8 
 
 21 52 22 
 
 0-3S 
 
 5 II 
 
 
 36 
 
 3 39 45 
 
 9 45 59 
 
 15 52 14 
 
 21 58 28 
 
 0.36 
 
 2 12 
 2 16 
 
 515 
 
 
 37 
 
 3 45 51 
 
 952 5 
 
 15 58 20 
 
 22 4 35 
 
 0.37 
 
 5 19 
 
 
 38 
 
 3 51 57 
 
 9 58 12 
 
 16 4 26 
 
 22 10 41 
 
 0.38 
 
 2 19 
 
 522 
 526 
 
 
 39 
 
 358 3 
 
 10 4 18 
 
 16 10 33 
 
 22 16 47 
 
 0-39 
 
 223 
 
 
 40 
 
 4 4 10 
 
 10 10 24 
 
 16 16 39 
 
 22 22 53 
 
 0.40 
 
 226 
 
 0.90 
 
 530 
 
 
 41 
 
 4 10 16 
 
 10 16 30 
 
 16 22 45 
 
 22 29 
 
 0.41 
 
 230 
 
 0.91 
 
 5 33 
 
 
 42 
 
 4 16 22 
 
 10 22 37 
 
 16 28 51 
 
 2235 6 
 
 0.42 
 
 234 
 
 0.92 
 
 5 37 
 
 
 43 
 
 42228 
 
 10 28 43 
 
 16 34 57 
 
 22 41 12 
 
 0.43 
 
 237 
 
 0-93 
 
 541 
 
 
 44 
 
 42835 
 
 10 34 49 
 
 16 41 4 
 
 22 47 18 
 
 0.44 
 
 241 
 
 0.94 
 
 544 
 548 
 
 
 43 
 
 43441 
 
 10 40 55 
 
 16 47 10 
 
 22 53 24 
 
 0.45 
 0.46 
 
 245 
 
 0-95 
 0.96 
 
 
 46 
 
 44047 
 
 1047 2 
 
 16 53 16 
 
 22 59 31 
 
 2 48 
 
 5 52 
 
 
 47 
 48 
 
 44653 
 4 53 
 
 1053 8 
 10 59 14 
 
 16 59 22 
 
 17 529 
 
 23 5 37 
 23 II 43 
 
 0.47 
 0.48 
 
 252 
 2 56 
 
 oip 
 
 5 55 
 
 5 59 
 
 6 3 
 
 
 49 
 
 4 59 6 
 
 II 5 20 
 
 17 II 35 
 
 23 17 49 
 
 0.49 
 
 259 
 
 0.99 
 
 
 50 
 
 5 5 12 
 
 II II 27 
 
 17 17 41 
 
 23 23 56 
 
 0.50 
 
 3 3 
 
 1. 00 
 
 6 6 
 
 
 51 
 
 5 II i& 
 
 II 17 33 
 
 17 23 47 
 
 2330 2 
 
 
 
 
 
 
 52 
 
 5 17 25 
 
 II 2339 
 
 17 29 54 
 
 2336 8 
 
 Ex: 
 
 imple : Gi 
 
 ven is'c 
 
 "o". 
 
 
 53 
 
 52331 
 
 II 2945 
 
 1736 
 1742 6 
 
 23 42 14 
 23 48 21 
 
 Th 
 
 i table giv 
 
 :s 
 
 
 
 54 
 
 52937 
 
 II 35 52 
 
 first f 
 
 or 14" 57" 
 or 2 
 
 i8» 2° 
 
 27' 
 
 
 55 
 
 5 35 43 
 
 II 41 58 
 
 17 48 12 
 
 23 54 27 
 
 then i 
 
 42 
 
 0.44 
 
 
 56 
 
 59 
 
 541 50 
 54756 
 
 5 54 2 
 608 
 
 6 615 
 
 II 48 4 
 
 11 54 10 
 
 12 17 
 12 623 
 
 17 54 19 
 
 18 025 
 18 631 
 18 12 V 
 
 24 033 
 24 639 
 24 12 46 
 24 18 52 
 
 Th 
 
 1 5' O" ( 
 is the 
 
 15 
 5 differenc 
 y — 2°27'. 
 required i 
 
 2 . 
 e 
 
 44=14" 
 nean tims 
 
 27.44 
 
 57°32'-S6 
 
 
 60 
 
 12 1229 
 
 18 18 44 
 
 24 24 58 
 
 
 Smithsonian Tables. 
 
 x6S 
 
Table 36. 
 
 LENGTH OF ONE DEGREE OF THE MERIDIAN AT DIFFERENT 
 
 LATITUDES. 
 
 [Derivation of table explained on pp. xlvi-xlviii.] 
 
 Latitude. 
 
 0° 
 
 I 
 
 2 
 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 i6 
 
 17 
 i8 
 
 19 
 20 
 
 21 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 27 
 28 
 29 
 
 30 
 
 31 
 32 
 33 
 
 34 
 
 35 
 
 36 
 
 ^2 
 38 
 
 39 
 
 40 
 
 41 
 42 
 
 43 
 
 44 
 
 45 
 
 Metres. 
 
 110568.5 
 1 10568.8 
 
 1 10569.8 
 110571.5 
 110573.9 
 
 1 10577.0 
 
 1 10580.7 
 1 1058 5.1 
 
 1 10590.2 
 
 1 1 0595.9 
 
 1 10602.3 
 110609.3 
 110617.0 
 1 10625.3 
 1 10634.2 
 
 II 0643.7 
 
 1 10653.8 
 1 10664.5 
 
 1 10675.7 
 I 10687.5 
 
 1 10699.9 
 
 1 107 1 2.8 
 1 10726.2 
 
 1 1 0740. 1 
 110754.4 
 
 1 10769.2 
 110784,5 
 
 1 10800.2 
 110816.3 
 
 1 10832.8 
 
 110849.7 
 
 1 10866.9 
 1 10884.4 
 
 1 10902.3 
 
 1 10920.4 
 
 1 10938.8 
 110957.4 
 
 1 10976.3 
 110995.3 
 111014.5 
 
 " 1033.9 
 
 1 1 1053.4 
 IH073.0 
 1 1 1092.6 
 111112.4 
 
 111132.1 
 
 Statute 
 Miles. 
 
 68.703 
 "-68.704 
 68.705 
 68.706 
 68.707 
 
 68.709 
 68.711 
 68.714 
 68.717 
 68.721 
 
 68.725 
 68.729 
 68.734 
 68.739 
 68.745 
 
 68.751 
 
 68.757 
 68.763 
 68.770 
 68.778 
 
 68.786 
 68.794 
 68.802 
 68.8ro 
 68.819 
 
 68.829 
 68.838 
 68.848 
 68.858 
 68.868 
 
 68.879 
 68.889 
 68.900 
 68.911 
 68.923 
 
 68.934 
 68.946 
 68.957 
 68.969 
 68.981 
 
 68.993 
 69.005 
 69.017 
 69.029 
 69.042 
 
 69.054 
 
 Geographic 
 
 Miles. 
 i' of the Eq 
 
 59-594 
 59-594 
 59-595 
 59-596 
 59-597 
 
 59-598 
 59.600 
 59.603 
 59.606 
 59.609 
 
 59.612 
 59.616 
 J9.620 
 59.625 
 59.629 
 
 59-634 
 59.640 
 S9.646 
 59-652 
 59.658 
 
 59-665 
 59.672 
 
 59679 
 59.686 
 59.694 
 
 59.702 
 59-710 
 59-719 
 59-727 
 59-736 
 
 59-745 
 59-755 
 59.764 
 
 59-774 
 59.784 
 
 59-794 
 59.804 
 59.814 
 59.824 
 59-834 
 
 59-845 
 59-855 
 59.866 
 59-876 
 59.887 
 
 59.898 
 
 Latitude. 
 
 45° 
 
 46 
 47 
 48 
 
 49 
 
 50 
 
 51 
 52 
 S3 
 54 
 
 55 
 
 56 
 57 
 58 
 59 
 
 60 
 
 61 
 62 
 
 64 
 
 65 
 
 66 
 67 
 68 
 69 
 
 70 
 
 71 
 72 
 
 73 
 74 
 
 75 
 
 76 
 77 
 78 
 79 
 
 80 
 
 81 
 82 
 
 f3 
 84 
 
 85 
 
 86 
 
 87 
 88 
 89 
 
 90 
 
 Metres. 
 
 111132.1 
 111151.9 
 111171.6 
 111191.3 
 111210.9 
 
 111230.5 
 111249.9 
 
 1 1 1269.2 
 
 11 1 288.3 
 
 1 1 1307.3 
 
 1 1 1326.0 
 111344.5 
 II 1362.7 
 1 1 1380.7 
 
 1 1 1398.4 
 
 111415.7 
 
 1 1 1432.7 
 
 1 1 1449.4 
 111465.7 
 111481.5 
 
 1 11497.0 
 111512.0 
 
 1 1 1526.5 
 
 I "540.5 
 111554.1 
 
 111567.1 
 
 II 1579-7 
 111591.6 
 I n 603.0 
 111613.9 
 
 H1624.1 
 111633.8 
 
 1 1 1642.8 
 111651.2 
 1 1 1659.0 
 
 11 1666.2 
 111672.6 
 111678.5 
 111683.6 
 I1I688.1 
 
 111691.9 
 111695.0 
 n 1697.4 
 111699.2 
 11 1 700.2 
 
 1 1 1700.6 
 
 Statute 
 Miles. 
 
 69.054 
 69.067 
 69.079 
 69.091 
 69.103 
 
 69.1 IS 
 69.127 
 69.139 
 
 69-151 
 69.163 
 
 69.175 
 69.186 
 69.198 
 69.209 
 69.220 
 
 69.230 
 69.241 
 69.251 
 69.261 
 69.271 
 
 69.281 
 69.290 
 69.299 
 69.308 
 69.316 
 
 69.324 
 69.332 
 69.340 
 69-347 
 69-354 
 
 69.360 
 69.366 
 69.372 
 
 69-377 
 69.382 
 
 69.386 
 69.390 
 69-394 
 69397 
 69.400 
 
 69.402 
 69.404 
 69.405 
 69.407 
 69.407 
 
 69.407 
 
 Geographic 
 
 Miles. 
 i' of the Eq. 
 
 59.898 
 59.908 
 
 59919 
 59.929 
 59.940 
 
 S9-9SJ 
 59.961 
 
 59-972 
 59.982 
 59-992 
 
 60.002 
 60.012 
 
 6o.022 
 60.032 
 60.041 
 
 60.051 
 60.060 
 60.069 
 60.077 
 60.086 
 
 60.094 
 60.102 
 60.110 
 60.118 
 60.125 
 
 60.132 
 60.139 
 60.145 
 60.151 
 60.157 
 
 60.163 
 60.168 
 60.173 
 60.177 
 60.182 
 
 60.186 
 60.189 
 60.192 
 60.195 
 60.197 
 
 60.199 
 60.201 
 60.202 
 60.203 
 60.204 
 
 60.204 
 
 Smithsonian Tables. 
 
 166 
 
Table 37. 
 LENGTH OF ONE DEGREE OF THE PARALLEL AT DIFFERENT 
 
 
 
 
 
 LATITUDES. 
 
 
 
 
 
 
 [Derivation of table explained on 
 
 p. xlix.] 
 
 
 
 
 
 
 Statute 
 
 Geographic 
 
 
 
 Statute 
 
 Geographic 
 
 
 Latitude. 
 
 Metres. 
 
 Miles. 
 
 Miles. 
 i' of the Eq. 
 
 Latitude. 
 
 Metres. 
 
 Miles. 
 
 Miles. 
 i' of the Eq. 
 
 
 0"= 
 
 111321.9 
 
 69.171 
 
 60.000 
 
 45° 
 
 78850.0 
 
 48.99s 
 
 42.498 
 
 
 I 
 
 II 1305.2 
 
 69.162 
 
 59.991 
 
 46 
 
 77466.5 
 
 48-135 
 
 41-753 
 
 
 2 
 
 111254.6 
 
 69.130 
 
 59.964 
 
 47 
 
 76059-2 
 
 47.261 
 
 40.994 
 
 
 3 
 
 1H170.4 
 
 69.078 
 
 59.918 
 59-855 
 
 48 
 
 74628.5 
 
 46.372 
 
 40.223 
 
 
 4 
 
 1 1 1052.6 
 
 69.005 
 
 49 
 
 73174-9 
 
 45-469 
 
 39-440 
 
 
 5 
 
 110901.2 
 
 68.911 
 
 59-773 
 
 50 
 
 71698.9 
 
 44-552 
 
 38.644 
 
 
 6 
 
 1 107 16.2 
 
 68.796 
 
 59-673 
 
 51 
 
 70200.8 
 
 43.621 
 
 37-837 
 
 
 7 
 
 1 10497.7 
 
 68.660 
 
 59-556 
 
 52 
 
 68681.1 
 
 42.676 
 
 37.018 
 
 
 8 
 
 1 10245.8 
 
 68.503 
 
 59.420 
 
 S3 
 
 67140.3 
 
 41.719 
 
 36.187 
 
 
 9 
 
 109960.5 
 
 68.326 
 
 59.266 
 
 54 
 
 65578.8 
 
 40.749 
 
 35-346 
 
 
 10 
 
 109641.9 
 
 68.128 
 
 59-095 
 
 55 
 
 63997.1 
 
 39-766 
 
 34-493 
 
 
 II 
 
 109290.1 
 
 67.909 
 
 58.905 
 
 56 
 
 62395.7 
 
 38-771 
 
 33-630 
 
 
 12 
 
 108905.2 
 
 67.670 
 
 58.697 
 
 57 
 
 60775.1 
 
 37-764 
 
 32-757 
 
 
 13 
 
 108487.3 
 
 67.411 
 
 58.472 
 
 58 
 
 59135-7 
 
 36-745 
 
 31-873 
 
 
 H 
 
 108036.6 
 
 67.131 
 
 58.229 
 
 59 
 
 57478-1 
 
 35-715 
 
 30-979 
 
 
 15 
 
 I07SS3-I 
 
 66.830 
 
 57.969 
 
 60 
 
 55802.8 
 
 34-674 
 
 3ao76 
 
 
 i6 
 
 107037.0 
 
 66.510 
 
 57.690- 
 
 61 
 
 541 10.2 
 
 33.622 
 
 29.164 
 
 
 '7 
 
 106488.S 
 
 66.169 
 
 57-395 
 
 62 
 
 52400.9 
 
 32.560 
 
 28.243 
 
 
 i8 
 
 105907.7 
 
 65.808 
 
 57.082 
 
 f3 
 
 50675-4 
 
 31.488 
 
 27-313 
 
 
 19 
 
 105294.7 
 
 65.427 
 
 56.751 
 
 64 
 
 48934-3 
 
 30.406 
 
 26-374 
 
 
 20 
 
 104649.8 
 
 65.026 
 
 56.404 
 
 65 
 
 47178.0 
 
 29-315 
 
 25.428 
 
 
 21 
 
 103973.2 
 
 64.606 
 
 56.039 
 
 66 
 
 45407.1 
 
 28.215 
 
 24-473 
 
 
 22 
 
 103265.0 
 
 64.166 
 
 55-657 
 
 67 
 
 43622.2 
 
 27.106 
 
 23.511 
 
 
 23 
 
 102525.4 
 
 63.706 
 
 55-259 
 
 68 
 
 41823.8 
 
 25.988 
 
 22.542 
 
 
 24 
 
 101754.6 
 
 63.227 
 
 54-843 
 
 69 
 
 40012.4 
 
 24.862 
 
 21.566 
 
 
 25 
 
 100953.0 
 
 62.729 
 
 54.411 
 
 70 
 
 38188.6 
 
 23.729 
 
 20.583 
 
 
 26 
 
 1 00 1 20.6 
 
 62.212 
 
 53'963 
 
 71 
 
 36353-0 
 
 22.589 
 
 '§■593 
 
 
 27 
 
 99257.8 
 
 61.676 
 
 53-498 
 
 72 
 
 34506.2 
 
 21.441 
 
 18.598 
 
 
 28 
 
 98364.8 
 
 61. 121 
 
 53-0I6 
 
 73 
 
 32648.6 
 
 20.287 
 
 17-597 
 
 
 29 
 
 97441.9 
 
 60.548 
 
 52.519 
 
 74 
 
 30780.9 
 
 19.126 
 
 16-590 
 
 
 30 
 
 96489.3 
 
 59-956 
 
 52.006 
 
 75 
 
 28903.6 
 
 17.960 
 
 15-578 
 
 
 31 
 
 9SS°7-3 
 
 59-345 
 
 51.476 
 
 76 
 
 27017.4 
 
 16.788 
 
 14.562 
 
 
 32 
 
 94496.2 
 
 58-717 
 
 50-931 
 
 '^l 
 
 25122.8 
 
 1 5.61 1 
 
 13-541 
 
 
 33 
 
 93456-3 
 
 58.071 
 
 50-371 
 
 78 
 
 23220.4 
 
 14.428 
 
 12.51S 
 11.486 
 
 
 34 
 
 92387.9 
 
 57-407 
 
 49-795 
 
 79 
 
 21310.8 
 
 13.242 
 
 
 35 
 
 91 291. 3 
 
 56726 
 
 49.204 
 
 80 
 
 19394.6 
 
 12.051 
 
 10.453 
 
 
 36 
 
 goi66.i 
 
 56.027 
 
 48.598 
 
 81 
 
 17472-4 
 
 10.857 
 
 9-417 
 8-378 
 
 
 37 
 
 89014.8 
 
 55-3" 
 
 47-977 
 
 82 
 
 15544-7 
 
 f^^i 
 
 
 38 
 
 87835.6 
 
 54-578 
 
 47-341 
 
 f3 
 
 13612.2 
 
 8.458 
 
 7-337 
 6.293 
 
 
 39 
 
 86629.6 
 
 -53-829 
 
 46.691 
 
 84 
 
 11675-5 
 
 7-255 
 
 
 40 
 
 85397.0 
 
 53-063 
 
 46.027 
 
 85 
 
 9735-1 
 
 6.049 
 
 5-247 
 
 
 41 
 
 84138.4 
 
 52.281 
 
 45-349 
 
 86 
 
 7791-7 
 
 4.841 
 
 4.200 
 
 
 42 
 
 82854.0 
 
 51-483 
 
 44.656 
 
 87 
 
 5845-9 
 
 3-632 
 
 3-151 
 
 
 43 
 
 81544.2 
 
 50.669 
 
 43-950 
 
 88 
 
 3898-3 
 
 2.422 
 
 2.101 
 
 
 44 
 
 80209.4 
 
 49.840 
 
 43-231 
 
 89 
 
 1949.4 
 
 1. 211 
 
 1. 05 1 
 
 
 45 
 
 78850.0 
 
 48-995 
 
 42.498 
 
 90 
 
 0.0 
 
 0.000 
 
 0.000 
 
 Smithsonian Tables. 
 
 167 
 
Table 38. 
 
 INTERCONVERSION OF NAUTICAL AND STATUTE MILES. 
 
 1 nautical mile* =6080.37 feet 
 
 Nautical Miles. 
 
 Statute Miles. 
 
 Statute Miles. 
 
 Nautical Miles. 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 I 
 9 
 
 1.1516 
 2.3031 
 
 4.6062 
 
 S-7S78 
 6.9093 
 8.0609 
 9.2124 
 10.3640 
 
 1 
 2 
 3 
 4 
 
 5 
 6 
 
 I 
 9 
 
 0.8684 
 1.7368 
 2.6052 
 34736 
 
 4.3420 
 5.2104 
 6.0788 
 6.9472 
 7.815s 
 
 Smithsonian Tables. 
 
 * As defined by the United States Coast and Geodetic Survey. 
 
 Table 39. 
 
 CONTINENTAL MEASURES OF LENGTH WITH THEIR METRIC AND 
 ENGLISH EQUIVALENTS. 
 
 The asterisk (*) indicates that the measure is obsolete or seldom used. 
 
 Measure. 
 
 El, Netherlands 
 
 Fathom, Swedish = 6 feet 
 
 Foot, Austrian,* 
 
 old French * 
 
 Russian 
 
 Rheinlandisch or Rhenish (Prussia, 
 
 Denmark, Norway)* 
 
 Swedish* 
 
 Spanish *=i vara 
 
 *Klafter, Wiener (Vienna) 
 
 *Line, old French = -j^ foot 
 
 Mile, Austrian post* =24000 feet. . . . 
 
 German sea 
 
 Swedish ^36000 feet 
 
 Norwegian = 36000 feet 
 
 Netherlands (mijl) 
 
 Prussian (law of 1868) 
 
 Danish 
 
 Palm, Netherlands 
 
 *Rode, Danish 
 
 *kuthe, Prussian, Norwegian 
 
 Sagene, Russian 
 
 *Toise, old French =6 feet 
 
 *Vara, Spanish 
 
 Mexican 
 
 Werst, or versta, Russian = 500 sagene 
 
 Metric Equivalent. 
 
 I metre. 
 
 1.7814 " 
 0.31608 " 
 0.32484 " 
 0.30480 " 
 
 0.3138s " 
 0.2969 " 
 0.2786 " 
 1.89648 " 
 0.22558 cm. 
 7.58594 km. 
 1.852 
 
 10.69 
 
 11.2986 
 I 
 7.500 
 
 7.5324 
 
 0.1 
 
 3.7662 
 
 3.7662 
 
 2.1336 
 
 1.9490 
 
 °-83S9 
 0.8 ^8o 
 1.0668 
 
 metre. 
 
 km. 
 
 English Equivalent. 
 
 3.2808 feet. 
 
 5.8445 " 
 
 1.0370 " 
 
 1.0657 " 
 
 I « 
 
 1.0297 " 
 
 0.9741 " 
 
 0.9140 " 
 
 6.2221 " 
 
 0.0888 inch. 
 
 4.714 statute miles. 
 
 1. 1508 " 
 
 6.642 " 
 
 7.02 " " 
 
 0.6214 " " 
 
 4.660 " « 
 
 4.6804 " 
 
 0.3281 feet. 
 12.356 " 
 12.356 " 
 
 7 
 
 6.3943 " 
 
 2.7424 " 
 
 2.7293 " 
 3500 " 
 
 Smithsonian Tables. 
 
 168 
 
Table 40. 
 ACCELERATION (g) OF GRAVITY ON SURFACE OF EARTH AND 
 DERIVED FUNCTIONS. 
 
 ^^9.77989-1-0.05221 sin' ^ 
 
 := 9,80599—0.02610 COS 2^ metres.* 
 ^= geographical latitude. 
 
 <p 
 
 ir 
 
 logr 
 
 >°S4 
 
 10g^;«- 
 
 '^: 
 
 
 Metres. 
 
 
 
 
 Mitres. 
 
 0° 
 
 9.7798 
 
 0-99033 
 
 8.70864-10 
 
 0.64568 
 
 0.99090 
 
 S 
 
 .7803 
 
 035 
 
 862 
 
 569 
 
 09s 
 
 10 
 
 .7814 
 
 040 
 
 857 
 
 572 
 
 106 
 
 IS 
 
 •7834 
 
 049 
 
 848 
 
 576 
 
 127 
 
 20 
 
 •7859 
 
 060 
 
 837 
 
 S82 
 
 152 
 
 25 
 
 •7893 
 
 075 
 
 822 
 
 589 
 
 186 
 
 30 
 
 .7929 
 
 091 
 
 806 
 
 597 
 
 222 
 
 35 
 
 .7969 
 
 109 
 
 788 
 
 606 
 
 264 
 
 40 
 
 .8014 
 
 129 
 
 768 
 
 616 
 
 309 
 
 45 
 
 .8060 
 
 149 
 
 748 
 
 626 
 
 355 
 
 5° 
 
 .8105 
 
 169 
 
 728 
 
 636 
 
 401 
 
 55 
 
 .8150 
 
 189 
 
 708 
 
 646 
 
 447 
 
 60 
 
 .8191 
 
 207 
 
 690 
 
 65s 
 
 488 
 
 65 
 
 .8227 
 
 223 
 
 674 
 
 663 
 
 525 
 
 70 
 
 .8261 
 
 238 
 
 659 
 
 670 
 
 559 
 
 75 
 
 .8286 
 
 249 
 
 648 
 
 676 
 
 584 
 
 80 
 
 .8306 
 
 258 
 
 639 
 
 680 
 
 605 
 
 85 
 
 •8317 
 
 263 
 
 634 
 
 683 
 
 616 
 
 90 
 
 .8322 
 
 26s 
 
 632 
 
 684 
 
 621 
 
 Smithsonian Tables. 
 
 * From The Solar Parallax and its Related Constants, by Wm. Harkness, Professor of Mathematics, U. S. N. ; 
 Washington : Government Printing OflBce, 1891. 
 t Tliis is length of seconds pendulum. 
 
 169 
 
Table 41 . 
 
 LINEAR EXPANSIONS OF PRINCIPAL METALS, IN MICRONS PER 
 METRE (OR MILLIONTHS PER UNIT LENGTH). 
 
 Name of metal. 
 
 Aluminum . . 
 Brass .... 
 Copper . . . 
 Glass .... 
 Gold .... 
 Iron, cast . . . 
 Iron, wrought 
 Lead .... 
 Platinum . . . 
 Platinum-iridium i 
 SUver .... 
 Steel, hard . . 
 Steel, soft . . . 
 
 Tin 
 
 Zinc 
 
 Expansion per 
 degree C. 
 
 20 
 
 19 
 17 
 9 
 IS 
 II 
 12 
 28 
 
 19 
 12 
 II 
 
 19 
 29 
 
 Expansion per 
 degree F. 
 
 II. I 
 
 10.S 
 
 9.4 
 
 6.1 
 
 6.7 
 
 iS-S 
 
 4.8 
 
 IO.S 
 
 6.1 
 10.S 
 16.1 
 
 Smithsonian Tables. 
 
 > Of Intematioiial Prototype Metres. 
 
 Table 42. 
 
 FRACTIONAL CHANCE IN A NUMBER CORRESPONDING TO A CHANGE 
 
 IN ITS LOGARITHM. 
 
 Computed from the formula, 
 
 AiV _ Alogjy 
 N ~ II. ' 
 
 II = modulus of commou logarithms = 0.43429448. 
 
 For 
 
 AlogiV 
 
 = I unit in 
 
 AJ\r 
 N 
 
 For 
 
 AlogiV 
 
 = 4 units in 
 
 AA^ 
 
 N 
 
 (in round numbers) 
 
 4th place 
 5th " 
 6th " 
 7th « 
 
 
 4th place 
 Sth " 
 6th " 
 7th " 
 
 nfW 
 
 Iftlot 
 
 loolinnr 
 
 4S4384C 
 
 Smithsonian Tables. 
 
 170 
 
APPENDIX. 
 
 CONSTANTS. 
 
 Numerical Constants. 
 Base of natural (Napierian) logarithms, 
 Log e, modulus of common logarithms, 
 Circumference of circle in degrees, 
 " " " in minutes, 
 
 " " " in seconds, 
 
 Circumference of circle, diameter unity. 
 
 Number. 
 
 Logarithm. 
 
 
 2* = 6.2831853 
 
 0.7981799 
 
 
 -^ = 1.0471976 
 
 0.0200286 
 
 
 1= 0.3183099 
 
 9.5028501 - 
 
 -10 
 
 ir2 = 9.8696044 
 
 0.9942997 
 
 
 Number. 
 = e =2.7182818 
 
 = A* = 0434294s 
 = 360 
 
 = 21600 
 = 1296000 
 
 = If =3.14159265 
 
 l/ifi = 0.1013212 
 
 Vt = 17724539 
 
 -ri = 0.5641896 
 V IT 
 
 The arc of a circle equal to its radius is 
 
 in degrees, p° = i8o/w 
 
 in minutes, p = 60 p° 
 
 in seconds, p" = 60 p' 
 For a circle of unit radius, the 
 
 arc of 1° = i/p° 
 
 arc of i' = i/p' 
 
 arc (or sine) of i"= i /p" 
 
 V7= 
 
 ■■ I.4I42I36 
 ■■ 1.7320508 
 
 Logarithm. 
 
 0.4342945 
 9-6377843 — 10 
 2.5563025 
 
 4-3344538 
 6. n 26050 
 0.4971499 
 
 9.0057003 — 10 
 0-2485749 
 9.7514251 — 10 
 
 0.1505150 
 0.2385607 
 
 57.29578° 1.7581226 
 
 3437-7468' 3-S362739 
 
 : 206264.8" 5.3144251 
 
 = 0.0174533 8.2418774 — 10 
 
 - 0.0002909 6.4637261 — 10 
 
 = 0.00000485 4.6855749 — 10 
 
 Geodetical Constants. 
 
 Dimensions of the earth (Clarke's spheroid, 1866) and derived quantities. 
 
 Equatorial semi-axis in feet, 
 
 in mUes, 
 Polar semi-axis in feet, 
 in miles, 
 
 (Eccentricity)^ =^^^ 
 Flattening = 2^:ii 
 
 = a = 20926062. 
 = a= 3963.3 
 = i = 208551 21. 
 = * = 3949-8 
 ^ = 0.00676866 
 
 7.3206875 
 3.5980536 
 7.3192127 
 3.5965788 
 
 7-8305030- 
 
 =/= 1/294.9784 7-5302098 — 10 
 
 = 24859.76 miles. 
 
 = 24901.96 " 
 
 = 196940400 square miles. 
 
 = 5.576± 0.016. 
 
 = 2.56 ±0.16. 
 
 Perimeter of meridian ellipse. 
 Circumference of equator. 
 
 Area of earth's surface, 
 Mean density of the earth (Harkness) 
 Surface density " " " 
 
 Acceleration of gravity (Harkness) : 
 
 g (cm. per second) = 980.60 (i — 0.002662 cos 2^) for latitude ^ and sea level. 
 
 g, at equator = 977.99 ; g, at Washington = 980.07 ; g, at Paris = 980.94 j 
 
 g, at poles = 983.21 ; g, at Greenwich = 981.17. 
 Length of the seconds pendulum (Harkness) : 
 
 / = 39.01 2540 -|- 0.208268 sin^ I/) inches = 0.990910 + 0.005290 sin^ (p metres. 
 
 Smithsonian Tables. 
 
 171 
 
APPENDIX. 
 
 CONSTANTS. -Continued. 
 
 Astronomical Constants (Harkness). 
 Sidereal year = 365.256 357 8 mean solar days. 
 Sidereal day = 23A ^Qi" 4.1100 mean solar time. 
 Mean solar day = 24* yn 56.^546 sidereal time. 
 Mean distance of the earth from the sun = 92 800 oco miles. 
 
 Physical Constants. 
 Velocity of light (Harkness) = 186 337 miles per second = 299 878 km. per second. 
 Velocity of sound through dry air = 1090 ^/I + 0.00367 t° C. feet per second. 
 Weight of distilled water, free from air, barometer 30 inches : 
 
 Weight in grains. Weight in grammes. 
 
 Volume. 62° F. 4O C. 6i° F. 4O C. 
 
 I cubic inch (determination of 1890) 252.286 252.568 16.3479 16.3662 
 
 I cubic centimetre (1890) 15-3953 'S4'2S 0.9976 0.9987 
 
 I cubic foot (1890) at 62° F. 62.2786 lbs. 
 
 A standard atmosphere is the pressure of a vertical column of pure mercury whose 
 height is 760 mm. and temperature 0° C, under standard gravity at latitude 45° 
 and at sea level. 
 I standard atmosphere = 1033 grammes per sq. cm. = 14.7 pounds per sq. inch. 
 Pressure of mercurial column i inch high = 34.5 grammes per sq. cm. = 0.491 
 pounds per sq. inch. 
 
 Weight of dry air (containing 0.0004 of i's weight of carbonic acid) : 
 
 I cubic centimetre at temperature 32° F. and pressure 760 mm. and under the 
 standard value of gravity weighs 0.001 293 05 gramme. 
 Density of mercury at 0° C. (compared with water of maximum density under atmos- 
 pheric pressure) = 13.5956. 
 Freezing point of mercury = — 38.°5 C. (Regnault, 1862.) 
 
 Coefficient of expansion of air (at const, pressure of 76o»"«) for 1° C. (do.) : 0.003670. 
 Coefficient of expansion of mercury for Centigrade temperatures (Broch) : 
 
 A = Aq (i — 0.000 181 792 t — 0.000000000 175 fl — .000000000035 llfi '")• 
 Coefficient of linear expansion of brass for 1° C, = 0.000 0174 to 0.000 0190. 
 Coefficient of cubical expansion of glass for 1° C, y = 0.000 021 to 0.000 028. 
 
 Ordinary glass (Recknagel) : at 10° C, 7 = 0.000 0255 ; at 100°, 7 = 0.000 0276. 
 Specific heat of dry air compared with an equal weight of water : 
 
 at constant pressure, £p = 0.2374 (from 0° to 100° C, Regnault). 
 
 at constant volume, A» = 0.1689. 
 
 Ratio of the two specific heats of air (Rontgen) : ^p /Kv = 1.4053. 
 
 Thermal conductivity of air (Graetz) : k = 0.000 0484 (i + 0.001 85 f, C.) ^^'"""' 
 
 cm. sec. 
 [The quantity of heat that passes in unit time through unit area of a plate of unit thickness, when its 
 opposite faces differ in temperature by one degree.] 
 
 Latent heat of liquefaction of ice (Bunsen) = 80.025 mass degrees, C. 
 Latent heat of vaporization of water = 606.5 — °'695 '° C. 
 
 Absolute zero of temperature (Thomson, Heat, Encyc. Brit.) : — 273.°o C. = 459.°4 F. 
 
 Mechanical equivalent of heat : * 
 
 I pound-degree, F. (the British thermal unit) = about 778 foot-pounds. 
 
 I pound-degree, C. = 1400 foot-pounds. 
 
 1 calorie or kilogramme-degree, C. = 3087 foot-pounds = 426.8 kilogram- 
 metres = 4187 joules (for g — 981 cm.). 
 
 Smithsonian Tables. 
 
 * Based on Prof. Rowland^s determinations. (Proc. A m. Acad, Arts and ScU^ 1880.) 
 
 172 
 
APPENDIX. 
 
 SYNOPTIC CONVERSION OF ENGLISH AND METRIC UNITS. 
 
 English to Metric. 
 
 Units of length. 
 
 Matric equivalents. 
 
 I inch. 
 I foot. 
 I yard. 
 I mile. 
 
 Units of area. 
 I square inch. 
 I square foot. 
 I square yard. 
 I acre. 
 I square mile. 
 
 41 » 
 
 Units of volume. 
 1 cubic inch. 
 I cubic foot. 
 I cubic yard. 
 
 Units of capacity. 
 
 I gallon (U. S.) = 231 cubic inches. 
 
 I quart (U. S.). 
 
 I Imperial gallon (British^. 
 
 277.463 cubic inches (i8go). 
 I bushel (U. S.) = 2150.42 cubic inches. 
 I bushel (British). 
 
 centimetres, 
 metre. 
 
 2.54000 
 0.304801 
 0.914402 " 
 
 1.60935 kilometres. 
 
 6.45163 
 
 929.034 
 0.8361 31 
 0.404687 
 2.59000 
 
 259.000 
 
 16.3872 
 0.028317 
 0.764559 
 
 64.7990 
 
 0-453593 
 28.3496 
 
 3i-'035 
 1. 01 605 
 0.907186 
 
 square centimetres. 
 
 square metre. 
 
 hectares. 
 
 square kilometres. 
 
 hectares. 
 
 cubic centimetres, 
 cubic metres or steres. 
 cubic metres or steres. 
 
 3.78544 litres. 
 0.94636 litres. 
 4.54683 litres. 
 
 35.2393 litres. 
 36.3477 litres. 
 
 milligrammes. 
 
 kilogrammes. 
 
 grammes. 
 
 grammes. 
 
 tonnes. 
 
 tonnes. 
 
 Logarithms. 
 
 0.404 83s 
 9.484016 — 10 
 9.961 137 — 10 
 0.206 650 
 
 0.809 669 
 2.968 032 
 9.922 274 — 10 
 9.607 1 20 — 10 
 0.413300 
 2.413 300 
 
 1. 214 504 
 8.452 047 — 10 
 9.883411 — 10 
 
 0.578 u6 
 9.976056 — 10 
 0.657 709 
 
 1.547027 
 1.560477 
 
 i.8n 568 
 9.656 666 — 10 
 1.452 546 
 1.492810 
 0.006 914 
 9.957696 — 10 
 
 Units of mass. 
 
 I grain. 
 
 I pound avoirdupois. 
 
 I ounce avoirdupois. 
 
 I ounce troy. 
 
 I ton ^2240 lbs.). 
 
 I ton (2000 lbs.). 
 
 Units of velocity. 
 
 I foot per sec. (0.6818 miles per hr.) = 0.30480 metres per sec. = 1.0973 km. per hr. 
 I mile per hr. (1.4667 feet per sec.) = 0.44704 metres per sec. = 1.6093 !""• P^"^ hr. 
 
 Units of force. 
 I poundal. 
 
 Weight of I grain (for g = 981 cm.). 
 Weight of I pound av. (for^= 981 cm.). 
 
 13825.5 dynes. 
 
 63.57 dynes. 
 
 4.45 X 10' dynes. 
 
 4.140 682 
 1.803 237 
 5-648 335 
 
 Units of stress — in gravitation measure. 
 I pound per square inch = 70.307 grammes per sq. centimetre. 
 I pound per square foot = 4.8824 kilogrammes per sq. metre. 
 
 Units of work- 
 1 foot-poundal. 
 
 -In absolute measure. 
 
 421 403 ergs. 
 
 1.846997 
 0.688 634 
 
 5.624 6g8 
 
 — in gravitation measure. 
 I foot-pound (for^= 981 cm.) = 1356.3 X 10* ergs = 0.138255 kilogram-metres. 
 
 Units of activity (rate of doing work). 
 I foot-pound per minute (for^= 981 cm.) = 0.022605 watts. , , , 
 
 1 horse-power (33 000 foot-pounds per min.) = 746 wa s = 1.01387 force de cheval. 
 
 Units of heat. 
 
 I pound-degree, F. 
 I pound-degree, C. 
 
 — 252 small calories or gramme-degrees, C. 
 = 1.8 pound-degrees, F. 
 
 Smithsonian Tables. 
 
 173 
 
APPENDIX. 
 
 SYNOPTIC CONVERSION OF ENGLISH AND METRIC UNITS. 
 
 Metric to English. 
 
 English equivalents. 
 
 Logarithms. 
 
 Units of length. 
 I metre (lo' microns). 
 
 I kilometre. 
 Units of area. 
 
 I square centimetre. 
 
 I square metre. 
 ti it 
 
 I hectare. 
 
 I square kilometre. 
 
 Units of volume. 
 
 I cubic centimetre. 
 
 I cubic metre or st6re. 
 (( t( ii 
 
 Units of capacity. 
 I litre (61.023 cubic inches). 
 
 I hectolitre. 
 
 Units of mass. 
 I gramme. 
 I kilogramme. 
 (( 
 
 (1 
 
 I tonne. 
 It 
 
 Units of velocity. 
 I metre per second, 
 ti t( (1 
 
 I km. per hr. (0.2778 m. per sec). 
 
 Units of force. 
 
 I dyne (weight of (981)-' grammes, iorg= 981 cm.) ■■ 
 
 Units of stress — In gravitation measure. 
 I gramme per square centimetre. o 014223 pounds per sq. inch. 
 I kilogramme per square metre. 0.204817 pounds per sq. foot. 
 
 I standard atmosphere. i47 pounds per sq. inch. 
 
 Units of work — In absolute measure, 
 I erg. 2.3730 X 10"* foot poundals. 
 
 I megalerg = 10° ergs j i joule = 10'' ergs. 
 
 — In gravitation measure. 
 I kilogramme-metre (£or^= 981 cm.) = 981 X lo^ ergs = 7.2330 foot-pounds. 
 
 Units of activity (rate of doing work). 
 I watt = I joule per sec. (= 44.2385 foot-pounds per minute, for ^= 981 cm.) = 0.10194 
 
 kilogramme-metre per sec, for ^ = 981 cm. 
 I force de cheval = 75 kilogramme-metres per sec. = 735J watts = 0.98632 horse-power. 
 
 Units of heat. 
 I calorie or kilogramme-degree = 3.968 pound-degrees, F. = 2.2046 pound-degrees, C. 
 I small calorie or therm, or gramme-degree = o.ooi calorie or kilogramme-degree. 
 
 39.3700 
 
 inches. 
 
 1-595 165 
 
 
 3-28083 
 
 feet. 
 
 0.515984 
 
 
 1. 09361 
 
 yards. 
 
 0.038863 
 
 
 0.62137 
 
 miles. 
 
 9-793350- 
 
 - 10 
 
 0.15500 
 
 square inches. 
 
 9.190 331- 
 
 - 10 
 
 10.7639 
 
 square feet. 
 
 1.03 1 968 
 
 
 I.I9599 
 
 square yards. 
 
 0.077 726 
 
 
 2.47104 
 
 acres. 
 
 0.392880 
 
 
 0.38610 
 
 square miles. 
 
 9.586 701 — 
 
 -10 
 
 0.0610234 cubic inches. 
 
 8.785496- 
 
 -10 
 
 35-3145 
 
 cubic feet. 
 
 1-547 953 
 
 
 1.30794 
 
 cubic yards. 
 
 0.1 16 589 
 
 
 0.26417 
 
 gallons (U. S.). 
 
 9.421 884 — 
 
 -10 
 
 1.05668 
 
 quarts (U. S.). 
 
 0.023 944 
 
 
 0.21993 
 
 Imp. gallons (British). 
 
 9.342 291 - 
 
 10 
 
 2.83774 
 
 bushels (U. S.). 
 
 0.452973 
 
 
 2.75121 
 
 bushels (British). 
 
 0-439 523 
 
 
 15-4324 
 
 grains. 
 
 1.188433 
 
 
 2.20462 
 
 pounds avoirdupois. 
 
 0-343334 
 
 
 35-2739 
 
 ounces avoirdupois. 
 
 1-547 454 
 
 
 32.1507 
 
 ounces troy. 
 
 1.507 190 
 
 
 0.98421 
 
 tons (2240 lbs.). 
 
 9.993086- 
 
 10 
 
 1.10231 
 
 tons (2000 lbs.). 
 
 0.042 304 
 
 
 3.2808 
 
 feet per second. 
 
 0.515984 
 
 
 2.2369 
 
 miles per hour. 
 
 0-349653 
 
 
 0.62137 
 
 miles per hour. 
 
 9-793350- 
 
 10 
 
 = 7.2330 X lo-* poundals. 
 
 {See def. p. 172.) 
 
 Smithsonian Tables. 
 
 174 
 
APPENDIX. 
 
 DIMENSIONS OF PHYSICAL QUANTITIES. 
 
 L = length ; M = mass ; T = time. 
 
 Quantity. 
 Area. 
 Volume. 
 Mass. 
 Density. 
 Velocity. 
 Acceleration. 
 Angle. 
 Angular Velocity. 
 
 Dimensions Quantity. 
 
 [L^] Momentum. 
 
 [L8] Moment of Inertia. 
 
 [M] Force. 
 
 [M L~*] Stress (per unit area). 
 
 [LT-i] Work or Energy. 
 
 [LT-2] Rate of Working (Power), 
 
 [o] H e a t. 
 
 [■j—i] Thermal Conductivity. 
 
 Dimensions. 
 [L M T-i] 
 [ML2] 
 [L M T-2] 
 [L-i M T-2] 
 [L2 M T-^] 
 [L2 M T-2] 
 [U M T-2] 
 [L-i M T-i] 
 
 In Electrostatics. 
 
 Quantity of Electricity. 
 
 Surface Density: quantity per unit area. 
 
 Difference of Potential: quantity of work required 
 to move a quantity of electricity ; (work done) -f- (quan- 
 tity moved). 
 
 Electric Force, or Electro-motive Intensity: 
 (quantity) -;- (distance^). 
 
 Capacity of an accumulator : e-i- £. 
 
 Specific Inductive Capacity. 
 
 In Magnetics. 
 
 Q u a n t i t y of- Magnetism, or Strength of Pole. 
 Strength or Intensity of Field: 
 
 (quantity) -^ (distance^). 
 Magnetic Force. 
 
 Magnetic Moment: (quantity) X (length). 
 Intensity of Magnetization: magnetic moment per 
 
 unit volume. 
 Magnetic Potential: work done in moving a quantity 
 
 of magnetism j (work done) -f- (quantity moved). 
 Magnetic Inductive Capacity. 
 
 Dimensions in 
 Symbol, electrostatic system. 
 
 e [L? M* T-i] 
 
 T [L-* Mi T-i] 
 
 £ [L* Mi T-i] 
 
 Cor g 
 k 
 
 m 
 S 
 
 m I 
 I 
 
 [L-» M* T-i] 
 
 [L] 
 
 [o] 
 
 Dimensions in 
 
 electro-magnetic 
 
 system, 
 
 [L^ M* T-i] 
 
 [L-* Mi T-i] 
 
 [L-i Mi T-i] 
 \\} Mi T-i] 
 [L-i Mi T-i] 
 
 In Electro-magnetics. Symbol. 
 
 Intensity of Current. i 
 
 Quantity of Electricity conveyed by current : e 
 
 (intensity) X (time). 
 Potential, or difference of potential : (work 
 
 done) -=- (quantity of electricity upon which 
 
 work is done). 
 Electric Force: the mechanical force act- 
 ing on electro-magnetic unit of quantity; 
 
 (mechanical force) -f- (quantity). 
 Resistance of a conductor : E-^i. R 
 
 Capacity: quantity of electricity stored up q 
 
 per unit potential-difference produced by it. 
 Specific Conductivity: the intensity of 
 
 current passing across unit area under the 
 
 action of unit electric force. 
 Specific Resistance: the reciprocal of r 
 
 specific conductivity. 
 
 Vox a. [LiMiT-i] 
 
 y- [o] 
 
 Dimensions in Name of 
 
 electro-magnetic practical unit, 
 
 [Li Mi T-i] AmpSre. 
 [Li Mi] Coulomb. 
 
 E [L? Mi T-2] 
 
 3B [Li Mi T^ 
 
 [LT-i] 
 [L-i T2] 
 
 [L-^T] 
 [L2 T-i] 
 
 Volt. 
 
 Ohm. 
 Farad. 
 
 Smithsonian Tables. 
 
 I7S 
 
INDEX. 
 
 Acceleration, dimensions of 175 
 
 of gravity, formula for 171 
 
 table of values of igo 
 
 Air, cubical expansion, specific heat, thermal 
 
 conductivity, and weight of 172 
 
 Airy, Sir George, treatise cited xcviii 
 
 Albrecht, Dr. Th., treatise cited Ixxx 
 
 Algebraic formulas xiii-xv 
 
 Alignment curve ]vi 
 
 Aluminum, linear expansion of 170 
 
 Ampere, dimensions of 17 c 
 
 Angles, equivalents in arcs xviii 
 
 sum of, in spheroidal triangle Ivii 
 
 Angular velocity, dimensions of 175 
 
 Annulus, circular, area of xxx 
 
 Antilogarithms, explanation of use of xcix 
 
 4-place table of 26, 27 
 
 Appendix 171-17S 
 
 Arcs, equivalents in angles xvii 
 
 of meridians and parallels xlvi-1 
 
 table of lengths of meridional 78-80 
 
 table of lengths of parallel 81-83 
 
 table of time equivalents 162 
 
 Are xli 
 
 Area, of circle xxx 
 
 table of values of 23 
 
 of surface of earth 1-lii 
 
 Areas, of continents Ixv 
 
 of oceans Ixv 
 
 of plane and curved surfaces xxix-xxxi 
 
 of zones and quadrilaterals of the 
 
 earth's surface 1-lii 
 
 tables of values of 142-159 
 
 of regular polygons xxx 
 
 Arithmetic means, progression, and series.. xiii 
 
 Astronomical constants 172 
 
 co-ordinates Ixvii 
 
 latitude xliv 
 
 time Ixxii 
 
 Astronomy Ixvii-lxxxii 
 
 references to works on Ixxxii 
 
 Atmosphere, mass of earth's Ixvi 
 
 standard pressure of 172 
 
 weight of unit of volume of 172 
 
 Average error, definition of Ixxxiv 
 
 Azimuth, astronomical and geodetic Ivii 
 
 computation of differences of Iviii-lxi 
 
 determination of Ixxix 
 
 Babinet, barometric formula of 160 
 
 Barometer, heights by jgo 
 
 Binomial series xiv 
 
 Brass, linear expansion of 170 
 
 Brunnow, F., treatise cited Ixxxii 
 
 Bushel, Winchester xxxv 
 
 equivalent in litres 2 
 
 Cable length xxxviii 
 
 Calorie, value of 172 
 
 Capacity, measures of, British xxxviii 
 
 Metric xli 
 
 Centare jjli 
 
 Chauvenet, Wm., treatise cited Ixxxii 
 
 Circumference, of circle xxviii 
 
 table of values of . 
 
 23 
 
 of earth xlix, 171 
 
 of ellipse xxix 
 
 C. G. S. system of units xlii 
 
 Clarke, General A. R., spheroid of xliii 
 
 treatise cited ixyi 
 
 Coefficient, of cubical expansion of air and 
 mercury 172 
 
 of linear expansion of metals 170 
 
 of refraction. . .'. Ixiii 
 
 Compression, of earth xliii 
 
 Computation, of differences of latitude, lon- 
 gitude, and azimuth lyiij 
 
 of mean and probable errors xcv 
 
 Conductivity, thermal, of air 172 
 
 Cone, surface of xxxi 
 
 volume of xxxii 
 
 Constants, astronomical 172 
 
 geodetical 171 
 
 numerical 171 
 
 of earth's spheroid xliv 
 
 Continental measures (table of British and 
 
 Metric equivalents) 168 
 
 Continents, areas of Ixv 
 
 average heights of Ixv 
 
 Conversion, of arcs into angles and angles 
 into arcs xvii 
 
 of British and Metric units. . .2, 3, 173, 174 
 Co-ordinates, astronomical Ixvii 
 
 for projection of maps liii-lvi 
 
 table of, scale 1/250000 84-91 
 
 table of, scale 1/125000 92-101 
 
X78 
 
 INDEX. 
 
 Co-ordinates (continued). 
 
 table of, scale 1/126720 102-109 
 
 table of, scale 1/63360 110-121 
 
 table of, scale 1/200000 122-131 
 
 table of, scale 1/80000 1 32-1 41 
 
 of generating ellipse of earth's spheroid . . xliv 
 
 Copper, linear expansion of 170 
 
 Cord (of wood), volume of xxxix 
 
 Correction, for astronomical refraction, table 
 
 of mean values of 161 
 
 to observed angle for eccentric position 
 
 of instrument Ixiii 
 
 to reduce measured base to sea level. . .Ixiv 
 
 Cosines, table of natural 28, 29 
 
 use of table explained c 
 
 Cotangents, table of natural 3°. 3' 
 
 use of table explained c 
 
 Coulomb, dimensions of 175 
 
 Cubature, of volumes xxxii 
 
 Cubes, table of 4-22 
 
 Cube roots, table of 4-22 
 
 Cylinder, surface of xxxi 
 
 volume of xxxii 
 
 Day, sidereal and solar Ixxii, 172 
 
 Degrees, number of, in unit radius xviii 
 
 of terrestrial meridian xlvi, 166 
 
 of terrestrial parallel xlix, 167 
 
 Density, mean, of earth Ixv 
 
 mean, of superficial strata of earth Ixv 
 
 of mercury 172 
 
 Departures (and latitudes), table of 32-47 
 
 mode of use of table explained c 
 
 Depths, average, of oceans Ixv 
 
 Determination, of azimuth Ixxix 
 
 of heights, by barometer 160 
 
 by trigonometric leveling Ixi 
 
 of latitude Ixxvii 
 
 of time Ixxiv 
 
 Difference, between astronomical and geo- 
 detic azimuth Ivii 
 
 of heights, by barometer i6o 
 
 by trigonometric leveling Ixi 
 
 Differences, of latitude, longitude, and azi- 
 muth, on earth's spheroid Iviii 
 
 table for computation of 70-77 
 
 Difierential formulas xxi 
 
 Dimensions, of earth xliii, 171 
 
 of physical quantities 175 
 
 Dip, of sea horizon Ixiii 
 
 Distance, of sea horizon Ixiii 
 
 of sun from earth 172 
 
 Doolittle, Prof. C. L., treatise cited Ixxxii 
 
 Earth, compression of xliii, 171 
 
 Earth (continued). 
 
 density of Ixv 
 
 dimensions of xliii, 171 
 
 ellipticity of xliii, 171 
 
 energy (of rotation) of Ixvl 
 
 equatorial perimeter of xliii, 171 
 
 flattening of xliii, 171 
 
 mass of Ixvi 
 
 meridian perimeter of xlix, 171 
 
 moments of inertia of Ixvi 
 
 shape of xliii 
 
 surface area of lit 
 
 volume of Ixv 
 
 Eccentricity, of ellipse xliii 
 
 of earth's spheroid xliv, 171 
 
 El, value of 168 
 
 Electric quantities, dimensions of 175 
 
 Electro-magnetic quantities, dimensions of . 175 
 
 Ellipse, area of xxx 
 
 equations to xliv 
 
 length of perimeter of xxix 
 
 Ellipsoid, volume of (see Spheroid) xxxiii 
 
 Ellipticity, of earth xliii, 171 
 
 Energy, dimensions of 175 
 
 of rotation of earth Ixvi 
 
 Equations, of ellipse xliv 
 
 of Prototype Kilogrammes xl 
 
 of Prototype metres xl 
 
 Error, in ratio of English yard to Metre, .xxxvii 
 
 Errors, probable, mean, average . .Ixxxiv, Ixxxviii 
 
 table of, for interpolated quantities. .Ixxxvi 
 
 theory of Ixxxiii 
 
 Everett, J. D., treatise cited xlii 
 
 Excess, spherical or spheroidal Iviii 
 
 Expansion, cubical, for air and mercury .... 172 
 linear, of principal metals 170 
 
 Farad, dimensions of 175 
 
 Fathom, length of xxxviii 
 
 Swedish 168 
 
 Flattening, of earth xliii, 171 
 
 Foot, Austrian 168 
 
 British xxxvii 
 
 French, Rhenish, Spanish, Swedish. . . . i68 
 
 Force, dimensions of 175 
 
 Formulas, algebraic xiii-xv 
 
 for differentiation xxi 
 
 for integration xxiii 
 
 for solution of plane triangles xviii 
 
 for solution of spherical triangles xx 
 
 trigonometric xv 
 
 Freezing point of mercury 172 
 
 Functions, trigonometric, of one angle xv 
 
 of two angles xvi 
 
 special values of xv 
 
 values in series xvii 
 
INDEX. 
 
 179 
 
 Gallon, British and wine xxxviii 
 
 Gauss's formulas for spherical triangles xxi 
 
 Geocentric latitude xliv 
 
 Geodesy xliii-lxvi 
 
 references to works on Ixvi 
 
 Geodetic azimuth Ivii 
 
 Geodetic differences of latitude, longitude, 
 
 and azimuth Iviii 
 
 Geodetic line Ivii 
 
 Geodetical constants 171 
 
 Geographical latitude xliv 
 
 Geographical positions, computation of .Iviii-lxi 
 
 Geoid, definition of xliii 
 
 Geometric means, progression xiii 
 
 Glass, linear expansion of 170, 172 
 
 Gold, linear expansion of 170 
 
 Gravity, acceleration of, formula for 171 
 
 table of values of 169 
 
 Gunter's chain, length of xxxviii 
 
 Harkness, Prof. Wm., memoir cited 
 
 Ixv, 169, 171, 172 
 
 Heat, dimensions of . ^ 175 
 
 latent, of liquefaction of ice 172 
 
 of vaporizatigg of water 172 
 
 mechanical equiv^nt of 172 
 
 Hectare ♦. xli 
 
 Heights, average, of continents Ixv 
 
 determination of, 1^ barometer 160 
 
 trigonometrically Ixi 
 
 Helmert, Dr. F. R., treatise on geodesy 
 
 cited Ixvi 
 
 treatise on theory of errors cited xcviii 
 
 Horizon, dip of sea Ixiii 
 
 Imperial pound and yard xxxiv 
 
 Integrals, definite xxvi 
 
 indefinite xxxiii 
 
 Interconversion, of Eilglish and Metric 
 
 units .' 2,3, 173, 174 
 
 of sidereal and solar time Ixxiii 
 
 tables for i 164, 165 
 
 Iron, linear expansion of 170 
 
 Joule, value of 174 
 
 Kilogramme, Prototype xxxiv 
 
 equations of xl 
 
 relation to pound xxxvi, xli 
 
 Kinetic energy, dimensions of.:. 175 
 
 of rotation of earth Ixvi 
 
 Klafter, Wiener, in terms of foot and 
 metre. 1, 168 
 
 Latitude, astronomical, geocentric, and re- 
 duced xliv 
 
 determination of Ixxvii 
 
 Latitudes and departures, table of 32-47 
 
 mode of use of table explained c 
 
 Lead, linear expansion of 170 
 
 Least squares, method of Ixxxvi 
 
 references to works on xcviii 
 
 Legendre's theorem for solution of sphe- 
 roidal triangles Ivii 
 
 Length, of arc of meridian xlvi 
 
 of arc of parallel xlix 
 
 of equator of earth 171 
 
 of meridian circumference of earth 171 
 
 of perimeter of ellipse xxix 
 
 of Prototype Metres Nos. 21 and 27 xl 
 
 of seconds pendulum, formula for 171 
 
 table of values of 169 
 
 Leveling, trigonometric 1x1 
 
 Line (French), value of 168 
 
 Lines, lengths of xxviii 
 
 on a spheroid Ivi 
 
 Linear measures, British xxxvii 
 
 Metric xli 
 
 tables for iiiterconversion of. .2, 3, 173, 174 
 
 Litre xli 
 
 Logarithms, anti-, 4-place table of 26, 27 
 
 explanation of use of xcix 
 
 4-place table of common 24, 25 
 
 of natural numbers, table of 4-22 
 
 relations of different xv 
 
 series for xiv 
 
 Maclaurin's series xxii 
 
 example of xxiii 
 
 Magnetic quantities, dimensions of 175 
 
 Maps, co-ordinates for projection of (see 
 
 Co-ordinates for projection of maps) liii 
 
 projection of . .' cii 
 
 Mass, of earth Ixv 
 
 of earth's atmosphere Ixvi 
 
 of Prototype Kilogrammes Nos. 4 and 
 
 20 xl 
 
 Mayer's formula for transit instrument . . . .Ixxv 
 
 Mean, arithmetic and geometric xiii 
 
 Mean distance of earth from sun 172 
 
 Mean error, definition of Ixxxiv 
 
 computation of xcv 
 
 Mean time Ixxii 
 
 table for conversion to sidereal time 164 
 
 Measures xxxiv 
 
 of capacity, British xxxviii 
 
 Metric xli 
 
 of length, British xxxvi; 
 
 Continental 168 
 
 Metric xli 
 
i8o 
 
 INDEX. 
 
 Measures (continued). 
 
 of surface, British xxxviii 
 
 Metric xli 
 
 tables for interconversion of. .2, 3, 173, 174 
 
 Mechanical equivalent of heat 172 
 
 Mechanical units, dimensions of 175 
 
 Mensuration xxviii-xxxiii 
 
 Mercury, density and cubical expansion of , . 172 
 
 Meridian, arcs of terrestrial xlvi 
 
 table of lengths of 78-80 
 
 circumference of earth xlix, 171 
 
 Method of least squares Ixxxvi 
 
 Metre, Prototype xxxiv 
 
 equations of Nos. 21 and 27 xl 
 
 relation to British yard xxxvi, xli 
 
 Metric system xl 
 
 Mile, Austrian 168 
 
 British (statute) xxxvii 
 
 Danish, German sea, Netherlands, Nor- 
 wegian, Prussian, Swedish 168 
 
 Nautical 168 
 
 Modulus of common logarithms xv 
 
 Moivre's formula xvi 
 
 Moment of inertia of mass, dimensions of. . . 175 
 
 Moments of inertia of earth Ixvi 
 
 Momentum, dimensions of 175 
 
 Napierian base (of logarithms) xiv, 171 
 
 Napierian logarithms xiv 
 
 Napier's analc^gies xx 
 
 Natural logarithms xiv 
 
 Nautical mile, table of equivalents in statute 
 
 miles 168 
 
 Numerical constants 171 
 
 Ohm, dimensions of 175 
 
 Palm, length of, Snglish xxxviii 
 
 Netherlands 168 
 
 Parallel, arcs of terrestrial xlix 
 
 table of lengths of 81-83 
 
 Pendulum, length of seconds 171 
 
 table of lengths of 169 
 
 Perch (of masonry) volume of xxxix 
 
 Perimeter, of circle xxviii 
 
 of ellipse xxix 
 
 of regular polygon xxviii, xxx 
 
 Physical constants 172 
 
 Physical geodesy, salient facts of Ixv 
 
 Physical quantities, dimensions of 175 
 
 Platinum, linear expansion of 170 
 
 Platinum iridium, linear expansion of 170 
 
 Polyconic projection of maps liii 
 
 graphical process of, explained cii 
 
 Polygons, regular, areas of xxx 
 
 lengths of lines of xxvii 
 
 Potential (electric), dimensions of 175 
 
 Pothenot's problem Ixiv 
 
 Pound, imperial, avoirdupois xxxiv 
 
 Power, dimensions of 175 
 
 Pressure, of atmosphere 172 
 
 Prism, volume of xxxii 
 
 Probable error, definition of Ixxxiv 
 
 computation of xcv 
 
 Projection of maps liii, cii 
 
 Prototype Kilogrammes and Metres xxxiv 
 
 Equations of xl 
 
 Quadrilaterals, of earth's surface, areas of 1 
 
 tables of areas of 142-159 
 
 Quantity, of electricity, dimensions of 175 
 
 Radii, of curvature xiv 
 
 Radius of curvature, of meridian, table of 
 
 logarithms of 48-56 
 
 of section normal to meridian, table of 
 
 logarithms of 57-^5 
 
 of section oblique to meridian, table of 
 
 logarithms of 66, 67 
 
 Radius vector of earth's surface 1 
 
 Rate of working (power), dimensions of 175 
 
 Raiio, of pound to kilogramme xxxvi 
 
 of specific heats of air 172 
 
 of yard to metre xxxvi 
 
 Reciprocals, of natural numbers, table of. .4-22 
 
 Reduced latitude xliv 
 
 Reduction to sea level of measured base line. Ixiv 
 
 References, to works on astronomy Ixxxii 
 
 to works on geodesy Ixvi 
 
 to works on the theory of errors xcviii 
 
 Refraction, astronomical, table of 161 
 
 example of computation of civ 
 
 coefficients of terrestrial Ixiii 
 
 Right ascension Ixxii 
 
 Rode, Danish 168 
 
 Ruthe, Prussian, Norwegian 16S 
 
 Sagene, Russian 168 
 
 Sea level (see Geoid), reduction of measured 
 
 base line to Ixiv 
 
 Sea surface, area of Ixv 
 
 Secondary triangulation, differences of lati- 
 tude, longitude, and azimuth in Ix 
 
 Series, binomial xiv 
 
 logarithmic xiv 
 
 of Maclaurin and Taylor xxii 
 
 trigonometric xvii 
 
 Sidereal day and year, length of 172 
 
INDEX. 
 
 i8i 
 
 Sidereal time Ixxii 
 
 table for conversion to mean time.. 165 
 
 Signs, of trigonometric functions xv 
 
 Silver, linear expansion of 170 
 
 Sines, table of natural 28, 29 
 
 explanation of use of c 
 
 Solar time Ixxii 
 
 table for conversion of mean solar 
 
 to sidereal 164 
 
 Solution, of plane triangles xviii 
 
 of spherical triangles xx 
 
 of spheroidal triangles Ivii 
 
 Span, length of xxxviii 
 
 Specific heat of air 172 
 
 Sphere, equal in surface with earth lii 
 
 equal in volume with earth lii 
 
 surface of xxxi 
 
 volume of xxxii 
 
 Spherical excess (see Spheroidal excess) Iviii 
 
 Spheroid, representing the earth xliii 
 
 surface of xxxi 
 
 volume of xxxiii 
 
 volume of earth's Ixv 
 
 Spheroidal excess Iviii 
 
 example of computation of ci 
 
 Spheroidal triangle Ivii 
 
 Square roots, table of 4-22 
 
 Squares, table of 4-22 
 
 Standards, of length and mass xxxiv 
 
 Steel, linear expansion of 170 
 
 Stere xli 
 
 Stress, dimensions of 175 
 
 units of 173. '74 
 
 Sums, of arithmetic and geometric progres- 
 sion, and special series xiii 
 
 Surfaces (see Areas) xxix 
 
 Surface measures, British xxxviii 
 
 Metric xli 
 
 tables for interconversion of. . .2, 3, 173, 174 
 
 Surface, of continents Ixv 
 
 of earth's spheroid lii 
 
 of oceans ixv 
 
 of sphere and spheroid xxxi 
 
 Surveyor's chain, length of xxxviii 
 
 Table for cf .iversion of arc into time 162 
 
 con-'^rsion of mean into sidereal time . .164 
 ■conversion of sidereal into mean time. .165 
 
 iconversion of time into arc 163 
 
 determination of heights by barometer. . 160 
 dnterconversion of British and Metric 
 
 units 2,3,173. 174 
 
 interconversion of nautical and statute 
 
 miles ....168 
 
 Table of acceleration of gravity and derived 
 quantities '^9 
 
 Table of {contimted). 
 
 antilogarithms, 4-place z6, 27 
 
 areas of quadrilaterals of earth's surface 
 of 10° extent in latitude and longi- 
 tude 142 
 
 1° extent in latitude and longi- 
 tude 144, 145 
 
 30' extent in latitude and longi- 
 tude 146-148 
 
 15' extent in latitude and longi- 
 tude 150-154 
 
 10' extent in latitude and longi- 
 tude 156-159 
 
 areas of regular polygons xxx 
 
 circumference and area of circle 23 
 
 constants, astronomical 172 
 
 geodetical 171 
 
 numerical 171 
 
 for interconversion of English and 
 
 Metric units 2, 3, 173, 174 
 
 Continental measures of length 168 
 
 co-ordinates for projection of maps — 
 
 scale 1/250000 84-gi 
 
 scale 1/125000 92-101 
 
 scale 1/126720 102-109 
 
 scale 1/63360 110-121 
 
 scale 1/200000 122-131 
 
 scale 1/80000 132-141 
 
 departures and latitudes 32-47 
 
 dimensions of physical quantities 175 
 
 errors of interpolated values from nu- 
 merical tables Ixxxvi 
 
 expansions (linear) of principal metals. . 170 
 formulas for solution of plane triangles, .xlx 
 fractional change in number due to 
 
 change in its logarithm 170 
 
 latitudes and departures 32-47 
 
 lengths of arcs of meridian 78-80 
 
 of arcs of parallel 81-83 
 
 of 1° of meridian 166 
 
 of 1° of parallel 167 
 
 linear expansions of metals 170 
 
 logarithms, 4-place 24. 25 
 
 anti-, 4-place 26, 27 
 
 of factors for computing spheroidal 
 
 excess 68, 69 
 
 of factors for computing differences 
 of latitude, longitude, and azi- 
 muth 70-77 
 
 of meridian radius of curvature. .48-55 
 of radius of curvature of normal 
 
 section S^^S 
 
 of radius of curvature of oblique 
 
 sections 66, 67 
 
 mean astronomical refraction 161 
 
 measures and weights — 
 
 British, of capacity xxxix 
 
l82 
 
 INDEX. 
 
 Table o£ (continued). 
 
 British, of length xzxvUi 
 
 British, of surface , xxxviii 
 
 British, of weight xxxix 
 
 Metric xli 
 
 tables for interconversion of 
 
 2. 3> 173. 174 
 
 natural cosines 28, 29 
 
 natural tangents 30> 31 
 
 radii of curvature, logarithms of, for 
 
 meridian section 48-SS 
 
 for normal section 56-65 
 
 for oblique section 66, 67 
 
 reciprocals, squares, cubes, square roots, 
 cube roots, and logarithms of natural 
 
 numbers 4-22 
 
 refraction, mean astronomical 161 
 
 signs of trigonometrical functions xv 
 
 values for computing areas and dimen- 
 sions of regular polygons xxx 
 
 for computing perimeter of ellipse xxix 
 of log i (l — 2m) and log (I — m) 
 used in trigonometric leveling . . .Ixii 
 weights and measures (see Table of 
 
 measures and weights) 2, 3, 173, 174 
 
 Table, traverse (see Traverse table) 32-47 
 
 Tangents, natural, table of 30> 31 
 
 use of table explained c 
 
 Taschenbuch, Des Ingenieurs xcix 
 
 Taylor's series '. xxii 
 
 Temperature, absolute zero of 172 
 
 of freezing mercury 172 
 
 Theory of errors , . .Ixxxiii-xcviii 
 
 references to works on .' xcviii 
 
 Thermal conductivity, dimensions of 175 
 
 of air 172 
 
 Three-point problem Ixiv 
 
 Time, determination of Ixxiv 
 
 equivalents in arc, table of 163 
 
 example of use of table civ 
 
 interconversion of sidereal and solar, 
 
 tables for 164, 165 
 
 Tin, linear expansion of 170 
 
 Toise, value in feet and metres 168 
 
 Ton, long and short xxxix 
 
 Tonne 173, 174 
 
 Tonneau xli 
 
 Trapezoid, area of xxix 
 
 Traverse table 32-47 
 
 explanation of use of c 
 
 Triangles, plane, solution of xviii 
 
 Triangles {continued). 
 
 spherical, solution of xx 
 
 spheroidal, solution of Ivii 
 
 Triangulation, primary and secondary, differ- 
 ences of latitude, longitude, and azimuth 
 in Iviii-lx 
 
 Trigonometric functions, of one angle xv 
 
 of two angles xvi 
 
 series for xvii 
 
 Trigonometric leveling Ixi 
 
 Units, British System xxxvii 
 
 C. G. S. System xlii 
 
 Metric System xl 
 
 standards of length and mass '.xxxiv 
 
 tables for interconversion of British and 
 Metric 2, 3, 173, 174 
 
 Useful formulas xiii-xxvii 
 
 Vara, Mexican and Spanish 168 
 
 Velocity, dimensions of 175 
 
 of light and sound 172 
 
 Versta, Russian 168 
 
 Vertical section curve on spheroid Ivi 
 
 Volt, dimensions of 175 
 
 Volume, of earth Ixv 
 
 of solids xxxii 
 
 Weight, of distilled water 172 
 
 Weights and measures (see Measures and 
 weights), tables for interconversion of 
 
 British and Metric -> 3> '73. 174 
 
 Werst, Russian 168 
 
 Work, dimensions of 175 
 
 Wright, Prof. T. W., treatise cited xcviii 
 
 Yard, imperial ... , , xxxiv 
 
 ratio of, tor N.xxxvi, xxxvii 
 
 Zachariae, G., treatise ' ' 
 Zenith distances, use o. 
 
 leveling 
 
 Zenith telescope, use of 
 
 Zero, of absolute temperature . . . 
 
 Zinc, linear expansion of 
 
 Zones, of earth's surface, area of . 
 
 .„xlvi 
 
i 
 
 ^ 
 v^