'if'. mm Mi I'JWIWX Wil'li',,,-, iii. "(iiiilir i' I'll' I ill E ,!. II! iiiiMfri iil I' Kll fill ifilfS i|iji'!,f,i^ii'l!;:'i .'I'' '11 7 'i I' pllil I "I, i"'l ill llll. ,,|M„jl| i|l 113 I I III'/! ,1 ! 1 ill M\ i'lll f ll tl ll 1 I I I I I II I ' 3 Mil, i 1 ( 1 1 ,1 ■: '>•■ i< i> PJ"|i| ,' flll'li if. t #1 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004444042 Cornell University Library TA 545.S66a Smithsonian geographical tables, 3 1924 004 444 042 854 SMITHSONIAN GEOGRAPHICAL TABLES PREPARED BY R. S. WOODWARD THIRD EDITION CITY OF WASHINGTON PUBLISHED BY THE SMITHSONIAN INSTITUTION 1906 The Riverside Press, Cambridge, Mass.,V. S. A. EliCtrotyped and Printed by H. O. Houghton & Ca ADVERTISEMENT TO THIRD EDITION. The second edition of the Smithsonian Geographical Tables issued in 1897 having become exhausted, a third edition is now printed with a few necessary changes made in the plates. Richard Rathbun, Acting Secretary. Smithsonian Institution, Washington City, August 6, igob. ADVERTISEMENT TO SECOND EDITION. The edition of the Smithsonian Geographical Tables issued in 1894 having become exhausted, a second edition is now printed with a few necessary changes made in the plates. S. P. Langley, Secretary. Smithsonian Institution, Washington City, October 30, iSgj. 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, 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 VllI 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 oi 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 . . . liii 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 . Ixvii 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. Determination 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 lxxi?c 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 Ta.bles 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. TAB LB FAGB 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 4-22 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 ID. 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) 7°-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, b, c, ... is -l-(« + ^ + , + ...); their geometric mean is {abc.f. A case of special interest is b. Arithmetic progression. If a is the first term, and a-\-d, a-\- 2 d, a + 3 b, {a±by=a"±na''-'b+ '^ ^^ ~ "^ «— " l)" For X <. 1, , n(n— i) , , « (« — i) (« — 2) , , {i±xf=i±nx +-^777-^ *' ± -^ riJT^ ■ *+••■• l—X-\-X^ — S^-{-X^—... 1 -\- X I 1 — X I = I -j- * + «° + *' + *^ + • • • (i + :r)* = I + J a; — i «^ + -iVa;' — xfj ar*+ . . . (i —xf = i—^x — ix^ — ^x^ — T:^^x* — ... f. Exponential and logarithmic series. For — 00 < ^ < 00, \- -t- [- -I- 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 £ =^ 2.718281828459 .... In the natural system the following series hold with the limitations indicated : ' 1 ' 1.2 '1.2.3 — 00 < ^ < 00 j -V* A/18 jyii ,y,6 log(i + ^) = ^-^+^-^+^-... >V' 'V" -V^ "V" 4.V VV fc^ fc*V log (i —;);)=— a: ^ ^ 23 * < I ; O < :» < 00 ; ^"^ ^ I "2^+y + * \2x-\-y) + * (n^qrj) + • • • I log y <(2;c+^)''. USEFUL FORMULAS. XV g. Relations of natural logarithms to other logarithms. B = base of any system, N-=. any number, L = log N\.o base B = log^iV^ / = log iV^ to base e = log^iV: Then JV=/=B'^, L = l log^« = il^og^B, log£tf= i/loge^ = M, say, which is called the modulus of the system whose base is B, In the common, or Briggean system, ^ = logio? = 0.43429448 .... log IX. — 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 i ^Vz iV3 cosine . . . + 1 — I + 1 iV3 iV2 i tangent . . . 00 00 iV3 I V3 cotangent . . 00 00 00 V3 I Ws c. Fundamental formulas. sin' a -\- cos" a = 1, cos a sec a = I, sin a tan li cot a ^ 1, sin u cosec a ^ i, tan a ^= cos a 2, cos a cot a = -T- I + tan' a = 33^^ = sec" a I + cot'' « '■ sm^ a sin a : cosec' o, versed sin a ^ 1 — cos a. XVI USEFUL FORMULAS. d. Formulas involving two angles. sin (a ± j8) :^ sin a cos /8 ± cos a sin p, cos (a ± ^) = COS a COS ;8 T sin u sin p. tan (a ± yS) = (tan a ± tan /3)/(i T tan a tan ^), cot (a ± /?) = (cot a cot /3 qp l)/(cOt a ± COt )8). sin a -(- sin /3 = 2 sin i(a -|- ;8) cos J(a — ;3), sin a — sin j8 = 2 cos J(a -f- iS) sin J(a — /3). cos a + cos j8 = 2 cos J^(a -)- yS) COS J(a — j8), cos a — COS /3 ^ — 2 sin J(a -|- j8) sin ^(a — fi). sin (a ± B) tan a ± tan S = ^^ ^, '^ COS a COS /3 sin 08 ± o) cot a ± cot S = ■ ^ ■ — i- ^ sin a sin p 2 sin a sin /3 = cos (a — yS) — cos (a -}- /8), 2 cos a cos P = cos (a — )8) + COS (a -j- ^), 2 sin a COS p = sin (« — )8) -)- sin (a -|- |8). sin a + sin S , , „■, , , „s sin a I sin is = t^" i(a + ^) cot i(a - 13), cos a + cos B , / . «v cosaIcosg =-"°tKa + ^)cotK°-^). e. Formulas involving multiple angles. sin 2 a = 2 sin a cos a, sin 3 a = 3 sin a cos' a — sin° o. cos 3 a := cos' a — 3 sin* a cos a. 2 sin" a = 2 cos" o— I, sin a I — cos t^"i"=i+COSa' sin 2 tan a tan 2 a = : — «—> cot 2 a I — tan"" a Vl + cos aj cot" a — I 2 cot a I - tan^d 2 tan i a Sm a = i — o-j — ' cos a : — — -. — 7 «-, — - I -|- tan" i a I + tan-* J a 2 sin" a = I — cos 2 a, 2 cos" a :^ I -|- COS 2 a, 4 sin' a 1= 3 sin a — sin 3 a, 4 cos' a ^ 3 cos a -j- COS 3 cu f. Exponential values. Moivre's formula. e =^ base of natural logarithms, /= V — I. ^= — i> *'= ~ ^> ^^= "> etc. cos ^ = i (^'^ + ^ "'")> sin a: = ^ (d*' — e -*"), cos ix = ^ (e-" -{- e'), sin «::»; = jV (^ -"^ _ ^). (cos X ± i sin a;)" = cos mx ± « sin mx. USEFUL FORMULAS. g. Values of functions in series. For X in arc the following series hold within the limits indicated, *' x'^ ■ ' sin « = a; — 2 — I 6 ' 120 5040 cos X = I — 1- 2 24 720 ' ' — CO ■<, X <. -\- CO. tan :*r = « + J ^» + T% ^5 ^ gVj ^' + • . . , sec a; = I + i a;" + ^ a?« + t¥s *" + • • • » cot X = l(i - i x^ - ^ x' - ^i-s x'^ - . . .), cosec a; = ^ (i + i ^'^ + 3I5 ^* + j^^ x" -\- . . .), TT < a; < -)- TT. arc sin X = X -\- i x^ -{- ^^ x^ -{- ^i^-s x'' -\- . . . , ■yO /yiO A"7 /yS arc tan ;c = ;c .... 3 ' 5 7^9 ' — I < X < -\- I. « =: sin ^ 4" i sin' x -{- ^-^ sin" x -\- yf ^j sin' x -{-.,, , — i-ir =: base, e = hypothenuse, y = 90°. a = c sin a = ^ cos ^ = 6 tan a = l> cot /?, b ■= c sin 13 = c cos a = a tan ^ = 3 cot a. ^ = ^a^^J^a^ cot a^ ^6" tan a = :| r" sin 2 a. Table for solution of oblique triangles. XIX Given. Sought. Formula. «, ^, <: a sinia-y/(^-*H^-^), .-H«+^ + ^), COSia = ^£0^), V s(s-a) A A = ^s(s-a){s-b){s-c). a, b, a P sin p^ b sin a/a. When a > b, 13 < go° and but one value results. When b > a, P has two values. 7 y=i8o°-(a + /3). c c ■= a sin y/sin a. A ^ = ^ a i5 sin y. a, a, p b i5 = a sin y3/sin a. y y = 180° - (a + ;8). c c = a sin y/sin a ^=: a sin (a -j- j8)/sin a. A ^ ^ ^ 3 i5 sin y = J a'' sin ;8 sin y/sin a. a, b, y a 3 sin y ^^^<--b-acosy -,/3 tani(a-^)_^_^^cot Jy. (x)} dx. Iix = ^ (ji), and dx = tft' (y) dy, p{x) dx =f/{ (y)} 4>' (y) dy. iff(^,y)d^ = f^dx. XXiv USEFUL FORMULAS. Since d(uv) = udv -f- vdu, I udv = uv — I vdu ; and if « = f(x) and v:= (x), f/i.) ^) d. =/(.) .^ (.) - J.^ (.)^) )~^ arc cot (^/«)* x -\- C, for a and ^ both negative, (_ ady — hx = i (- «*)"* log (_^^)i_[_^;t: + C, for «^ negative. C(a + xydx = iix(a + xj + ^ « log {^ + (« + xj} + C. C(a^ — xf dx=\x{a'- x'y + i a' arc sin f + C. !"(« + ^«)* dx = l{a-\- bx)ilb + C. * 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. XXV i (a -\- 2 l>x -\- c«")* dx = ^ (6 -\- ex) (a-\- 2 dx-\- cx^fc -\.\{ac — F)lc C(a -\-2bx-\- cxF)-^ dx + C. C(a + 6x)-i dxz=2(a-\- bxfib + C. Jia + ^x) (a + dx)-i dx=^(3ai-2aP-\-l3 bx) {a + bxyilr" + C. I (a' — a;')"' dx=z ±. arc sin — (- C, = T arc cos " + C, /a4-x\i , „ = 2 arc tan I — ■ — I + C. \a — x) ' C{a + xY^ dx = log {x-\-(a-\- xj} + C, = ilog a: + (a + ■^'')* ar — (a x'y :, + c. C(a -{-2 bx-\- cx^-i dx = "7^ log {b-{-cx-\- (ac-\- bcx + ^x'f)-\- C, for o, I . b + cx , „ , - - ^ arc sin (^—^^ + C, for ^ (x) dx == I <^ {x) dx -\- \ <^ (x) dx -\- . . . I ((i — x) dx. o o If tf) (x) := (— x), an " even function " of x, a o a I ^ (x) dx=^i (x) dx. o — a ■ — a li fl> (x) = — fji (— x), an " odd function " of x, o a -\-a l (x) dx=z\ C {x) dx> £ (b- a), a a formula useful in determining approximate values of integrals. See, e. g., section 6, d. b If « = \^{x) dx, a O I 00 / dx _ r_^_— I o 00 USEFUL FORMULAS. XXVII O 00 OO 00 fe -^' dx = \ Vtt, Ce-""" dx=\ V(jr/a«). o o Ce-"'^' x^^'dxz^ 1.3.5... (2 n - 1) a" (2 a)-'^-^^Wv. o 00 Ce-"' «-» dx = V(Tr/»). o I sill mx sin nx dx ■=z | cos mx cos «a: ^a; ^ o. o o when m and « are unequal integers. IT J sin »2a; cos nx dx = ^ _ i > for m and « integers and m — n odd, o = o, for m and « integers and m— n even. IT TT I sin° »«;»: = :r (« + *) (i + i «= 4- ^ «^ + ^K «" + . . ■) •=. It {a -\- b^ q, say, where q stands for the series in n. The values of q cor- responding to a few values of n are: — « ? « ? 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-2ISS I.O 1.273a 2. Areas. a. Area of plane triangle. (See table on p. xix.) b. Area of Trapezoid. bi = upper base of trapezoid, bi = lower base of trapezoid, a = altitude of trapezoid, or perpendicular distance between bases. Area := J (^1 + bs) a. XXX MENSURATION. c. Area of regular polygon. A = area, r, R=. radii of inscribed and circumscribed circles, s = length of any side, n ^ number of sides, /3 = angle subtended by j = 36o°/«. A = nr^\xa.\^ = \nR^sm.^ = \ns^ cot \^. Table of Values. n /J ^ A R s 3 120° 0.4330 s"^ 1.2990 i?^ 0.5774 f 1.7321^ 4 90 1. 0000 2.0000 0.7071 I.4I42 S 72 1-7205 2.3776 0.8507 I.I756 6 60 2.5981 2.5981 I.OOOO I.OOOO 7 Si^ 3-6339 2.7364 I.I524 0.8678 8 45 5.8284 2.8284 1.3066 0.7654 9 40 6.I8I8 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 I-93I9 0.5176 13 281*,^ 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 16 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, 'i! 'a = smaller and greater radii of an annulus. Area of circle z= tt r^ ■=. \ -^ d^, 7r = 3.14 159 265, log 57 = 0.49714987. Area of sector =zar^, for a in arc, = a- r"" (a/360), for a in degrees. Area of annulus = it (r^ — r^). e. Area of ellipse. a, b ■= semi axes respectively e = eccentricity = (a^ — V^^/a = {(a + b){a-b)yia. MENSURATION, IjXxi Area of ellipse = Tr a d, = TT a" cos ^, if tf = sin Surface of spherical triangle == r" «, for £ in arc, = T^ ijp", for £ in seconds, p" = 206 264.8", log p" = 5.31 442 513. g. Surface of right cylinder. r = radius of bases of cylinder, A = altitude of cylinder. Area cylindrical surface = 2 tt r A. Total surface = 2 ir r (r -\- A). h. Surface of right cone. r = radius of base, A = altitude, J = slant height. Conical surface z=Trrs = 'jrr(A^-\- r")*. Total surface = tt r (s -\- r). i. Surface of spheroid. a, b =■ semi axes, e = eccentricity = {(a -\- i) (a — b)Yla. Surface of oblate spheroid = 2 ir a'' i i + log { I \ = ^^ a' (i - i e' - ^ ^ - ^ e' - . . .). Surface of prolate spheroid = 2 irab < (i— *^^ eccentricity of generating ellipse, £1 — ^ / = » the flattening, ellipticity, or compression, a — 6 a-\-6 I — « d = a \J 1 —e^ = a(i —/) ^ a i-\- n = — j — = 2 (« — «^ + «' — «*+•• •)• Xliv GEODESY, / 4 n (I + «y = 4 (« — 2 «° -|- 3 «' — 4 «* -|- . e^ ^ ^ ^ ~?~ T '^ ~^ '^ ~%"^ lb '^ ■ • I — s/i - ^' _^4.jliSi!_|_7£!i '*— i_l_y/r^i-?— 4 ^ 8 "1^ 64 ^128^ — The numerical values of the most useful of these quantities and their logarithms are — log a = 20 926 062 feet, 7.3206875, b = 20 855 121 feet, 7.3192127, e^= 0.00676866, 7.8305030 — 10, m = 0.00339583, 7-5309454 — 10, n = 0.00169792, 7.2299162 — lo. 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 ■^ + ^s- = I, (i) a and 6 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 6, J/ = ^ sin e, ^^J which give (i) by the elimination of 6. 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. 1 by the notation given below. GEODESY. xlv In order to express the analytical relations between the different latitudes let n ^ = the astronomical latitude, i/f = the geocentric latitude, 6 = the reduced latitude. Then, referring to equations (i) and (2) under ' section 4 above, and to Fig. i, it appears that dx dy ' tan <^ : a'^'y ' — + b'^x Fig.l. tan i/r — y tane = f. ox Hence b"" tan i/f = — 2" tan <^ = (i — ^ tan <^, tan e = (i — e^^ tan <^ = (i - ir")-* tan 1^. ^ — i/f = »2 sin 2 (^ — J^ »2'^ sin 4 <^ -j- • • • • <^ — 6 = « sin 2 ^ — J «^ sin 4 <^ -1- . . . . For the adopted spheroid and log (i — ^) = 9.9970504, (ji — iff (in seconds) = 7oo."44 sin 2 — i."i9 sin 4 <^, . 6. Radii of Curvature. p„ = radius of curvature of meridian section of spheroid at any point whose latitude is ^ = I'D, 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. pm = a (i—e^(i- e^ sin'' <^)-i, p„ = a(i-e^ sin'' <^)-*, I cos' a , sin' a P« Pn ^ (i -| 3^ cos' <^ cos' a) (i — e' sin' )*. log (i - ^' sin' <^)-» = + log (i + n) — fj, n cos 2^ -[- J yii «' cos 4 -\- [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<^ 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 ^ = o, log Pn= 7-3214243 - 7373-9 _± 6^ = 7.320687s = 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 — l) , -\- <{,,), (i) In these, c^i and ^2 are the latitudes of the ends of the arc, A J/ is the required length, and p„ is the meridian radius of curvature for the latitude <^ of the middle point of the arc. The formula for AM implies that A<^ is expressed in parts of the radius. If A<^ is expressed in seconds, minutes, or degrees of arc, the for- mula becomes — Meridional distance i^M in feet. A ,^— Pm A<)!) (in seconds) 206264.8 ' Pm A<^ (in minutes) ~ 3437-747 p„ A<^ (in degrees) . ~ 57.29578 ' (2) log (1/206264.8) = 4.6855749 - 10, log (1/3437-747) = 6.4637261 — lO, log (1/57-29578) = 8.2418774 - 10. ^11 #2 = end latitudes of arc, A^ = 0, — 0^, p„ = meridian radius of curvature for = ^(^j + 0,) ; for log p„ 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 J «= A<^'' cos 2 . t^M, which vanishes for <^ = 45°, and which for A<^ := 5° is at most TT^'sntr ^^t or about II feet. Numerical example. Suppose — ^ = 37° 29' 48."i7. «^i = 35° 48' 29-"89. Then <^ = K<^. + '^i) = 36°39' °9."o3, A(^ = <^2 - <^i = 1° 41' i8."28, = 6o78."28. From the first of (2) cons't. log 4.6855749 — 10 Table 10, log p„ 7.3193112 log A<;f» 3.7837807 t^M=. 614705 feet, log AJ/ 5.7886668 The values of ^Mior 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 : — AJ/'= Ao A — Ai cos 2 sin A<^ + A2 cos 4^ sin 2A<^ — As cos 61^ sin 3A<^ (3) + At cos 81^ sin 4A^ In this, AJ^ , and A(^ have the same meanings as above, and Ao, Ai, . . . are functions of a and * or of a and n. Thus, in terms of a and n, Ao = a(i-{- n)-' (i + J „= + ^ «* + ... ), ^1 = 3^^ (i + «)~^ (« - i «'-■•■ ). ^3 = |f «(i+«)-i(«»-...), Introducing the adopted values of a and «, these constants become — log. .^0 = 20 890 606 feet, 7.3199510, Ai^ 106 411 feet, 5.0269880, A2= 113 feet, 2.0528, A3 = 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 An- In the use of (3) it will generally be most convenient to express A^ in degrees, and in this case Aq must be divided by the number of degrees in the radius, viz. : 57.2957795 [i. 7581226]. Applying this value and writing the logarithms of Ao, Ai, etc., in rectangular brackets in place of A^, Ai, etc., (3) becomes Meridional distance Ail/ in feet. AM^ [5.5618284] A<^ (in degrees) — [5.0269880] cos 2t^ sin A^ -|- [2.0528] cos 4(f> sin 2A<^ (4) 20 = 02 + 01. A^ = 0, — < 01, = 45° and A<^ = 90°, since all of the remaining terras vanish. Numerical examples. — 1°. Suppose 1^1 = 0° and ^2 = 45°' Then 2^ = 45°. A-^ = 4S°- cons't 45 log. 5.5618284 1.6532125 ist term + 16 407 443 feet ist term 7.2150409 cos 2<^ 9.8494850 — 10 sin A<^ 9.8494850 — 10 cons't 5.0269880 2d term — 53 205.7 ^^^^ ^'^ term 4.7259580 The third terra 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 16354237 feet. 2°. Suppose Then <^i = 45° and <^2 = 90°. 2-^= 135". A<^= 4s». The numerical values of the terms will be the same as in the previous example, but the sign of the second term will he. 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 A^ = 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 = p„ cos , and the entire length of the parallel is — 2 TT r = 3 TT p„ cos ., .. , , = '^ AX (in degrees). log (2 77/1296000) = 4.6855749 — io> log (2 77/21600) = 6.4637261 — 10, log (2 77/360) = 8.2418774 — 10. Aj, \, = 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 Example. — Suppose <^ = 35°, and AX = 72°. Then from the third of (9) log. cons't 8.2418774 — 10 Table 11, p„ 7.3211716 cos<^ 9-9133645 — 10 ^^ 1-8573325 A^= 21 564 827 feet, A^ 7-333746o * The best formula for computing the entire length of a meridian curve is this : TT (a + *) (I + i «2 + A «< + . . .), in which a, b, and n are the same as defined in section z. For the values here adopted — log. (i + i «' + • - •( 0.0000003 (a + b) 7.6209807 ir 0.4971499 length 8.1181309 The length of the perimeter of the generating ellipse, or the meridian circumference of the 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 (i - 2«» sin" 4> + ^ sin» c^)* (i - «' sin" ^)-». log p = log ^_|: ,^^;^ + /i (w - «) cos 24> — \ f>.{m^ — n^ cos 4^ + J A* (»«• — «') cos 6 For the adopted spheroid log (p in feet) ::= 7.3199520 -f- [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„ (/0, is (see Fig. i), dZ =2 IT r p^ dcj> = 2 IT p^ p„ cos d. By means of the relations in section 6 this becomes j^r 2 / 2N cos <^ d dZ = 2 IT d' (i —e^ 7 o ■ , ,,. 2 I — g" d je sin <^) (0 The integral of this between limits corresponding to <^i and ^2, or the area of a zone bounded by parallels whose latitudes are 1^1 and ^2 respectively, is Z-=ir d' X — €'■ e sin <^2 e sin <^j I — «" sm' <^2 I — «" sin= <^i + i Nap. log ('+^siny (i - . sin «^,) IS ^ s (i — g sm ^2) (i 4- ^ sin <^i) (2) GEODESY. To get the area of the entire surface of the spheroid, make <^i = — ^ tt and <^j = -}- i T in (2). The result is Surface of spheroid = 2 tt »= i -|- Nap. log L _g ) • (s) For numerical applications it is most advantageous to express (3) in a series of powers of e. Thus, by Maclaurin's theorem, I — — c" "■•')■ (^) 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 \ si^d e sin <^2 ^nd express the result in terms of multiples of the half sum and half difEerence of <^i and <^2. Thus, (2) readily assumes the form Z =z 2 IT a^ {1 — e^\ (sin 2 — sin ^i) + - «" (sin° <^2 — sin' <^i) -f- • • • I- From this, by substitution and reduction, there results wherein „ 1 Ci cos ^ sin J A<^ — C2 cos 3(^ sin | A<^ ) ^ — 2 'r j _|_ Q cos s^ sin f A^ - . j <-S; 4> = i(2 + "Pi ). ^ = 't>2 — "^i. ^,, (p2 = latitudes of bounding parallels, AA. = difference of longitude of bounding meridians. * '^1, '2. "^s ^^^ obtained from C,, Cj, Cj respectively by dividing the latter by the number of degrees in the radius, viz : 57.29578. lii GEODESY. Numerical examples. — i°. Suppose <;fii = o, <^2 ^ 9°° and AX =: 360°. Then (7) should give the area of a hemispheroid. The calculation runs thus : log. log. log. c\ 5-7375398 C2 2.79173 Ci 9.976 - 10 cos <^ 9.8494850 — 10 cos 3 9!) 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 — ist term ■=i -\- 98358591 2dterm = -- 111429 3d term = -|- 169 Q =■ sum = 98470189 • Twice this is the area of the spheroidal surface of the earth ; i. e., 196 940 378 square miles. 2°. The last result may be checked by (4). Thus, (y- + ^ + ■ • . j = 0.00225928 log (i — Y - • • • ) = 9-9990177 log a^ = 7.1961072 log 4 IT = 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 ^TyTyiirTrTr P^^rt 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. r-i = radius of sphere having same volume as the earth's spheroid, ^2 = radius of sphere having same surface as that spheroid. GEODESY. liii »2 = a{r- 3 IS ~ is ,..)- a — ri = ^ae^(i-\-^e^-\-...)^ 0.00113 a, about. ^2 — ri = 5I5 ae* -[-... = 0.00000 1 a, about. II. Coordinates 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 EJ^ (Fig. 2) will appear in projection as the arc of a circle £0E (Fig. 3) whose radius OG^l is equal to the slant height of the tangent cone EFG (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 AA, and denote the angle at the centre G subtended by the developed arc OP by a. IJV GEODESY. Then from Fig. 3 it appears that X = I sin a, y=. 2 I sin° ^a. But from Figs. 2 and 3, '^=PnCOt<^, la = r A\ = p„ AA. COS <(>, whence a = AA. sin sin (AX sin ), , » y^2p„cot sin'' J (A\ sin ^). Numerical example. — Suppose <^ = = 40° and AX = 25° ^ 90000". Then log 90000" = 4-9542425, log sin 40° = 9.8080675 — 10, log S78So-"88 = 4.7623100; AX sin.(^ = 16° 04' io."88, J (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 11 7.3212956 cot <^ 0.0761865 p„, Table II 7.3212956 2 0.3010300 X 6.8396581 .y 5-9893731 ^ = 6 912 865 feet y = 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 if) and sin'' ^(AX sin <^) and reach results of a much simpler form. Thus, sin (AX sin tf)) = AX sin — i(AX sin <^)' -j- • - - , . sin y - ^(AX sin ^)* + . . . ; whence, to terms of the second order, a; = p„ AX cos ^ [i — J(AX sin t^)"],. , ^ y = iPn (AX)'' sin 2<^ [i - T^(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 4>y . x, and the value of y will be too great by an amount somewhat less than ■i''3(AX sin <^)" . y. 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 (j>y for a few values of the arguments AX and (j>. GEODESY. Values of J(AX sin <^)*. Iv «^ A\ 20° 40° 60° o I 1/1680OO 1/47700 1/26260 2 1/42000 1/119OO 1/6560 3 1/1870O 1/5300 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 A\ = 2°, and <^ = 40°, and for a scale of two miles to one inch (i/i 26720), is 53.063 inches less 1/11900 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 AX does not exceed 1°, it is amply sufiicient, therefore, to use x-^ Pn AA. cos ^, y = i Pn (AX)2 sin 24>. (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 A\ in minutes and x and y in inches. For this purpose, sup- posing log p„ to be taken from Table 11, if s be the scale of the map, or scale factor, equations (3) become — Co-ordinates j; and j/ in inches for scale s. p„ S AX cos <^, 3437-747 AX 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 and^ 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 : Jvi GEODESY. log. log. cons't 7.54291 — 10 cons't 3.4046 — 10 Pn 7-32130 Pn 7-3213 s 5.19818 — 10 J 5.1982 — 10 IS 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 J/ = 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 AA. and y as (AA.)'', 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 ; (f) the geodesic line ; and (c) the alignment curve. Imagine two points in the surface of a spheroid, and denote them by J'l and I\ respectively. The vertical plane at I\ containing /'g aiid the vertical plane at ^2 containing /i give vertical section curves or lines. The curves cut out by these two planes coincide only when I'l and ^Pg ^re in a meridian plane. The geodesic line is the shortest line joining I'l and /a, and 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 /^ and i'j- 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 (c) 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. I^et ai.2 = azimuth of vertical section at T'l through I\, «2.x = azimuth of vertical section at P2 through J\, 61, 62 = reduced latitudes of Pi and T'a respectively, Si, 82 = angles of depression at J^i and P^ respectively of the chord joining these points. Then the characteristic property of the vertical section curve joining /'land/j is sin ttij cos Oi cos Sj = sin (aj.j — 180°) cos 62 cos Sj. GEODESY. ivil The azimuths a^.-, and oj.i, 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.3 = azimuth of geodesic line at /j, a'2.1 = azimuth of geodesic line at /ji Oi] 62 = reduced latitudes of -fi and P^ respectively. Then the characteristic property of the geodesic line is sin ai.2 cos 61 = sin (i8o°— ogj) cos 0^ = cos 60, where 60 is the reduced latitude of the point where the geodesic through /j and /j is at right angles to a meridian plane. The difference between the astronomical azimuth ai.2 and the geodesic azimuth a'i.2 is expressed by the following formula : "1.2 — "'1^ (in seconds) = y^ p" ^' {-) cos^ <^ sin 2ai.2> where s = length of geodesic line I'l F^, a = major semi-axis of spheroid, e^ eccentricity of spheroid, p" = 2o6264."8, 4> = astronomical latitude of Fi, ai.2 = azimuth (astronomical or geodesic) of F^ F^, log tV p"( -) '=■ 7-4244 — 20, for a in feet. Thus, for <^ = o and ai.2 = 45°, for which cos'' (j) sin 201.2 = i, the above for- mula gives "■1.2 — «'i.2 = o-"o74, for J = 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 S 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, £, C, respectively ; the cor- responding angles of the plane triangle by a, j3, y (as on p. xviii) ; and the sides common to the two triangles by a, b, c. Then S ■= \ ab ^va. y ■=. \ be ^ivi a ■=■ \ ca sin /3. a = A — \^, ^ — B — \^, y=C— 1£. Tables 13 and 14 give the values of log {jl'lzp^^ 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 P^ respec- tively. Let J = length of geodesic line joining P^ and P^, <^i, (^2 = astronomical latitudes of P^ and P^, Aj, X2 = longitudes of Pi and P^ AA, = X2 — Ai, ai.2 = azimuth of P^ P^ {s) at P^, 02.1 = azimuth of P^ Pi (s) at P^, e =■ eccentricity of spheroid, Pm. Pn = principal (meridian and normal) radii of curvature at the point Pi. Then for the longest sides of measurable triangles on the earth the following formulas will give <^2, ^2> and 0^,1 in terms of <^i, Aj, ai.2, and s. The azimuths are astronomical, and are reckoned from the south by way of the west through 360°. 180° — ai.2, and 02.1 = 180° -\- a", for ai.2 180' ^^i{-+lT^^ (^J'cos» i COS= a'} (2) ^ = k TZrp cos^ ^1 sin 2 a' (3) * For the solution of very large triangles and for a full treatment of the theory thereof, consult Die Mathematiscken und Physikalischen Theoriem der H'oheren Geodasie, von Dr. F. R. Helmert. Leipzig, 1880, 1884. GEODESY. lix tan K-" + AX + = g|- !^^°I " 1^ T i cot ^ a' cos i(9o — ^1 + 1?) ^ (4) tan Ka" - AX + = ;;" ffi°: -tli cot ^ a' sin ^(90 — i + 7i) " 'l^-'^^ = i £|^^^4^)0 + TV.^COS^Ka"-a')}. (s) To express 97, ^, 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 rj" and 4"> or 1; and ^ in seconds, we may write with an accuracy generally sufficient log r," = log (p"s/p:) + I J^ (^J' cos'' , COS» a', (6) log r = log I (TiT^, + log {(v'r COS» <^i sin 2 a7, (7) where /j. in (6) is the modulus of common logarithms. For units of the 7th deci- mal place of log ij" we have for the adopted spheroid . 1 /* ^ — ^i) ^"d 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 + ^ ^ 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 k = sin <^ sin 8 -\- cos <^ cos 8 cos t, cos A cos A ^ — cos ^ sin 8 -|- sin cos 8 cos f, (i) cos A sm A = cos 8 sin /. When it is desired to compute both A and A by means of logarithms, the most convenient formulas are, m sin M :=^ sin 8, ,, tan 8 m cos J/= cos 8 cos t, cos r ,_ , tan fcosM , ^ smA = m cos (<^ - M), tan A = gin (^ -M) ' ^^^ cos /i COS A =m sin (<^ — JIf), ^^^^ ^ __ ^o^ -^ _ cos A sin A = cos 8 sin f, tan ((^ — M)' A > 180° when ( > 180° and A < 180° when f < 180°. For the computation of A and z separately, the following formulas are useful : sin / tan A = cos tan 8(1 — tan ^ cot 8 cos i) (3) a sin f I — l> cos /' where a = sec <^ 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 & will be constant for a given place and date. cos z = cos (^ ~ 8) — 2 cos (j) cos 8 sin' ^ t, sin' i z = sin' i (sfc ~ 8) -f cos cos 8 sin' ^ t, , \ (<^~ 8) = (^-S, for<^ >S ^^-^ = 8 — <^, for .^< 8. ASTRONOMY. IxiX For logarithmic application of (4) we may write m^ = cos <^ cos 8, «' = sin° \{<^ ~ S), tan J^= ^ sin 1 1, (s) n ." , n m . 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 8 cos / = cos ^ sin ^ -|- sin <^ cos h cos A, (i) cos 8 sin ^ = cos h sin A. For logarithmic computation by means of an auxiliary angle Mone may write »2 sin M= cos A cos A, taniJ/:= cot A cos A, m cos J/'= sin h, • ff • /J Ttr\ J. J tan ^ sin Jlf , v sm 8 = »? sm (d> — J/), tan t ■=. -. j^— , (2) ^ ' cos (<^ — My cos 8 cos t-= m cos (<^ — M), cos 8 sin ^ ^ cos ^ sin A, tan 8 = tan (<^ — M'\ cos /. d. Hour angle and azimuth in terms of zenith distance. , cos z — sin A sin 8 cos t = — ^ cos cos ^^^, sin_(a^^^)_cos_(o^) ^^i(^_^8 + ^). cos er cos (a- — z) . sin Zl^_^L(2L=:^, ^ = n<^ + 8 + ^)- cos o- sm ( sin z sin ^ = cos <^ sin t ; whence tan iV:= cot cos t, . ■ tan / sin iV , , tan sm «r = -: — j^^. — — r, (2) * sm (8 + N) ^ ' tan z cos f = cot (8 + -i^)- A similar adaptation results for the last three of equations ,(i) by interchanging 8 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 t, they are best given by the Gaussian relations sin \ z sin ^{A + ^) = sin ^ ^ cos i(<^ + 8), sin \ z cos \(A -\- q)^cos\t sin J(^ — 8), , . cos I z sin J(^ — ^) ^ sin ^t sin ^(^ + 8), cos ^ 2 cos J(^ — ^) ^ cos J ^ cos J(^ — 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 <^ ^°' ' = tiJT' cos 8 . . sm A = T> (i) cos <^ '• ' sin rf) cos z = -7— "s" sin o When all of the quantities t, A, and z are to be computed the following formulas are more advantageous : — K^ = sin (8 + <^) sin (8 - ^), sm t = 7 — : — s' COS A = T' sm z ■= —. — k> (2 ) cos sin 8 cos i^ sin S '■ ' J tan a = — i;;-'-' cos sm 8 sm 8 =< ^ For special accuracy the following group will be preferred : — tanH^= '?"g7S - * sm (^ -|- 8) tan i^-tanK-^ + S)' tan^ (45" - i ?) = tan J(.^ + 8) tan iC^ - 8). h. Hour angle and azimuth of a star when 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-= — :; : cos <^ ' or by *^" ^^— cos(<^ + 8)' tann^= rffi°:-i+^i - ^ tan ^(90 — ^ — 8) On account of refraction, the values of / and A given by these formulas are subject to the following corrections, to wit : — R . . tan . The following values derived from (i) are of interest as showing the dependence of z and ^ on / in special cases : — (dz\ (dA\ \dt) \dt) cos S For a star in the meridian = o, = '^rrv sin 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, E = 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 = '^ — - — = 1.002738. 365.2422 /' = rl=z /+ (r - i) /= 7+ 0.002738 I r= r-i 7' = /' - (i - r-') r = I' - 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"' o4.'o9i solar time, 24* mean time = 24* 03" s6.'555 sidereal time. d. Interconversion of sidereal and mean solar time. T„ ^ mean time at any place, Tg = 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 -{- T„ expressed in sidereal time. T^ = (T, — 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 T,. T. = a-{-f, t=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 = T-\- b.T. 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 7J = a, the right ascension of the star ; and if T is the observed clock time of the transit, A7'=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.! Let a = azimuth constant, b = inclination or level constant, c = collimation constant. a is considered plus when the instrument points east of south ; l> is plus when the west end of the rotation axis is the higher ; and c 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 <^ =: latitude of the place, 8 = declination of star observed, a = right ascension of star observed, 7"= observed clock time of star's transit, AT=: the clock correction at an assumed epoch 7J, r = the rate of the clock, or other timepiece, M sin (d) — 8) ■^ = J^ \, s ^ = the " azimuth factor," cos d ' £ = S2l(±^ ^ the " level factor." cos o ' C = j = the " collimation factor." cos d Then, when a, b, c are small (conveniently less than lo' each, and in ordinary practice less than i' each), T-\- b^T-\- Aa -\- Bb ■\- Cc -\- r (T- TJ) = 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 ^T-^Aa-\-Cc-\-r{T-T^) = 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 ^T. 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 A7'+ 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 (J) and c small), this simple sort of observation with a theodolite will give A 7* 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- othei 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 suffice 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 oi 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 SC may be made small by suitably selected stars, since two of the A'^ and C's are positive and two negative. The value \ S(a — T) is thus always a close approximation to AT'for the epoch Tq^\ 1,T, when %A and %C approximate to zero. But if these sums are not small, 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 LT, 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 Zi = the observed zenith distance of the star, Ji = the refraction, z = the true zenith distance of the star, = z, + Ji, a, 8, = the right ascension and declination of the star, / = hour angle of star at time of observation, T= observed time when Zi is measured, A 7'= correction to timepiece, <^ = latitude of place. Then the hour angle / may be computed by „ sin (o- — (^) sin (o- — S) 1 / jL I s I s tan" i / = ^ ^-^ — 7 — ^^ — r — ^, o- = Ufk J_ 8 _L. 2). ^ cos 0- cos (a — z) "'^^ ' ' ^ • 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 AT is given by AT=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 Zj 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 .md 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 refrao 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 S 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). Th«; 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 knov^n time. h = measured altitude corrected for refraction, Tg = observed sidereal time, a,^=z right ascension and declination of star, t = hour angle of star, <^ = latitude of place. Then (^ may be computed by means of the following formulas : — t=T,-a, t,„ o tan 8 ^„^ sin h sin j8 tan ^ = -, cos y = -. — g-t:, cos r sin 8 <^ = /8 ± y. In the application of these /3 may be taken numerically less than 90°, and since t may also be taken less than 90°, ^8 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 : — <^ = ^ — / 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 \f' 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 S„ denote the declinations of these stars respec- tively. Then «, = <^ — S„ ^» = 8» — <^, whence ,^ = J(8. + 8„) + i(?,-z„)- It appears, therefore, that this method requires only that the difTerence (z, — z„) 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^ — «„) 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 (2, — 0„) being measured on the circle ; and a determination of latitude within 5" or less is thus easy with small theodolites of the best class (i. e., with those whose circles read to 10" or less by opposite verniers or microscopes). 6. Determination of Azimuth. a. By observation of a star at a known time. T, ■= sidereal time of observation, a, 8 = right ascension and declination of star observed, t = hour angle of star, = T, — a, = 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, l> = tan cot 8, - a sin f (i) t^" ^ = - I - ^ cos / The angle A will fall in the same semicircle as /, and A is thus determined by its tangent without ambiguity. The quantities a and 6 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 o , . ^ . , sin «■ . . ; A/, — sm A cot z Ad, -; — - As, sin z ' ^' sm z ' respectively, where z and ^ are the zenith distance and parallactic angle of the star. Hence the effect of Aif will vanish for a star at elongation ; the effect of A(fi 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 4, the azimuth Ag, and the altitude A^ of a star at elongation are given by the formulas (2) of section 2,/. Those best suited to the purpose are ^^ = sin (8 + ^) sin (8 - <^), IT , cos 8 , sin ^'^' > ■ * To the same order of approximation one may write in tlie first term of this expression (2 sin^ ^ ^f\ ■ which latter is the most convenient form when tables giving log ■■ — -r, — - for the argument A/ 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 Formcln und HUlfstafeln fur Geographische Orts- bcstimmungen, von Dr. Th. Albrecht. Leipzig; Wilhelm Engelmann, 4to, 2d ed., 1879. ASTRONOMY. Ixxxi This last formula gives AA in seconds of arc when Aj? 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 (A^y and (Ar)° 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. "8)^ and tan 4 in the denominator of the second term, it will be very small unless A^ 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 A/ < 40*" and when observations are made in middle latitudes. By reference to formulas (2) of section 2,/, it is seen that sin S cos 8 sin^ 8 cos 8 sin 4 cos -^ ' sin 8 cos 8 sin" 8 cos" 8 sin ^" ' X" = sin (8 + ) sin (8 - <^).* b. By an observed altitude of a star. A = true altitude of star observed ; i. e., the observed altitude less the refrac- tion, (^ = latitude of place, p = polar distance of star, A = azimuth of star. tan" \A = sin(^-0)sin(cr-/^) cos o- cos (cr — /) The most advantageous position of the star, on account of possible error in the observed value of ^, 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 J> < 90° — ; it will cross the prime vertical when/ lies between 90° — <^ and 90° ; and it will neither elongate nor cross the prime vertical when/ >9o°, or when the declination (8) of the star is negative. c. 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

(^)dc. If e vary continuously between equal positive and negative limits whose magnitude is a, the sum of all the probabili- ties <^(e)(/£ must be unity, or (e) de = I. s* For the case of tabular logarithms, etc., alluded to above, ^(i) ^ c, 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-\- (v2 — v^ t=Vi(i. — t) -\- v^ t, where Vi and v^ are the tabular values, and t the interval between v^ and the derived value V, ^(c) has the following remarkable forms when the extra decimals (practically the first of them) in (v^ — Vi) t are retained : — <^(£) = . ° "^ ^ for values of e between — J and — (i — 0> (I — t) t = — — for values of e between — (i — f) and + ft — ^)> (^) = , ^ ~ ^ for values of c between + (i ~ ^) ^^^ + i- (I — r; / 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 (wj — v^ t is rounded to the nearest unit of thie last tabular place, <^(i) 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 A 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 th& 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„, 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 : — — «P o -\- ^ + « J*<^(.) d, = J4,{i) d. = f4>(e) de = J^(£) d. = i, — a — e^ o -]- £p J_ J. ^'^ -\- a -\- a ^m = f4>(A ^ di, e„ = jr.^(£) . 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=Vi-\-{v2 — Wi) f where v^ and v^ are consecutive tabular values and / is the interpolating factor. For the species of interpolated value wherein the quantity (wa — v-i) 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 : — £p ::= i (i — ^) for t between o and \, = i — i V2/ (i — /) for t between \ and §, = J ^ for ^ between f and i. V — {\ — 2 ff U 96(1-/)^ s I — (i - 2/)' ^ , £. = 7 -jr-T for t between o and *, " 24 (i — /) ^ '" = - — 7^^ Ts-r for / between \ and i. 24 (i — /) ^ ■* 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 / 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 (z/j — 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 suifice here to give the practical results which flow 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 iox t^ \, \, \, . . . ■^. 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. IxXXVi THEORY OF ERRORS. Characteristic Errors of Interpolated Logarithms, etc. Interpolating factor t Probable error Mean error Average error Maximum actual error I 0.250 0.289 0.250 h i .292 .408 •m I * .256 •347 .287 1 4 .276 .382 •313 I i .268 •370 •303 -h * .277 •38s ■zn , I \ .274 •380 •3" H \ .279 ■389 .318 I i .278 .386 .316 a 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 whose 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 /, q, r, . . . . Let the corrections to the observed quantities be denoted by ^x, A_j/, Az, . . . ; so that the corrected quantities are x -\- A^c, y -\- ^y, z + Az, . . . . Let the disposable quantities whose values are to be determined along with the correc- tions be denoted by ^, % C, . . . . Then, the theoretical conditions which must be satisfied hy x -\- i^x, y -\- A_y, z -|- Az, . . . and by f, 17, ^, . . . may be symbolized by K {^, r,, i, . . . X -\- ^x, J/ -f Aj, 4- Aa, . . .) = o. (4) Subject to the conditions specified by the n equations (4), we must also have p (A»)" -|- q (Aj')* -\- r {p^zf + • . . = 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-}-A.x-{-y-j-Ay-\-z-^iiZ^ 180° + spherical excess, or ■ ■ ■ , Ax^ Ay -\- Az = 180° 4" spherical excess — (x -\-y -{- z) = ^, say. This is the only condition of the form (4). The problem is completely stated, then, in the two equations Ax -\- Ay-\- Az = c f {Aocf -\- q {Ayf -\- r {Aif = 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 =:p {^xf -\- q {Ayy -\- r (c — Ax — Ay^. The conditions for a minimum of u are : — 9u 9Ax 9u = (p -{- r) Ax -\- rAy — re ^ o, :=. rAx -\- (^ -\- r) Ay — rt: = o ; 9Ay and these give, in connection with the value Az ^ c — Ax — Ay, Ax=Q, Ay=Q; Az=^. where Q = ' + - + - p^ q^ r When the weights are equal, or when / ^ ^ = r, the corrections are — Ax ■=■ Ay ■=. Az ■= \ e. 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 $ and 17, the latter being the smaller. Let the observed values of the sum (i + '?) be denoted by xi, X2, . . . x^ and their weights pi, A, ■ • • A respectively. Likewise, call the observed values of the difference (^ — 17), yi, y^, . . . y^ and their weights fi, ?2 • • • ?» respectively. Then there will hs m -\- n equations of the type (4), namely : — ^ + 17 - (^1 + A^i) = o, ^-\-r, — {x2-\- i^x^ = o, (a) and the minimum equation is u = A (AxO» + A (^x,y + . . . + s-i (Aji)" + 4'2 (Ajs)' + . . . = a min. (b) The equations of group (a) give A.XI = ^ -\- rj — xj, Ax2 = i-\-r] — Xi, (c) Ayi = ^ -V - yi, ^yi = ^ — v— yi, • ' • ', and these values in (b) give u=A(^ + v- *i)' + • ■ ■ + ^1 (^ - v-y.y-\-- ■ • (d) Thus it appears that all conditions will be satisfied if f and 17 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 ^ and 77 must be found from ll = [/ + ?] ^ +[/- rf ^ - l>* + ^.y] = o, (e) -^ = [/-?] f +[/ + ?] V -[/•*- ?J'] = 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 +00 : €p = 0.6745 e„. = 0.8453 'a- (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, . . . x^=^ observed values of unknown quantity, A A • • ■ /m = the weights of x^, ^2, . . . »i, Vi, . . . v^ = most probable corrections to Xi, X2, . . . X = most probable value of the unknown quantity, m = the number of independent observations. Then the conditional equations (4) are X — Xt_-=^ Vi, X ^— X^ — V^f •^ *m ^— ^m ; the minimum equation (5) is A»i' + A^'2'' + . . . = iJv'^ = \J{x - a^OT = a min., where « = i, 2, . . . /«, and _ P\Xl +/2^2 + • • -Pn^m _ \j>x\ A+A + ..-A "[/>]■ When the weights are equal, A =P% = • • ■ =pm> and M or the arithmetic mean of the observed values. Weight of Jc = [/] when the/'j are unequal, = m when the/V are equal. Mean error of an observed value of weight unity = y _-' for unequal weights, . l\vv\~ . , . , = Y __ • for equal weights. / \ pvvX Mean error of an observed value of weight p^y , _ \ . for unequal weights. / [pvv] Mean error of a; = y -, — _ ^r- ■, for unequal weights, — i/ — IZ_J — ^ for equal weights. \ m{m — 1) 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 Average error = j , _ \ f^ ^°^ unequal weights. [v] = ./ / — s for equal weights. \m (m — 1) ^ ° 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 ^,i},K--.. 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 F{i, 17, ^ x-\- ^x):=o•, that is, each of them contains but one observed quantity x along with several disposable (disposable in satisfying the minimum equation) quantities f, 17, ^ . . . . The process of solution in this case consists in eliminating the corrections tyxi, 1^X2, . . . from the above conditional equations, substituting their values in the minimum equation (5), and then placing the differential coefficients of u with respect to i, rj, ^ .,. . separately equal to zero. There will thus result as many independent equations as there are unknown quantities of the class in which f, 1/, t ■ . ■ fall, the remaining unknown quantities Axi, Ax^, . . . , or the corrections to the observed values, are then found from the conditional equations. In many applications it happens that the conditional equations F(^, v, C • • • x-\- Ax) = o, are not of the linear form. But they may be rendered linear in the following manner. First, eliminate the quantities x -\- Ax from the conditional equations. The result of this elimination may be written /(^. V, ^ • • ■) — X — Ax = o. Secondly, put ^=fo Af, where fo> Vo, • ■ ■ are approximate values of f, 17, ... , found in any manner, and A|, Arj, . . . are corrections thereto. Then supposing the approximate values THEORY OF ERRORS. XCI loi %» • ■ . SO close that we may neglect the squares, products, and higher powers of Af, Arj, . . . , Taylor's series gives /(4%, fo, ...)+ sf ^^ + |Ja, + ||^a^+. ^^-A^=o, which is linear with respect to the corrections Af, Ai;, . , . . For brevity, and for the sake of conformity with notation generally used, put n = x — /(fo, Vo> &•■■)> V = A^, x = ^t, y = i^% « = A^, — Then the conditional equations will assume the form ax -\- by -\- cz -\- . . . — n = v, and if they are m in number they may be written individually thus : — aix -\- b^ -\- c-^z -\- . . . — ni = Vi, "m -{- ^m + "^m + ■ • ■ — «m = Z'm- The minimum equation (5) becomes u = [pv'^ = \p{ax ■\- by -\- cz -\- . . . — «)^ ; so that placing -^, -y^, -^, . . . 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 (Gaiissian) notation of least squares : — \^ad\x -\- \pal)]y -\- \pac\ z -\- . . . — [^an] ^ o, [pab] +\jbb-] +[pbc] -{-...-[j,i„] = o, (b) \jac] -^lpbc-\ -\-\Jcc-\ +...-[/^]=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/ (^0, %, f . • •)> 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 coefifiGient 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 my, z, . . . are found in a similar manner. XCU 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 + • ■ • = •?!, «2 + ^2 + "^2 + • • • ^ -^2. Multiply each of these, first, by its pa ; secondly, by its//5, etc., and then add the products, The results are [paa\ + \J>ab'\ -\- [J)ac\ -|- . . . = \_pas\ \Jab] + \jbb-\ + [pbc] + ... = [pbs] 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] + [pbn] + [Jen] + . . . = {psn\, which must be satisfied if the absolute terms are correct. Checks on the computation of x, y, z, . . . from (b) and of v^, v^, • . . from (a) are furnished by \_pav] = o, \_pbv\ ==■ 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, £, C, . . . respectively. Then the solution of (b) will be expressed by X=:aiA-^M + yiC+..., 7 = "2 + A + 72 + • • • , (C) « z= 03 4-^3 -j- 73 + . . . , in which oj, j8i, yi, . . . are numerical quantities ; and weight of ^ ^ — > weight of ^ = 7T> (,d) Pi weight of 2 = — ) To compute mean (and hence probable) errors the following formulas apply : — m = the number of observed quantities « = number of equations of condition, fjL =; number of the quantities x, y, z, . . . c„ = mean error of an observed quantity («) of weight unity, tp = corresponding probable error = 0.6745 e„. \ m — ju, THEORY OF ERRORS, XCiii for unequal weights, __ / \yv\ £^^ equal weights, y m — \x. Mean error of any observed quantity («) of weight/ = -^> Mean error of ^ = e„ y/^^ Mean error of j' =: e„ \f^^^ Mean error of 2: = «„ v'i^, . . . , where oj, ^ji 7s( • • • a-re 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 f, 17, ^, . . . . 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 plus their respective corrections mus"- 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 ^, r/, (, . . . suppressed, or If(xi -\- i\ Xi, X2 -\- A X2, Xs -{- \ Xs, . . . ) =:: o, 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 9jF 9F F(x^, x„x,...)-\-Q^^Xy■^-^^^x, = ... = o, since the method supposes that the squares, products, etc., of Aarj, A*2 • • • fnay be neglected. The last equation is then linear with respect to the corrections A^i, Ajcj . • . which it is desired to find. For brevity put F{xi, X2, Xi . . .)^ q-i, a known quantity, 9F _ 3F_ 5F __ 5^ — ''" 9x, — ''" dx, — «3, • . ■ . Then the conditional equations will be of the type ai£^Xi + a^^Xi -\- ajA^g -|- . . . -f- ^1 = 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^ -{- A^i, x^ -\- A^vg, . . . must satisfy. Suppose there are A such relations, and let the differential coefficients 3^/3^i, QJ^/Qx^, ... for the sec- ond relation be denoted by ii, 4, h, • • • ', io^ the third relation by Ci, c^, Cg, . . . , etc. Then all of the conditional equations may be written thus : a^iiXi -\- «2A«2 + aa^Xi + . . . + ?i = o. *i -\-l>2 -\-h + • • ■ + ?2 = o, (a) Ci + ^2 + ^3 -j- • • • + ?8 = o, . . . } the number of these equations being k. Call the weights of the observed quantities Xx, x^, . . . p^, p2, . . . . Then, sub- ject to the conditions {a) we must have (in accordance with (5)) u =pl\x,y + A(A^27 + . . . = [p{^xY\ {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 Aa;'s in {b) ; in other words, if m denote the number of observed quantities, m > k, ior \i m ^ k 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 Aa:'s as there are equations (a), or k such values, and substitutes them in (b). The remaining {m — k) values of ^x 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 Ax's) of the sum of these products from half the differential of {b). The result of these operations is iidu= {/lAxi _ (ai(2i -f ^i(22 + ^iG + • ■ •)} '^^^i + {Pi^2 -(' S^— sin cos/ (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 dbserved .angle ; and 3d, of the corrected or adjusted angle ? Let X, y, z denote the observed angles, /, q, r their weights, and A^, A_y, As the corresponding corrections. Then, as shown on p. Ixxxvii, A* -|- Aji/ -[- A« = f := 180° -(- sph. excess — {x -\-y -{• z) = error of closure of triangle, Q = '- - + - + - Ax =2, Ay— Q, Ag ~ Q, p q r * As remarked by Sir George Airy in liis Theory of Errors, t M = modulus of common logarithms. THEORY OF ERRORS. XCVU For brevity, put g = i8o° -|- spherical excess, h = - - + - + - Then 0.= h {g — X — y — z) =^ he, t>.x = -i^g - X - y - z), P X -\- ^x = —{g — X— y — z)A^x, P 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 A^, Aj, A^ are subject), pit^x)"" + q{b.y)^ H- KM' = f-= h^. Hence, the squares of the mean errors of the observed angles x, y, z, their weights being/, q, r respectively, are hc"^ he" he^ respectively. To get the mean error of a correction, Aa: for example, formula {a) gives A r= A(A*) = - 1(^, + ^, + O, and the corresponding expressions for the actual errors of Aj/ and Az are found from this by replacing phy 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 t^x = {hcjpf, " " " ^y = {hcjqf, " " " ^ — {hejrf. Likewise, the formula for the actual error of Je + A^ is A V=^ ^.(x + Aa:) ^ 1 1 — -\e^ — '^e„ — '!^„ h h - —e,, — —i P ' P and the corresponding expressions for the actual errors oi y -\- ^y and z -\- ii.z are found by interchange of q and r with/. Thus the squares of the mean errors of the adjusted angles are : — for(. + A.), y{"~~p)' for {y + ^y), ^ (i - ^), for(. + A.), ■7-'(i-^)- XCviii THEORY OF ERRORS. In case the weights are equal, or in case p = q = r, h-=\, and there result, — Square of mean error of observed angle = J c\ " " " " correction to observed angle z=z ^ c\ " « " " " adjusted angle = S ^, 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 \\J2, h ^^^ W^- 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., izmo, 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 '"* (nth 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 1. We thus find 6394 for the number corresponding to the mantissa 8410. To 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 difiEerence ^ 26. Proportional part for 7' (7 X 2.6) • . 18. sine 28° 17' 0.4738. tangent 28° 10', Table 8 O.S3S4. 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 ZcosB 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 Z = 4837, in any unit, and -ff = 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^ = 39oo.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' " B " 47 52 C " 47 04 Mean latitude 47 41 Angle C= 51° 22' 5s" log sin C 9.89283 — 10 log a (feet) 5.64401 log l> (feet) 5.58681 log factor, Table 13, for 47° 41' 0.37176 Spheroidal excess = 3i."29o, log i. 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 tb 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' i7."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" 3°36-23 " •54 " s" 506.04 " .09 " o."6 12.41 60.72 " 7-oS " sum. .01 " 34.09 v 60 ^ 12.41 " Distance = 70407.10 " When the degree of precision required is as great as that of the example just eii 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<^ = 695."6 2.8423596 = 41° 34' os.''6, p„ (Table 10) 7.3196820 cons't 4-6855749 Length ::= 70407.10 feet 4.8476165 i 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. Table 19. scale : 260000 20, I lasooo 21, 1 (2 miles to i inch) 22, I 033(10 (i mile to I inch) 23, 24, I 200000 I 80000 > unit = millimetr 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 hne 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 D, 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, ig.o6, and 28.59 inches. These distances are to be laid off on NN' in both directions from AB. At the points L, M, N, IJ, 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. EXPLANATION OF SOURCE AND USE OF TABLES. Clll 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, U'vci 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, TX should be equal to WY, CN io CN', EVX.0EW, etc.* 1 9 X I V y s T G S F JB JV M l! E L M N D Q w C JP V 35' 50' 4(S 3(i 30' Id 34° Fig.4. 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.t This table is taken from the " Smithsonian Meteorological Tables " (the first volume of this series). The ijianner of using it is explained in connection with the table. * It should be noted that CN'\& not equal to £ F, jV and F referring here to points on the developed parallels. f Ccmftes 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' 12. "7 Refraction = 1' 24."3 +%X i."i = i 24-5 True altitude of star =34 iS 48.2 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 followiug examples illustrate their use : — (i.) To convert 68° 29' 48."8 into time we have from Table 32 — 68° = 4" 32"" 00" 29' = I 56 48" = 3.20 o."8 = .OS Equivalent in time = 4 33 59.25 (2.) To com ert ^^ 43"" 28.'8 into arc we have from Table 33 — s" = 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. Hi. 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 1/ loooooth part. EXPLANATION OF SOURCE AND USE OF TABLES. CV and {g/f^) 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/ioo 000th part. * Washington, Government Printing Office, 1891, GEOGRAPHICAL TABLES Table i . FOR CONVERTING U. S. WEIGHTS AND MEASURES.* CUSTOMARY TO METRIC. LINEAR. CAPACITY. Fluid Fluid Inches to Feet to Yards to Miles to millilitres ounces to Quarts to Gallons to metres. metres. Itilometres. or cubic milli- litres. litres. ceuti- metres. litres. I = 25-4001 0-304801 0-914402 1-60935 I = 3-70 29-57 0-94636 3-78543 2 = 50-8001 0-609601 1-828804 3-21869 2 = 7-39 59- « 5 1-89272 7-57087 3 = 76-2002 0-914402 2-743205 4-82804 3 = 11-09 88-72 2-83908 11-35630 4 = 101-6002 1-219202 3-657607 6-43739 4 = 'f^g 118-29 3-78543 I5-14174 127-0003 I •524003 1-828804 4-572009 8-04674 5 = 18-48 147-87 473179 18-92717 6 := 152-4003 5-48641 1 9-65608 6 = 22-18 177-44 5-67815 22-71261 7 = 177-8004 2-133604 6-400813 11-26543 7 = 25-88 207-02 6-62451 26-49804 8 = 203-2004 2-438405 7-315215 12-87478 8 = 29-57 236-59 7-57087 30-28348 9 = 228-6005 2-743205 8-229616 I4-484I2 9 = 33-27 266-16 8-51723 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 tc grammes. Avoirdu- pois pounds to kilo- grammes. Troy ounces to grammes. J __ 6-452 9.290 0-836 0-4047 1 ^-. 64.7989 28-3495 0-45359 31-10348 2 = 12-903 18-581 27-871 1-672 0-8094 2 = 129-5978 56-6991 85-0486 0-90719 62-20696 3 = I9-355 2-508 1-2141 3 = 194-3968 1-36078 93-31044 4 = 25-807 37-i6i 3-344 1-6187 4 = 259-1957 113-3981 I -8 1437 124-41392 32-258 46-452 4- 18 1 2-0234 323-9946 141-7476 2-26796 186-62088 5 = 38-710 SS742 5-017 2-4281 6 = 388-7935 170-0972 2-72156 7 ^ 45-161 65-032 5I53 2-8328 7 = 453-5924 198-4467 3-17515 217-72437 8 = 51-613 74-323 6-689 3-2375 8 = 518-3914 226-7962 3-62874 248-82785 9 = 58-065 83-613 7-525 3-6422 9 = 583-1903 255-1457 4-08233 279-93133 CUBIC. I Gunter's chain = 20-1168 metres. Cubic inches to cubic centi- metres. Cubic feet to cubic metres. Cubic yards to cubic metres. Bushels to hectolitres. J -- 16-387 0-02832 0-765 0-35239 • 2 = 32774 0-05663 1-529 0-70479 I sq. statute mile =^ 259-000 hectares. 3 = 49-161 0-08495 2-294 1-05718 I fathom = 1-829 metres. 65-549 O-I1327 3-058 1-40957 I nautical mile = 1853-25 metres. 5 = 81-936 O-14158 3-823 I-76196 I foot = 0.304801 me re, 9-4840158 log. 6 = 98-323 0-16990 4-587 2-11436 I avoir, pound = 453-5924277 gram. 7 = 114-710 0-19822 5-352 2-46675 15432-35639 grams = I kilogramme. 8 = 131-097 0-22654 0-25485 6-1 16 2-81914 9 = 147-484 6-88 1 3-17154 The only authorized material standard of customary length is the Troughton scale belonging to this office, whose length at 59°.62 Fahr. conforms to the British standard. The yard in use in 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. TBe grain 1 Toy 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. Office of Standard Weights and Measures, and republished here by permission of Superintendent of Coast and Geodetic Survey. Smithsonian Tables. 2 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 miles. or cubic centi- metres to fluid drams. Centi- litres to fluid ounces. Litres to quarts. Deca- litres to gallons. Hecto- litres to bushels. ,= 39-3700 3.28083 I -09361 1 0-62137 ,- 0-27 0-338 1-0567 2-6417 2-8377 2 = 787400 6-56167 2-187222 1-24274 2 = 0-54 0-676 2-1 134 5-2834 5-6755 3 = iiS-iioo 9-84250 3-280833 I-86411 3 = 0-8 1 1-014 3-1700 7-9251 8-5132 4 = 157-4800 13'I2333 4-374444 2-48548 4 = 1-08 1-353 4-2267 10-5668 11-3510 196-8500 16-40417 5-46S056 3-10685 1-35 1-691 5-2834 13-2085 14-1887 6 = 236-2200 19-6S500 6-561667 3-72822 6 = 1-62 2-029 6-3401 15-8502 18-4919 17-0265 7 = 275-5900 22-96583 7-655278 8-748889 4-34959 7 = 1-89 2-367 7-3968 19-8642 8 = 314-9600 26-24667 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. j_ 0-1550 10-764 I-196 2-471 ,= 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 '''iV' 3 = 0-04630 46297-07 10-5822 6-61387 4 = 0.6200 43-055 4-784 9-884 4 = 0-06 J 73 61729-43 14-1096 8-S1849 0-7750 53-819 5-980 Iz-355 5 "=■ 0-07716 77161-78 17-6370 11-02311 6=^ 0.9300 64-583 7-176 14-826 6 ^ 0-09259 92594-14 21-1644 - 13-22773 *j-=. 1-0850 75-347 8-372 17-297 7 = 0-10803 108026-49 24-6918 15-43236 8 = 1-2400 86-1 11 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 =: 0-0610 61-023 35-314 1-308 ,= 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 = 661-39 881-85 6613-9 96-4522 4 = 0-2441 244-094 141-258 5-232 4 = 8818-5 128-6030 0-3051 305-117 176-572 6-540 5 = I102-31 11023-1 160-7537 - 6 = 0-3661 366-140 211-887 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 10-464 8 = 1763-70 17637-0 257-2059 9 = 0-5492 549-210 317-830 I1-771 9 = 1984-16 19841-6 289-3567 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 intercompared, 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 ill 1866. ,.,,., , ,. The International Standard Metre is derived from the Mttre des Archives, and its length is defined by the dis- tance between two lines at 0° Centigrade, on a platinum-iridium bar deposited at the International Bureau of Weights and Measures, , ., , . ...... j, , , The International Standard Kilogramme is a mass of platinum-indium deposited at the same place, and its weight in vacuo is tlie 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. Smithsonian Tables. 3 Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. n 1 000. J rfl »8 v» \n log. « 1 1000.000 I I 1.6000 1. 0000 0.00000 2 500.000 4 8 1. 4142 1.2599 0.30103 3 333.333 9 27 I.732I 1.4422 0.47712 4 250.000 16 64 2.0000 1.5874 0.60206 5 200.000 25 125 2.2361 I.7100 0.69897 6 166.667 36 216 2.4495 1.8171 0-77815 7 142.857 49 343 2.6458 1.9129 0.84510 8 125.000 64 512 2.8284 2,0000 0.90309 9 III. Ill 81 729 3.0000 2.0801 0.95424 10 100.000 100 1000 3-1623 2-1544 1. 00000 i: 90.9091 121 1331 3-3166 2.2240 1. 041 39 12 83-3333 144 1728 3.4641 2.2894 1.07918 '3 76.9231 169 2197 3.6056 2.3513 I.I 1394 14 71.4286 196 2744 3-7417 2.4101 1.14613 15 66.6667 225 3375 3-8730 2.4662 1.17609 i6 62.5000 58.8235 289 4096 4.0000 2.5198 1.20412 '? 4913 4-1231 2.5713 1.23045 i8 55-5556 324 5832 4.2426 2.6207 1.25527 1.27875 19 52.6316 361 6859 4.3589 2.6684 20 50.0000 400 8000 4-4721 2.7144 1.30103 21 47.6190 441 9261 4.5826 2.7589 1.32222 22 45-4545 484 10648 2.8020 1.34242 23 43-4783 529 12167 4.7958 2.8439 1-36173 24 41.6667 576 13824 4.8990 2.884s 1.38021 25 40.0000 625 15625 5.0000 2.9240 1.39794 26 38.4615 676 17576 5.0990 2.9625 1.41497 ^? 37.0370 729 19683 5.1962 3.0000 I -43 1 36 28 35-7143 784 21952 5-2915 3.0366 1.44716 29 34.4828 841 24389 5-3852 3.0723 1.46240 30 33-3333 32.2581 900 27000 5-4772 3.1072 1.47712 31 961 29791 5-5678 3-1414 1.49136 32 31.2500 1024 32768 5-6569 3.1748 1.50515 1.51851 33 30-3030 1089 35937 5.7446 3-2075 34 29.4118 1156 39304 5.8310 3-2396 1.53148 35 28.5714 1225 4?!7S 5-9161 3.27 1 1 1.54407 36 27.7778 1296 46656 6.0000 3-3019 1.55630 1.56820 31 27.0270 1369 50653 6.0828 3.3322 38 26.3158 1444 54872 6.1644 3.3620 1-57978 39 25.6410 I52I 59319 6.2450 3.3912 1.59106 40 25.0000 1600 64000 6.3246 3.4200 1.60206 41 24.3902 i68i 68921 6.4031 3.4482 1.61278 42 23.8095 1764 74088 6.4807 3.4760 1.62325 43 23.2558 1849 79507 6-5574 3-5034 1.63347 44 22.7273 1936 85184 6.6332 3-5303 1.64345 45 22.2222 2025 91125 6.7082 3-5569 1. 65321 46 21.7391 2Il6 97336 6.7823 3.5830 1.66276 47 21.2766 2209 103823 6-8557 6.9282 3.6088 1.67210 48 20.8333 2304 1 10592 3-6342 1.68124 49 20.4082 2401 I 17649 7.0000 3-6593 1.69020 50 20.0000 2500 125000 7.0711 3.6840 1.69897 SI 19.6078 2601 ,132651 7-1414 3.7084 1.70757 52 19.2308 2704 140608 7.21H 3-7325 1.7 1600 53 18.8679 2809 148877 7.2801 3-7563 1.72428 54 18.5185 2916 157464 7-3485 3-7798 1.73239 Smithsonian Tables. Table 3. VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. » 1000.J «« «8 ^« V« log. « 55 18.1818 3025 3136 166375 7.4162 3.8030 1.74036 S6 17.8571 175616 7-4833 3.8259 3-8485 1.74819 57 17-5439 3249 185 193 7-5498 1.75587 S8 17.2414 3364 I95II2 7.6158 3.8709 1-76343 59 16.9492 3481 205379 7.6811 3-8930 1.7708s 60 16.6667 3600 2i6nno 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 1.83251 1.8388s 69 14.4928 4761 328509 8.3066 4.1016 70 14.2857 4900 343000 8.3666 4.1213 1.84510 71 14.0845 5041 3579H- 8.4261 4.1408 1.85126 72 13.8889 5184 373248 8.4853 4.1602 1-85733 73 13.6986 5329 389017 8.5440 4-1793 4.1983 1.86332 74 13-5135 5476 405224 8.6023 1.86923 75 13-3333 5625 421875 8.6603 4.2172 1:87506 76 13-1579 5776 438976 8.7178 4.2358 1.88081 77 12.9870 5929 456533 lv^°. 4-2543 1.88649 78 12.8205 6084 474552 8.8318 4.2727 1.89209 79 12.6582 6241 493039 8.8882 4.2908 1.89763 80 12.5000 6400 512000 8.9443 4.3089 1.90309 1.90849 81 12.3457 6561 531441 9.0000 4.3267 82 12.1951 12.0482 6724 551368 90554 4-3445 1.91381 83 6889 571787 9.1104 4.3621 1.91908 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 f7 11.4943 7569 658503 681472 9-3274 4.4310 1-93952 88 11.3636 7744 9.3808 4.4480 1.94448 89 r 1.2360 7921 704969 9.4340 4.4647 1-94939 90 11. nil 8100 729000 9.4868 4.4814 1.95424 91 10.9890 8281 ''Wi 9-5394 4-4979 1-95904 92 10.8696 8464 778688 9-5917 4-5144 1.96379 93 10.7527 l^.n 804357 830584 9-6437 4-5307 1.96848 94 10.6383 8836 9-6954 4.5468 1-973 13 95 10.5263 9025 ^57375 9.7468 4.5629 1.97772 96 10.4167 9216 884736 9.7980 4.5789 1.98227 10.3093 9409 912673 9.8489 4.5947 1.98677 98 10.2041 9604 941 192 9.899s 4.6104 1-99123 99 lO.IOIO 9801 970299 9-9499 4.6261 1.99564 100 10.0000 lOOOO lOOOOOO 10.0000 4.6416 2.00000 lOI 9.90099 I020I I03030I 10.0499 4.6570 2.00432 102 9.80392 10404 1061208 10.0995 4.6723 2.00860 •03 9-70874 10609 1092727 10.1489 4.6875 2.01284 104 9.61538 10816 1124864 10.1980 ^ 4.7027 2.01703 105 9.52381 11025 II57625 10.2470 4.7177 2.02119 106 9-43396 II 236 II91016 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 II88I 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 lOOO.g- «2 «5 V« %n log. 7i 110 9.09091 12100 1331000 10.4881 4-7914 2.04139 HI 9.00901 12321 1367631 10-5357 10.5830 4-8059 2.04532 112 8.92857 12544 1404928 4.8203 2.04922 "3 8.84956 12769 1442897 10.6301 4.8346 2.05308 114 8-77193 12996 I481544 10.6771 4.848S 2.05690 115 8.69565 13225 1520875 10.7238 4.8629 2.06070 ii6 8.62069 13456 13689 1560896 10.7703 4.8770 2.06446 117 8.54701 1601613 10.8167 4.8910 2.06819 ii8 8.47458 13924 1643032 10.8628 4.9049 2.07188 iig 8.40336 14161 1685159 10.9087 4.9187 2-07555 120 Ifi'^l 14400 1728000 10.9545 4-9324 2.07918 121 1 4641 I77I561 1 1 .0000 4.9461 2.08279 122 8!i9672 14884 1815848 11.0454 4-9597 2.08636 123 8.13008 15129 1860867 11.0905 4-9732 2.08991 124 8.06452 15376 1906624 "■135s 2.09342 125 8.00000 15625 I953I25 11.1803 5.0000 2.09691 126 7-93651 15876 200.0376 11.2250 5-0133 2.10037 127 7.87402 16129 2048383 11.2694 5.0265 2.10380 128 7.81250 16384 2097152 2146689 "-3137 5-0397 2.10721 129 7-75194 16641 11-3578 5.0528 2.11059 130.. 7.69231 16900 2197000 11.4018 5.0658 5.0788 2.11394 131 7-63359 17161 2248091 11.4455 2.11727 132 7.51880 17424 2299968 11.4891 5.0916 2.12057 2.12385 133 17689 2352637 11-5326 5-i°45 134 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 1 1.7047 5-1551 2.13672 138 7.24638 19044 2628072 11.7898 5.1676 2.13988 139 7.19424 19321 2685619 5.1801 2.14301 140 7.14286 19600 2744000 11.8322 5-1925 2.14613 141 7.09220 19881 2803221 11-8743 5.2048 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 3112136 12.0830 5.2656 2-16435 ^% 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 5-3133 2.17609 •151 6.62252 3442951 12.2882 5-3251 2.17898 2.I8184 152 6.57893 23104 35I1808 12.3288 5-3368 153 6.53595 23409 3581577 12.3693 5-3485 2.18469 1 54 6.49351 23716 3652264 12.4097 5-3601 2.18752 155 ■6.45161 24025 3723875 12.4499 5-3717 2.19033 156 . 6.41026 24336 3796416 1 2.4900 5-3832 2.19312 ^\ 6-36943 24649 3869893 12.5300 5-3947 2.19590 158 6.3291 1 24964 3944312 12.5698 5.4061 2.19866 159 6.28931 25281 40:9679 12.6095 5-4175 2.20140 160 6.25000 25600 4096000 12.6491 5.4288 2.20412 161 6.21 1 18 25921 4173281 12.6886 5.4401 2.20683 162 6.17284 26244 4251528 12.7279 5-4514 2.20952 163 6-13497 26569 26896 4330747 12.7671 5.4626 2.2I2I9 164 6.09756 4410944 12.8062 5-4737 2.21484 Smithsonian Tablesi Table 3. VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. « 1000.^ «2 k3 ^» ^ log. fc 165 166 169 6.06061 6.02410 5.98802 5-95238 5-91716 27225 27889 28224 28561 . 4492125 4574296 4657463 4741632 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 5177717 5268024 13.0384 13.0767 13.1149 13.1529 13.1909 5-5397 5-5505 5-5613 5-5721 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 5-55556 5.52486 5-49451 5.46448 5-43478 32400 32761 33124 33489 33856 5832000 5929741 6028568 6128487 6229504 13.4164 13-4536 13.4907 13-5277 13-5647 5.6462 5-6774 5.6877 2.25527 2.25768 2.26007 2.26245 2.26482 3B5 186 III 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 i3-7i'3 13-7477 5.6980 5-7083 5-7185 5-7287 5-7388 2.26717 2.26951 2.27184 2.27416 2.27646 190 191 192 193 194 5.26316 5.23560 5-20833 5-18135 5.15464 36100 37249 37636 6859000 6967871 7077888 7189057 7301384 13.7840 13.8203 13.8564 13.8924 13.9284 5-7489 5-7590 5.7690 s-7790 5.7890 2.27875 2.28103 2.28330 2.28556 2.28780 195 196 199 5.12821 5.10204 5.07614 5-05051 S-02513 38025 38809 39204 39601 7414875 7529536 7645373 7762392 7880599 13.9642 14.0000 r4-0357 14.0712 14.1067 5.8186 5-8285 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 41616 8000000 8120601 8242408 8365427 8489664 14.1421 14.1774 14.2127 14.2478 14.2829 5.8480 5-8771 5.8868 2.30103 2.30320 2-30535 2.30750 2.30963 205 206 208 209 4.87805 4-85437 4.83092 4.80769 4-78469 42025 42436 42849 43264 43681 8615125 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 4.58716 4.56621 46225 46656 47089 47524 47961 993837s 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 Tables. Table 3 VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. « 1000.^ «2 «8 rin "?« log. « 220 4-5454S 48400 10648000 14.8324 6.0368 2.34242 221 4.52489 48841 10793861 14.8661 6.0459 2.34439 222 4.50450 49284 10941048 14.8997 6.0550 2-34635 223 4.48431 49729 I 1089567 14-9332 6.0641 2.34830 224 4.46429 50176 11239424 14.9666 6.0732 2-35025 225 4.44444 50625 I 1390625 15.0000 6.0822 2.35218 226 4.42478 51076 1 1 543176 15-0333 6.0912 2.35411 227 4.40529 51529 11697083 15.0665 6.1002 2.35603 228 4.38596 51984 11852352 12008989 15-0997 6.1091 2.35793 229 4.36681 52441 15-1327 6.ii8o 2.35984 230 4-34783 52900 12167000 15.1658 6.1269 2.36173 231 4.32900 53361 53824 12326391 15.1987 ^•'358 2.36361 232 4-3I034 12487168 1 5-23 '5 6.1446 2.36549 233 4.29185 54289 12649337 15.2643 ^■'P* 2.36736 234 4.27350 54756 I 28 I 2904 15.2971 6.1622 2.36922 235 4-25532 55225 12977875 15-3297 6.1710 2.37107 236 4.23729 55696 13144256 15.3623 t-^w 2.37291 237 4.21941 56169 13312053 15-3948 6.1885 2-37475 238 4.20168 56644 13481272 15.4272 6.1972 2.37658 239 4.18410 57121 13651919 15-4596 6.2058 2.37840 240 4.16667 57600 13824000 15.4919 6.2145 2.38021 241 4.14938 58081 13997521 15.5242 6.2231 2.38202 242 4-13223 58564 14172488 '5-5563 6.2317 2.38382 243 4-11523 4.09836 59049 14348907 15.5885 6.2403 2.38561 244 59536 14526784 15.6205 6.2488 2-38739 245 4.08163 60025 14706125 15-6525 6.2573 .6.2658 2.38917 246 4.06504 4.04858 60516 14886936 15.6844 2.39094 247 61009 15069223 15.7162 6.2743 2.39270 248 4.03226 61504 15252992 15.7480 6.2828 2-39445 249 4.01606 62001 15438249 15.7797 6.2912 2.39620 250 4.00000 62500 15625000 15.8114 6.2996 2.39794 251 3.98406 63001 I581325I 15.8430 6.3080 2.39967 252 3.96825 63504 16003008 15.8745 6.3164 2.40140 253 3-95257 64009 16194277 15.9060 6-3247 2.40312 254 3-93701 64516 16387064 15-9374 6-3330 2.40483 255 3.92157 65025 16581375 15.9687 6-3413 2.40654 256 3.90625 6^36 16777216 16.0000 6.3496 2.40824 257 3.89105 16974593 16.0312 6-3579 2.40993 258 3-87597 66564 17173512 16.0624 6.3661 2.41162 259 3.86100 67081 17373979 16.093s 6-3743 2.41330 260 3.84615 67600 17576000 16.1245 6.3825 2.41497 261 3-83142 68121 17779581 16.1555 6.3907 2.41664 262 3.81679 68644 17984728 16.1864 6.3988 2.41830 263 3.80228 69169 18191447 16.2173 6.4070 2.41996 264 3.78788 69696 18399744 16.2481 6.4151 242160 265 3-77358 70225 18609625 16.2788 6.4232 2.42488 266 3-75940 70756 71289 18821096 16.3095 6.4312 267 3-74532 19034163 16.3401 6-4393 2.42651 268 3-73134 71824 19248832 16.3707 6-4473 2.42813 269 3-71747 72361 19465109 16.4012 6-4553 2.42975 270 3^70370 72900 19683000 16.4317 6-4633 2.43136 271 3.69004 73441 I99O25U 16.4621 6-4713 2.43297 272 3.67647 73984 20123648 16.4924 6.4792 2-43457 273 3.66300 74529 20346417 16.5227 6.4872 2.43616 274 3.64964 75076 20570824 16.5529 6.4951 2.43775 Smithsonian Tables. Table 3. VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. » 1000.^ «2 «8 */« v« log. n 275 276 277 278 279 3.63636 3.62319 3.61011 3-S97I2 3-58423 75625 76176 76729 77284 77841 20796875 21024576 21253933 21484952 21717639 16.5831 16.6132 16.6433 16.6733 16.7033 6.5030 6.5108 6.5187 6.5265 6-5343 2.43933 2.44091 2.44248 2.44404 2.44560 280 281 282 283 284 3-57143 3-55872 3-54610 3-53357 3-521 13 78400 78961 79524 80656 21952000 2218804I 22425768 22665187 22906304 16.7332 16.7631 16.7929 16.8226 16.8523 6.5421 6.5499 6.5577 6.5654 6-5731 2.44716 2.44871 2.45025 2.45179 2.45332 285 286 289 3-50877 3-49650 3-48432 3-47222 3.46021 81225 81796 82369 82944 83521 23149125 23393656 236399°3 23887872 24137569 16.8819 16.91 1 5 1 6.941 1 16.9706 17.0000 6.5808 6.5885 6.6039 6.61 1 5 2.45484 2.45637 2.45788 2.45939 2.46090 290 291 292 293 294 3-44828 3-43643 3.42466 3.41297 3-40136 84100 84681 85264 IS 24389000 246421 7 I 24897088 25153757 25412184 17.0294 17.0587 17.0880 17.1172 17.1464 6.6191 6.6267 6.6343 6.6419 6.6494 2.46240 2.46389 2.46538 246835 295 296 299 3-38983 3-37838 3.36700 3-35570 3-34448 87025 87616 88209 88804 89401 25672375 25934336 26198073 26463592 26730899 17.1756 17-2047 17-2337 17.2627 17.2916 6.6569 6.6644 6.6719 2.46982 2.47129 2.47276 2.47422 2.47567 300 301 302 303 304 3-33333 3.32226 3-31126 3-30033 3-28947 90000 90601 91204 91809 92416 27000000 27270901 27543608 27818127 28094464 17.3205 17-3494 17-3781 17.4069 17-4356 6.6943 6.7018 6.7092 6.7166 6.7240 2.47712 2.47857 2.48001 2.48144 2.48287 305 306 308 309 3.27869 3.26797 3-25733 3-24675 3-23625 93636 94249 94864 95481 28372625 28652616 28934443 29218112 29503629 17.4642 17.4929 17.5214 17.5499 17-5784 6.7313 6.7460 6.7606 2.48430 2.48572 2.48714 2.48855 2.48996 310 3" 312 313 314 3.22581 3-21543 3-20513 3.19489 3.18471 96100 96721 97344 97969 98596 29791000 30080231 30371328 30664297 30959144 17.6068 17.6352 17.6635 17.6918 17.7200 6.7679 6.7752 6.7824 6.7897 6.7969 2.49136 2.49276 2.49415 2.49554 2.49693 315 316 317 318 319 3.17460 3.16456 3-1 5457 3.14465 3.13480 99225 loff IOII24 IOI761 3125587s 31554496 31855013 32157432 32461759 17.7482 17.7764 17.8045 17.8326 17.8606 6.8041 6.8113 6.8185 6.8256 6.8328 2.49831 2.49969 2.50106 2.50243 2.50379 320 321 322 323 324 3.12500 3- 11527 3- 10559 3.09598 3.08642 102400 I 03041 103684 104329 104976 32768000 330761 61 33386248 33698267 34012224 17.8885 17.9165 17.9444 17.9722 18.0000 6.8399 6.8470 6.8541 6.8612 6.8683 2.50515 2.50651 2.50786 2.50920 2.5105s 325 326 329 3.07692 3.06748 3.05810 3-04878 3-03951 105625 106276 106929 107584 IO824I 34328125 34645976 34965783 35287552 3561 I 289 18.0278 18.0555 18.0831 18.1108 18.1384 6.8753 •6.8824 6.8894 6.8964 6.9034 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. n 1000.J «2 '■ i „8 ^/« J» log. ,t 330 331 332 333 334 3.03030 3.021 15 3.01205 3.00300 2.99401 108900 I 09561 I 10224 I 10889 111556 35937000 36264691 36594368 36926037 37259704 18.1659 18.1934 18.2209 18.2483 18.2757 6.9104 6-9174 6.9244 6.9313 6.9382 2.518a 2.51983 2.52114 2.52244 2-52375 335 336 339 2.98507 2.97619 2.96736 2,95858 2.94985 II2225 112896 "3569 114244 I 14921 37595375 37933056 38272753 38614472 38958219 18.3030 18.3303 18.3S76 18.3848 18.4120 6-9451 6.9521 6.9727 2.52504 2.52634 2.52763 2.52892 2.53020 340 341 342 343 344 2.941 18 2-93255 2.92398 2-91545 2.90698 1 1 5600 I16281 116964 117649 "8336 39304000 39651821 40001688 40353607 40707584 18.4391 18.4662 18.4932 18.5203 18.5472 6.9864 6.9932 7.0000 7.0068 2.53148 2-53275 2-53403 2.53529 2.53656 345 346 348 349 2.8985s 2.89017 2.88184 2.87356 2.86533 119025 119716 120409 1 21 104 121801 41063625 41421736 41781923 42144192 42508549 18.5742 18.6011 18.6279 18.6548 18.6815 7.0136 7.0203 7.0271 7-0338 7.0406 2.53782 2.53908 2-54033 2.54158 2.54283 350 351 352 353 354 2.85714 2.84900 2.84091 2.83286 2.82486 122500 123201 123904 124609 I25316 42875000 43243551 43614208 43986977 44361864 18.7083 18.7350 18.7617 18.7883 18.8149 7-0473 7.0540 7.0607 7.0674 7.0740 2.54407 2-54531 2-54654 2-54777 2.54900 355 356 357 358 359 2.81690 2.80899 2.801 1 2 2-7933° 2.78552 126025 126736 127449 128164 128881 44738875 451 18016 45499293 45882712 46268279 18.8414 18.8680 18.8944 18.9209 18.9473 7.0807 7-0873 7.0940 7.1006 7.1072 2.55023 2-55145 2.55267 2.55388 2.55509 360 361 362 363 304 2.77778 2.77008 2.76243 2.75482 2-74725 129600 130321 131044 131769 132496 46656000 47045881 47437928 47832147 48228544 18.9737 19.0000 19.0263 19.0526 19.0788 7-1 138 7.1204 7.1269 7-1335 7.1400 2.55630 2-55751 2-55871 2.55991 2.56110 365 366 367 368 369 2-73973 2.73224 2.72480 2.71739 2.71003 133225 133956 134689 135424 I3616I 48627125 49027896 49430863 49836032 50243409 19.1050 19.1311 19-1572 19-1833 19.2094 7.1466 7-1531 7.1726 2.56229 2.56348 2.56467 2.5658s 2.56703 370 371 372 373 374 2.70270 2.69542 2.68817 2.68097 2.67380 136900 '37641 138384 I39I29 139876 50653000 51064811 51478848 5i895"7 52313624 19-2354 19.2614 19.2873 19.3132 19-3391 7.1791 7.1855 7.1920 7-1984 7.2048 2.56820 2-56937 2-57054 2.57171 2.57287 375 376 377 378 379 2.66667 2.65957 2.65252 2.64«o 2.63853 140625 141 376 142129 142884 143641 52734375 53157376 53582633 54010152 54439939 19.3649 19.3907 19.4165 19.4422 19.4679 7.2112 7.2177 7.2240 7.2304 7.2368 2-57403 2-57519 2-57634 2-57749 2.57864 380 382 3f3 384 2.63158 2.62467 2.61780 2.61097 2.60417 144400 I4516I 145924 146689 147456 54872000 55306341 5p'i87 56623104 19.4936 19-5192 19.5448 19.5704 19-5959 7-2432 7.2495 7-2558 7.2622 7-2685 2.57978 2.58092 2.58206 2.58320 2-58433 Smithsonian Tables. 10 Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. n 1000.^ »a «8 V» V log. « 385 2.59740 148225 57066625 ig.6214 7.2748 2.58546 386 2.59067 148996 57512456 19.6469 7.2811 2.58659 387 2.58398 149769 57960603 19.6723 7.2874 ^■5?77i 38S 2-57732 150544 58411072 19-6977 7-2936 2.58883 389 2.57069 151321 58863869 19.7231 7.2999 2.58995 390 2.56410 1 52100 59319000 19.7484 7-3061 2.59106 391 2-55754 152881 59776471 19-7737 7-3124 2.59218 392 2.55102 153664 60236288 19.7990 7.3186 2-59329 393 2-54453 154449 60698457 61 162984 19.8242 7-3248 2-59439 394 2-53807 155236 19.8494 7-33^0 2-59550 39S 2.53165 156025 6162987s 19.8746 7-3372 2.59660 396 2-52525 1 56816 62099136 19.8997 7-3434 2.59770 397 2.51889 157609 62570773 19.9249 7-3496 2.59879 398 2.51256 158404 63044793 19.9499 7-3558 2.59988 399 2.50627 159201 63521 199 19-9750 7.3619 2.60097 400 2.50000 160000 64000000 20.0000 7-3681 2.60206 401 2-49377 160801 64481201 20.0250 7-3742 2.60314 402 2.48756 161604 64964808 20.0499 7-3803 2.60423 403 2.48139 162409 65450827 20.0749 7.3864 2.60531 404 2.47525 163216 65939264 20.0998 7-3925 2.60638 405 2.46914 164025 66430125 20.1246 7.3986 2.60746 406 2.46305 164836 66923416 20.1494 7-4047 2.60853 407 2.45700 165649 67419143 20.1742 7.4108 2.60959 408 2.45098 166464 67917312 20.1990 7.4169 2.61066 409 2-44499 167281 68417929 20.2237 7.4229 2.61172 410 2-43902 168100 68921000 20.2485 7.4290 2.61278 411 2-43309 168921 69426531 20.2731 7-4350 2,61384 412 2.42718 169744 69934528 20.2978 7.4410 2.61490 413 2.42131 170569 70444997 20.3224 7.4470 2.6159s 414 2.41546 171396 70957944 20.3470 7-4530 2.61700 415 2.40964 172225 7147337s 20.3715 7-4590 2.61805 416 2.40385 1738^9 71991296 20.3961 7.4650 2.61909 417 2.39808 72511713 20.4206. 7-4710 2.62014 418 2.39234 174724 73034632 20.4450 7.4770 2.62118 419 175561 73560059 20.4695 7-4829 2.62221 420 2.38095 176400 74088000 20.4939 7-4889 2-62325 421 2-37530 177241 74618461 20.5183 7-4948 2-62428 422 2.36967 178084 ^5^6^67 20.5426 7.5007 2.62531 423 2.36407 178929 20.5670 7.5067 2.62634 424 2.35849 179776 76225024 20.5913 7.5126 2-62737 425 2-35294 180625 76765625 20.6155 7.5185 2.62839 426 2-34742 181476 77308776 20.6398 7-5244 2.62941 427 2.34192 182329 77854483 20.6640 7-5302 2.63043 428 2.3364s 183184 78402752 78953589 20.6882 7-5361 2.63144 429 2.33100 184041 20.7123 7-542Q 2.63246 430 2.32558 184900 79507000 20.7364 7-5478 2-63347 431 2.32019 185761 80062991 20.7605 7-5537 2.63448 432 2.31481 186624 80621568 20.7846 7-5595 2.63548 433 2.30947- 187489 81182737 20.8087 7-5654 2.63649 434 2.30415 188356 81746504 20.8327 7-5712 2.63749 435 2.29885 189225 82312875 20.8567 7-5770 2.63849 436 z-29358 190096 82881856 20.8806 7-5!?^ 2-63949 437 2.28833 190969 83453453 20.9045 7.5886 2.64048 438 2.2831 1 191844 84027672 20.9284 7-5944 2.64147 439 2.27790 192721 84604519 20.9523 7.6001 2.64246 Smithsonian Tables. Table 3. VALUES OF .UES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, C ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS CUBE n 1000.^ «2 «» ^„ \n log. « 440 2.27273 193600 85184000 20.9762 7.6059 2.64345 441 2.26757 194481 85766121 21.0000 7.6117 2-64444 442 2.26244 195364 8635088S 21.0238 7.6174 2.64542 443 2.25734 196249 86938307 21.0476 7^62^9 2.64640 444 2.25225 197 136 87528384 21.0713 2.64738 445 2.24719 198025 88121125 88716536 21.0950 21.1187 7.6346 2.64836 446 2.24215 198916 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 91 I 25000 21.2132 7-6611 2.65321 451 2.21730 203401 91 733851 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 94196375 21.3307 7.6914 2.65801 456 2.19298 207936 208849 94818816 21.3542 7.6970 2.65896 ^^l 2.18818 95443993 21.3776 7.7026 2.65992 458 2.18341 2.17865 209764 96071912 21.4009 7.7082 2.66087 459 210681 96702579 21.4243 7.7138 2.66181 460 2.17391 211600 97336000 21.4476 7.7194 2.66276 461 2.16920 212521 97972181 21.4709 7.7250 2.66370 462 2.16450 2.15983 213444 9861 u 28 21.4942 7.7306 2.66464 2.66558 463 214369 99252847 21.5174 7.7362 464 2.15517 215296 99897344 21.5407 .7.7418 2.66652 465 2.15054 216225 100544625 21-5639 7.7473 2.66745 466 2.14592 217156 218089 101194696 21.5870 7.7529 2.66839 467 2-14133 I 01 847 563 21.6102 7.7584 2.66932 468 2.1367s 219024 102503232 21.6333 7-7639 2.67025 469 2.13220 219961 103161709 21.6564 7.7695 2.671 17 470 2.12766 . 220900 103823000 21.6795 7-7750 2.67210 471 2.12314 221841 1 04487 HI 21.7025 7-7805 2.67302 472 2.1 1864 232784 105154048 21.7256 21.7486 7.7860 ^%S 473 2.11416 223729 105823817 7-7915 474 2.10970 224677 106496424 21.7715 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 477 2.09644 227529 J0853I333 21.8403 7.8134 2.67852 478 2.09205 2.08768 228484 109215352 21.8632 7.8188 2.67943 479 229441 109902239 2I.886I 7-8243 2.68034 480 2.08333 230400 1 10592000 21.9089 7.8297 2.68124 481 2.07900 231361 1 1 1 284641 21.9317 7-8352 2.68215 482 2.07469 232324 111980168 21.9545 7.8406 2.6830s 483 2.07039 233289 1 1 2678587 21.9773 7-8460 2.6839s 484 2.06612 234256 "3379904 22.0000 7.8514 2.6848s 485 2.06186 235225 114084125 22.0227 7.8568 2.68574 486 2.05761 236196 114791256 22.0454 22.0681 7.8622 2.68664 ^^ 2-05339 237169 "5501303 7.8676 2.68753 488 2.04918 238144 116214272 22.0907 7-8730 2.68842 489 2.04499 2391 21 1 1 6930169 22.1133 7-8784 2.68931 490 2.04082 240100 1 17649000 22.13 w 22.1585 22.1811 7-8837 2.69020 491 2.03666 241081 I I 837077 I 7.8891 2.69108 492 2.03252 242064 1 19095488 7-8944 2.69197 2.69285 493 2.02840 243049 "9823157 120553784 22.2036 7-8998 494 2.02429 244036 22.2261 7-9051 2-69373 Smithsonian Tables. 12 Table 3. VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOCARITHhAS OF NATURAL NUMBERS. « 1000.^ «2 »» <» V« log. « 495 2.02020 245025 121287375 122023936 22.2486 7.9105 2.69461 496 2.01613 246016 22.2711 7-9158 , 2.69548 497 2.01 207 247009 122763473 22.2935 7.9211 2.69636 498 2.00803 248004 123505992 22.3IW 22-3383 7.9264 2.69723 499 2.00401 249001 124251499 7-9317 2.69810 500 2.00000 250000 125000000 22.3607 7-9370 2.69897 501 1. 99601 251OOI 125751501 22.3830 7.9420 2.69984 502 1.99203 252004 126506008 22.4054 7-9476 2.70070 S°3 1.98807 253009 127263527 22.4277 7.9528 2.70157 504 1.98413 254016 128024064 22.4499 7.9581 2.70243 505 1.98020 255025 128787625 22.4722 7-9634 2.70329 S06 1.97628 256036 129554216 22.4944 7.9686 2.70415 5°7 1.97239 257049 130323843 22.5167 7-9739 2.70501 508 1.96850 258064 I3IO965I2 22.5389 22.5610 7-9791 2.70586 509 1.96464 259081 131872229 7-9843 2.70672 510 1.96078 260100 132651000 22.5832 7.9896 2-70757 5" 1.95695 261 1 21 133432831 22.6053 7.9948 2.70842 512 1.95312 262144 134217728 22.6274 8.0000 2.70927 513 1.94932 263169 175005697 22.6495 8.0052 2.71012 514 1-94553 264196 105796744 22.6716 8.0104 2.71096 515 1.94175 1.93798 265225 136590875 22.6936 8.0156 2.71181 S16 266256 137388096 22.7156 8.0208 2.71265 5'? 1-93424 267289 138188413 22.7376 8.0260 2.71349 518 1.93050 268324 138991832 22.7596 8.0311 2-71433 S19 1.92678 269361 139798359 22.7816 8.0363 2-71517 520 1.92308 270400 140608000 22.8035 8.0415 2.71600 S2I 1.91939 271441 14I420761 22.8254 8.0466 2.71684 522 1.91571 272484 142236648 22.8473 8.0517 2.71767 523 1.91205 273529 143055667 22.8692 8.0569 2.71850 524 1.90840 274576 143877824 22.8910 8.0620 2.71933 525 1.90476 275625 144703125 22.9129 8.0671 2.72016 526 1.90114 145531576 22.9347 8.0723 2.72099 527 1-89753 277729 146363183 22.9565 8.0774 2.72181 528 1.89394 278784 147197952 148035889 22.9783 8.0825 2.72263 529 1.89036 279841 23.OOUO 8.0876 2.72346 530 1.88679 280900 148877000 23.0217 8.0927 2.72428 531 1.88324 281961 I4972I29I 23-0434 8.0978 2.72509 S32 1.87970 283024 150568768 23.0651 8.1028 2.72591 533 1.87617 284089 I5I4I9437 23.0868 8.1079 2.72673 534 1.87266 285156 152273304 23.1084 8.1130 2.72754 535 1.86916 286225 I53I30375 23.1301 8.1 180 2.72835 536 1.86567 287296 153990656 23-1517 8.1231 2.72916 537 1.86220 288369 154854153 23-1733 8.1281 2.72997 538 1.85874 289444 155720872 23.1948 ^•'332 2.73078 539 1.85529 290521 156590819 23.2164 8.1382 2.73159 540 1.85185 291600 157464000 23-2379 §•'^33 2-73239 541 1.84843 292681 1 5834042 1 23.2594 23.2809 8.1483 2.73320 542 1.84502 293764 159220088 §•'533 2.73400 543 1.84162 294849 160103007 23.3024 8.1583 2.73480 544 1.83824 295936 160989184 23.3238 8.1633 2.73560 545 1.83486 297025 161878625 23-3452 8.1683 2.73640 546 1.83150 298116 162771336 23.3666 8.1733 2.73719 547 1.82815 299209 163667323 23.3880 f-'r3 2-73799 548 1.82482 300304 164566592 23.4094 §•'§33 2.73878 549 1.82149 301401 165469149 23-4307 8.1882 2.73957 Smithsonian Tables, 13 Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. K looo.i- «2 »8 vf« v« log. « 550 S5I 552 553 554 i.8i8i8 1.81488 1.81159 1.80832 1.80505 302500 303601 304704 305809 306916 166375000 167284I5I 168196608 1 691 1 2377 170031464 23.4521 23-4734 23-4947 23.5160 23-5372 8.1932 8.1982 8.2031 8.2081 8.2130 2.74036 2.741 IS 2.74194 2.74273 274351 555 559 1.80180 1.79856 1-79533 1.79211 1.78891 308025 309136 310249 31 1364 31 2481 170953875 I71879616 172808693 I7374III2 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 560 562 564 17857 1 1.78253 177936 1.77620 177305 313600 314721 316969 318096 175616000 176558481 177504328 178453547 179406144 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 275051 2.75128 565 566 - 569 1.76991 1.76678 1.76367 1.76056 175747 319225 320356 321489 322624 323761 180362 I 25 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 2755" 570 571 572 573 574 175439 175131 1.74825 1.74520 1.74216 324900 326041 327184 328329 329476 185193000 186169411 187 149248 188132517 1891 19224 23.8747 23-8956 23.9165 23-9374 23-9583 8.2913 8.2962 8.3010 8.3059 8.3107 2.75587 2.75664 2.75740 2.75815 2.75891 575 576 5" 578 579 1 7391 3 1. 7 361 1 173310 1.73010 1.72712 330625 331776 332929 334084 335241 I 9010937 5 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 2.761 18 276193 2.76268 580 581 582 583 584 1.72414 1.72117 1.71821 1.71527 17 1 233 336400 337561 338724 339889 341056 195112000 1961 22941 1 97 1 37368 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 f 589 1.70940 1.70648 1.70358 1.70068 1.69779 342225 343396 344569 345744 346921 200201625 201 230056 202262003 203297472 204336469 24.1868 24.2074 24.2281 24.2487 24.2693 8-3730 8-3777 8.3825 2.76716 2.76790 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 2.7708s 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 21 1708736 212776173 213847192 214921799 24.3926 24-4131 24-4336 24-4540 24-4745 8.4108 8-4155 8.4202 8.4249 8.4296 277452 2.77525 2.77597 2.77670 277743 600 601 602 603 604 1.66667 1.66389 1.66113 '■65837 1-65563 360000 361201 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. n 1000.^ „2 «8 in %n log. n 605 606 607 608 609 1.65289 1.6 5017 1.6474s 1.64474 1.64204 366025 367236 368449 369664 370881 22I445125 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 ^§^6 1-63399 1-63132 1.62866 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 2.78604 2.78675, 2.78746 2.78817 615 616 617 618 619 1.62602 1-62338 1.62075 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 388129 389376 238328000 239483061 240641848 241804367 242970624 24.8998 24.9199 24.9399 24.9600 24.9800 8.5270 8.5316 ^•536? 8.5408 8-5453 2-79239 2.79309 2-79379 2.79449 2.79518 625 626 627 628 629 1.60000 1-59744 1.59490 1.59236 1.58983 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 2-79934 2.79657 2.79727 2.79796 2.79865 630 632 633 634 1.58730 1.58479 1.58228 1.57978 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 2.80003 2.80072 2.80140 2.80209 635 636 638 639 1.57480 1-57233 1.56986 1.56740 1.56495 403225 404496 405769 407044 408321 256047875 257259456 258474853 259694072 260917119 25.1992 25.2190 25-2389 25.2587 25.2784 8-5952 8.5997 8.6132 2.80277 2.80346 2.80414 2.80482 2.80550 640 641 642 643 644 1.56250 1.56006 1-55763 I-555ZI 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 i-SS°39 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 2.8iogo 2.81158 2.81224 650 652 653 654 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 2.81291 2.81358 2.81425 2.81491 2.81558 655 656 658 659 1.52672 1.52439 1.52207 1.51976 1-S1745 429025 430336 431649 432964 434281 281011375 282300416 283593393 284890312 286191I79 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 COAAMON LOGARITHMS OF NATURAL NUMBERS. n innn.^ n^ »» in \n log. « 660 66i 662 663 664 1.51286 1-51057 1.50830 1.50602 435600 436921 438244 439569 440890 287496000 288804781 2901 17528 291434247 292754944 25.6905 25.7099 25.7294 25.7488 25.7682 8.7066 8.7110 ?-7'54 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 149477 44222s 4448^9 446224 447561 294079625 295408296 296740963 298077632 299418309 25.7876 25.8070 25.8263 25-8457 25.8650 8.728s 8.7329 8.7373 8.7416 8.7460 2.82282 2.82347 2.82413 2.82478 2-82543 670 671 672 674 1.49254 1.49031 1.48810 1.48588 1.48368 448900 450241 451584 452929 454276 300763000 302111711 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 461041 30754687s 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 2.8299s 2.83059 2.83123 2.83187 680 681 682 683 684 1-47059 1.46843 1.46628 1.46413 1.46199 462400 463761 467856 314432000 315821241 317214568 318611987 320013504 26.0768 26.0960 26.1151 26.1343 26.1534 8.8023 8.8066 8.8io8 2.83251 2-83315 2.8337S 2.83442 2.83506 685 686 688 689 1.45985 1-45773 1.45560 1-45349 1. 45 1 38 469225 470596 471969 473344 474721 321419125 322828856 324242703 325660672 327082769 26.1725 26.1910 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 690 691 693 694 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 IS 2.8388c 2.83948 2.8401 1 2.84073 2.84^6 695 696 699 1.43881 1.43678 1.43472 1.43266 1.43062 483025 484416 485809 488601 335702375 ■ ^3«3 340068392 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 2.84323 2.84386 2.84448 700 701 702 703 704 1.42857 1.42653 1.42450 1.42248 1.4204s 490000 491401 492804 494209 495616 343000000 344472101 345948408 347428927 348913664 26.4575 26.4764 26.4953 26.5141 26.5330 8.8790 8.8833 8.887 s 8.8917 8.8959 2.84510 2.84572 2.84757 705 706 708 709 1.41844 1.41643 1.41443 1. 41 243 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 8.9085 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 506944 508369 509796 357911000 359425431 360944128 362467097 363994344 26.6458 26.6646 26.6833 26.7021 26.7208 8,9211 8.9253 8.929s 2.85126 2.85187 2.85248 2.85309 2.85370 Smithsonian Tables. 16 Table 3> VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. n loop.'- «2 «8 vf« \n log. « 715 716 717 718 719 1.39860 1.39665 1.39470 1.39276 1.39082 511225 512650 514089 3fS5f5875 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 8.9711 8.9752 8.9794 2-85733 2.85794 2.85854 2.85914 2.85974 725 726 729 I-3793I I-3774I 1-37552 1-37363 1-37174 525625 527076 528529 529984 531441 381078125 382657176 384240583 385828352 387420489 26.9258 26.9444 26.9629 26.9815 27.0000 8.9876 8.9918 8.9959 9.0000 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 39061 7891 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 735 736 739 1.36054 1-35870 1.3568s I-35SOI 1-35318 540225 541696 543169 544644 546121 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 -3477 1 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" 555025 556516 558009 559504 561001 41349362s 41 5160936 416832723 418508992 420189749 27.2947 27.3130 27-3313 27.3496 27.3679 9.0654 9.0694 9-0735 9.0816 2.87216 2.87274 2.87332 2.87390 2.87448 750 75' 752 7S3 7S4 1-33333 1-33156 1.32979 1.32802 1.32626 562500 564001 565504 567009 568516 421875000 423564751 425259008 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 755 756 759 1.32450 1.32275 1.32100 1.31926 1.31752 570025 571536 573049 574564 576081 430368875 432081216 433798093 435519512 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 579121 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 2.88195 2.88252 2.88309 765 766 769 1.30719 1.30548 1.30378 1.30208 1.30039 5882^9 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.^ »2 k5 in ^ log. « 770 771 772 773 774 1.29870 1.29702 1.29534 1.29366 1. 29199 592900 594441 595984 597529 599076 456533000 4583I4OII 460099648 4618899I7 463684824 27.7489 27.7669 27.7849 27.8029 27.8209 9-1657 9.1696 9.1736 9-1775 9.1815 2.88649 2.88705 2.88762 2.88818 2.88874 773 776 777 778 779 1.28700 1-28535 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 780 782 783 784 1.28205 1. 28041 1.27877 1-27714 1-27551 608400 609961 61 1 524 613089 614656 474552000 476379541 4782II768 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 785 786 787 788 789 1.27389 1.27226 1.27063 1.26904 1.26743 616225 617796 619369 620944 622521 483736625 485587656 487443403 489303872 491 169069 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^ 2.89708 790 791 792 793 794 1.26582 1.26422 1.26263 1. 26103 1.25945 624100 625681 627264 628849 630436 493039000 49491 367 1 496793088 4986772W 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 795 796 799 1.25786 1.25628 1.25471 1-25313 1.25156 632025 633616 635209 636804 638401 502459875 504358336 506261573 508169592 510082399 28.1957 28.213s 28.2312 28.2489 28.2666 9.2638 9.2677 9.2716 9-2754 9-2793 2.90037 2.90091 2.90146 2.90200 2.90255 800 801 802 803 804 1.25000 ;:2f6^ 1-24533 1.24378 640000 641601 643204 644809 646416 512000000 5I39224OI 5I7781627 5I9718464 28.2843 28.3019 28.3196 28.3373 28.3549 9.2832 9.2870 9.2909 9.2948 9.2986 2.90309 2.90363 2.90417 2.90472 2.90526 805 806 807 808 809 1.24224 1.24069 I.239I6 1.23762 1.23609 648025 649636 652864 654481 521660I25 523606616 525557943 527514II2 529475129 28.3725 28.3901 28.4077 28.4253 28.4429 9-3025 9.3063 9.3102 9.3140 9-3179 2.90580 2.90687 2.90741 2.90795 810 811 812 814 1-23457 1.23305 I-23I53 1. 23001 1.22850 656100 657721 659344 660969 662596 53I44IOOO 53341 I73I 535387328 537367797 539353144 28.4605 28.4781 28.4956 28.5132 28.5307 9.3217 9-3255 9-3294 93332 9-337° 2.90849 2.90902 2.90956 2.91009 2.91062 815 816 817 818 819 1.22699 1.22549 1.22399 1.22249 I.22IO0 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 2.91 1 69 2.91222 2.91275 2.91328 820 821 822 823 824 1.21951 1. 21803 I.21655 I. 21 507 I-2I359 672400 577329 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 2.91 38 1 2.91434 2.91487 2.9:540 2-91593 Smithsonian Tables. iS Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. n 1000.^ «2 rfi ^„ =v/« log. n 825 826 827 828 829 I.2I2I2 1.21065 1. 20919 1.20773 1.20627 680625 682276 683929 685584 687241 561 51 5625 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 I.204S2 1.20337 1.20192 1.20048 I.19904 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 f37 838 839 1.19760 I.I9617 1. 19474 I.I9332 I.1919O 700569 702244 703921 582182875 584277056 586376253 588480472 590589719 28.8964 28.9137 28.9310 28.9482 28.9655 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 I.18906 1.18765 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 1% 849 I.I8343 1. 1 8203 1.18064 1.17925 1.17786 714025 715716 717409 719104 720801 603351125 605495736 607645423 609800192 611960049 29.0689 29.0861 29.1033 29.1204 29.1376 9-4541 9.4578 9-461 S 9.4652 9.4690 2.92686 2.92737 2.92788 2.92840 2.92891 850 852 fS3 8S4 1.17647 1.17509 1.17371 1-17233 1.17096 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 2.9309s 2.93146 855 856 1% 859 1.16959 1.16822 1.16686 1.16550 1.16414 731025 732736 734449 736164 737881 62502637s 627222016 629422793 631628712 633839779 29.2404 29.257s 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 1.16144 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 865 866 869 1.15607 1-15473 1-15340 I.I 5207 1.1507s 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 f7' 872 873 874 1.14943 1.14811 1.14679 1.14548 1.14416 756900 758641 760384 762129 763876 658503000 66077631 I 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 2.941 51 875 876 In 878 879 1.14286 1-14155 1.14025 765625 767376 769129 770884 772641 66992187s 672221376 674526133 676836152 679151439 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, C ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS CUBE n 1000.^ «« «' i» \n log. « 880 88i 882 883 884 1. 1 3636 1-13507 I-I3379 1.13250 1. 13122 774400 776161 777924 779689 781456 681472000 68iif8968 688465387 690807104 29.6648 29.6816 29.6985 297153 29.7321 9.5828 9.5865 9.5901 9-5937 9-5973 2-94448 2.94498 2.94547 2.94596 2.94645 885 886 887 888 889 1. 12740 1.12613 1.12486 783221 786769 788544 790321 69315412s 695506456 697864103 700227072 702595369 29.7489 29.7658 29.7825 297993 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 29.8831 29.8998 9.6190 9.6226 9.6262 9.6298 96334 2-94939 2.94988 2.95036 2.9508s 2.95134 895 896 897 898 899 1.11732 1.11607 1.11483 i-"359 1.11235 801025 802816 804609 806404 808201 716917375 7>9323i36 721734273 724150792 726572699 29.9166 299333 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 1.1086s 1.10742 1.10619 Sioooo 811801 813604 815409 817216 729000000 731432701 733870808 736314327 738763264 30.0000 30.0167 30-0333 30.0500 30.0666 96549 9.6585 9.6620 9.6656 9.6692 2.95424 2.95472 2.95521 3.95569 2.95617 905 906 907 909 1.10497 '•I037S 1.10254 1.10132 I.IOOII 819025 826836 822649 824464 826281 741217625 743677416 746142643 74861 331 2 751089429 30.0832 30.0998 30.1164 30-1330 30.1496 9.6727 9-6763 9-6799 9.^34 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 1.09170 1.09051 1.08932 1.08814 837225 839056 842724 844561 76606087s 768575296 771095213 773620632 776151559 ■ 30-2490 30.265s 30.2820 30.2985 30-3150 9.7082 9.7118 9-7153 9.7188 9.7224 2.96142 2.96190 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 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 1.0787s 1.07759 1.07643 855625 857476 859329 861 184 863041 791453125 794022776 796597983 30.4138 30.4302 30.4467 30.4631 30-4795 9-7435 9-7470 9-7505 9.7540 97575 2.96614 2.96661 2.96708 2196802 930 931 932 933 934 1.07527 1.07411 1.07296 1.07181 1.07066 868624 870489 872356 804357000 806954491 809557568 812166237 814780504 30-4959 30.5123 30.5287 30-5450 30.5614 9.7610 9.7645 9.7680 9-7715 9-7750 2.96848 2.9689s 2.9703s Smithsonian Tabues. Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS. 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 . 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 9S8 989 1.06952 1.06838 1.06724 1.06610 1.06496 1.06383 1.06270 1. 06 1 57 1.0604s 1.05932 1.05820 1.05708 I-OS597 1.05485 1-05374 1.05263 1.05152 1.05042 1.04932 1.04822 1. 047 1 2 1.04603 1.04493 1.04384 1.04275 1. 041 67 1.04058 1.03950 1.03842 1-03734 1.03627 1.03520 I -0341 3 1.03306 1.03199 1.03093 1.02987 1. 0288 1 1.02775 1.02669 1.02564 1.02459 1.02354 1.02249 1.02145 1. 02041 1.01937 1.01833 1.01729 1.01626 1-01523 1.01420 1.01317 1.01215 I.0I1I2 874225 876096 877969 879844 88172I 883600 887364 889249 891136 893025 894916 896809 898704 900601 902500 904401 906304 908209 910116 912025 913936 915849 917764 919681 921600 923521 925444 927369 929296 931225 933156 935089 937024 938961 940900 942841 944784 946729 948676 950625 952576 954529 956484 958441 960400 962361 964324 966289 968256 970225 972196 974169 976144 978121 817400375 820025856 822656953 825293672 827936019 830584000 833237621 835896888 838561807 841232384 843908625 846590536 849278123 851971392 854670349 857375000 860085351 86280140S 865523177 868250664 870983875 873722816 876467493 879217912 881974079 884736000 887503681 890277128 893056347 895841344 898632125 901428696 904231063 907039232 909B53209 91 2673000 915498611 918330048 92I167317 924010424 926859375 929714176 932574833 935441352 938313739 941192000 94407614I 946966168 949862087 952763904 955671625 958585256 961 504803 964430272 967361669 \jn 30.5778 30-5941 30.6105 30.6268 30.6431 30.6594 30-6757 30.6920 30.7083 30.7246 30-7409 30-7571 30-7734 30.7896 30.8058 30.8221 30.8383 30.8545 30.8707 30.8869 30.9031 30.9192 30-9354 30.9516 30.9677 30-9839 31.0000 31.0161 31.0322 31.0483 31.0644 31.0805 31.0966 31.1127 31.1288 31.1448 31.1609 31.1769 31.1929 31.2090 31.2250 31.2410 31.2570 31.2730 31.2890 31-3050 31.3209 31-3369 3'-35^8 31,3688 31-3847 31.4006 31.4166 31-4325 31.4484 V« 9-7785 9.7819 9-7854 9.7889 9.7924 9-7959 9-7993 9.8028 9.8063 9.8097 9.8132 9.8167 9.8201 g.8236 9.8270 9.8305 9-8339 9-8374 9.8408 9.8443 9.8477 9.8511 9.8546 9.8580 9.8614 9.8683 9-8717 9.8751 9,8785 9.8819 9.8854 9.8888 g.8922 9.8956 9.8990 9.9024 9,9058 9.9092 9.9126 9.9160 9.9194 9.9227 9.9261 9.9295 9.9329 9-9363 9.9396 9.9430 9.9464 9.9497 9-9531 9.9565 9.9598 9.9632 log. « 2.97081 2.97128 2.97174 2.97220 2.97267 2-97313 2-97359 2.97405 2.97451 2.97497 2.97543 2.97589 2.97635 2.97681 2.97727 2.97772 2.97818 2.97864 2.97909 2-97955 2.98000 2.98046 2.98091 2.98137 2.98182 2.98227 2.98272 2.98318 Z.98363 2.98408 2.98453 2.98498 2.98543 2.( " 2.< 2.98677 2.98722 2.98767 2,98811 2.98856 2.98900 2.98945 2.98989 2.99034 2.99078 2.99123 2.99167 2.9921 1 2.99255 2.99300 2.99344 2.99388 2.99432 2.99476 2.99520 SwiTHSOMiAN Tables. 21 Table 3. VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, C ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS CUBE » iooo.i- »2 „8 v« \» log. ft 990 991 992 993 994 995 996 997 998 999 1000 I.OIOIO 1.00705 1.00604 1.00503 1.00402 1.00301 1.00200 I.OOIOO I. ooooo 980100 982081 984064 986049 988036 990025 992016 994009 998001 1000000 970299000 973242271 976191488 979146657 982107784 985074875 988047936 991026973 99401 1 992 997002999 1 000000000 31-4643 31.4802 31.4960 3I-5II9 31.5278 31-5436 31-5595 31-5753 31-59" 31.6070 31.6228 9.9666 9.9699 9-9733 9.9766 9.9800 & 9.9900 9-9933 9.9967 10.0000 2.99564 2.99651 2.99695 2.99739 2.99782 2.99826 2.99870 2.99913 2.99957 3.00000 Smithsonian Tables. 22 CIRCUMFERENCE AND AREA OF CIRCLE DIAMETER d. IN Table 4 TERMS OF ltd v d ijtd^ 10 II 12 13 IS 16 17 19 20 21 22 23 24 26 27 28 29 3° 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 81.681 84.823 87.965 91.106 94.248 97-389 100.53 103.67 106.81 109.96 113.10 116.24 119.38 122.52 78.5398 95-033: 113.097 132.732 153-938 176.715 201.062 226.980 254.469 2S3.529 314-159 346.361 380.133 415.476 452-389 490.874 530.929 572-555 615.752 660.520 706.858 754.768 804.248 855.299 907.920 962.113 1017.88 1075.21 1134.11 1194.59 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 56 57 58 59 60 61 62 63 64 65 66 67 68 69 125.66 128.81 131-95 135.09 138.23 141-37 144.51 147.65 150.80 153.94 157.08 160.22 163.36 166.50 169.65 172.79 175-93 179.07 182.21 188.50 191.64 194.78 197.92 201.06 204.20 207.35 210.49 213.63 216.77 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 ^3 84 85 86 87 90 91 92. 93 94 95 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 70S8.22 7238.23 7389.81 7542.96 7697-69 Smithsonian Tables. 23 Table 5. LOGARITHMS OF NUMBERS. N. 12 3 4 5 6 7 8 9 Prop. Parts. 12 3 4 t> b 7 8 9 10 II 12 13 14 0000 0043 0086 0128 0170 0414 0453 °492 0531 0569 0792 0828 0864 0899 0934 1139 1173 1206 1239 1271 1461 1492 1523 1553 1584 0212 0253 0294 0334 0374 0607 0645 0682 0719 0755 0969 1004 1038 1072 1106 1303 1335 1367 1399 1430 1614 1644 1673 1793 1732 4 8 12 4811 3 710 3 6 10 369 17 21 25 15 19 23 14 17 21 13 16 19 12 15 18 29 33 37 26 30 34 24 28 31 23 26 29 21 24 27 15 i6 11 19 1761 1790 1818 1847 1875 2041 2068 209s 2122 214S 2304 2330 2355 2380 2405 2553 2577 2601 2625 2648 2788 2810 2833 2856 2878 1903 1931 1959 1987 2014 2175 2201 2227 2253 2279 243° 2455 2480 2504 2529 2672 2695 2718 2742 2765 2900 2923 2945 2967 2989 368 3 5 8 2 5 7 257 247 11 II 10 9 9 14 13 12 12 II 15 14 13 20 22 25 18 21 24 17 20 22 16 19 21 16 18 20 20 21 22 23 24 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 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 246 246 246 246 245 8 8 8 7 7 II 10 10 9 9 13 12 12 II II 15 14 14 13 12 17 19 16 18 15 17 1517 14 16 25 26 2^ 29 3979 3997 4014 4031 4048 4150 4166 4183 4200 4216 4314 433° 4346 4362 4378 4472 4487 4502 4518 4533 4624 4639 4654 4669 4683 4065 4082 4099 41 I 6 4133 4232 4249 4265 4281 4298 4393 4409 4425 4440 4456 4548 4564 4579 4594 4609 4698 4713 4728 4742 4757 2 3 5 2 3 5 235 2 3 5 I 3 4 7 I 6 6 8 8 7 10 10 9 9 9 12 II II II 10 14 15 13 15 13 14 12 14 12 13 30 31 32 33 34 4771 4786 4800 4814 4829 4914 4928 4942 4955 4969 5051 5065 5079 5092 5105 5185 5198 5211 5224 5237 5315 5328 5340 5353 5366 4843 4857 4871 4S86 4900 4983 4997 50" 5024 5038 5119 5132 5145 5159 5172 5250 5263 5276 5289 5302 5378 5391 5403 5416 5428 I 3 4 I 3 4 I 3 4 I 3 4 I 3 4 6 6 5 5 5 7 7 I 6 8 8 8 10 10 9 9 9 II 13 II 12 II 12 10 12 10 11 35 36 39 5441 5453 5465 5478 5490 5563 5575 5587 5599 56" 5682 5694 5705 5717 5729 5798 5809 5821 5832 5843 591 1 5922 5933 5944 5955 5502 5514 5527 5539 5551 5623 5635 5647 5658 5670 5740 5752 5763 5775 5786 5855 5866 5877 5888 5899 5966 5977 5988 5999 6010 I 2 4 I 2 4 I 2 3 I 2 3 I 2 3 5 5 5 5 4 6 6 6 6 5 7 7 7 7 7 8 8 8 10 II 10 II 9 10 9 10 9 10 40 41 42 43 44 6021 6031 6042 6053 6064 6128 6138 6149 6160 6170 6232 6243 6253 6263 6274 6335 6345 635s 6365 6375 643s 6444 6454 6464 6474 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 I 2 3 I 2 3 I 2 3 I 2 3 I 2 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 8 7 7 7 7 9 10 11 45 46 47 48 49 6532 6542 6551 6561 6571 6628 6637 6646 6656 6665 6721 6730 6739 6749 6758 6812 6821 6830 6839 6848 6902 6gii 6920 6928 6937 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 I 2 3 I 2 3 I 2 3 I 2 3 I 2 3 4 4 4 4 4 5 5 5 4 4 6 6 5 5 5 7 I 6 6 8 9 7^^ 50 SI 52 53 54 6990 6998 7007 7016 7024 7076 7084 7093 7101 7110 7160 7168 7177 7185 7193 7243 7251 7259 7267 7275 7324 7332 7340 7348 7356 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 I 2 3 I 2 3 I 2 2 I 2 2 122 3 3 3 3 3 4 4 4 4 4 5 S 5 5 5 6 6 6 6 6 7 i 7 8 7 7 6 7 N. 12 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 Smithsonian Tables, 24 LOGARITHMS OF NUMBERS. Table 5. N. 12 3 4 5 6 7-8 9 Prop. Parts. 1 2 3 4 b 6 7 8 9 55 56 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 74^6 7474 7520 7528 7536 7543 7551 7597 7604 7612 7619 7627 7672 7679 7686 7694 7701 7745 7752 7760 7767 7774 I 2 I 2 I 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 S 4 4 5 5 5 5 5 6 7 6 7 6 7 60 61 62 64 7782 7789 7796 7803 7810 7853 7860 7868 787s 7882 7924 7931 7938 7945 7952 7993 8000 8007 8014 802I 8062 8069 807s 8082 8089 7818 7825 7832 7839 7846 7889 7896 7903 7910 7917 7959 7966 7973 798o 7987 8028 8035 8041 8048 8055 8096 8102 8109 81 16 8122 2 2 2 2 2 3 3 3 3 3 4 4 3 3 3 4 4 4 4 4 S 5 5 5 5 6 6 6 6 6 6 5 6 65 66 69 8129 8136 8142 8149 8156 8195 8202 8209 8215 8222 8261 8267 8274 8280 8287 §325 8331 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 2 2 2 2 2 3 3 3 3 2 3 3 3 3 3 4 4 4 4 4 5 5 5 4 4 5 ^ 5 6 70 71 72 73 74 8451 8457 8463 8470 8476 8513 8519 8525 8531 8537 8573 8579 8585 8591 8597 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 8675 8681 8686 8722 8727 8733 8739 874s 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 6 5 5 5 5 5 5 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 8915 8949 8954 8960 8965 8971 9004 9009 9015 9020 9025 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 4 5 4 5 4 5 80 81 82 84 9031 9036 9042 9047 9053 9085 9090 9096 9101 9106 9138 9143 9149 9154 9159 9191 9196 9201 9206 9212 9243 9248 9253 9258 9263 9058 9063 9069 9074 9079 9112 9117 9122 9128 9133 9165 9170 9175 9180 9186 9217 9222 9227 9232 9238 9269 9274 9279 9284 9289 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 5 4 5 4 5 4 5 4 5 85 86 87 88 89 9294 9299 9304 9309 9315 9345 9350 9355 9360 93^5 9395 9400 9405 9410 9415 9445 9450 9455 946o 9465 9494 9499 9504 9509 95 13 9320 9325 9330 9335 9340 9370 9375 9380 9385 9390 9420 9425 9430 9435 9440 9469 9474 9479 9484 9489 9518 9523 9528 9533 9538 I I 1 2 2 2 2 2 2 3 3 2 2 2 3 3 3 3 3 4 4 3 3 3 4 5 4 5 4 4 4 4 4 4 90 91 92 93 94 9542 9547 9552 9557,9562 9590 9595 9600 9605 9609 9638 9643 9647 9652 9657 9685 9689 9694 9699 9703 9731 9736 9741 9745 9750 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 I I I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 95 96 97 98 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 I I I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 3 4 N. 12 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Smithsonian Tables. 25 Table 6. ANTILOCARITHMS. L. 12 3 4 5 6 7 8 9 Prop. Parts. 8 9 1 2 U 4 5" " / .00 .01 .02 •03 .04 1000 1002 1005 1007 1009 1023 1026 1028 1030 1033 1047 1050 1052 1054 1057 1072 1074 1076 1079 1081 1096 1099 I 102 I 104 I 107 1012 Z014 1016 1019 1021 1035 1038 1040 1042 1045 1059 1062 1064 1067 1069 1084 1086 1089 1091 1094 1109 1112 1114 1117 1119 I ] 1 1 1 I I I I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 .05 .06 ■°l .08 .09 II22 II25 II27 II30 II32 II48 1151 II53 II56 II59 II75 II78 I180 I183 I186 1202 1205 1208 I2II I2I3 >Z30 1233 1236 1239 1242 1135 1138 1140 1143 1146 1161 1164 1167 1169 1172 1 189 1 191 1 194 1 197 1 199 i2i6 1219 1222 1225 1227 1245 1247 1250 1253 1256 I I I 1 I I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 .10 .11 .12 ■13 .14 1259 1262 1265 1268 I27I 1288 1 291 1294 1297 1300 I318 I32I 1324 1327 1330 1349 1352 I3S5 1358 1361 1380 1384 1387 1390 1393 1274 1276 1279 1282 1285 1303 1306 1309 1312 1315 1334 1337 1340 1343 1346 1365 1368 1371 1374 1377 1396 1400 1403 1406 1409 I I I I I I I I 2 I 2 I 2 I 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 2 3 3 3 3 3 .15 .16 ■17 .18 .19 1413 1416 1419 1422 1426 1445 1449 14W I4S5 1459 1479 1483 i486 1489 1493 1514 1517 1521 1524 1528 1549 1552 1556 1560 1563 1429 1432 1435 1439 1442 1462 1465 1469 1472 1476 1496 1500 1503 1507 1 510 1531 1535 1538 1542 IS4S 1567 1570 1574 1578 1 581 I I 1 I 1 J I 2 I 2 I 2 [ I 2 [ 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 .20 .21 .22 •23 •24 1585 1589 1592 1596 1600 1622 T626 1629 1633 1637 1660 1663 1667 1671 1675 1698 1702 1706 1710 1714 1738 1742 1746 1750 1754 1603 1607 161 I 1614 1618 1641 1644 1648 1652 1656 1679 1683 1687 1690 1694 1718 1722 1726 1730 1734 1758 1762 1766 1770 1774 I I I I 1 2 ( 2 2 2 2 2 2 .22 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 3 4 .25 .26 .27 .28 .29 1778 1782 1786 1791 1795 1820 1824 1828 1832 1837 1862 1866 187 1 1875 1879 1905 1910 1914 1919 1923 1950 1954 1959 1963 1968 1799 1803 1807 181 I 1816 1841 1845 1849 1854 1858 1884 1888 1892 1897 1901 1928 1932 1936 1941 1945 1972 1977 1982 1986 1991 I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 3 4 3 4 4 4 4 4 .30 •31 ■32 •33 •34 1995 2000 2004 2009 2014 2042 2046 2051 2056 2061 2089 2094 2099 2104 2109 2138 2143 2148 2153 2158 2188 2193 2198 2203 2208 2018 2023 2028 2032 2037 2065 2070 2075 2080 2084 2113 2n8 2123 2128 2133 2163 2168 2173 2178 2183 2213 2218 2223 2228 2234 I I I I 2 2 2 2 2 2 2 2 ! 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 S .35 •36 % •39 2239 2244 2249 2254 2259 2291 2296 2301 2307 2312- 2344 2350 2355 2360 2366 2399 2404 2410 2415 2421 2455 2460 2466 2472 2477 2265 2270 2275 2280 2286 2317 2323 2328 2373 2339 2371 2377 2382 2388 2393 2427 2432 2438 2443 2449 2483 2489 2495 25CX3 2506 ! 2 3 I 2 3 J 2 3 Z 2 3 I 2 3 3 3 3 3 3 4 4 4 4 4 4 5 4 S 4 S 4 S 5 S .40 .41 .42 •43 •44 2512 2518 2523 2529 2535 2570 2576 2582 2588 2594 2630 2636 2642 2649 2655 2692 2698 2704 2710 2716 2754 2761 2767 2773 2780 2541 2547 2553 2559 2564 2600 2606 2612 2618 2624 2661 2667 2673 2679 2685 2723 2729 2735 2742 2748 2786 2793 2799 2805 2812 ! 2 3 2 2 3 ! 2 3 2 3 3 2 3 3 4 4 4 4 4 4 4 4 4 4 S S .45 .46 •47 .48 •49 2818 2825 2831 2838 2844 2884 2891 2897 2904 29 I I 2951 2958 2965 2972 2979 3020 3027 3034 3041 3048 3090 3097 3105 3112 3119 2851 2858 2864 2871 2877 2917 2924 2931 2938 2944 2985 2992 2999 3006 3013 3055 3062 3069 3076 3083 3126 3133 3I4I 3148 3155 2 3 3 2 3 3 2 3 3 2 3 4 2 3 4 4 4 4 4 4 5 S 5 5 S 6 6 6 6 L. 12 3 4 5 6 7 8 9 1 2 : i 4 5 6 7 8 S Smithsonian Tables. 26 ANTILOCARITHAAS. Table 6. L. 12 3 4 5 6 7 8 9 1 2 Prop. Parts. 3 4 5 6 7 8 9 .50 •S" ■52 •S3 •54 3162 3170 3177 3184 3192 3236 3243 3251 3258 3266 33" 3319 3327 3334 3342 3308 3396 3404 3412 3420 3467 3475 3483 3491 3499 3199 3206 3214 3221 3228 3273 3281 3289 3296 3304 3350 3357 3365 3373 3381 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 5 6 7 5 ^ '' 1 ^ 7 6 6 7 6 6 7 .55 ■56 .58 •59 3548 3556 3565 3573 3581 3631 3639 3648 3656 3664 3715 3724 3733 3741 3750 3S02 3811 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 ^ 7 7 678 6 7 8 678 678 .60 .61 .62 .64 3981 3990 3999 4009 4018 4074 4083 4093 4102 41 I I 4169 4178 4188 4198 4207 4266 4276 4285 4295 4305 4365 4375 4385 4395 44o6 4027 4036 4046 4055 4064 4121 4130 4140 4150 4159 4217 4227 4236 4246 4256 4315 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 5 4 5 4 5 4 5 4 5 6 6 6 6 6 678 7 8 9 7 8 9 7 8 9 789 .65 .66 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 5000 I 2 I 2 I 2 I 2 I 2 3 3 3 3 3 4 5 4 5 4 5 1i 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 537° 5383 5395 54o8 5420 5495 5508 5521 5534 5546 5070 5082 5093 5105 51 17 5188 5200 5212 5224 5236 5309 5321 5331 5346 5358 5433 5445 5458 547° 5483 5559 5572 5585 5598 5610 I 2 I 2 I 2 I 3 1 3 4 4 4 4 4 5 6 7 7 8 8 9 II 8 10 II 9 10 II 9 10 II 9 10 12 .75 ■76 .78 79 5623 5636 5649 5662 5675 5754 5768 578l 5794 5808 5S88 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 • 3 I 3 I 3 I 3 I 3 4 4 4 4 4 5 7 5 7 5 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 IS .85 .86 .87 .88 .89 7079 7096 7112 7129 7145 7244 7261 7278 7295 731 1 7413 7430 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 7705 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 85H 8531 8551 8570 8590 8710 8730 8750 8770 8790 8035 8054 8072 8091 8iio 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 9 810 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 9:62 9183 9204 9333 9354 9376 9397 9419 9550 9572 9594 9616 9638 9772 9795 9817 9840 9863 9016 9036 9057 9078 9099 9226 9247 9268 9290 931 1 9441 9462 9484 9506 9528 9661 9683 9705 9727 9750 9886 9908 9931 9954 9977 2 4 2 4 2 4 2 4 2 5 6 6 7 7 7 8 10 12 811 13 9" 13 9 II 13 9 II 14 15 17 19 1517 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 Smithsoniam Tables. 27 Table 7. NATURAL SINES AND COSINES. Natural Sines. Prop. Angle. 0' 10' 20' 30' 4^ 50' 60' Angle. Parts lor 1'. 0° .0000 00 .0029 09 .0058 18 .0087 27 .011635 .0145 44 .0174 52 89° 2.9 I .017452 .0203 6 .0232 7 .0261 8 .0290 8 .03199 .03490 88 2.9 2 .0349 .0378 I .0407 I .0436 2 .0465 3 .0494 3 ■05234 87 2.9 3 .0523 4 .0552 4 .0581 4 .0610 5 •06395 .0668 5 .0697 6 86 2.9 4 .0697 6 .0726 6 •0755 6 .0784 6 .0813 6 .0842 6 .08716 85 2.9 5 .0871 6 .0900 5 .0929 5 .0958 5 .0987 4 .10164 •1045 3 84 2.9 6 •1045 3 .12187 .10742 .1103 I .1132 .11609 .11898 .12187 f3 2.9 ■ 7 .1247 6 .12764 •i3°5 3 .1478 •1334 •1363 ■1392 82 2.9 8 .1392 .1421 .1449 •iS°7 •1536 .1564 81 2.9 9 .1564 •'593 .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 2.9 2.8 12 .2079 .2108 .2136 .2164 .2193 .2221 .2250 77 13 .2250 .2278 .2306 •2334 •2363 .2391 •^^ng 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 'Z .2924 .2952 •2979 .3007 •3035 .3062 .3090 72 2.8 18 .3090 .3118 ■3145 •3173 .3201 .3228 .3256 71 2.8 19 .3256 •3283 •33" •3338 •3365 •3393 .3420 70 2.7 20 .3420 •3448 ■m •3502 •3529 •3557 ■3584 69 2.7 21 •3584 .3611 •3665 .3692 •3^0^ ■3746 68 2.7 22 •3746 •3773 .3800 •3827 •3854 •3881 •3907 P. 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 •43°5 •4331 •4358 •4384 64 2.6 26 •4384 .4410 •4436 .4462 .4488 •4514 .4540 63 2.6 27 ■4540 .4566 .4592 •4617 •4643 .4695 .4848 62 2.6 28 !4848 .4720 •4746 •4772 ■4797 •4823 61 2.6 29 .4874 .4899 .4924 .4950 •4975 .5000 60 2.5 30 .5000 .5025 .5050 •5075 .5100 .5125 •5150 59 2.5 31 .5150 •517s .5200 ■5225 .5250 ■5275 •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 35 ■5736 •5760 •5783 •5807 •5831 •5854 .5878 54 2.4 36 .5878 .5901 •5925 •5948 •5972 •5995 .6018 53 2-3 37 .6018 .6041 .6065 .6088 .6111 .6134 .6157 52 2-3 38 .6157 .6180 .6202 .6225 .6248 .6271 .6293 51 2-3 39 .6293 .6316 ■6338 .6361 •6383 .6406 .6423 50 2-3 . ; 40 .6428 .6450 ■6583 .6472 .6494 .6517 •6539 .6561 49 2.2 ' 41 ..6561 .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 SO' 50' 40' 30' 20' 10' c Angle. Smithsonian Tables. Natural Cosines. 28 Table 7. NATURAL SINES AND COSINES. Natural Sines. Prop. Angle. 0' 10' 20' 30' 40' 50' 60' Angle. Parts !or 1'. 45° .7071 .7092 .7112 ■7133 •7153 ■7173 •7193 44° 2.0 46 •7193 •7214 ■7234 ■7254 .7274 .7294 ■7314 43 2.0 47 •7314 ■7333 ■7353 ■7373 ■7392 .7412 •7431 42 2.0 48 •7431 •7451 .7470 .7490 .7509 •7528 ■7547 41 1.9 49 •7547 •7566 ■7585 .7604 .7623 .7642 .7660 40 1.9 50 .7660 .7679 .7698 .7716 ■7735 •7753 ■7771 39 1.9 5' .7771 .7790 .7S08 .7826 .7844 .7862 .7880 38 1.8 52 .7880 .7898 .7916 ■7934 •7951 .7969 .7986 37 1.8 S3 .7986 .8004 .8021 .8039 .8056 .8073 .8090 36 '■7 54 .8090 .8107 .8124 .8141 .8158 •817s .8192 35 '■7 55 .8192 .8208 .8225 .8241 .8258 .8274 .8290 34 1.6 56 .8290 .8307 •8323 ■8339 ■8355 •8371 .8387 33 1.6 5? •^387 .8403 .8418 •8434 .8450 .8465 .8480 32 1.6 58 .8480 .8496 .8511 .8526 .8542 •8557 .8572 31 1^5 59 ■8572 .8587 .8601 .8616 ■8631 .8646 .8660 3° '■5 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 ^3 .8910 •8923 .8936 .8949 .8962 •8975 .89S8 26 13 64 .8988 .9001 .9013 .9026 •9038 .9051 .9063 25 '■3 65 .9063 .9075 .9088 .9100 .9112 .9124 •9135 24 1.2 66 •9135 .9147 •9159 .9171 .9182 .9194 .9205 23 1.2 67 .9205 .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 ■9537 •9546 •9555 •9563 17 0.9 73 •9563 ■9572 ■9588 .9596 ■9605 .9613 16 0.8 74 .9613 .9621 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 •975° •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 o^5 80 .9848 •9853 .9858 .9863 .9868 ■9872 .9877 9 0.5 Si .987,7 .98S1 .9886 .9890 .9894 .9899 ■9903 8 0.4 82 •9903 .9907 ■99" .9914 .9918 .9922 .9925 7 0.4 f^ •9925 .9929 ■9932 •9936 •9939 .9942 •9945 6 0-3 84 •9945 •9948 ■9951 •9954 ■9957 •9959 .9962 5 0^3 85 .9962 .9964 .9967 .9969 .9971 •9974 .9976 4 0.2 86 .9976 .9978 .9980 .9981 •9983 •9985 .9986 3 0.2 87 .9986 • .9988 ■9989 .9990 ■9992 •9993 ■9994 2 0.1 88 •9994 ■9995 .9996 •9997 ■9997 •999S .9998 I 0.1 89 .9998 ■9999 •9999 1. 0000 1. 0000 1. 0000 1. 0000 0.0 60' 50' 40' 30' 2(y 10' 0' Angle. Smithsonian Tables. Natural Cosines. 29 Table 8 NATURAL TANGENTS AND COTANGENTS. Natural Tangents. Angle. C 10' 20' 30' 40' 50' 60' Angle. Prop. FaitB for 1'. 0° .0000 .0029 I .0058 2 ■00873 .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 ?? 2.9 3 .0524 1 ■0553 3 .0582 4 .0611 6 .0640 8 .0670 .06993 86 2.9 4 .06993 .0728 5 ■07 57 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 ■"39 4 .11688 .11983 .12278 =3 2.9 •J .12278 .12574 .12869 .13165 .1346 •1376 .1405 82 3-0 8 .1405 •143s .1465 ■1495 .1524 ■1554 .1584 8i 3^0 9 .1584 .1614 .1644 ■1673 ■1703 ■1733 ■1763 80 3^0 10 •1763 ■1793 .1823 ■1853 .1883 .1914 .1944 79 3^0 II .1944 .1974 .2004 ■2035 .2065 .2095 .2126 78 3^0 12 .2126 .2156 .2186 .2217 .2247 .2278 ■2309 ''I 3-1 '3 .2309 ■2339 .2370 .2401 .2432 .2462 .2493 76 3-1 14 •2493 •2524 •2555 .2586 .2617 .2648 .2679 75 3^1 15 .2679 .2711 .2742 •2773 .2805 .2836 .2867 74 3^1 i6 .2867 .2899 .2931 .2962 .2994 .3026 ■3057 73 3.2 17 •3057 .3089 .3121 ■3153 ■3378 .3217 ■3249 72 3.2 i8 •3249 •3281 ■3314 •3346 ■34" ■3443 71 3^2 19 ■3443 ■3476 •3508 ■3541 ■3574 .3607 .3640 70 3-3 20 .3640 ■3673 .3706 ■3739 ■3772 •380s ■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 •424s ■4279 ■4314 •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-^ 26 •4877 ■4913 .4950 .4986 .5022 •5059 ■5095 ^3 3^6 27 •5095 ■5132 .5169 .5206 ■5243 .5280 ■5317 62 H 28 •5317 ■5354 ■5581 ■5392 ■5430 .5467 ■5505 ■5543 61 3-? 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 .6830 .6619 .6661 .6703 .6745 56 4.2 34 .6745 .6787 .6873 .6916 .6959 .7002 55 4.3 35 .7002 .7046 .7089 ■7133 .7177 .7221 .7265 54 4.4 36 ■7265 ■7310 ■7355 .7400 ■7445 .7490 •7536 ■ 53 4.5 37 ■7536 ■7813 .7^81 .7^60 .7627 ■7673 •7720 .7766 52 4.6 38 .7907 ■7954 .8002 .8050 5' 4.7 39 .8098 .8146 .8195 ■8243 .8292 ■8342 .8391 50 4.9 40 .8391 .8441 .8491 .8541 ■8591 .8642 ■8693 49 5.0 41 .8693 .8744 .8796 .8§47 .8S99 .8952 .9004 ■ 48 5-2 42 .9004 .9057 .9110 .9163 .9217 .9271 ■9325 47 5.4 43 ■9325 .9380 ■9435 .9490 %i'^ .9601 ■9657 46 5-5 44 ■9657 ■9713 .9770 .9827 .9884 .9942 1. 0000 45 5.7 60' 50' 40' 30' 20' 10' 0' Angle. Smithso viAN Tab LES. Natural Cotange nts. 30 Table 8. NATURAL TANGENTS AND COTANGENTS. Natural Tangents. Angle. 0' 10' 20' 30' 40' 50' 60' Angle. Prop. Parts for 1'. 45° 1. 0000 1.0058 1.0117 1.0176 1.0235 1.0295 1-0355 44° 5-9 46 I-0355 1.0416 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 1.1106 1.U71 1.1237 1-1303 1. 1369 1-1436 I.I 504 41 6.6 49 I.I 504 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. 3 190 1.3270 37 Z-9 S3 1.3270 1-3351 1-3432 1-3514 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 8.6 55 1.4281 1-4370 1.4460 1.4550 1. 4641 1-4733 J. 4826 34 9.1 56 1.4826 1.4919 1.5013 1. 5108 1.5204 1-5301 1-5399 33 9.6 57 1-5399 I-S497 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 1.8291 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 2.0057 2.0204 2-0353 2.0503 26 14.6 64 2.0503 2.0655 2.0809 2.0965 2.1123 2.1283 2.1445 25 15-7 65 2.1445 2.1609 2-1775 2.1943 2.2998 2.2113 2.2286 2.2460 24 16.9 66 2.2460 2.2637 2.2817 2-3183 2-3369 2-3559 23 18.3 67 2-3559 2.375° 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 2.6051 21 21.7 69 2.6051 2.6279 2.651 1 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 7^ 3.0777 3.1084 3-1397 3.1716 3.2041 3-2371 3-2709 '? 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 3-732' 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 4-3315 13 77 4-3315 4-3897 4.4494 4-5'07 4-5736 4.6382 4.7046 12 78 4.7046 4-7729 4-8430 4-9152 4-9894 5.0658 5-1446 II 79 5.1446 5-2257 5-3093 5-3955 5.4845 5-5764 5-6713 10 80 5-6713 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 7-1 154 8 82 7.1 154 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 1 1 .4301 5 85 1 1. 430 1 11.8262 12.2505 1 2.7062 13.1969 13-7267 14.3007 4 86 14.3007 14.9244 1 5.6048 16.3499 17.1693 18.0750 19.081 1 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 QD 60' 50' 40' 30' 20' 10' 0' Angle. Smithsonian Tables. Natural Cotangents. 31 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. 1 3 e 6 u 1 C ° ] 2 i Q ■3 a 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 3.00000 0.00000 2.99954 0.05235 0.069S0 2.99817 0.10470 3 4 4.00000 0.00000 3-99939 3-99756 0.13960 4 5.00000 0.00000 4.99923 0.08726 4-99695 0.17450 5 60 6 6.00000 0.00000 5.99908 0.1 047 1 S-99634 0.20940 6 7 7.00000 o.oocoo 6.99893 0.12216 6-99573 0.24430 7 8 8.00000 0.00000 7.99878 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 0.02 18 1 0.99922 0.03925 1 2 1.99998 0.00872 1.99952 0.04363 1.9984s 2.99768 0.07851 2 3 2.99997 0.01308 2.9992S 0.06544 0.1 1777 3 4 3.99995 0.01745 3.99904 0.08725 3.99691 0.15703 4 '5 4-99995 0.02 181 4.99881 0.10907 4.99614 0.19629 5 45 6 5-99994 0.02617 5-99857 0.13089 5-99537 0.23555 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 0.35333 9 1 0.99996 0.00872 0.99965 0.02617 0.99904 0.04361 1 2 1.99992 0.01745 1.99931 0.0523s 1.99809 0.08723' 2 3 0.02617 2.99897 0.07853 2.99714 0.13085 3 4 3.99984 0.03490 3.99862 0.10470 3.99619 0.17447 4 3° 4.99981 0.04363 4.99828 0.13088 4.99524 0.21809 30 6 5-99977 0.0523s 5-99794 0.15706 0.18323 5-99428 0.26171 6 7 6-99973 0.06108 6.99760 6.99333 0-30533 7 8 7.99969 0.069SI 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 °-99953 - 0.03053 0.99884 0.04797 1 2 1.99982 0.02617 2.99860 0.06107 1.99769 0-09595 2 3 2.99974 03926 0.09 1 61 2.99654 0-14393 3 4 3-99965 0.0523s 3-99813 0.12215 3-99539 0.19191 4 45 5 4-99957 0.06544 4-99766 0.15269 0.18323 4.99424 0.23989' 15 6 5-99948 0.07853 5.99720 5-99309 0.28786 6 7 6.99940 0.09162 6-99673 0.21376 6,99193 0.33584 7 8 7-99931 O.I 047 1 7.99626 0.24430 7.99078 0.38382 8 9 8.99922 O.I 1780 8.99580 0.27484 8.98963 0.43180 9 a w 3 n n g en o Dep. Lat. Dep. Lat. Dep. Lat. s B 1 8 9° 8 B° 8 70 Smithsonian Tables. 32 TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. Table 9,. -Continued. 1 5 3 4 5 ■s c 2 tn Q i c 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 0.13951 1.99238 0.17431 2 3 2.99589 0.15700 ^,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 0.41868 6.98294 0.48829 6.97336 0.61008 7 8 7.98904 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 0.1 1338 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 0.1 569 1 0.19169 2 3 2.99440 0.18314 2.99075 0-23537 2.98618 0.28753 3 4 3-99253 0.24419 3.98766 0-3I383 3-98158 0.38338 4 3° 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 0.1 656 1 1.98993 0.20037 2 3 2-99357 0.19620 2.98969 0.24842 2.98490 0.30056 3 4 3-99143 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 5.969S1 0.60112 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 g a Dep. Lat. Dep. Lat. Dep. Lat. S - 3' i 3 9 P 3 c 8 6° 8 5" 8 i° Smithsonian Tables. 33 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 3 6 u 6 7 8 u 1 Lat. Dep. Lat. Dep. Lat. Dep. 1 0.99452 0.10452 0.99254 0.12186 0.99026 0.13917 0.27834 1 2 0.20905 0-31358 1.98509 0-24373 1.980 a 2.97080 2 3 2.98356 2-97 763 0.36560 0.41751 3 4 3.97808 0.41811 3.97018 0.487^7 3.96107 0.55669 4 4.97261 0.52264 4.96273 0.60934 4-95134 0.69586 \ 60 6 5-96713 0.62717 5-95519 0.73121 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 1. 988 1 1 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 0.43546 3.96801 0.50479 3.95860 0-57397 4 15 S 4.97028 0.54433 4.96002 0.63099 4-94825 0.71746 5 45 6 5-96433 0.65320 5.95202 0.7019 0.88339 5-93790 0.86095 6 7 6-95839 0.76206 6.94403 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-I3S79 8.90686 1-29143 9 1 0-99357 0.1 1 320 0.99144 1.98288 0.13052 ■ 0.98901 0.14780 1 2 1.98714 0.22640 0.26105 1.97803 0.29561 2 3 2.9807 1 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 s 4.96786 0.56601 4.95722 0.65263 0.78315 0.91368 4.94508 0.73904 30 6 5.96143 0.67921 5.94866 5-93409 0.88685 6 7 6.95500 7-94857 0.79242 6.9401 1 6.92311 1.03466 7 8 0.90562 1. 01882 7-93155 1.04420 7.91212 1. 18247 8 9 8.94214 8.92300 1-17473 8.901 14 1.33028 9 1 0.99306 0.11753 0.99086 0.13485 0.98836 0.15212 1 2 1. 9861 3 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 s 4-96534 0.58768 4-95432 0.67425 4.94180 0.76061 IS 6 5-95841 0.70522 5-94519 0.80910 5.93016 0.91274 6 7 6.95147 0.82276 6.93606 0-94395 6.91853 7.90689 1.06486 7 8 7-94454 0.94029 7.92692 1.07880 1.21698 8 9 8.93761 1.05783 8.91779 1-21365 8.89525 1.36911 9 a Dep. Lat. Dep. Lat. Dep. Lat. % S' c 1 5- B 8 J° 8 2° a 1° Smithsonian Tables. 34 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. i a c u C3 .a Q 9° 10 ° 11 i a c Lat. Dep. Lat. Dep. Lat. Dep. 1 0.98768 0.15643 0.98480 0.17364 0.98162 0.19081 1 2 1-97537 0.31286 1. 96961 0.34729 1.9632s 2.94488 0.38162 2 3 2.96306 0.46930 2.95442 0.52094 0-57243 3 4 3-95°75 0.62573 3-93923 0.69459 3.92650 0.76324 4 4-93844 0.78217 4.92403 0.86824 4.90813 0.95405 5 60 6 5.92612 0.93860 5.90884 1.04188 5.88976 1.14486 6 7 6.91381 1.09504 6.89365 1-21553 6.87139 1.33566 7 8 7.90150 1.25147 7.87846 1.38918 Z-?53oi 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 I. 961 57 0.39018 2 3 2.96098 0.48222 2.95212 0-53383 2.94235 0.58527 3 4 3-94798 0.64297 3.93616 0.71177 3-92314 0.78036 4 IS 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 qo8q7 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 0.98325 0.18223 0.97992 0.19936 1 2 2|58^S 0.33009 1.96650 0.36447 1.95984 0.39873 2 3 0.49514 2.94976 0.54670 2-93977 0.59810 3 4 3-94514 0.66019 3-93301 0.72894 3.91969 0-79747 4 3° 4.93142 0.82523 0.99028 4.91627 0.91117 4.89962 0.99683 5 30 6 5.91771 5.89952 1-09341 5-87954 1. 1 9620 6 7 6.90399 1-15533 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 0.98555 0.16935 0.98245 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 5 4.92778 0.84675 4.91225 0.93262 4.89522 1.01820 5 IS 6 5-91333 1.01610 5-89470 1.11914 5-87427 1.22185 6 7 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 1-5241 5 8.84205 1.67871 8.81140 1.83277 9 5 in' Dep. Lat. Dep. Lat. Dep. Lat. ft 1 8 0° 7 90 78° 9 til Smithsonian Tables. 35 Table S TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 10 o a i In 12° 13° 14° 8 .2 i s .3 S p Lat. Dep. Lat. Dep. Lat. Dep. a S 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 ^■2l°H 0.72576 3 4 3.91259 0.83164 3.89748 0.89980 3.881 18 0.96768 4 S 4.89073 1-03955 4.87185 1-12475 4.85147 1. 20961 5 60 6 5.86888 1.24747 5.84622 1.34970 S.82177 1-45153 6 7 6.84703 1-45538 6.82059 1-57465 6.79206 ' 1-69345 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 1.94675 0.45840 1.93846 0.49230 2 3 2.93169 0.63653 2.92013 0.68760 2.90769 0.73845 3 4 3.90892 0.84871 3-89351 o.gi68o 3.87692 0.98461 4 15 4.88615 1.06088 4.86689 1. 1 4600 4.84615 1.23076 5 45 6 5-86338 1.27306 5.84027 1-37520 kH^r 1.47691 6 7 6.84061 1.48524 6.81365 1.60440 6.78461 1.72307 7 8 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 1.95259 0.43288 1.94474 0.46689 1.93629 0.50076 2 3 2.92888 0.64932 2.917 1 1 0.70033 2.90444 0.751 14 3 4 3.90518 0.86576 3.88948 0-93378 3-87259 1.00152 4 3° 4.88148 1.08220 4.86185 1. 167.22 4.84073 5.80888 1. 25190 5 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 8 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. 01 840 4 45 5 4.87671 1. 10348 4.85671 1. 1 8843 4-83523 1-27301 IS 6 5.85205 1.32418 5.82805 1.42611 5.80227 I. 52761 6 7 6.82739 1.54488 6-79939 1.66380 6.76932 1.78221 7 8 l-^°m 1-76557 7-77073 I. 90 I 48 7-73636 2.03681 8 9 8.77808 1.98627 8.74207 2.13917 8.70341 2.29141 9 o En 1 Dep. Lat. Dep. Lat. Dep. Lat. a P 3" B 7 7° 7 6° 7 5° Smithsonian Tables. 36 _ Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. i 3 C ii 1 .a 15° 16° 17° .2 P i 3 a Lat. Dep. Lat. Dep. Lat. Dep. 1 0.96592 1.9318s 0.25881 0.96126 0.27563 0.95630 0.29237 1 2 0.51763 1.92252 0.55127 0.82691 1. 91 260 0.58474 0.87711 2 3 2.89777 0.77645 2.88378 2.86891 3 4 3-86370 1.03527 3.84504 1. 10254 3.82521 1.16948 4 5 4.82962 1.29409 4.80630 1.37818 4.78152 1.46185 5 60 6 6.76148 1-SS291 1.81173 5-76757 1.65382 5-73782 1.75423 6 7 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 8.65135 2.48073 8.C0674 2.63134 9 1 0.96478 0.26303 0.96005 0.27982 0.95502 0.29654 1 2 1.92957 0.52606 1. 920 10 0.55965 1.91004 0.59308 2 3 2.89436 0.78909 2.88015 0.83948 2.86506 0.88962 3 4 3.85914 1.05212 3.84020 1.11931 3.82008 1.18616 4 IS s 4-82393 1.84121 4.80025 1.39914 4.77510 1.4S270 45 6 S-78872 5.76030 1.67897 5.73012 1.77924 6 7 6-7S3SI 6.72035 1.95880 6.68514 2.07579 7 8 7.71829 8.68308 2.10424 7.68040 2.23863 7.64016 2.37233 8 9 2.36728 8.64045 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 0.80171 1.91764 0.56803 1.90743 0.60141 2 3 2.89089 2.87646 0.85204 2.86115 0.902 1 1 3 4 3.85452 1.0689s 3-83528 1. 1 3606 3.81486 1.20282 4 3° 5 4.81815 5.78178 1-33619 4.79410 1.42007 4.76858 1.50352 1.80423 5 30 5 1.60343 S-75292 1.70409 5.72230 6 7 6.74541 1.87066 6.71 174 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 2.70635 9 1 0.96245 0.27144 0.95757 0.28819 0.95239 0.30486 1 2 1.92491 0.54288 0.81432 1.91514 0.57639 1.90479 0.60972 2 3 2.88736 3.84982 2.87271 0.86458 2.85718 0.91459 3 4 1.08576 3.83028 4-78785 1.15278 3.80958 1,21945 4 4S 5 4.81227 1.35720 1.44098 4.76197 1.52432 5 is 6 5-77473 6.73718 1.62864 5-74542 1.72917 5-71437 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 g 53' Dep. Lat. Dep. Lat. Dep. Lat. n i 1 74 1° 7: J° 7S !° Smithsonian Tables. 37 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. A 18° 19° 1 20° 8 a .1 CO .2 3 :g 5 Lat. Dep. Lat. Dep. Lat. Dep. fi S 1 0.95105 0.30901 0:61803 0.94551 0.32556 0.93960 1.87938 O.34202 1 2 i.go2ii 1-89103 0.651 13 0.68404 2 3 2.85316 0.92705 2-83655 3-78207 0.97670 2.81907 3-75§77 1 .02606 3 4 3.80422 1.23606 1.30227 1.36808 4 5 4.75528 1-54508 4-72759 1.62784 4.69846 1.71010 s 60 6 5-70633 1.85410 5-673" 1.95340 5-63815 2.05212 6 7 6-65739 2.16311 6.61863 2.27897 6.57784 2.39414 7 8 7-60845 2.47213 7.56414 2.60454 7-SI754 2.73616 8 9 8-55950 2.781 1 5 '8.50966 2.9301 1 8.45723 3.07818 9 1 0-94969 0.31316 0.94408 0.32969 °i& 0.34611 1 2 1-89939 0.62632 1.88817 0-65938 0.98907 1.31876 0.69223 2 3 2.84909 0-93949 2.83226 2.81457 1.0383s 1.38446 3 4 3-79879 1.25265 3-77635 3.75276 4 IS 4.74849 \f,f^ 4.72044 1.64845 4-69095 1.73058 5 45 6 5.69819 S-66453 1. 978 1 4 5-62914 2.07670 •J 6.64789 2.19214 6-60862 2.30783 6-56733 2.44281 7 8 7-59759 2.81847 Z-SS?7i 2.63752 7-50553 2.76893 8 9 8.54729 8.49680 2.96721 8.44372 3' "505 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 0.95 1 91 2.82792 1.00142 2-81001 1.05062 3 4 3-79329 1.26921 3.77056 1-33522 3.74668 1.40082 4 3° 5 4.74161 1.58652 1.90382 4.71320 1.66903 4.68336 i-75'03 5 30 6 5-65584 6-59849 2.00284 5.62003 2.10124 6 7 t.(>^-X 2.22113 2.33664 6.55670 2-45145 7 8 7-58658 2-53843 7-541 1 3 2.67045 7-49337 2.80165 8 9 8.53491 2-85574 8.48377 3.00426 8.43004 3-15186 9 1 0-94693 0-32143 0.941 17 0-33791 0-93513 1-87027 0.35429 0.70858 1.06287 1 2 1.89386 0.64287 1.88235 0-67583 2 3 2,84079 0.96431 2.82352 1-01375 2.80540 3 4 3.78772 1.28575 3.76470 i!6g958 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 2-57151 2-89295 7-52940 2.70333 7.48108 2-83432 3.18861 8 9 8-52237 8.47058 3.04125 8.41621 9 Minutes. 2 (A Dep. Lat. Dep. Lat. Dep. Lat. 1 % % 71° 70° 6 90 . ''^1 Smithsonian Tables. 38 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. i 8 21° 22° 23° i 1 Lat. Dep. Lat. Dep. Lat. Dep. .2 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 2.76151 1.17219 3 4 3-73432 1-43347 3-70873 1.49842 3.68201 1.56292 4 o 5 4.66790 1.79183 4.63591 1-87303 4.60252 1-95365 2-34438 60 6 5.60148 2.15020 5.56310 2.24763 5.52302 6 7 7146864 2.50857 2.86694 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 1-13594 2.75637 1.18423 3 4 3.72803 1.44975 3.70216 1-51459 3-67516 1.57897 4 tS 5 4.66004 1.81219 4.62770 1.89324 4-59395 1.97372 5 45 6 5.59204 2.17462 5-55324 2.27189 5.51274 2.36846 6 7 6.52405 2.53706 2.89950 6.47878 2.65054 6.43153 2.76320 7 8 7.45606 8.38807 8!32986 3.02918 7-35032 3-15795 8 9 3.26194 3-40783 8.26912 3.55269 9 1 0.93041 0.36650 0.92388 0.38268 0.91706 0.39874 1 2 1.86083 0.73300 1.84776 0.76536 1.14805 1. 8341 2 0.79749 2 3 2.79125 1.09950 2.77164 2.75118 1.19624 3 4 3.72167 1.46600 3.69552 1-53073 3.66824 1.59499 4 30 4.65208 1.83250 4.61940 1.91341 4-58530 1-99374 5 30 6 5.58250 2.19900 5-54328 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 3-44415 8.25354 3.58874 9 1 0.92881 0.37055 0.92220 0.38671 0.91531 0.40274 1 2 1.85762 2.78643 0.74111 1.84440 0.77342 1.83062 0.80549 1.20824 2 3 1.11167 2.76660 •1.16013 2.74593 3 4 3-71524 1.48222 3.68880 1.54684 3.66124 1.61098 4 4S 4.64405 1.85278 4.61100 1-93355 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 g Dep. Lat. Dep. Lat. Dep. Lat. « g en 8 1 6 8° 67° 6 6° Smithsonian Tables. 39 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 8 24 t° 2S ° 2e p 1 c 1 .s to 5 Lat. Dep. Lat. Dep. Lat. Dep. S 1 0.91354 0.40673 0.90630 0.42261 0.89879 1.79758 0.43837 1 2 1.82709 0.81347 1.81 261 0.84523 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 4-56772 2.03368 4-53153 2.11309 4-49397 2.19185 1 60 6 5.48127 2.44041 5-43784 2-53570 2.95832 5.39276 2.63022 6 7 6.39481 2.84715 6.34415 6.29155 3.06859 7 8 7.30836 7.25046 3-38094 sioigH 3.50696 8 9 8.22190 3.66062 8.15677 3-80356 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 IS 4.55881 2.05359 4.52227 2.13284 4-48436 2.21144 1 45 6 547057 2.46431 5-42673 6.33118 ^•■55941 5-38123 2.65373 y 6.38233 2.87503 2.98598 6.27810 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 3-8391 1 8.07185 9 1 0.90996 0.41469 0.90258 0.43051 0.89493 0.44619 1 2 1.81992 0.82938 1.80517 o.85i02 i.7§986 0.89239 2 3 2.72988 1.24407 2.70775 1.29153 2.68480 '-33859 1.78479 3 4 3-63984 1.65877 3.61034 1.72204 3-57973 4 3° 5 4.54980 2.07346 4.51292 2.15255 2.58306 4-47467 2.23098 5 30 6 S-4';976 2.48815 5-41551 6.31809 5.36960 2.67718 6 7 6.36972 2.90285 3-01357 6.26454 3-12338 7 8 7.27969 3-3"754 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. 8 1 628 0-83732 1.80139 0.86889 1.78595 0.90019 2 3 2.72442 1.25598 2.70209 1-30333 2-67893 1.35029 3 4 3-63257 1.67464 3.60279 1.73778 3-57191 1.80039 4 45 5 4.54071 2.09330 4-50349 2.17222 4.46489 2.25049 5 '5 6 5-44885 2.51196 5.40418 2.60667 S-35787 2.70059 6 7 6.35700 2.93062 6.30488 3.04111 6.25085 3.15068 7 8 7.26514 3-34928 7.20558 3-47556 7-14383 3.60078 8 9 8.17328 3-76794 8.10628 3.91000 8.03681 4.05088 9 3' Dep. Lat. Dep. T,at, Dep. L... d 1 en 65° 64° 63° Smithsonian Tables. 40 Table TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 1 i 27° 28° 29° ■J) 1 % 5 Lat. Dep. Lat. Dep. Lat. Dep. p iS 1 0.89100 0-45399 0.88294 0.46947 0.87462 0.48481 1 2 1.78201 0.90798 1.76589 0.93894 1.74924 0.96962 2 3 2.67301 1.36197 2.64884 ^■i°Hl 2.62386 3.49848 1-45443 3 4 3.56402 1. 8 1 596 3-53179 1.87788 1-93924 4 O 4-45503 2.26995 4-41473 2.34735 4-37310 2.42405 5 60 6 5-34603 2.72394 5.29768 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 7.06358 3-75577 6.9969(1 3.87848 8 9 8.01905 4.08591 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 15 4.44508 2.28937 4-40445 2.36660 4.36248 2.44310 5 45 6 5-33410 2.74724 5-28534 2.83992 5-23497 2.93172 6 7 6.2231 1 3.2051 1 6.16623 3-31324 6.10747 3-42034 7 8 7.11213 3.66299 7-04712 3.78656 4.25988 6.97996 3.90896 8 9 8.001 1 5 4.12086 7.92801 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 1.7407 1. 0.98484 2 3 2.66:03 1.38524 2.63645 I-43H7 2.61 106 1.47727 3 4 3.54804 1.84699 3-51526 1.90863 3.48142 1.96969 4 3° 4-43505 2.30874 4-39408 2.38579 4-35177 2.4621 1 5 30 6 5.32206 2.77049 5.27290 2.8629s 5.22213 2.95454 6 7 6.20907 3.23224 6.15171 3-34011 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 4.4318 1 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 4-42493 2.32807 4-38363 2.40494 4-34099 2.48108 5 15 6 5.30992 2.79368 5.26036 2.88593 5.20919 2.97729 6 7 6. 1 949 1 3-25930 6.13708 3-36692 3.84791 6.07739 3-47351 7 8 7.07990 3.72491 7.01381 6.94559 3-96973 8 9 7.96488 4.19053 7.89054 4.32889 7.81378 4.46594 9 a g Dep. Lat. Dep. Lat. Dep. Lat. g 3' s 55' 1 E3 c n 62° 61° 60° Smithsonian Tables. 41 Table 9. TRAVERSE TABLE, DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. i c s c 30° 31° 32° 8 in 1 ii (5 Lat. Dep. Lat. Dep. Lat. Dep. Q IS 1 0.86602 0.50000 0.85716 0.51503 0.84804 0.52991 i.o;9§3 1-58975 1 2 1.73205 1. 00000 1-71433 1.03007 1.69609 2 3 2.59807 1.50000 2.57150 1-545" 2.54414 3 4 3.46410 2.00000 3.42866 2.06015 3-39219 2.11967 4 4.33012 2.50000 4.28583 2.57519 4.24024 2.64959 5 60 5.19615 3.00000 5.14300 3.09022 5.08828 3-17951 6 7 6.06217 3.50000 6.00017 3.60526 678438 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 1.70982 0.51877 0.84572 0-53361 1 2 1.72767 1.007 54 1-03754 1-69145 1.06722 2 3 2.59150 I-5II32 2.56473 1-55631 2.53718 1.60084 3 4 3-45534 2.01509 2.51887 3.41964 2.07509 3.38291 2.13445 4 15 4-31917 4.27456 2.59386 4.22863 2.66807 5 45 6 5-18301 3.02264 5.12947 3.11263 5-07436 3.20168 6 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 7-77451 4-53396 7.69420 4.66895 7-6" 55 4-80253 9 1 0.86162 0-50753 0.85264 0.52249 0-84339 0.53730 1 2 1-72325 1. 01 507 1.70528 1.04499 1.68678 1.07460 2 3 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 5 4.30814 2.53769 4.26320 2.61249 4.2169s 2.68650 3.22380 30 6 5.16977 3-04523 5.11584 3-13499 5-06034 6 7 6.03140 3-55276 5.96948 3-65749 5-90373 3.76110 7 8 6.89303 4.060TO 6.82112 4.17998 6-74713 4.29840 8 9 7.75466 4.56784 7.67376 4.70248 7.59052 4-83570 9 1 fy?sr. 0.51 129 0.85035 0.52621 0.84103 0-54097 1 2 1.02258 1.70070 1.05242 1.68207 1. 08 1 94 2 3 2.57821 1-53387 2.55105 1.57864 2.52311 1.62292 3 4 3-43762 2.04517 3.40140 2.10485 3-36415 2.16389 4 45 1 4.29703 2.55646 4.25176 2.63107 4-20519 2.70487 15 6 5-15643 3.0677s 5.10211 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 1 01 Dep. Lat. Dep. Lat. Dep. Lat. 1 5£ i° 5) J° 5' f° Smithso MIAN 1 rABLES. 42 TRAVERSE TABLE. "^"^'■^ ®' DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. c 8 "to 33° 34° 33° a5 1 1 g 5 Lat. Dep. Lat. Dep. Lat. Dep. Q s 1 0.83867 0.54463 0.82903 0.55919 0.81915 0-57357 1 2 1-67734 1.08927 1.65807 1. 1 1838 1.63830 1.14715 2 3 2.51601 t-63391 2.4871 1 1.67757 2-45745 1.72072 3 4 3-35468 2.17855 3-31615 4.14518 2.23677 3.27660 ^■I'^M^ 4 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 S-87069 6.70936 3.81247 5.80326 6.63230 3-91435 5-73406 4-01503 7 8 4-357" 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 1.64487 1-65318 1. 12560 1.63328 1. 1 5429 2 3 2.50885 2.47977 1.68841 2.44992 '•73143 2.30858 3 4 3-34SH 2.19317 3-30636 2.25121 3.26656 4 IS 5 4.18143 2.74146 4.13295 2.81402 4.08320 2.88572 5 45 6 5.01771 3-28975 4-95954 3-37682 4.89984 3-46287 6 7 5.85400 3-83805 4-38634 5-78613 3-93963 5.71649 4.04001 7 8 6.69028 6.61272 4-50243 6-53313 4.61716 8 9 7.52657 4-93463 7-43931 5.06524 7-34977 5.1943a 9 1 0.83388 0.55193 0.82412 0.56640 0.814H 0.58070 1 2 1.66777 1.10387 1.64825 1.13281 1.62823 1.16140 2 3 2.50165 1.65581 2.47237 1. 6992 1 2.44234 1.74210 3 4 3-33554 2.20774 3.29650 2.26562 3.25646 2.32281 4 3° 5 4.16942 2.75968 4.12063 2.83203 4.07057 2.90351 5 30 6 5-00331 3.3II62 3-86355 4-94475 5.7688^ 3-39843 4.88469 3.48421 6 7 5.83720 3-96484 5.69880 6.51292 4.06492 7 8 6.67108 4.41549 6.59300 4-53124 4.64562 8 9 7.50497 4-96743 7-41713 5.09765 7-32703 5-22632 9 X 0.83147 0.55557 0.82164 5-56999 0.81157 0.58425 1 2 1.66294 1.11114 1.64329 1.13999 1.62314 1. 16850 2 3 2.49441 1.66671 2.46494 1.70999 2.43472 1.75275 3 4 3-32588 2.22228 3.28658 2.27998 3.24629 2.33700 4 45 4.98882 2.77785 4.10823 2.84998 4.05787 2.92125 5 15 6 3.33342 4.92988 3.41998 4.86944 3-50550 6 7 5.82029 0.65176 7-48323 3.88899 5-75152 3-98997 5.68101 4.08975 7 8 4.44456 6.57317 4.55997 6.49260 4.67400 8 9 5.00013 7.39482 5.12997 7.30416 S-25825 9 g « Dep. Lat. Dep. Lat. Dep. Lat. % 3' 1 i 9 a 1 5 6° 5 5° 54° Smithsonian Tables. 43 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. 1 a s 8 2 36° 37° 38° 1 .J5 i .a S Q Lat. Dep. Lat. Dep. Lat. Dep. P 1 1 0.80901 0.58778 0.79863 0.60181 0.78801 0.61566 1 2 I. 61 803 '■I75S7 1.59727 1.20363 1.57602 1.23132 2 3 2.42705 '•76335 2.39590 1.80544 2.36403 1.84698 3 4 3.23606 2-3S"4 3-19454 2.40726 3.15204 2.46264 4 S 4.04508 2.93892 3-993 '7 3.00907 3.94005 3.07830 5 60 6 4.85410 3.52671 4.79181 3.61089 4.72806 3-69396 6 7 5.6631 1 4-11449 4.21270 5-51607 4-30963 7 8 6.47213 4.70228 6.38908 4.81452 6.30408 4.92529 8 9 7.281 IS 5.29006 7.18771 5-41633 7.09209 S-54095 9 1 0.80644 0.59130 1.18261 0.79600 0.60529 1-78531 0.61909 1 2 1.61288 1.59200 1.21058 1.81588 1.57063 1.23818 2 3 241933 1.77392 2.38800 2-35595 1.85728 3 4 3-Z2577 2.36523 3.18400 2.421 17 3. 1 41 26 2.47637 4 IS 5 4-°3"? 2-95654 3.98001 3.02647 3.92658 3*9547 5 45 6 4.83866 3-54785 4.77601 3.63176 4.7 1 190 3-71456 6 7 5.6451 1 4.13916 5.57201 4.23705 5.49721 4-33365 7 8 64515s 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 30 s 4.01928 2.9741 1 3.96676 3.04380 3-91304 3.1 1 257 5 30 o 4.82314 3-56893 4.7601 1 3.65256 4-69564 3-73508 6 7 5.62609 4-16375 4.75858 5-55347 4.26132 5.47825 4-35760 4.9801 1 7 8 6.43085 6.34682 4.87009 6.26086 8 9 7-23471 5-35340 7.14017 5-47885 7-04347 5.60263 9 1 0.80125 0.59832 0.79068 0.61 221 0.77988 0.62592 1 2 1.60250 1. 1 9664 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 '953 2.50369 4 45 s 4.00626 2.99162 3-95344 3.06108 3.89942 3.12961 15 6 4.80752 3-58994 4-74413 3-67330 4.67930 3-75554 4.38146 6 7 S.60877 4.18827 5-53482 4-28552 5-45919 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 5-6333' 9 S s Dep. Lat. Dep. Lat. Dep. Lat. ?• 3 5" d a CO P 1 CD 3' G 5 3° 5 2° 5 1° Smith SO NIAN Tables. 44 TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. - Table 9. -Continued. i .1 1 (5 3£ 1° 40° 41° CO 5 i a .a Lat. Dep. Lat. Dep. Lat. Dep. 1 0.77714 0.62932 0.76604 0.64278 0.75470 0.65605 1 2 1.55429 1.25864 1.88796 1.53208 i.28«7 1.92836 1. 50941 1.31211 2 3 2-33H3 2.29813 2.26412 1. 9681 7 3 4 3.10858 2.51728 3.06417 2.57115 3.01883 2.62423 4 5 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 6 7 5.44002 4.40524 6.1283s 4-49951 5.28296 4.59241 7 8 6.21716 S-°34S6 5.14230 6.03767 5.24847 8 9 6.99431 5.66388 6-89439 5.78508 6.79238 S-90453 9 1 0-77439 0.63270 0.76323 0.6461 2 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 3-097S7 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-S"?4 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 5-81511 6.76656 5-9341 1 9 1 0.77162 0.63607 0.76040 0.64944 0.7489s 0.66262 1 2 1.54324 1.2721S 1.52081 1.29889 1.49791 '■3^524 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° 3.85812 3.18039 3.80203 3-24724 3-74477 3-31310 5 30 6 4.62974 3.81646 4.56243 3.89668 5.24268 3-97572 6 7 S-40137 4.45254 5.32284 4.54613 4-63834 7 8 6.17299 6.08324 5-19558 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 I-S3768 1.27887 1-51513 1.30552 1.49211 1-33176 2 3 2.30652 1.91831 2.27269 1.95828 2.23817 2.98422 1.99764 3 4 3-07 S36 2-55775 3.03026 2.61 104 2.66352 4 45 3.84420 3.19719 3-78782 2.26380 3.73028 3-32940 5 15 6 4.61 30s 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 5.96845 5-32705 8 9 6-91957 5-75495 6.81808 S-87484 6.71451 5-99293 9 1 Dep. Lat. Dep. Lat. Dep. Lat. 3 P s- n 5 0° 4 90 4 8° Smithsonian Tables. 4S ^*"'"'^ ^" TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED. rt 42° 43° 44° .9 i B a ii Q Lat. Dep. Lat. Dep. Lat. Dep. i 1 0.74314 0.66913 0.73135 0.68199 0-71933 0.69465 i-3|93' 1 2 1.48628 1.33826 1.46270 1-36399 1.43867 2 3 2.22943 2.00739 2.19406 2-04599 2. 1 5801 2.08TO7 3 4 2.97257 2.67652 2.92541 2.72799 2.8773s 2.77863 4 iVs^ll 3-34565 4.01478 3-65676 3-40999 3-59669 3-47329 1 60 6 4.38812 4.09199 4.31603 4.16795 7 5.20201 4.68391 5- 1 1947 4-77398 5-03537 4.86260 1 8 11^^^^ 5-35304 5.85082 6.58218 5-45598 5-75471 6-47405 5.55726 5.25192 8 9 6.02217 6.13798 9 1 0.74021 0.67236 0.72837 0.68518 0.71630 1.43260 0.69779 1-39558 1 2 1.48043 1-34473 1-45674 1-37036 2 3 2.22065 2.01710 2.18511 2.05554 2.14890 2-09337 3 4 2.96087 2.68946 2.91348 2-74073 2.86520 2.791 16 4 15 3.70109 3-36183 3.64185 3.42591 3-58151 4.29781 3-48895 1 45 6 4.44130 4.03420 4.37022 4.11109 4.18674 b 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 6.2801 1 9 1 0.73727 0-67559 0.72537 0.68835 0.71325 1.42650 0.70090 1.40181 1 2 1-47455 2.21 183 1.35118 1.45074 1.37670 2 3 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 30 S 3-68638 3-37795 3.62687 3-44177 3.56625 3-50454 1 30 6 4.42366 4-05354 4.35224 4-I30I2 4.27950 4-20545 4-90636 7 5.16094 4-72913 5-07762 4.81848 4-99275 7 8 5.89821 5.40472 &fe 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 0.691 51 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 2.1 1204 3 4 2.93729 2.71520 2.88945 2.7660s 2.84074 2.81605 4 45 3.67161 3-39400 3.61182 3-45756 3-55092 3.52007 ^ 'S 6 4-40593 4.07280 4-33418 4.14907 4.261 1 1 4.22408 7 5.14025 5.87458 4.75160 5.05654 4.84059 4.97129 4.92810 7 8 5-43040 5.77891 5-53210 5.68148 5.632H 8 9 6.60890 6.10920 6.50127 6.22361 6.39166 6.33613 9 Dep. Lat. Dep. Lat. Dep. Lat. a a- 1' s 1 . g 1 p R 4 70 4 6° 4 S° R p Smithsonian Tables. 46 Table 9. TRAVERSE TABLE. DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued. 1 45° i Q Lat. Dep. 1 0.70710 0.70710 1 2 1.41421 1. 41 42 1 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 Dep. Lat. cn 1 4. 5° Smithsonian Tables, 47 Yable 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ FEET. [Derivation ot table explained on p. xlv.] IN ENGLISH Lat 0= 1° 2° 3° 4° S° 6° 7° 8° 9° 10° P. P. 0' I 2 3 4 1 i 9 10 11 12 "3 '5 i6 \l '9 20 21 22 23 24 11 27 28 29 30 31 32 33 34 u 37 38 39 40 ■41 42 43 44 11 47 48 49 60 5' S2 S3 54 55 56 57 58 59 60 7.317 7379 7.317 7392 7.317 7433 7.317 7SOO 7.317 7593 7.317 7714 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 7435 7436 7437 7438 7438 7439 7440 744' 75° ■ 7503 7504 7506 7507 7508 75 lo 7511 7513 7595 7597 7599 7600 7602 7604 7606 7608 7610 7716 77'9 7721 7723 7726 7728 7730 7732 7735 7864 7866 7869 7872 787s 7877 7880 ^?J 7885 8037 8040 8043 8046 8050 8053 8055 8059 8062 8237 8240 8244 8247 8251 8255 8258 8262 826s 8462 S466 8470 8474 8478 8482 8485 8490 8494 8713 8718 8722 8727 8731 8735 8740 8744 8749 ro 20 SO 40 lo° .2 •3 •5 -.1 1.0 7379 7397 7442 7514 7612 7737 7888 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 745 1 75'5 7517 7518 7S20 752' 7522 7524 7525 7527 7614 7616 7618 7619 7621 7623 7625 7627 7629 7739 7742 7744 7746 7749 775" 7753 7755 7757 7891 7894 7896 7899 7902 7905 7908 7910 79'3 8068 8071 807s 8078 808 1 8084 8087 8091 8094 8273 8276 8280 8283 8287 8291 8294 8298 8301 8502 8506 8510 8518 8523 8527 853 J 8535 8758 8762 8757 8771 S775 8780 8785 8789 8794 2 10 20 30 40 ■3 •7 I.O '■3 '•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 7S30 7531 7533 7534 7535 7537 7538 7540 754' . 7633 7635 7637 7638 7640 7642 7644 7645 7648 7762 7765 ■ 7767 7770 7772 7774 7777 7779 7782 7919 7922 7924 7927 7930 7933 7936 7938 794' 8100 8104 S107 8110 81 14 8117 8120 8123 8.27 8309 8312 8316 8320 8324 8327 833' 8335 8338 8543 8547 8551 855s 8559 8564 8568 !57; 8576 8803 8807 8812 8816 8821 8825 8830 883s 8839 3 ID 20 30 40 50 5o •5 1.0 1-5 2.0 '■S 30 7382 7409 7463 7543 7650 7784 7944 8130 8342 8580 8844 7382 73S3 7383 7383 7384 7384 73S4 7384 738s 7410 7410 741 1 74 12 74'3 7413 7414 7415 7415 7464 7465 7465 7467 7469 7470 747> 7472 7473 7545 7546 7548 7549 7551 7533 7554 7556 7557 7652 7654 7656 7658 7661 7663 7665 7667 7669 7786 77'?9 7791 7794 7796 7799 7801 7804 7806 7947 7950 7953 7956 7959 7961 7964 7967 7970 8133 8137 8140 8144 8147 8150 8154 8,57 8161 8346 8350 8353 8357 8351 836s 8369 8372 8376 8584 8588 8593 8597 86o« 8605 8609 8614 8618 8849 88^3 8858 8862 8867 8872 8876 8881 8885 4 10 20 30 40 50 6a •7 i '■3 2.0 2-7 3-3 4.0 7385 7416 7474 7559 7671 7809 7973 8164 8380 8622 8890 7^81 7386 7386 7387 7387 7387 7387 7388 7388 74 '7 7418 7418 7419 7420 742' 7422 7422 7423 7<75 ",76 478 7479 7480 7482 7483 7486 7561 7562 7564 7565 7567 7569 7571 7573 7574 7673 7675 7677 7679 7682 7684 7686 7688 7690 7811 7814 7815 7819 7821 7824 7826 7829 7831 7976 7979 7982 7985 7988 7991 7994 7997 8000 8167 8171 8174 8178 8181 8184 8188 8191 8 '95 8384 8388 8392 8396 8400 8403 8407 84" 84'5 8626 8631 8635 8639 8543 8648 8652 8656 8551 889s 8899 8904 8909 8914 8918 8932 6 10 20 30 40 so 60 .8 '■7 2-5 3-3 4.2 5-0 7424 7487 7576 7692 7834 8003 8.98 8419 856s 8937 7388 73S9 7389 7390 7390 7390 7391 7391 7392 742s 7426 7427 7428 7429 7429 7430 7431 7432 7488 7489 7490 7491 7493 7494 7496 7497 7498 7578 7579 7581 7583 7584 7585 7588 7590 759" 7694 7696 7699 7701 7703 7705 7707 7710 7712 7837 7839 7842 7845 7848 7850 7853 7855 7858 8006 8009 8012 801S 8019 8022 8025 8028 8031 8201 8205 8208 8212 8215 8219 8222 8226 8229 8423 8427 8431 8435 8439 8442 8446 8450 8454 8669 8674 8678 8583 8687 8691 8696 8700 87°5 8942 8947 8951 8956 8961 8966 8971 7392 7433 7500 7593 77'4 7861 8034 8233 8458 8709 898s Slum ISONIAN Tables , — " — ' 48 LOGARITHMS OP MERIDIAN RADIUS OF CURVATURE p„ FEET. tDerivation of table explained on p. xlv.] Table 10. IN ENGLISH Lat. II" 12= 13° .4° 15° 16° 17° 18° 19° 20° P.P. 0' I 2 3 4 I 7 8 9 10 II 12 •3 14 IS l6 17 i8 '9 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 tl 49 60 SI 52 S3 54 SS 56 % 59 60 7.317 8985 7.317 9285 7.317 9611 7.317 9960 7.318 0333 7.318 0730 7.318 1149 7.318 1591 7.318 2054 7.318 2539 4 8990 8995 8999 9004 9009 9014 9019 9023 9028 9290 9296 9301 9306 9312 9317 9322 9327 9333 9617 9622 9628 9533 9639 9645 9650 9656 9661 9966 9972 997S 9984 9990 9996 *0002 *ooo8 *ooi4 0340 ':346 0353 03 59 0366 0372 0379 0385 0392 0737 0744 075° 0757 0764 0771 0778 0784 0791 ..56 1 163 1171 1178 Ii8s 1192 1 199 1207 1214 '599 1606 1614 1621 1629 1637 1644 1652 1659 2062 2070 2078 • 2086 2094 2102 2110 2118 2126 2547 2556 25S4 2580 2589 2597 2605 2614 10 20 30 40 so 60 .7 "•3 2.0 2-7 3-3 4.0 6 9033 9338 9667 *OO20 0398 0798 1221 1667 2134 2622 9038 9043 9048 9°53 9058 9062 9067 9072 9077 9343 9349 9354 9359 9365 9370 9375 9380 93S6 9673 .967S 9684 9690 9696 9701 9707 9713 9718 »0026 *0032 *oo39 •0045 *oo5i *ooS7 *oo63 *0070 *o076 0404 041 1 0418 0424 0430 0437 0443 0450 0456 0805 08J2 0SI9 0S26 0833 0839 0846 0853 0S60 1228 1236 1243 1250 1258 1265 1272 1279 1287 1690 1697 1705 1713 1720 1728 "735 2142 2150 2158 2166 2174 2182 2190 2198 2206 2630 2639 2647 2655 2663 2672 2680 2688 2697 10 20 30 40 50 60 .8 1-7 2-5 3-3 4-2 S-o 6 9082 9391 9724 •0082 0463 0867 1294 "743 2214 2705 9087 9092 9097 9102 9107 9112 91 17 9122 9127 9396 9402 9407 94'3 9418 9423 9429 9434 9440 9730 9736 9741 9747 9753 9759 9765 9770 9776 *oo88 *oo94 *OIOI *ci07 *oii3 *on9 *OI2S '0132 *oi38 0470 0476 0483 0489 0496 0503 0509 0516 0522 0881 0888 0895 0902 0909 0916 0923 0930 1301 ■ 309 1316 1323 1330 133S 1345 1360 ■75" ml •774 1781 1789 1797 1805 1812 2222 2230 2238 2246 2254 2262 2270 2278 2286 2713 2722 2730 2739 2747 2755 2764 2772 2781 10 20 30 40 50 60 I.O 2.0 3.0 4.0 5-0 6.0 7 9132 9445 9782 *oi44 0529 0937 1367 1820 2294 2789 9137 9142 9147 9152 9"57 9162 9167 9172 9177 94SO 9456 9461 9467 9472 9477 9488 9494 9788 9794 9800 9806 9812 98.7 9823 9829 9835 *oi5o »ois6 *oi63 *oi69 *oi75 *oi8i *oi87 *oi94 *0200 0536 0542 0549 OS5S 0562 0569 0575 0582 0588 0944 0951 0958 0965 0972 0979 0986 , 0993 1000 1382 1389 1397 1404 141 1 1419 1426 1434 1828 1835 1843 185 1 '1858 1866 1874 1882 1889 2302 2310 2318 2326 2334 2343 2351 2359 2367 2797 2806 2814 2823 2831 2840 2848 2857 2865 10 20 30 40 so 60 1.2 2-3 3-5 4-7 5-8 7.0 8 9182 9499 9841 *0206 0595 1007 1441 1897 2375 2874 9187 9192 9197 9202 9207 9213 9218 9223 9228 9505 95 10 9516 9521 9527 9533 9538 9544 9549 9847 9853 9859 9865 9871 9876 9882 9888 9894 *02I2 *02I9 *0225 *023I *0238 »0244 ♦0250 *0256 *o263 0602 0608 0615 0622 0629 0635 0642 0649 0655 1014 I02I 1028 103 s 1042 1050 1057 1064 107 1 1448 1456 1463 1471 1479 i486 1494 1501 1509 1905 I9'3 1920 1928 1936 1944 1952 1959 1967 2383 239' 2400 2408 2416 2424 2432 2441 2449 2882 2891 2899 2908 2916 292s 2933 2942 2950 10 20 30 40 so 60 1-3 2.7 4.0 5-3 6.7 8.0 9 9233 9553 9900 ♦0269 0662 1078 1516 1975 2457 2959 9238 9243 9249 9254 9259 9264 9269 927s 92S0 9561 9566 9572 9577 9583 9589 9594 9600 9605 9906 9912 9918 9924 9930 9936 9942 994S 9954 *o288 •029s *030i *0307 *o3i4 *0320 *o327 0669 0676 0682 06S9 0696 0703 0710 0716 0723 1085 1092 1099 1 106 HI3 II2I II28 "35 1142 1524 ■531 1539 1546 ■554 1561 1569' 1576 1584 1983 1991 ■999 2007 2014 2022 2030 2038 2046 246s 2473 2482 2490 , 2498 2506 2514 2523 2531 2968 2976 2985 2993 3002 301 1 3019 302S 3036 10 20 30 40 so 60 i-S 3-0 t.l 7-5 90 9285 9611 9960 •0333 0730 1149 1591 2054 2S39 304s Smithsonian Tables. 49 Table 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm FEET. [Derivation of table explained on p, xlv.] IN ENGLISH Lat. 21° 22° 23° 24° 2f 26° 27° 28° 29° 30° p.p. 0' I 2 3 4 1 I 9 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 li 11 29 30 31 32 33 34 11 39 40 41 42 43 44 45 46 47 48 49 60 SI 52 53 54 55 56 11 59 60 7.318 3045 7.318 3570 7.318 411S 7.318 4678 7.318 5259 7.318 5858 7.318 6474 7.318 7105 7.318 775' 7.318 8412 8 3°53 3062 3070 3096 3105 3113 3122 3579 3588 3597 3606 3614 3623 3632 3641 3650 4124 4133 . 4142 4152 4161 4170 4'79 4189 4198 4688 4697 4707 4716 4726 4735 4745 4754 4764 5269 5279 5289 5299 5309 53 '9 5328 5338 5348 5868 5878 5889 5899 5909 5919 5929 5939 5949 6484 6494 6505 nil 6536 6546 6557 6567 7116 7126 7'37 7.48 7'58 7169 7180 7190 7201 7762 7784 7795 7806 7817 7828 7839 7850 8423 8434 8445 1% 8479 8490 Sjoi 8512 10 20 30 40 50 60 J.3 2.6 4.0 8.0 3131 3659 4207 4774 5358 5960 6578 7212 7860 8523 3139 3148 3'57 3165 3174 3183 3191 3200 3209 3668 Itll 3695 3704 3713 3722 3731 3740 4216 4226 4235 4244 4254 4263 4272 4282 4291 4783 4793 4802 4812 4822 4831 4841 4851 4860 5368 5378 5388 5398 5408 5417 5427 5437 5447 5970 5980 5990 6000 6011 6021 6031 6041 6051 6588 6599 6609 6620 6630 6640 til] 6672 7222 7233 7244 7254 7265 7276 7287 7297 7308 nil 7893 7904 7915 7926 7937 7948 7959 8557 8568 8579 8591 8602 8613 8624 9 10 20 30 40 lo >-5 3.0 ti 7-5 9.0 3217 3749 4300 4870 5457 6062 6682 7319 7970 8635 3226 3235 3244 3252 3261 3270 3278 3287 3296 3758 3767 3776 3785 3794 3804 3813 3822 3831 4310 43>9 4328 4338 4347 4356 4365 4375 4384 ti8l 4899 4908 4918 4928 4937 4947 4957 5467 5487 5497 5507 55-7 5527 5537 5547 6072 6082 609Z 6102 6113 6123 6133 6143 6154 . 6693 6703 6714 6724 6735 674s lite 6777 7329 7340 7351 7362 7383 7394 7405 7416 7981 7992 8003 8014 8025 8036 8047 8058 8069 8647 8658 8669 8680 8691 8703 8714 8725 8736 10 10 20 30 40 50 60 '•7 33 S.o tl lO.O 3305 3840 4394 4966 5557 6164 6787 7426 8080 8747 3313 3322 3331 3340 3349 3357 3366 3384 3849 3858 3867 3876 388s 3894 3904 3913 3922 4403 4413 4422 4431 4441 4450 4460 4469 4479 4976 4986 4996 5005 S015 5025 5034 5044 5054 5567 5587 5597 5607 5617 5627 5637 5647 6174 6l8s 6195 6205 6215 6226 6236 6246 6256 6798 6808 6819 6829 6840 6851 6861 6872 6882 7437 7448 7459 7469 7480 7491 7502 7513 7523 8091 8102 81 13 8124 l^ii 8179 8759 8815 8826 8838 8849 11 10 20 30 40 50 60 1.8 3-7 7-3 9.2 II.O 3393 3931 4488 5064 5657 6267 6893 7534 8190 8860 3401 3410 3419 3428 3437 3446 3463 3472 3940 3949 3958 3967 3977 3986 3995 4004 4°'3 4498 4507 4516 4526 4535 4545 4554 4564 4373 S073 5083 5093 5103 5112 5122 5132 5'42 5'5i 5667 nil 5697 5707 5717 5727 5737 5747 6277 6287 6298 6308 6318 6329 6339 6349 6360 6903 6914 6924 693s 6946 6956 6967 7545 7556 7567 7578 7588 7599 7610 7621 7632 8201 8212 8223 8234 8246 8257 8268 8871 8883 8894 8905 8916 8928 8939 8950 8962 12 10 zo 30 40 50 60 2.0 4.0 6.0 8,0 lO.O 12.0 348. 4022 4583 S161 5757 6370 6999 7643 8301 8973 3490 3499 3508 3516 352s 3534 3S43 3552 3561 4032 4041 4050 nt 4078 4086 4096 4105 4592 4602 461 1 4621 4630 4640 4649 4659 4668 S171 5181 519' 5200 5210 5220 5230 5240 5250 5767 5777 5787 5808 5818 ■5828 5838 5848 6380 6391 6401 6411 6422 6432 5442 7009 7020 7030 7041 IX 7073 7084 7094 7653 7664 7675 7686 7697 7708 7719 7729 7740 8312 8323 8334 8345 8356 8368 8379 8390 S401 9007 9018 9030 9041 9052 9064 907s 3570 4115 4678 5259 5858 6474 710S 775' 8412 9086 Smiths ONIAN 1 ABLES. ^-^ SO Table 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH FEET. [Derivation of table explained on p. xlv,] 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 7.318 7.319 0' 1 2 3 9086 9773 0472 1182 1902 2631 3369 4114 4866 5623 11 9098 9109 9120 9785 9796 9807 0484 0495 0507 "94 1206 12 18 1914 1926 193S 2643 2656 266S 3381 3394 3406 4126 4139 4151 4878 4891 4904 5636 5649 5661 4 1 9132 9143 9154 9819 9831 9843 0519 0531 0542 1230 1241 1253 1950 1962 "974 2680 2692 2705 3418 3431 3443 4164 4176 4189 4916 4929 4941 5674 5687 5699 10 1.8 3-7 5.5 7-3 9.2 Il.O 7 S 9 10 11 12 13 9166 9177 9189 9854 9866 9877 0566 0577 1265 1277 .289 1986 1999 2011 2717 2729 2741 3455 3468 3480 4201 4214 4226 4954 4966 4979 5712 S725 5737 30 40 9200 9889 0590 1301 2023 2753 3492 4239 4992 5750 9211 9223 9234 9900 9912 9924 0601 0613 0625 I3'3 1325 1337 2035 2047 2059 2766 2778 2790 3505 3517 3530 4251 4264 4276 5004 5017 5029 5763 5775 5788 '4 15 16 9245 nil 9935 9947 9958 0637 0648 0660 1349 1361 1373 2071 2083 2095 2803 2815 2S27 3542 3567 4289 4301 4314 5042 5055 5067 5801 5813 5826 19 20 21 22 23 9280 9291 9302 9970 9982 9993 0672 0684 0696 138s ■397 1409 2108 2120 2132 2839 2852 2864 3579 3592 3604 4326 4339 4351 5080 5092 5105 5839 5851 5864 12 9314 *ooos 0707 1421 2144 2S76 3616 4364 5"8 5877 9325 9337 9348 *ooi6 •0028 *0040 0719 0731 0743 1433 1445 1457 2156 2168 2i8o 288S 2901 2913- 3629 3641 3634 4376 4389 4401 5130 5143 5156 5890 5902 S9'.5 24 11 9360 9371 9382 *bo5i *oo63 *oo7S 0778 1469 1481 1493 2192 2205 2217 2925 2938 2950 3666 367S 3691 4414 4426 4439 S168 5181 5193 5928 5940 5953 10 2.0 11 29 30 31 32 33 9393 9405 9417 *oo86 *oo98 *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 20 30 40 50 60 4.0 6.0 8.0 10.0 12.0 9428 *OI2I 0826 1541 2265 2999 3741 4489 5244 6004 9440 9451 9463 »oi33 *0i44 •0156 0837 1553 1565 1577 2278 2290 2302 3011 3024 3036 3753 3765 3778 4502 4514 4527 5256 5269 5282 6017 6029 6042 34 9474 9485 9497 *oi68 *oi79 *oigi 0S73 088s 0897 1589 1601 1613 2314 2326 2338 3048 3060 3°73 3790 3803 381s 4539 4552 4564 5294 5307 5320 6055 6067 6080 11 39 40 41 42 43 9508 9S20 9531 *0203 *0214 *0226 0908 0920 0932 1625 1637 1649 2351 2363 2375 3085 3097 3110 3828 3840 3852 4577 4589 4602 5332 5358 6093 6106 6118 13 9543 *0238 0944 1661 2387 3122 3865 4614 S370 6131 9577 •0273 0956 0968 0980 1673 16S5 1697 2399 2411 2424 3134 3147 3159 3877 3890 3902 4627 4640 4652 53S3 5395 5408 6144 6156 6169 44 9589 9600 9612 *o285 ♦0296 »03o8 0992 1003 IOI5 1709 1721 1733 2436 2448 2460 I7sl 3196 3^15 3927 3939 4665 4677 4690 5421 5433 5446 6182 6195 6207 10 2.2 s 10.8 13.0 47 48 49 60 51 53 9623 9635 9646 •0320 *033i *0343 1027 1039 105 1 1745 1757 1769 2472 2485 2497 3208 3221 3233 3952 3964 3977 4702 4715 4727 5459 5484 6220 6233 6245 30 40 50 bo 9658 *0355 1063 1781 2509 3245 3989 4740 5497 6258 9669 96S1 9692 *0366 *o378 *0390 1075 1087 1098 I8I7 2521 2533 2546 3258 3270 3282 4002 4014 4027 4753 4765 4778 5509 5522 5535 6271 6284 6296 54 11 9704 9715 9727 *0402 *04i3 •0425 IIIO 1122 "34 1829 I84I 1854 2558 2570 2582 3295 3307 3319 4039 4052 4064 4790 4803 4815 5547 5S6o 5573 6309 6322 6335 59 60 9739 975° 9762 *0437 *0449 •0460 1146 1158 1170 1866 1878 1890 2594 2607 2619 3332 3344 3356 4077 4089 4101 4828 4841 4853 5598 56" 6347 6360 6373 9773 *0472 1 182 1902 2631 3369 4114 4866 5623 6385 Smithsonian Tables. SI Table 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm FEET. [Derivation of table explained on p. xlv.] IN ENGLISH Lat. 41° 42° 43° 44° 45° 46° 47° 48° 49° SO" P.P. 7.319 7.319 7.319 7.319 7.319 7.320 7.320 7.320 7.320 7.320 0' I 2 3 6385 7152 7921 8692 9464 0236 1007 1776 2543 3306 6398 6411 6424 7164 7177 7190 7933 7946 7959 8704 8717 8730 9476 9489 9502 0248 0261 0274 1020 "033 1045 1789 1802 1815 2556 2569 2581 3319 333^ 3344 4 5 6 6436 6449 6462 7203 7216 7228 7972 7985 7998 8743 8769 9515 9528 954' 0287 0300 0313 1058 1071 1084 1827 184a 1853 2594 2607 2619 3357 3369 3382 I 9 10 II 12 13 6475 6487 6500 7241 7254 7267 8010 8023 8036 8782 8794 8807 9554 9566 9579 0326 0338 0351 1097 mo 1122 1866 1879 1892 2632 264s 2658 3395 3407 3420 12 6513 7280 8049 8820 9592 0364 "35 1904 2670 3433 6526 6538 6551 7292 7305 7318 8062 8075 8087 8833 8846 8859 9605 9618 9631 0377 0390 0403 1148 1161 1 174 1917 J930 ■943 2683 2696 2709 3445 3458 347' ■0 20 30 2.0 4.0 6.0 '4 6564 6577 6589 7331 7344 7356 8100 8113 8126 8872 8884 8897 9644 0416 0429 0441 1 187 1 199 1212 ■955 1968 1981 2721 2734 2747 3483 3496 3509 40 50 60 8.0 ■0.0 12.0 19 20 21 22 23 6602 6615 6628 7369 7382 7395 8139 8152 8165 8910 8^3^ 9682 9695 9708 0454 0467 0480 1225 1238 1251 ■994 2007 2019 2760 2785 3521 3534 3547 6640 7408 8177 8949 9721 0493 1264 2032 2798 3559 ^11 6679 7420 7433 7446 S190 8203 8216 8962 9734 9747 9760 0506 0519 0531 1276 1289 1302 2045 2058 2071 2811 2823 2836 3572 3585 3597 24 25 26 6692 6704 6717 7459 7472 7485 8229 8242 8254 9000 9013 9026 9772 9785 9798 0544 0557 0570 '3'| 1328 134' 2083 2096 2109 2849 2861 2874 3610 3623 3635 27 28 29 30 32 33 6730 6743 6755 7497 7510 7523 8267 8280 8293 9039 9052 9065 98.1 9824 9837 0583 0595 0609 1353 1366 1379 2122 2134 2147 2887 2900 2912 3648 3661 3673 13 6768 7536 8306 9077 9850 0621 1392 2160 2925 3686 6781 6794 6806 7549 7561 7574 8319 8332 8344 9090 9103 9116 9862 9888 0634 0647 0660 1405 1418 1430 2173 2j86 2^98 2938 2950 2963 3699 371^ 3724 34 11 6819 6832 6844 7600 7613 8357 9129 9142 9155 9901 9914 9927 0699 1456 1469 221 1 2224 2237 3001 3736 3749 3762 10 20 30 40 50 60 3.2 H 10.8 13.0 37 38 39 40 41 42 43 6858 6870 6883 7626 7638 7651 839S S409 8422 9168 9180 9193 994° 9953 0711 0724 0737 1482 1494 1507 2249 2262 2275 3014 3027 3039 3787 3800 6896 7664 8434 9206 9978 0750 1520 2288 3052 3812 6909 6921 6934 7677 7690 7702 8447 8460 8473 9219 9232 9245 9991 •0004 *ooi7 0763 0776 0788 1533 1546 1559 2301 23.3 2326 3078 3090 3825 3838 3850 44 45 46 6947 6960 6973 77-5 7728 7741 8486 8499 8512 9258 9270 9283 •0030' •0043 •0055 0801 0814 0827 ■571 ■584 ■597 2339 2352 2364 3^03 3116 3128 3863 387s 3888 49 60 51 52 S3 6985 6998 701 1 7754 7767 7779 8524 8537 8550 9296 9309 9322 *oo68 •0081 •0094 0840 0853 0866 1610 1623 ■635 2377 2390 2403 3>4l 3I66 390" 39<3 3926 7024 7792 8563 9335 *oi07 0878 1648 2415 3^79 3938 7036 7049 7062 7805 7818 7831 It 8692 9348 936' 9373 *0I20 *oi33 •0146 0891 0904 0917 1661 2428 2441 2454 3 '92 3205 3217 3951 3964 3976 54 7100 7844 7856 7869 8614 8627 8640 9386 9399 9412 •0158 •0171 •0184 0930 0943 0955 1699 1712 1725 2466 2479 2492 3230 3243 3255 3989 4002 4014 57 58 59 60 7"3 7126 7139 7882 7895 7908 8653 8666 8679 9425 9438 9451 •0197 *02I0 •0223 0968 0981 0994 1738 '763 2505 2517 2530 3268 3281 3293 4027 4039 4052 7152 7921 8692 9464 •0236 1007 1776 2S43 3306 4065 Smithsonian Tables. 52 Table 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH FEET. [Derivation of table explained on p. xlv.] Lat. sr° 52° 53° 54° 55° 56° 57° S8° 59° 60° P.P. 7.320 7.320 7.320 7.320 7.320 7.320 7.320 7.320 7.320 7.321 0' I 2 3 4065 48.7 5564 6303 7034 7756 8467 9168 9857 0534 13 4077 4090 4102 4829 4842 4854 S57.S 5589 5601 6315 6327 6340 7046 7058 7070 7768 7780 7792 8479 8491 8j02 9180 9191 9203 9868 9880 9891 0545 0556 0567 4 5 6 41 IS 4127 4140 4867 4879 4892 5613 5625 5638 6352 6364 6376 7082 7094 7107 7804 7815 7827 8514 8526 8538 9214 9226 9238 9903 9914 9925 0578 0589 0601 2.2 9 10 II 12 13 4152 4165 4177 4904 4917 4929 5650 5662 5675 6388 6401 6413 7119 713' 7143 7839 lilt 8573 9249 9261 9272 9937 9948 9960 0612 0623 0634 20 30 40 50 60 4-3 8.7 10.8 13.0 4190 4942 5687 6425 7155 7875 8585 9284 9971 064s 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 4240 4253 4266 4992 5004 5017 5737 5749 5761 6474 6486 6498 7203 7215 7227 7923 7934 7946 8632 8643 865s 9330 9341 9353 *ooi6 *0027 *oo39 0689 0701 0712 17 ig 19 20 21 22 23 4278 4291 4303 5029 5042 5054 5774 5786 5799 6510 6523 6535 7239 725" 7263 7958 7970 7982 8667 8679- 8690 9364 9376 9387 *oo5o *oo6i *oo73 0723 0734 0745 4316 5067 5811 6547 7275 7994 8702 9399 •0084 0756 4328 4341 4353 5079 5092 5104 5823 5848 6559 6571 6584 7287 7299 7311 8006 8018 8030 8714 872s 8737 9410 9422 9433 •0095 *oi07 *oii8 0767 0778 0789 24 2I 4366 4378 4391 5117 S129 5141 5860 6620 7323 7335 7348 8042 8053 8065 8749 8760 8772 9445 *OX29 *oi40 *OI52 0800 0812 0823 12 27 28 29 30 31 32 33 4403 4416 4428 5179 5897 5909 5922 6632 664s 6657 7360 7372 7384 81Q1 8784 8796 8S07 9479 9491 9502 *oi63 *oi74 *oi86 0S34 084s 0856 20 30 40 50 60 t.l 8.0 10.0 12.0 4441 5I9I 5934 6669 7396 8113 8819 9514 *oi97 0867 I466 4479 5203 5216 5228 5946 5959 597" 6681 6693 6706 7408 7420 7432 8125 8137 8148 8831 8842 8854 9525 9537 9548 *0208 *02I9 *023I 0878 0889 0900 34 35 36 4491 4504 4517 5241 11^ 5983 5995 6008 6718 6730 6742 7444 7456 7468 8160 8172 8184 8866 8877 8889 9560 9571 9583 ♦0242 *Q253 *0264 0911 0922 0933 39 40 4' 42 43 4529 4542 4554 5278 5291 5303 6020 6032 6045 6754 6767 6779 7480 7492 7504 8196 8207 8219 8901 8913 8924 9594 9606 9617 *Q275 *0290 0944 11 4567 53 '6 6057 6791 7516 8231 8936 9629 ♦0309 0977 4579 4592 4604 5328 5341 5353 6069 6082 6094 6803 6815 6828 7528 7540 7552 8243 825s 8266 8948 8959 8971 9640 9652 9663 *0320 *0332 •0343 0988 0999 1010 44 45 46 4617 4629 4642 5366 5378 5390 6io6 6118 6131 6840 6852 6864 7564 7576 7588 8278 8290 8302 8982 8994 9006 9697 *03S4 •0365 *0377 102 1 1032 1043 10 1.8 49 50 51 52 53 4679 5403 54'5 5428 6143 6155 6168 6876 6889 6901 7600 7612 7624 8314 8325 8337 9017 9029 9040 9709 9720 9732 *0388 *0399 *o4ii >o54 1065 1076 20 30 40 lo 3-7 5-5 7-3 9-2 II. 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 till 9064 9°7S 9087 9777 *0433 *0444 •0456 1098 1 109 1 120 54 11 4742 4754 4767 5490 5502 5514 6229 6241 6254 6961 6973 6986 7684 7695 7708 8420 9098 9110 9122 9789 9800 981 1 •0467 ♦0489 1131 1 142 "53 59 60 4779 4792 4804 5527 5539 5552 6266 6278 6391 6998 7010 7022 7720 7732 7744 8432 8443 8455 9133 9145 915S 9823 9834 9846 *b5oo *05I2 •0523 1 164 1175 1186 4817 5564 6303 7034 7756 8467 9168 9857 *o534 1 197 Smithsonian Tablcs. SS Table 10. LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p„ FEET. IN ENGLISH [Derivation of table explained on p. xlv.] Lat. 61° 62° 63" 64° 6s= 66° 67° 68° 69° 70° p.p. 0' z 2 3 4 5 6 9 10 II 12 13 M 15 i6 17 i8 "9 20 21 22 23 24 25 26 :i 29 30 3' 32 33 34 11 37 38 39 40 4' 42 43 44 :i 49 so 51 52 53 54 P % 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 I2|I 1262 1273 1284 1295 1856 1866 1877 1888 1898 1909 1920 193' 1941 2489 2500 2510 2521 2531 2541 2552 2562 2573 3107 3117 3127 3'37 3158 3168 3>78 3188 3708 37"8 3728 3738 3747 3757 3767 3787 4292 4301 43" 4320 4330 4340 4349 4876 4885 4894 4904 49'3 4922 4932 5405 5414 5423 5432 544° 5449 5458 5467 5476 5933 5941 5950 5958 , 5976 5984 5993 6001 6440 6448 6457 6465 6506 10 20 30 40 "IS 1.8 3-7 5-5 7.3 9.2 11.0 1306 1952 2583 3 '98 3797 4378 4941 5485 6010 65-4 13 17 1328 ■338 1349 1360 "371 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 nil 4387 4397 4406 4416 4425 4435 4444 4454 4463 495° 4959 4969 4978 4987 4996 5005 5015 5024 5494 5503 5512 5521 5529 5538 5547 5556 5565 6018 6027 6035 6044 6061 6069 6078 6086 6522 6530 6539 6547 till 6571 6580 6588 10 1414 2058 26S7 3299 3895 4473 5033 5574 6095 6596 10 20 30 40 1^ ■•7 3-3 !■" 8.3 lO.O 1425 1436 1447 1458 1468 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 3380 3390 3905 3915 3924 3934 3944 3954 3964 3973 3983 4482 4492 4501 45" 4520 4530 4539 4549 4558 5042 5051 5060 5069 5097 5106 5"5 5583 5592 5600 i^ 5627 5636 5644 5653 6103 6112 6120 6129 6'37 6146 6.54 6163 617. 6604 6612 6621 6629 6637 6645 6653 6662 6670 e 1523 2164 2790 3400 3993 4568 5124 5662 618a 6678 '534 1545 ■555 1566 nil »599 1609 1620 217s 2185 2196 2206 2217 2228 2238 2249 2259 2800 2811 282 s 2831 2841 2852 2862 2872 2883 34IO 3420 3430 3440 3450 3460 3470 3480 3490 4003 4012 4022 4032 4041 4051 4061 4071 4080 4587 4596 4606 4615 4624 4634 4643 4653 5133 5142 5IS' 5160 5'69 5179 5188 5197 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 675' 10 20 30 40 •■5 3-0 ti 7-S 9.0 163 1 2270 2893 3500 4090 4662 5215 5750 6264 6759 8 1642 1674 1684 1695 1706 1717 1727 2280 2291 2301 2312 2322 2333 2343 2364 2903 2913 2924 2934 2944 2954 2964 2985 35'0 3520 3530 3540 3549 3559 3569 3579 3589 4100 4109 4119 4128 4138 4148 4157 4167 4176 4671 4681 4690 4695 4708 4718 4727 4736 4746 5224 5233 5242 5260 5270 5279 5288 5297 5759 5767 5776 5785 5793 5802 5811 5820 5828 6272 6281 6289 6298 6306 6314 6323 633' 6340 6767 6775 6783 6791 ^^ 6815 6823 6831 10 20 30 40 50 6a "■3 2.6 4,0 li 8.0 1738 2375 2995 3599 4186 4755 5306 5837 6348 6839 1749 1759 1770 1781 1791 1802 1813 1824 1834 2385 2396 2406 2417 2427 2437 2448 2458 2469 300s 3015 3026 3036 3046 3056 3066 3609 3619 3629 Itlt 3658 3668 3678 3688 4196 4205 4215 4224 4234 4244 4253 4263 4272 4764 4774 4783 48II 4820 4829 4839 53>5 5324 5333 5342 535- 5360 5369 5378 5387 5846 5854 5863 5880 5889 5898 5907 5915 6356 6365 6373 6382 6390 6398 6407 64/5 6424 6847 lUl 6871 tm 6895 6903 69II 1845 2479 3097 3698 4282 4848 5396 5924 6432 6919 Smiths ONIAN 7 *BLEa> 54 LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE FEET. [Derivation o£ table explained on p. xlv>] Table 10. IN ENGLISH Lat. 71° 72° 73° 74° 75° 76° 77° 78? 79° 80° P.P. 0' 1 2 3 4 5 6 I 9 10 II 12 13 ■4 \l \l 19 20 21 22 23 M 25 26 27 28 29 30 31 32 33 34 35 36 11 39 40 41 42 43 44 11 :i 49 60 5' 52 S3 54 11 11 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 6958 6966 6974 6982 6990 7392 7400 74°7 7415 7422 7430 7437 7445 7452 7836 7843 785. 7858 7865 7872 7879 7887 7894 8258 8265 8271 8278 8285 8292 8299 8305 8312 8636 8563 8669 8576 8682 8688 869s 8701 8708 9031 9037 9043 9049 9055 9061 9067 9073 9079 9388 9394 9399 940s 94" 9416 9422 9427 9709 9714 9720 9725 9730 9735 9740 9746 9751 0012 0017 0021 0026 0031 0036 cx)4i 0045 0050 0288 0293 0297 0302 0306 0310 0315 0319 0324 10 20 30 40 50 60 1-3 2.6 4.0 !•' 8.0 6998 7460 7901 8319 8714 908s 9433 9755 0055 0328 7 7006 7014 7021 7029 7037 7°45 7053 7060 7068 7467 7482 7490 7497 7505 7512 7520 75^7 7908 7915 7922 7929 7936 7944 7951 7958 7965 8326 8332 8339 8346 8353 8359 8366 8373 8379 8720 8727 8733 8739 8745 8752 8758 8764 8771 909 > 9097 9103 9.09 9115 9121 9127 9133 9139 9438 9444 9449 9455 9460 9456 9471 9477 9482 9761 9766 977' 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 5° 60 1.2 2.3 3.5 4-7 5.8 7.0 7076 7535 7972 8386 8777 9145 9488 9807 0102 0371 7084 7092 7099 7107 7115 7 "3 713 1 7138 7146 7542 7550 7557 7565 7572 75S0 7587 7595 7602 7979 7986 7993 8000 S007 8014 802 1 8028 803s 8393 8399 8406 8413 8419 8426 8433 8440 8446 8783 8790 8796 8802 8808 88.5 8821 8827 8834 9151 9157 9>53 9169 9174 9180 9186 9192 9198 9493 9499 9504 9510 9515 9521 9526 9532 9537 9812 9817 9822 9827 9832 9838 9848 9853 0107 OIII 01 16 0120 01 25 0130 o>34 0139 0143 0375 0379 0384 0388 0392 0396 0400 0405 0409 6 10 20 30 40 IS I.O 2.0 3.0 4.0 l:S 7154 7610 8042 8453 8840 9204 9543 9838 0148 0413 7162 7170 7177 7185 7193 7201 7209 7216 7224 7617 7625 7632 7639 7646 7654 7661 7668 7676 8049 8056 8063 8070 8077 80R4 8091 8098 8105 8460 8465 8473 8479 8486 8493 8499 8506 8512 8845 8852 8859 8865 8871 8877 8883 8890 8896 9210 9216 9221 9227 9233 9239 9245 9250 9256 9548 9554 9559 9555 9570 9575 «!; 9586 9592 9873 ^f 9883 9888 9893 9898 9903 0153 0157 0162 oi56 0171 0176 0180 0185 0189 0417 0421 0426 0430 °434 0438 0442 0447 0451 5 10 20 30 40 IS .8 1-7 2-5 3-3 4-2 50 7232 7683 8lT2 85 19 8902 9262 9597 9908 0194 0455 7240 7247 7255 7263 7270 7278 7286 7^94 7301 7690 7798 7705 7712 7719 7727 7734 7741 .7749 8119 8126 8133 8<40 8147 8154 8161 8168 8175 8526 8532 8539 8545 8552 8559 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 9535 9640 9645 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 IS ■7 1-3 2.0 2.7 3-3 4.0 7309 7756 8182 8585 8964 9320 9651 9958 0240 0496 7317 7324 7332 7339 7347 7355 7362 7370 7377 7763 777' 7778 7785 7792 7800 7807 7814 7S22 8189 8196 S203 8210 8216 8223 8230 8237 8244 8598 8604 861 1 8617 8624 8630 8537 8643 8970 8988 8994 9001 9007 9013 9019 9326 9331 9337 9343 9348 9354 9360 9365 9371 9556 9662 9667 9672 9677 9683 9688 9693 9699 9963 9968 9973 9978 9982 9987 9992 »9997 *0002 0244 0249 0253 0258 0262 0266 0271 0275 0280 0500 0504 0508 0512 0516 0520 0524 0528 0532 738s 7829 8251 8650 9025 9377 9704 *ooo7 0284 0535 Smithsonian Tables. ss ^LOCARITHMS OF MERIDIAN RADIUS OF CURVATURE p„, IN ENGLISH FEET. [Derivation of table explained on p. xlv.] Lat. 81° 82° 83° 84° 8s° 86° 87° 88° 89° P.P. 0' I 2 3 4 I I 9 10 II 12 13 M 15 i6 \l ■9 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 \l 49 60 51 52 53 54 55 56 11 59 60 7.322 0536 0540 0544 0548 0552 0556 0560 0564 0568 0572 7.322 0763 7.322 0963 7.322 1138 7.322 1285 7.322 J407 7.322 ■ 501 7.322 ■ 569 7.322 i6og 4 0766 0770 0773 0777 0780 0784 07S7 0791 0794 0966 0969 0972 0975 0978 0982 ^^1 0991 1141 "43 1146 114S 1151 1 154 1156 1 159 ii6i 1287 1289 1292 1294 1296 129S 1300 1303 ■305 1409 1410 1412 1414 ■4IS 1417 1419 1421 1422 1502 ■ 504 '505 ■506 1507 ■ 509 ■S^o ■ 5" ■5^3 1570 1571 ■57' 1572 ■573 ■574 ■575 ■575 ■ 576 1609 1610 1610 1611 1611 1611 1612 1612 1613 ID 20 30 40 50 60 •7 ■•3 2.0 2.7 3-3 4.0 0376 0798 0994 1 164 1307 1424 IS^4 ■577 1613 0580 □588 0592 0595 0599 0603 0607 0611 0801 0805 0808 0812 0815 0819 0822 0826 0829 0997 1000 1003 1006 1009 1012 1015 1018 1021 1 167 1169 1172 1 174 1 180 1 182 1 185 1 187 1309 1311 i3>4 1316 1318 1320 1322 1325 1327 1426 ■427 1429 ■43' ■432 ■434 1436 ■43S ■439 ■515 1517 ■s^s 1519 ■ 520 ■ 522 ■ 523 ■ 524 ■ 526 ■ 578 ■579 ■579 ■ 5S0 ■ 581 1582 ■ 583 ■ 583 ■584 1613 1614 1614 1615 1615 1615 1616 1616 1617 3 0615 0833 1024 1 190 1329 1441 ■527 ■ 585 1617 10 20 30 40 .5 I.O ■•5 2.0 2-5 3.0 0619 0623 0626 0630 0634 0638 0642 0645 0649 0836 0840 0843 0846 0849 0853 0856 0859 0863 1027 1030 1033 1036 1039 1042 1045 1048 1051 1192 "95 "97 1200 1202 1205 1207 1210 1212 1331 1333 1335 1337 1339 1341 1343 1345 1347 ■443 1446 1447 ■449 ■45' 1452 ■454 ■455 1528 1529 1530 '53^ 1532 ■ 534 '"1 ■536 ■537 1586 1586 ■ 587 1588 .588 ■589 ■ 590 ■ 591 1591 1617 1617 1618 1618 • 1618 161S 1618 1619 1619 2 0653 0S66 IOS4 121S 1349 ■457 1538 1592 1619 0657 0660 □664 0668 0671 0675 0679 0683 0686 0S69 0879 0882 0886 08S9 0892 0896 1057 1060 1062 1065 1068 107 1 1074 1076 1079 1217 1220 1222 1225 1227 1229 1232 1234 1237 1351 ■353 1355 ■357 '359 1361 1363 1365 ■367 ■459 1460 1462 1463 ■465 1467 1468 1470 1471 ■539 ■540 ■541 ■542 ■543 ■545 1546 ■547 1548 ■593 ■593 ■594 ■595 ■595 1596 1598 1619 1619 1620 1620 1620 1620 1620 1621 1621 10 20 30 40 •3 •7 1.0 ■ ■3 '•7 2.0 o6go 0899 1082 1239 ■369 ■473 ■549 ■599 1621 0694 0697 0701 0705 0708 0712 0716 0720 0723 0902 0906 0909 0912 091S 0919 0922 0925 0929 1085 1088 1090 1093 1096 1099 1 102 1104 I107 124I 1244 1246 1249 I25I 1253 1256 1258 1261 1371 ■373 ■375 ■ 377 1378 ■380 1382 ■384 1385 ■474 1476 ■477 ■479 1480 1481 1483 ■484 i486 ■550 ■55' ■ 552 ■553 ■554 ■555 1556 ■557 ■558 ■599 1600 1600 1601 1601 1602 1602 1603 1603 1621 1621 162 1 1621 162 1 1622 1622 1622 1622 1 0727 0932 I no 1263 ■388 ■487 ■559 1604 1622 10 . 20 30 40 To .2 .3 • 5 :l 1.0 0731 0734 0738 0741 °745 0749 0752 0756 0759 0935 0938 0941 0944 0947 0951 09S4 0957 0960 1113 1116 .118 1121 1 124 1127 I130 1 132 "35 1265 1267 1270 1272 1274 1276 1278 1281 1283 ■ 390 1392 ■394 1396 ■397 ■399 1401 ■403 1405 1488 1490 1491 ■493 1494 ■495 ■497 1498 1500 1560 ■ 561 1562 ■563 1564 1565 ■ 566 1567 ■ 568 1604 1605 ■605 l6c5 1606 1607 1607 1608 1608 1622 1622 1622 1622 1622 1623 J623 1623 1623 - 0763 0963 .138 1285 1407 ■ 501 1569 1609 1623 Gmithsonian Tables. 56 Table 1 1 . LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION p„ IN ENGLISH FEET. [Derivation of table explained on p. xlv.] Lat. 0° 1° 2° 3° 4° s° 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 0' I 2 3 6875 6S80 6893 6916 6947 6987 7036 7094 7160 7235 7319 687s 687s 6875 6S8o 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 6S94 6894 6895 69.8 6918 6918 6950 6950 6951 6990 6991 6992 7040 7041 7041 7098 7099 7100 7165 7166 7167 7240 7241 7243 7325 7326 7327 9 10 II 12 13 6875 6875 6881 6881 6881 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 687s 6881 6896 6920 6953 6995 7045 7104 7172 7248 7333 6875 6875 6881 6881 6882 6896 6897 6897 6920 6921 6921 6954 6955 6955 6996 6996 6997 7046 7047 7048 7105 7106 7107 7173 7'74 7176 7249 7251 7252 7334 7336 7338 14 i6 687s 6876 6876 6882 6882 6882 6898 6898 6898 6922 6922 6923 6956 6956 6957 6998 6999 6999 7049 7050 7050 7108 7109 7111 7177 7178 7179 7254 7255 7256 7339 7341 7342 10 20 30 40 50 60 .2 •3 ■5 •7 .8 I.O 11 ■9 20 21 22 23 6876 6876 6876 6882 5883 6883 6899 6899 6goo 6923 6924 6924 6957 6958 6959 7000 7001 7001 7051 7052 7053 7112 7113 7114 7180 7182 7183 7258 7259 7261 7343 7345 7346 6876 6883 6900 6925 6959 7002 7°S4 7115 7184 7262 7348 6876 6876 6876 6883 6883 6884 6900 6901 6901 6925 6926 6926 6960 6960 6961 7003 7004 7004 7°S5 7056 7057 7116 7117 7118 7185 7186 7188 ■7263 7265 7266 7350 7351 7353 24 25 26 6876 6876 6876 6884 6884 6884 6goi 6902 6902 6927 6927 6928 6962 6962 6963 7006 7007 7058 7059 7060 7119 7120 7122 7189 7190 7191 7268 7269 7270 7354 7356 7358 27 28 29 30 31 32 33 6875 6876 6876 6884 6885 6885 6902 6go2 6903 6928 6929 6929 6964 6965 6;65 7008 7008 7009 7061 7062 7063 7123 7124 712s 7192 7'94 7195 7272 7273 7275 7359 7361 7362 6876 6885 6903 6930 6966 7010 7064 7126 7196 7276 7364 6877 6877 6877 6885 6886 6886 6903 6904 6904 6930 6931 6931 6967 6967 6968 701 1 7012 7013 7067 7128 7129 7197 7199 72CX) 7277 7279 7280 7366 7367 7368 34 35 36 6877 6877 6877 6886 6887 6887 6905 6905 6905 . 6932 6932 6933 6969 6969 6970 7014 70'5 7015 7068 7069 7070 7130 7131 7133 7201 7202 7204 7282 7283 72S4 7370 7371 7373 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 7>35 7136 7205 7206 7208 7286 7287 7289 7374 7376 7377 2 6877 6888 6907 6935 6973 7019 7073 7137 7209 7290 7379 10 20 3° .3 ■7 1.0 1-3 6877 6877 6877 6883 6888 6889 6907 6908 6909 6936 6936 6937 6974 6974 6975 7020 7021 7021 7074 7075 7076 7138 7139 7140 7210 7212 7213 7291 7293 7294 7381 7382 7384 44 49 60 ^3 6877 6878 6878 6878 6878 6878 6889 6889 6889 6889 68;o 6890 6909 6910 6910 6910 6911 691 1 6937 6938 6938 6939 6939 6940 6976 6976 6977 6978 6979 6979 7022 7023 7024 7025 7025 7026 7077 7078 7079 7080 7081 7082 7141 7142 7144 7146 7147 7214 7216 7217 7218 7219 7221 7296 7297 7298 7300 7301 7303 7385 7387 7389 7390 7392 7393 50 60 1-7 2,0 6878 6890 691 1 6941 6980 7027 7083 7148 7222 7304 7395 6878 6878 6879 6890 6891 6891 6gi2 6912 6913 6942 6942 6943 6981 6981 69S2 7028 7029 7030 7084 7085 7086 7149 7150 7152 7223 7225 7226 7305 7307 7308 7397 7398 7400 54 11 6879 6891 6892 6892 6913 6914 6914 6943 6944 6944 6983 6983 6984 7031 7032 7032 7090 7153 7154 715s 7227 7228 723° 7310 73" 7313 7401 7403 7405 57 58 59 60 6S80 6892 6892 6S93 691 5 6945 6945 6946 6985 6986 6986 7033 7034 7035 7091 7092 7'S6 7158 7159 7231 7232 7234 7314 7316 7317 7406 7408 7409 68S0 6893 6916 6947 6987 7036 7094 7160 7235 7319 741 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° 15° 16° 17° 18° 19° 20° p.p. 0' I 3 3 4 7 8 9 10 11 12 13 H IS i6 »7 l8 19 20 21 22 23 24 25 26 '7 28 29 30 31 32 33 34 u 37 38 39 40 41 42 43 44 45 46 47 48 49 60 SI 52 S3 54 S5 % 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 859s 1 7413 7414 7416 7417 7419 7421 7422 7424 7425 7S>3 7514 7Si6 7518 7519 7521 7523 7525 7526 7621 7623 7625 7627 7628 7630 7632 lilt 7738 7740 7742 7744 7746 7748 7750 7752 7754 7862 7864 7867 7869 7871 7873 llll 7994 7997 7999 8001 8003 8006 8008 8010 8013 8134 8.37 8>39 8142 8144 8146 8149 8151 8154 8282 8284 8287 8289 8292 829s 8297 8300 8302 8437 8439 8442 8444 8447 8450 8452 8455 8457 8598 8601 8603 86o6 8609 86i2 8615 8617 8620 10 20 30 40 .2 •3 •5 -.1 1.0 7427 7528 7638 7756 7882 8015 8156 8305 8460 8623 7429 7430 7432 7433 743S 7437 7438 7440 7441 7530 7532 7533 7S3S 7537 7539 7S4I 7S42 7544 7640 7642 7644 7646 7647 7649 7651 7653 7655 77S8 7760 7762 7764 7766 7768 7770 7772 7774 7884 7886 7888 7890 7892 7895 7897 7899 7901 8017 8020 8022 8024 8026 8029 8031 8033 8036 8158 8161 8163 8166 8168 8170 8.73 8175 8178 8307 8310 8312 8315 8317 8320 8322 832s 8327 8463 til 8471 8473 8476 8479 8482 8484 8626 8629 8631 8634 8637 864a 8643 8648 2 7443 7546 7657 7776 7903 8038 8180 8330 8487 8651 7445 7446 7448 745° 7451 7453 74SS 7457 7,458 7548 7550 75SI 7553 75S5 7557 7SS9 7560 7562 7659 7661 7663 7665 7666 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 8049 8052 io^6* 8059 8182 8185 8187 8190 8192 819s 8197 8200 82^2 8333 till 8340 8343 8346 8348 8351 8353 8490 8492 849s 8498 8500 8503 8506 8509 8511 8654 8657 8659 8662 8665 8668 8671 8673 8676 10 20 30 40 lo .3 •7 I.O 1.3 1-7 2.0 7460 7564 7675 7797 7925 8061 8205 8356 8514 8679 7462 7463 7465 7465 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 78.2 7816 7927 7929 79J2 7934 7936 793S 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 8376 8378 8517 8519 8522 8525 mi P! 8538 8682 8685 8687 8690 ^^ 8699 8701 8704 3 7476 7582 7696 7818 7947 8085 8229 8381 8541 8107 7478 7479 7481 7483 7484 7486 7488 7490 7491 7584 "ft 7588 7590 7591 7593 7595 7597 7599 J698 770D 7702 7704 7706 7708 7710 7712 7714 7820 7822 7824 7826 7828 7831 7S33 783 s 7837 7949 7952 7954 7956 7958 7961 7963 7965 7968 8087 8090 8092 8094 8096 8099 8101 8103 8106 8231 8234 8236 8239 8241 8244 8246 8249 8251 8384 8386 8389 8391 8394 8397 8399 8402 8404 8544 8546 8549 8552 8554 8S57 8560 8563 8565 8710 8713 871S 8718 8721 8724 8727 8729 8732 10 20 30 40 50 60 ■5 1.0 1-5 2.0 2.S 3.0 7493 7495 7497 7498 7500 7502 7504 7506 7507 7509 7601 7716 7839 7970 8108 • 8254 8407 8568 8735 7603 7605 7606 760S 7610 7612 7614 7615 7617 7718 7720 7722 7724 7726 7728 7730 7732 7734 7841 7843 7845 7847 7849 7852 7854 7856 7fs8 7972 7974 7977 7979 7981 7983 ^^8i 7c,9o 8110 81 13 8115 8118 8120 8122 8125 8127 8130 8256 8259 8261 8264 8266 8269 8271 8274 8276 8410 8412 8415 8418 8420 8423 8426 8429 8431 8571 nil 8584 8587 8590 8592 8738 8741 8743 8746 8749 8752 875s lllL 7SII 76.9 7736 7860 7992 8.32 8279 8434 8595 8763 Smiths 3NIAN T ABLES. 58 LOGARITHMS OF Table 11, RADIUS OF CURVATURE OF NORMAL SECTION p, IN ENGLISH FEET. [Derivation of table explained on p. xlv.] Lat. 21° 22° "■ 24° 25° 26° 27° 28° 29° 30° p. P. 0' X 2 3 4 I 7 8 9 10 ZI 13 »3 '4 15 l6 '7 i8 "9 20 21 22 23 24 11 11 29 30 31 33 33 34 35 36 37 38 39 40 41 4> 43 44 ti tl 49 60 SI S3 S3 S4 II 11 59 60 7.320 8763 7.320 8939 7.320 9120 7.320 9308 7.320 9502 7.320 9701 7.320 9907 7.321 0117 7.321 0333 7.321 0553 a 8766 8769 8772 !"5 8778 8780 8784 8786 8789 8942 8945 8948 895- 8953 8956 8959 8962 8965 9"3 9126 9.39 9133 9136 9139 9142 9145 9148 9311 9314 9318 9331 9324 9337 9330 9333 9337 9505 9508 9S>3 95-5 95 '8 9521 9525 9528 9531 9705 9708 9712 97'S 9718 9723 9735 9728 9732 9910 99>3 9917 9920 9924 9927 993" 9934 9938 012I 0124 O12S OI31 0135 0138 0143 OI4S 0149 0336 0340 0343 0347 035 ■ 0354 0358 0361 0365 0556 0560 0564 0567 0571 OS7S 0579 0582 0586 8793 8968 9151 9340 9535 9735 9941 o"53 0369 0590 10 20 30 40 50 60 ■3 •7 I.O 1-3 1-7 3.0 8795 8804 8807 SSio 8812 88li 8971 8974 8977 8980 8983 8986 8989 8992 8995 9154 9157 gi6o 9163 9167 9170 9'73 9176 9"79 9343 9346 9349 9353 9356 9359 9363 9365 9368 9538 954' 9545 9548 955" 9554 9558 9561 9564 9739 9742 9745 9749 9752 9756 9759 9763 9766 9945 9948 9952 9955 9959 9962 9966 9969 9973 0156 0159 0163 0167 0170 0174 0177 0181 0185 0372 0376 0380 0383 0387 0391 0394 0398 0403 0594 0597 0601 0605 0608 0612 0616 0620 0633 3 8821 8998 9182 9373 9568 9769 9976 0188 040s 0627 8824 8827 8830 8833 8836 8839 8841 %** 8847 9001 9004 9007 goto 9013 9016 9020 9<'33 9026 '9'85 9188 9191 9195 9198 9201 9204 9207 9210 9375 9378 9381 9384 9388 9391 9394 9398 9401 9571 9574 9578 9581 9584 9588 9591 9594 9598 9773 977S 9779 9783 9785 9790 9793 9795 9800 9980 9983 9987 9990 9994 9997 *OOOI *ooo4 *ooo8 0193 0195 0199 0203 0206 0210 03I3 0217 0220 0409 0413 0416 0420 0434 0427 0431 0435 0438 063 c 0635 0638 0642 0646 0649 0653 0657 0661 10 20 3° 40 1: ■S 1.0 1-5 3.0 3 5 3.0 8850 9029 9213 9404 9601 9803 *OOII 0224 0442 0664 8853 8856 8859 8862 8865 8868 8871 8874 8877 9032 9035 9038 9041 9044 9047 9050 9053 9056 9216 9220 9323 9226 9229 9233 923s 9338 9242 9407 9411 9414 9417 9420 9424 9427 9430 9433 9604 9608 9611 9614 961S 9631 9624 9628 9631 9807 981a 9814 9817 9830 9824 9827 9831 9834 *ooi5 *ooi8 *0032 *oo25 *0029 *oo32 ♦0036 •0039 *cx>43 0228 0231 0335 0238 0242 0346 0249 0253 0256 0445 0449 0453 0457 0460 0464 0468 0471 0475 0668 0672 0676 fell 0687 0691 0694 0698 4 8879 9059 924s 9437 9634 9838 *oo46 0260 0479 0702 8882 888s 8888 8891 8894 S897 8900 8903 8906 9062 906s 906S 9071 9074 9077 9080 9083 9086 9248 9251 9254 9257 9260 9264 9267 9270 9273 9440 9443 9446 9450 9453 9456 94S9 9463 9466 9638 9641 9644 9648 9651 9654 9658 9661 9664 9841 9848 9851 985s 9858 9863 9865 9869 *oo5o *oo53 *ooS7 *oo6o *oo64 »oo67 »oo7, *oo74 ♦0078 0264 0267 0271 0274 0278 0382 028s 0289 0293 0482 0486 0490 0493 0497 0501 050s 0508 0512 0706 0710 0713 0717 ■0721 0725 0728 0732 0736 10 20 30 40 •7 1-3 3.0 3-7 3-3 4.0 8909 9089 9275 9469 9668 9873 »oo82 0196 0516 0740 8912 8915 8918 8931 8924 8927 8930 8933 8936 9093 9095 9099 gio2 910S 9108 9111 91 14 9117 9279 9286 9289 9292 9295 9298 93?3 93°5 9472 9476 9479 9482 9485 9489 9492 9495 9498 9671 9678 9681 9685 9688 9691 9695 9698 9875 9886 98S9 9893 9896 9900 9903 *oo85 *ooS9 *0092 *bo96 *oo99 0103 *oio6 *OTIO *oii3 0300 0303 0307 03II 03 >4 0318 0322 °3ZS 0329 0519 0533 0527 0530 0534 0538 0542 054s 0549 0743 0747 0751 0755 0759 0762 0766 0770 0774 8939 9120 9308 9S03 9701 9907 *oii7 0332 0553 0777 «*- Smithsonian Tables. 59 Table 1 1 . LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION Pn IN ENGLISH FEET. [Derivation of table explained on p. xlv.] Lai. 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' X 0777 1006 ■239 1476 1716 '959 2205 2453 2704 2956 0781 lOIO 1243 1480 1720 '963 2 209 2457 2708 296^ 3 0785 IOI4 1247 .484 1724 ■967 2213 2462 2712 2965 3 0789 1018 125 1 1488 1728 '97 ■ 22^7 2466 2716 2969 3 4 1 0793 0796 1022 1026 1255 1259 1492 1496 1732 1736 '975 ■979 2221 2226 2470 2474 2721 2725 2973 2978 odoo 1029 1263 1500 1740 ■983 2230 2478 2729 2982 10 ■5 z 0804 1033 1267 1504 1744 1988 2234 248J 2733 2986 20 I.O 8 o8o3 1037 1271 1508 1748 ■992 2238 2487 2737 2990 30 1-5 9 10 II 081 1 104 1 "275 1512 ■752 ■996 2242 249' 2742 2994 40 60 2.0 2-S 3.0 081S I04S 1279 1516 1756 2000 2246 2495 2746 2999 0819 1049 1282 1520 1760 2004 2250 2499 2750 3003 12 0823 1053 1286 ■524 ■ 764 2008 2254 2503 2754 3007 ■3 0827 IOS7 1290 1528 ■768 20^2 2259 2507 2758 3o^i 14 0830 1060 1294 1532 1772 2o^6 2263 2512 2763 3016 «S °Pi 1064 1298 1536 1776 20ZO 2267 2316 2767 3020 l6 0838 1068 1302 1540 1780 2024 iiyi '2520 2771 3024 17 0842 1072 J 306 '544 ■784 2028 22 75 2524 2775 3028 i3 0846 1076 1310 1548 1789 2033 2279 2528 2779 3032 * ■9 20 21 0S49 1080 '3'4 ■ 552 ■793 2037 2283 2532 2784 3037 0853 1084 131S 1556 ■797 2041 2287 2537 2788 3041 0857 1087 1322 1560 ■ 8o^ 2045 2292 254^ 2792 3045 22 0861 Togi 1326 'S^ ■ 805 2049 2296 2545 2796 3049 23 0865 1095 ■330 1568 ■809 2053 2300 2549 2800 3054 4 24 25 0869 0872 1099 1103 1334 1337 ■ 572 1576 ■813 ■ 817 2057 2061 2304 2308 25-3 2557 2805 2809 3058 3062 26 0S76 1107 <34l 1580 l82^ 2065 23^2 2562 2813 3066 10 ■7 '1 088a iiii 1345 1584 ■825 2069 2316 2 65 2817 3071 20 '•3 28 0884 1115 1349 1588 ■ 829 2073 232^ 2570 2822 3075 30 40 1^ 3.0 29 30 31 0888 1118 1353 1592 ■833 2077 232s 2574 2826 3079 2-7 3-3 4.0 0891 089s 1122 I3S7 1596 ■837 2082 2329 2578 2830 3083 1126 13'ii 1600 i84^ 2086 2331 2583 2834 3087 32 0899 1130 1365 1604 ■845 2090 2337 2587 2838 3092 33 0903 ■ 134 1369 1 60S ■ 849 2094 234^ 2591 2843 3096 34 0907 1.38 1373 1612 ■853 2098 2345 2595 2847 3100 ^1 0910 1 142 J377 l6i5 1857 2I02 2350 2599 285 ■ 3104 36 0914 1146 . 1381 1620 i86^ 2106 2354 2603 2855 3109 H 0918 1150 1385 1624 ■ 865 2110 2358 2608 2859 3>^3 38 0922 1153 '389 1628 ■ 870 2^^4 2362 26^2 2864 3"7 39 40 41 0926 >"57 1393 1632 ■ 874 2^^9 236S 2S16 2868 3I2I 0930 ii6[ 1397 1636 1878 2123 2370 2620 2872 3'26 0933 ii6s 1401 1640 1882 2^27 2374 2624 2876 3'30 42 0937 1 169 1405 1644 ■ 886 2^3i 2379 2629 2880 3'34 43 0941 1 173 1409 1648 ■ 89a 2135 2383 2633 2885 3'38 6 44 0945 0949 1 177 Ii8i 1412 1416 1652 1656 \1'4 2139 2'43 2387 239' 2637 2641 2889 2893 3143 3'47 46 0953 n8s 1420 1660 ■902 2147 2395 2'45 2897 3'5' 10 .8 *l 0956 1 189 1424 'Iti ■906 2I5I 2399 2649 2902 3'SS 20 30 40 50 60 - '-7 2.5 3.3 4.2 5-0 48 09^0 1192 1428 1668 19^0 2156 2403 2634 2906 3'6o 49 60 51 0964 i,g6 1432 1672 1914 2160 2408 2658 2910 3 '64 0968 1200 ■436 1676 ■ 9^8 2164 2412 2662 2914 3 '68 0972 1204 1440 1680 1922 2168 24^6 2666 2918 3172 3'77 52 0976 1208 1444 1684 ■ 326 2172 2420 2670 2923 53 0979 1212 1448 1688 IJ3" 2176 2424 2675 2927 3181 54 °9S3 1216 1452 169Z '935 2180 2428 2679 2931 3185 11 0387 1220 1456 1696 '939 2^84 2433 2683 2935 3189 0991 1224 1460 1700 I 43 2188 2437 2687 2940 3193 11 099s 1228 .464 1704 '947 2193 244^ 2691 2944 3198 3202 0999 1231 1468 1708 ■951 2197 2445 2696 2948 59 60 1003 1235 1472 1712 '955 220^ 2449 2700 2952 3206 1006 1239 1476 1716 '959 2205 2453 2704 2956 3210 Smiths ONIAN 1 '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° 50° P.P. 7.321 7.321 7.321 7.321 7.321 7.321 7.321 7.321 7.321 7.321 1 2 3 3210 3466 3722 3979 4236 4494 4751 5007 5263 5517 3215 3219 3223 3470 3474 3479 3726 3731 3735 3983 3988 3992 4241 4245 4249 4498 45°2 4507 4760 4764 5012 5016 5020 5267 5271 5276 5522 5526 5530 4 5 6 3227 3232 3236 3483 3487 3491 3739 3744 3748 3996 4001 4005 4254 4258 4262 45" I 4515 4520 4768 4772 4777 5024 5029 5"33 5280 5284 5288 5534 5538 5543 I 9 10 II 12 13 3240 3244 3249 3496 3500 3504 3752 3756 3761 4009 40'3 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 3361 3266 3513 3517 3521 3769 3774 3778 4026 4031 4035 4284 4288 4292 454" 4545 4550 4798 4802 4807 5054 5059 5063 5310 5314 5318 5564 5568 5572 ■4 'S i6 3270 3274 3278 3526 3530 3534 3782 3786 3791 4039 4043 4048 4297 4301 4305 4554 4558 4562 481 1 481S 4819 5067 5071 5076 5322 5327 5331 5576 5581 5585 10 .7 >9 20 21 22 23 32S3 3287 3291 353S 3543 3547 3795 3799 3803 4052 4056 4061 4309 4314 4318 4567 4571 4575 4824 4828 4832 5080 5084 5088 5335 5339 5344 5589 5593 5598 30 40 2.0 2-7 3-3 4.0 3295 3551 '3808 4065 4322 4580 4837 5°93 5348 5602 3300 3304 3308 3560 3564 3812 3S16 3821 4069 4073 4078 4327 4331 4335 4584 4588 4592 4841 4845 - 4849 5097 5101 5 -05 5352 5356 5361 5606 5610 5614 24 25 26 3312 3317 3321 3568 3573 3577 3825 3829 3833 4082 4086 4091 4339 4348 4597 4601 4605 4854 5110 5114 5118 5365 5369 5373 . 5619 5623 5627 27 28 29 30 31 32 33 3325 3329 3334 35S1 3585 3590 3838 3842 3846 4095 4099 4104 4352 4361 4610 4614 4618 4866 4871 4875 5 '23 5127 5131 5378 5382 5386 5631 5636 5640 3338 3594 3851 4108 4365 4622 4879 5135 5390 5644 3342 3347 3351 3598 3602 3607 3855 4..2 4tio 4121 4369 4374 4378 4627 4631 4635 4884 4888 4892 5140 5'44 514S 5395 5399 5403 5648 5652 5657 34 35 36 3355 3359 3364 361 1 3615 3620 3868 3872 3876 4125 4129 4134 4382 4387 4391 4640 4896 4901 4905 S152 5157 5161 5407 5412 5416 5661 5665 5669 11 39 40 41 42 43 3368 3372 3376 3632 3881 388s 3889 4138 4142 4146 4395 4399 4404 4652 4657 4661 4909 4913 4918 5165 5169 5174 5420 5424 5428 5673 5678 5682 6 10 20 30 40 5° .8 ■■7 2-5 3-3 4-2 5.0 3381 3637 3893 4151 4408 4665 4922 5178 5433 5686 3385 3389 3393 3641 3645 3649 3898 39°2 3906 4155 4159 4164 4412 4417 4421 4670 4674 4678 4926 4931 4935 5182 5186 5191 5437 5441 5445 5690 5694 5699 44 45 46 3398 34°2 3406 3654 3658 3662 39" 3915 3919 4168 4.72 4176 4425 4430 4434 4682 4687 4691 4939 4943 4948 5195 5199 5203 5450 5458 5703 5707 57" 47 48 49 60 51 52 53 3410 3415 3419 3667 3671 3675 3923 3928 3932 4,81 4.85 41S9 4438 4442 4447 4695 4700 47°4 4952 4956 4960 5208 5212 5216 5462 5467 5471 5716 5720 5724 3423 3679 3936 4194 4451 4708 496s 5220 5475 5728 3427 3432 3436 36S4 36SS 3692 3941 3945 3949 4i98 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 3705 3953 3958 3962 4211 4215 4219 4468 4472 4477 4725 4730 4734 4982 4986 4990 5237 5242 5246 5492 5496 5500 5745 5749 5753 59 60 3453 3457 3462 3709 3714 3718 3966 3971 3975 4224 422S 4232 4481 4485 4490 4738 4742 4747 4995 4999 5003 5250 5254 5259 5505 5509 5513 5758 5762 5766 3466 3722 3979 4236 4494 4751 5007 5263 5517 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.3 Lat SI" S2° 53° 54° 55° 56° 57° S8° 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' I 2 3 5770 6021 6270 6517 6760 7CX3I 7238 7472 7701 7927 6 5774 5778 5783 6025 6029 6034 6274 6278 6282 6521 6525 6529 6764 5768 6772 7005 7009 7° '3 7242 7246 7250 7476 7480 7483 7705 7709 7712 7931 7934 7938 4 1 5787 5791 5795 6038 6042 6045 6286 6290 6295 6533 6537 6541 ^7^ 6785 7017 7021 7025 7254 7257 7261 7487, 7491 , 7495 7716 7720 7724 7942 7945 7949 8 I 9 10 II 12 '3 5799 5804 5808 6050 6055 6059 6299 6303 6307 6545 6549 6553 6789 6793 6797 7029 7033 7037 7265 7269 7273 7499 7S02 7506 7728 7731 7735 7953 7957 7960 20 30 40 1^ 1-7 2-5 3-3 4-2 5.0 5812 6063 6311 6557 6801 7041 7277 7510 7739 7964 5816 5820 5825 6067 6071 6075 631S 6319 6324 6561 656s 6569 6805 6809 6813 7045 7049 7053 7281 7285 7289 75 "4 7S>8 7522 7743 7747 7750 7968 7971 7975 14 i6 5829 5833 5837 6079 6083 6088 6328 6332 6336 6573 6577 6582 68.7 6821 6825 7060 7064 7293 7296 7300 7526 7529 7533 7754 7758 7762 7979 7982 7986 '7 i8 19 20 21 22 5841 5846 5850 6092 6096 6100 6340 6345 6349 6586 6590 6594 6829 6833 6837 7068 7072 7076 7304 7308 73'2 7537 7541 7545 7766 7769 7773 7990 7994 7997 5854 6104 6353 6598 6841 7080 7316 7549 7777 8001 Hi 5867 6108 6112 6117 6357 6361 6365 6602 6606 6610 6845 tltl 7084 7088 7092 7320 7324 7328 7552 7560 7781 8005 8008 8012 24 26 5871 5875 5879 6121 6125 6129 6369 6614 6618 6623 Ml? 686s 7096 7100 7104 7332 7335 7339 7572 7792 7796 7800 8016 8019 8023 4 11 29 30 31 32 33 5892 6133 6138 6142 6382 6386 6390 6627 6631 6635 6869 6873 6877 7108 7112 7116 7343 7347 7351 7576 ■7579 7583 7804 7807 781 1 8027 8031 8034 ID 20 30 40 50 60 •7 1-3 2.0 2-7 3-3 4.0 5896 6146 6394 6639 6881 7120 7355 7587 7815 8038 5900 59°4 5909 6150 6154 6158 6398 6402 6405 6643 6651 6885 6889 6893 7124 7128 7132 7359 7363 7367 7591 7595 7598 7819 7822 7825 8042 8045 8049 34 35 36 5913 5917 5921 6162 6i65 6171 6410 6414 6419 6659 6663 6897 6901 6905 7136 7139 7143 7371 7374 7378 7602 7606 7610 7830 7833 7837 8056 8060 37 38 39 40 41 42 43 5925 593° 5934 617s 6179 6183 6423 6427 6431 6667 6671 667s 6909 6913 6917 7147 7151 7155 7382 7386 7390 7614 7617 7621 7841 ^4 8064 8068 8071 3 5938 6187 643s 6679 6921 7159 7394 7625 7852 8075 5942 5946 5951 6igi 6195 6200 6439 6443 6447 6683 6687 6691 6925 6929 6933 7163 7167 7171 7398 7402 7406 7629 7633 7636 7856 786a 7863 8086 44 45 46 5955 5959 5963 6204 6208 6212 6451 6695 6699 6704 6937 6941 6945 7>75 7'79 7183 7410 7413 7417 7640 7867 7871 787s 8089 8093 8097 47 48 49 BO 51 52 53 5967 5972 5976 6216 6221 6225 6464 6468 6472 6708 6712 6716 6949 6953 6957 7187 7191 7195 7421 7425 7429 7652 765s 7659 7886 8loo 8104 8107 20 30 40 50 60 1.0 1-5 2.0 2.5 3-0 5980 6229 6476 6720 6961 7199 7433 7663 7890 8111 5984 5988 5992 6233 6237 6241 6480 6484 6488 6724 6728 6732 6969 6973 7203 7207 7211 7437 744' 7445 7667 7671 7674 7894 7897 7901 81 15 8118. 8l22 54 11 5996 6000 6005 6245 6249 62S4 6492 6496 6501 6736 6740 6744 6977 6981 6985 7215 7218 7222 7449 7452 7456 nil 7686 79°5 7908 7912 8126 8129 8133 11 59 60 6009 6013 6017 6258 6262 6266 650s 6509 6513 6748 6752 6756 6989 6993 6997 7226 7230 7234 7460 7464 7468 7690 7693 7697 7916 7920 7923 8'37 8141 8,44 6021 6270 6517 6760 7001 7238 7472 7701 7927 8148 Smithsonian Tables. 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° 6s° 66° 67° 68° 69° 70° P.P. 0' I 2 3 4 i I 9 10 II 12 13 14 '5 i6 \l ■9 20 21 22 23 24 li 11 29 30 31 32 33 34 11 37 38 39 40 41 42 43 44 4S 46 % 49 60 51 52 53 54 55 56 57 58 59 60 7.321 8148 7.321 8364 7.321 8575 7.321 8781 7.321 8982 7.321 9176 7.321 9365 7.321 9548 7.321 9724 7.321 9893 4 8152 8155 8159 8162 8166 8170 8173 8368 8371 8375 8378 8382 8386 8389 8393 8396 ?578 8582 8585 8589 8606 8788 8791 8795 8798 8801 8805 8808 8812 M5 8989 8992 8995 8998 go02 9008 9012 9179 9182 9186 9189 9192 9195 9198 9202 9205 9368 «37i 9374 9377 9380 9384 9387 9390 9393 9551 9554 9557 9560 9562 9565 9568 957' 9574 9727 9730 9732 9735 9738 9741 9744 9746 9749 9896 9898 9901 9904 9906 9909 9912 99'5 9917 8184 8400 8610 8815 9015 9208 9396 9577 9752 9920 10 20 30 40 IS ■7 '■3 2.0 2.7 3-3 4.0 8188 8191 8195 8198 8202 8206 8209 8213 8216 8403 8407 841a 8414 8417 8421 8424 8428 8431 8613 8617 862a 8624 8627 8631 Itlt 8641 8818 8822 8825 8829 8832 8835 8S39 8842 8846 9018 9021 9025 9028 9031 9034 9037 9041 9044 9211 9214 9218 9221 9224 9227 9230 9234 9237 9399 9402 9405 9408 9411 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 8220 8435 8645 8849 9047 9240 9427 9607 9781 9948 8224 8227 8231 8238 8242 8246 8250 8253 8438 S442 844s 8449 8452 8456 8459 8463 8466 8648 8652 8655 8659 8662 8665 8669 8672 S676 88;2 8856 88S9 8862 8865 8869 8872 8875 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 9803 9806 995' 9953 9956 9959 9961 9964 9967 9970 9972 10 20 30 40 50 60 •5 I.O '•5 2.0 2.5 3.0 8257 8470 8679 8882 9080 9272 9457 9636 9809 9975 8261 8264 826S 8271 8275 8279 8282 8286 8289 8473 8477 8480 8484 8487 8491 8498 8501 S682 8686 S689 f^ 8699 8703 8706 8710 8885 8889 8892 8896 8899 8902 8906 8909 8913 9083 9086 9090 9093 9096 9099 9102 9106 9109 9275 9278 9281 9284 9287 9291 9294 9297 9300 9460 9463 9466 9469 9472 9475 9478 9484 9639 9642 9645 9648 9651 9654 9657 9660 9663 9812 9815 9S17 9820 9823 9S26 9829 983' 9834 9978 9980 9983 9986 9988 999' 9994 9997 9999 2 8293 8505 8713 8916 9112 9303 9487 9666 9837 *0002 8296 8300 8303 8307 8310 8314 8317 8321 8324 8508 8512 8513 8519 8522 8526 8529 ?"3 8536 8716 8720 8723 8727 8730 8733 8737 8740 8744 8919 8923 8926 8929 8932 ■8936 8939 8942 8946 9115 9118 9122 912s 9128 913 1 9'34 9138 9141 9306 9309 9312 9315 9318 9322 9325 9328 9331 9490 9493 9496 9499 9502 9506 9509 9512 9515 9669 9672 9675 9683 9686 9689 9692 9840 9843 9845 9848 9851 9854 9857 9859 9862 *ooos *ooo7 *0OIO *ooi3 *oois tfooiS *002I *ao24 •0026 10 20 30 40 50 60 •3 •7 1.0 1-3 1-7 2,0 8328 8540 8747 . 8949 9144 9334 9518 9695 9865 *0029 1 8332 833s 8339 P*l 8346 835° 8353 8357 8360 8543 8547 8550 8554 nil 8^68 8571 8750 8754 8757 8761 8764 8767 8771 8778 8952 8956 8959 8962 8965 8969 8972 897s 8979 9 '47 9150 9154 9'57 9160 9163 9166 9170 9"73 9337 9340 9343 9346 9349 9353 9356 9359 9362 9521 9524 9527 9530 9533 9536 9539 9542 9545 9698 9701 9704 9707 9709 9712 97'5 9718 9721 9868 ^7' 9873 9876 9879 9882 9887 9890 *oo32 *oo34 *0037 «0O39 *CX342 *oo45 *°°47 *oo5o *0052 8364 8575 8781 8982 9176 9365 9548 9724 9893 ♦0055 Smithsonian Tables. 63 Table 1 1 , LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION p„ IN ENGLISH FEET. [Derivation 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' I 2 3 0055 0210 03S9 0499 0632 0757 0875 0984 1085 "77 005S 0060 0063 0213 0215 0218 0361 0364 0366 0501 0504 0506 0634 0636 0639 0759 0761 0763 0S77 0879 0880 0986 09S7 0989 1087 1088 logo 1178 1 180 1 181 4 I 7 8 9 10 II 12 13 0066 0068 0071 0074 0077 0079 0220 0223 0226 0228 0231 0233 0369 0371 0373 0376 0378 0381 0508 0510 OS '3 0515 o5'7 0520 0641 0643 0645 0647 0650 0652 0765 0767 0769 0771 0773 077s 0882 0884 0886 0888 0889 0891 0991 0992 0994 0996 0998 0999 1091 1093 1095 1096 1098 '099 1 183 1184 1 186 1 187 1189 1 190 3 10 20 30 40 1: •5 I.O 1-5 2.0 2-5 3-0 0082 0236 0383 0522 0654 0777 0893 ICOI IIOI 1 192 0085 0087 0090 0238 0241 0243 0390 0524 0526 0529 0656 0658 0660 0779 0781 0783 0895 0897 0899 1003 1004 1006 1 102 1104 1 105 "93 "95 1 196 14 0092 0095 0098 0246 0248 0251 0392 0394 0397 0531 0533 0535 0662 0664 0667 078s 0787 0789 0901 0902 0904 1008 1009 lOII 1 107 1108 IIIO 1 198 "99 120Q 17 i8 19 20 21 22 23 0100 0103 0105 0253 0256 0258 0399 0401 0404 0537 0540 0542 0669 0671 0673 0791 0793 079s 0906 0908 0910 I0I3 IOI5 IOI6 nil III3 III4 1202 1203 1205 0108 0261 0406 0544 0675 0797 0912 IOI8 1 1 16 1206 OIII 01 13 0II6 0263 0266 0268 0408 041 1 0413 0546 0549 0551 0677 0679 0681 0799 0801 0803 0914 0916 0917 1020 I02I 1023 III8 III9 II2t 1207 1209 1210 24 25 26 ?7 28 29 30 31 32 33 □118 0121 0124 0126 0129 0I3I 0271 0273 0276 0278 0281 0283 0416 0418 0420 0423 0425 0428 0553 °5S5 0558 0560 0562 0565 0683 0685 0688 0690 0692 0694 0805 0807 0809 081 1 0813 0815 0919 0921 0923 0925 0926 0928 1025 1026 1028 1030 1032 1033 II22 II26 1 127 1 129 1 130 1212 12 13 1214 1216 1217 12 19 2 10 20 30 40 5° 60 •3 .7 1.0 1-3 1-7 2.0 0134 02S6 0430 0567 0696 0817 0930 1035 II32 1220 1221 1223 1224 0137 0139 0142 0288 0291 0293 0432 043s 0437 0569 0571 0574 069S 0700 0702 0819 0821 0823 0932 0934 °935 1037 1038 1040 1 133 "35 1136 34 11 0144 0147 0150 0296 0298 0300 0439 0441 0444 0576 0578 0580 0704 0706 070S 0825 0S26 0828 0937 0939 0941 1042 1043 1045 1 138 1 139 1141 1226 1227 1228 37 38 39 40 41 42 43 0152 DISS 0157 0303 0305 0308 0446 0448 0451 0582 0585 0587 0710 0712 0714 0830 0832 0S34 0943 0944 0946 1047 1049 1050 1 142 1 144 1 145 1230 123 1 1233 0160 0310 0453 0589 0716 0836 0948 1052 1 147 1234 0162 0165 0167 0312 03 IS 0317 0455 0458 0460 0591 0593 0596 0718 0720 0722 0838 0840 0842 0950 0952 0953 1054 105s 1057 1148 1150 ■151 "35 1237 ■ 238 44 11 47 48 49 60 5- 52 53 0170 0172 0175 0177 oiSo 0182 0185 0320 0322 0324 0327 0329 0332 0462 0464 0467 0469 0471 0474 0598 0600 0602 0604 0607 o6og 0724 0726 0729 0731 0733 0735 0844 0846 0848 0850 0852 0854 0955 °957 0959 0961 0962 0964 1058 1060 1062 1063 1065 1066 "53 '■54 1156 IIS7 1159 1160 _ 1240 1241 1242 1244 1245 1247 1 10 20 30 40 50 60 .2 •3 ■5 -.1 1.0 0334 0476 0611 0737 0856 0966 1068 1 162 124S 0187 oigo 0192 0336 ■ 0339 0341 0478 □481 0483 0613 0615 0617 0739 0741 0743 0858 0860 0862 0968 0970 0971 1070 107 1 1073 1 163 1165 1166 1249 1251 1252 54 11 0195 0197 0200 0344 0346 0349 0485 0487 0490 o6ig 0621 0624 074s °747 0749 0864 0865 0867 0973 0975 0977 1075 1076 JO78 1168 1169 1171 1253 1254 1256 11 59 60 0202 0205 0207 0351 0354 0356 0492 0494 0497 0626 0628 0630 0751 °753 07S5 0869 0871 0873 0979 0980 0982 1080 1082 1083 I172 1 174 "75 "57 1260 0210 0359 0499 0632 0757 0875 0984 .085 "77 1261 Smiths iONIAN ' Fables. 64 Table 1 1 , LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION p„ IN ENGLISH FEET. [Derivation of table explained on p. xlv.j Lat. 81° 82° 83° 84° 8s° 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' 2 3 1261 1337 1403 1461 1511 ■551 '583 '605 1619 1262 1264 1265 1338 1339 1340 1404 1405 1406 .462 J463 1464 ■ 512 ■ 512 "5'3 1552 ■552 ■553 '583 '584 1584 1605 1606 i6b6 i6ig 1619 1619 4 5 6 1266 1267 1269 1341 1342 1344 1407 1408 1410 1465 1465 1466 1514 1514 ■5^5 ■553 ■554 "555 ■58s 1585 1585 1606 1606 1607 1619 1619 1620 I 9 10 11 12 13 1270 1271 1273 1345 1346 1347 1411 1412 I4^3 1467 1468 1469 IS. 6 1517 IS^7 1555 1556 1556 1586 1586 ■ 587 1607 1620 1620 1620 2 1274 134S 1414 1470 1518 '557 ■587 1608 1620 1275 1277 1278 1349 1350 1352 1415 1416 1417 147 1 1472 1473 1519 1519 1520 1558 ■558 '559 'Si 1588 i6o8 i6og 1609 1620 1620 1620 14 •5 i6 1279 1280 1282 1353 1354 1355 1418 1419 1420 1474 ■474 ■475 ■ 521 ■521 1522 '559 1560 1561 ■589 1589 1589 1609 1609 1610 1620 1620 1621 to •3 19 20 21 22 23 1283 1284 12S6 1356 1358 1359 142 1 1422 1423 1476 1477 .478 ■523 1524 ■524 1561 1562 1562 1590 ■59° 1591 1610 x6io 1611 1621 162 1 1621 20 40 1^ •7 I.O '■3 '•7 2.0 1287 1360 1424 1479 ■525 1563 '59' 161 1 1621 1288 1290 1291 .36. 1362 1363 ■425 1426 1427 1480 1481 .481 1526 1526 1527 '563 1564 1564 ■59' 1592 '592 1611 1611 1612 162 1 1621 1621 24 25 26 1292 1293 1295 1364 .365 1367 1428 1429 143° 1482 1483 1484 1528 1528 1529 '565 ■565 ■ 566 ■593 '593 '593 1612 1612 1612 162 1 1621 1622 27 28 29 30 31 32 33 1296 1297 1299 1368 1369 1370 ■43 1 1432 ■433 1485 148s i486 1530 '531 1531 1566 1567 1567 '594 '594 '595 l6l2 I6I3 I6I3 1622 1622 1622 1300 1371 1434 1487 1532 1568 ■595 I6I3 1622 1301 1302 1304 1372 1373 ■374 ■435 1436 1437 1488 1489 1489 1533 ■533 1534 1568 ■569 '569 1596 1596 1613 I6I3 I6I4 1622 1622 1622 34 35 36 1306 1307 1375 1376 1378 ■438 1438 1439 ■49° 1491 1492 ■535 ■535 ■ 536 '57° 1570 '57' ■597 '597 1597 1614 I6I4 I6I4 1632 1622 1623 37 38 39 40 41 42 43 1308 13 10 1311 1379 1380 1381 1440 1441 1442 ■493 1493 1494 1537 1538 1538 '57' 1572 1572 1598 1598 '599 16x4 1615 16IS 1623 1623 1623 1 10 20 30 40 50 6a .3 ■3 •5 1.0 1312 1382 ■443 1495 ■539 ■573 '599 I6IS 1623 1313 i3'S 1316 1383 1384 1385 ■444 1446 1496 1497 1497 1540 ■54° 1541 1573 ■574 ■574 '599 1600 1600 1615 I6I5 1616 1623 1623 1623 44 45 46 1317 1318 1320 1386 1387 1389 1447 1447 1448 1498 1499 1500 1541 1542 ■543 '575 '575 1576 1600 1600 1601 I6I6 i6i6 1616 1623 1623 1623 47 48 49 50 51 52 53 1321 1322 1324 1390 1 391 1392 ■449 1450 1451 1501 1501 1502 ■543 ■544 ■544 1576- 1577 '577 i6oi 1601 1602 1616 1617 1617 1623 1623 1623 1325 ■393 ■452 1503 1545 '578 1602 1617 1623 1326 1327 1329 1394 ■395 1396 1453 ■454 ■455 1504 1505 1505 1546 ■ 545 1547 1578 '579 1579 1602 1603 1603 1617 1617 161S 1623 1623 1623 54 55 56 1330 1331 1332 1397 1398 ■399 1456 1456 1457 1506 1507 1508 ■547 1548 ■549 '580 '580 1581 1603 '603 1604 1618 i6i8 161S 1623 1623 1623 11 59 60 1333 133s 1336 1400 1401 1402 ■458 1459 1460 1509 1509 1510 1549 "550 1550 158' '582 1582 1604 1604 1605 j6i8 1619 1619 1623 , 1623 1623 1337 1403 1461 1511 ■551 '583 160S 16x9 1623 SmiTHSoNrAN Tables. 65 l^ABLE 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° o° 6.80237 6.80242 6.80248 6.80254 6.80260 6.80266 6.80272 6.80279 6.80285 6.80292 S 10 239 244 254 244 250 259 250 264 256 261 270 262 267 276 268 III 274 280 285 294 287 292 300 $ 20 30 266 282 300 271 287 305 277 292 309 282 297 314 288 302 319 293 308 324 299 313 330 305 319 335 3" 325 340 3'7 331 346 3S 40 45 320 364 324 345 367 329 350 371 333 354 375 338 358 379 343 362 383 348 387 353 372 391 358 377 396 363 382 400 50 386 407 427 389 410 430 392 413 432 396 416 435 399 420 438 403 423 442 407 426 445 411 43° 448 415 434 451 419 437 455 65 70 75 445 461 473 476 478 468 480 455 470 482 458 473 484 461 475 487 464 478 489 467 481 492 470 484 494 80 85 90 483 489 490 485 490 492 487 492 494 489 494 496 491 496 498 493 498 500 495 5°' 502 498 5°3 504 500 505 507 S02 507 509 Azimuth. LATITUDE. 32° 33° 34° 35° 36° 37° 38° 39° 40° 4.° o" 6.80299 6.80306 6.80313 6.80320 6.80327 6.8033s 6.80342 6.80350 6.80357 6.8036s 5 ID 15 300 305 313 307 312 320 314 326 322 326 333 329 333 340 336 340 348 344 348 355 351 359 II 366 370 376 20 25 30 324 337 352 330 337 349 364 343 355 370 350 ^76 382 364 371 382 394 401 38s 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 5° 1^ 423 441 458 428 445 462 432 449 465 436 469 441 457 472 445 461 476 450 465 480 484 459 487 464 478 491 6S 70 75 473 486 497 476 489 500 480 492 502 483 495 505 486 498 508 489 SOI Sio 493 504 513 496 507 516 500 510 519 503 S'4 522 80 85 90 505 51° 5" 507 512 514 510 5>4 5.6 5'2 5>7 518 515 519 521 517 522 523 520 524 526 523 527 528 525 529 53' 528 532 533 Smithsonia N Table ■. 66 Table 1 2. LOGARITHMS OF RADIUS OF CURVATURE p^ (IN METRES) OF SECTION OF EARTH'S SURFACE INCLINED TO MERIDIAN AT AZIMUTH a. [Formula forpo giver 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 5 10 15 374 382 385 391 389 393 399 397 400 406 404 408 413 412 415 420 4: 45 4: 428 3 43c 8 435 438 442 443 445 450 20 25 3° 392 402 413 408 420 406 415 426 413 422 433 420 429 439 427 436 446 4: 4^ 4. i4 441 ^2 44S )2 458 448 456 465 471 35 40 45 426 440 454 446 459 438 464 444 457 470 450 462 475 480 4f )2 46S 4 47S S5 49c 474 485 495 480 490 500 5° 468 482 495 499 478 490 502 482 487 499 510 492 503 514 45 5' 51 )6 501 )8 512 8 522 506 5'^ 526 510 520 530 65 7° 75 507 517 525 510 520 528 514 523 530 517 526 534 520 529 536 524 532 539 5= 5: 5^ 8 531 56 539 t2 545 534 542 548 538 545 551 8o 85 90 53' 536 534 537 538 536 540 541 539 542 544 542 545 546 544 548 549 5"^ 5. 5. \-7 55c )0 553 )i 554 553 55| 556 559 LATITUDE. Azimuth. 52° 53° 54° 55° 56° 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 15 450 453 457 460 464 46s 467 471 472 474 478 479 481 485 486 488 492 493 495 500 502 50s 507 509 5" 20 25 30 462 469 477 469 476 484 49° 483 489 496 489 495 502 496 501 S08 5°S 508 514 509 514 519 515 520 525 35 40 45 496 5°5 492 501 510 498 506 515 5°3 512 520 5°9 517 525 515 522 53° 520 527 534 525 532 539 531 537 543 SO 515 524 533 520 528 537 524 533 541 528 537 544 533 54i 548 537 545 552 542 548 555 546 55? 558 555 II2 65 70 75 541 548 554 545 551 557 548 554 559 551 560 565 5f $1 561 566 570 564 569 573 567 572 575 80 85 90 1 561 5^' 5^3 564 S66 566 S68 569 568 570 571 571 573 574 573 575 576 57f 578 578 580 Smithsonian Tables. 67 Table 13. LOGARITHMS OF FACTORS t^FOR COMPUTING SPHEROIDAL EXCESS OF TRIANGLES. UNIT = THE ENGLISH FOOT. [Derivation and use ol table explained on p. Iviii.] log. factor and log. factor and log. factor and log. factor and change per minute. <* change per minute. « change per minute. « change per minute. 0° 0.37498 20° 0.37429 40° 0-37255 „ 60° 0.37056 * — 0.00 — 0.12 — 0.18 — 0.IS I 498 — 0.02 21 422 — 0.12 41 244 — 0.17 61 047 -0.IS 2 497 — 0.02 22 415 — 0.12 42 234 — 0.17 62 038 — 0.13 3 496 — 0.02 23 408 — 0.12 43 224 — 0.17 63 030 -0.13 4 495 — 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 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 l^ii8 68 991 — 0.12 9 484 — 0.07 29 360 — 0.15 49 162 — 0.17 69 984 —0.12 10 480 — 0.07 30 351 — 0.15 50 152 — 0.17 70 977 — O.IO II 476 — 0.07 31 342 — 0.15 51 142 -0.17 71 971 — O.IZ 12 472 — 0.07 32 333 — 0.17 52 132 — 0.17 72 964 —0.08 13 468 — 0.08 33 323 — 0.15 S3 122 — 0.17 73 959 — O.IO 14 463 — 0.07 34 314 — 0.17 54 112 — 0.15 74 953 „ — 0.08 IS 459 — O.IO 35 304 — 0.15 55 103 — 0.17 75 948 — 0.08 16 453 „ — 0.08 36 29s — 0.17 56 093 — 0.17 76 943 „ — 0.08 17 448 — O.IO 37 285 — 0.17 S7 083 — 0.15 77 938 — 0.07 18 442 O.IO 38 275 — 0.17 58 074 — 0.15 78 934 — 0.07 19 436 — 0.12 39 26s — 0.17 59 06s — 0.15 79 930 — 0.07 20 429 — 0.12 40 *5S „ — 0.18 60 056 — 0.IS 80 926 Smithsonian Tables. 68 Table 14. LOGARITHMS OF FACTORS _e^ p„p; 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 « change per minute. * change per minute. change per minute. fli *i=ci 02 ii ca Ol *i = a oz h ei o°oo' 7.99669 7-99374 — 00 — 00 0.372 I0°00' 7-996SS 7-99369 9.621 9.926 0.398 10 ^ 374 7.839 f-'37 0.372 lO 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 8.739 0.372 40 654 369 9.650 9-9SS 9.963 0.402 5° 669 374 8.538 8.836 0.372 50 653 369 9.657 0.403 I 00 669 374 f-!i7 f-9'5 0.372 II 00 653 3ff 9-663 9.970 0.404 10 ffg 374 8.684 8.982 0.372 10 652 368 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 8.793 9091 0.373 30 651 368 9.683 9.990 0.406 40 668 374 sisio 9-137 0.373 40 651 3^ 9.690 9-997 0.407 5° 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 O.OIO 0.409 10 668 373 SI 9.251 9.283 0-373 10 649 367 9.708 0.016 0.410 20 668 373 0.373 20 649 367 9.714 0.023 0.412 3° 668 373 9.01s 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 5° 668 373 9.069 9-368 0.374 50 647 367 9.732 0.041 0415 300 668 373 9.094 9.393 0-374 1300 646 3^ 9.738 0.048 0.416 10 ^7 373 9.1 18 9.417 0.37 s 10 646 366 9-744 0.054 0.417 20 667 373 9.140 9.439 0.375 20 64s 366 9-749 0.060 0.418 30 667 373 9.161 9.460 0-375 30 645 366 9-755 0.065 0.419 40 ^^ 373 9.182 9-481 0.376 40 644 366 9.761 0.071 0.420 SO 667 373 9.201 9500 0.376 SO 644 36s 9766 0.077 0.422 4 00 667 373 9.220 9.519 0.376 14 00 643 365 9.771 0.083 0.088 0.423 10 666 373 9-237 9-537 0.377 10 642 36s 9.777 0.424 20 666 373 9.254 9.554 0.377 ■ 20 642 36s 9.782 0.094 0.425 30 666 373 9271 9.570 0.377 30 641 365 9-787 O.IOO 0.426 40 666 373 9.287 9.586 0.378 40 640 364 9.792 0.105 0.428 5° 666 373 9.302 9.602 0.378 50 640 364 9.798 O.I 1 1 0.429 500 665 373 9.317 9.617 0.379 15 00 639 364 & O.I 16 0.430 10 ^5 373 9.331 9.631 0.379 10 6^ 364 0.I2I 0.431 20 665 372 9-345 9-645 0.379 20 363 9.813 0.127 0-433 30 ^5 372 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 0-43S 50 664 372 9-384 9.685 0.381 SO 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 10 635 362 9.837 0.153 0.158 0.439 20 663 372 9.420 9.721 0-383 20 634 362 9.841 0.441 30 ^3 372 9-432 9.732 0.383 30 633 362 9.846 0.163 O.I^ 0.442 40 ^3 372 9-443 9-744 0.384 40 632 362 9.851 0-443 50 662 372 9-453 9-755 0.384 SO 632 361 9.855 o-'73 0.445 700 662 372 9464 9-765 0.385 1700 631 36J 9.860 0.178 0.446 10 662 371 9-474 9.776 0.386 10 630 361 9.864 0.182 0.448 20 662 371 9-484 9.786 0.386 20 630 36. 9.869 0.187 0.449 30 661 371 9.494 H^^ 0.387 30 629 360 9|73 9.878 0.192 0450 40 66i 371 9.504 9.806 °-3?7 40 628 360 0.197 0.452 SO 661 371 9.513 9.816 0.388 SO 627 360 9.882 0.202 0453 800 660 371 9.523 9.825 0.389 1800 627 360 9.886 0.206 0.45s 10 660 371 9-532 9.834 0.389 10 626 359 9.890 0.21 1 0.450 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 ^^^ 370 9-558 9.861 0.392 40 624 359 9-903 0.225 0.461 SO 658 370 9.566 9.870 0.392 50 623 358 9.907 0.229 0.463 900 658 370 9-575 9-!?? 0-393 1900 622 358 9-9' I 0.234 0.464 10 fiF 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 20 620 358 9.919 0.243 0.467 30 A^l 370 9-M 9-903 0.396 30 620 357 9-923 0.248 0.469 40 656 ^JS. 9.606 9910 0.396 40 619 357 9.927 0.252 0.470 50 656 369 9.614 9.918 0-397 SO 618 357 9-93' 0.256 0.472 10 00 6SS 369 9.621 9.926 0.398 20.00 617 357 9-935 0.261 0.474 Smithsonian Tables, 70 Table 15i 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,] * fli 6i = ci oa 62 Ci 'P ai *a = C2 an h C2 29°00' 7.99617 7-993S7 9-93S 0.261 0.474 30°oo' 7-99558 7-99337 0.13s 0.138 0.496 0-593 lO 6i6 356 9-939 0.265 0-475 10 557 337 0.500 0-595 20 615 356 9-943 0.270 0-477 20 556 336 0.141 0-503 0.598 3° 61S 356 9-947 0.274 0.479 30 SSS 336 0.144 0.507 0.600 40 014 355 9-951 0.278 0.480 40 554 335 0.146 0.511 0.603 5° 613 3SS 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 6n 355 9.962 0.291 0.486 10 550 334 0.15s 0.522 0.610 20 610 354 9.966 0.29s 0.487 20 549 334 0.158 0.525 0.612 30 609 354 9-970 0.299 0.489 30 548 333 o.i6i 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 H 353 9.981 0.312 0.494 3200 545 332 0.169 0-540 0.622 10 606 353 9.984 0.316 0-496 10 544 332 0.172 0-S43 0.624 20 60s 353 9.988 0.320 0.498 20 542 332 0.175 0-547 0.627 30 604 352 9-991 0.324 0.500 30 541 331 0.177 0.550 0.629 40 603 352 9-995 9.998 0.328 0.502 40 540 331 0.180 0.554 0.632 SO 602 352 0-332 0.503 SO 539 330 0.183 0-558 0.634 2300 601 351 0.002 0-33^ 0-505 3300 538 330 0.186 0.561 0.637 10 600 351 0.00s 0.340 0.507 10 537 330 0.188 0.56s 0.639 20 600 351 0.009 0-344 0.509 20 535 329 0.191 0.568 0.642 ' 30 S99 350 0.012 0.348 0.511 30 534 329 0.194 0.572 0.644 40 598 350 0.016 0-352 0-513 40 533 328 0.197 0-S7S 0.647 SO 597 350 0.019 0.356 0.515 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 S9S 349 0.026 0-364 0.518 10 529 327 0.205 0.586 0.655 20 594 349 0.029 0.368 0.520 20 528 327 0.208 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 0.663 SO 591 348 0.039 0.380 0.526 SO 525 326 0.216 0.600 0-665 2500 590 347 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 20 588 347 0.049 0.392 0.532 20 521 324 0.224 0.61 1 0-673 30 587 346 0.052 0.396 0-S34 30 520 324 a2^6 0.615 0.676 40 586 346 0.056 0.399 0.536 40 S19 324 a229 o.6t'8 a679 SO 585 346 0.059 0.403 0.538 SO 517 323 0.232 0.622 0.681 2600 584 345 0.062 0.407 0.540 3600 516 323 0.234 ■0-625 ' 0.684 10 583 34S 0.065 0.068 0.41 1 0-543 10 51S 322 D.237 0.629 0.687 20 582 345 0.415 0-54S 20 514 322 0.239 0.632 0.689 30 S8i 344 0.072 0.418 0.547 30 512 322 0.242 0.636 0.692 40- S8o 344 0.075 0.422 0-549 40 S" 321 0.24s 0.640 0.69s SO 579 344 0.078 0.426 0.551 SO 510 321 0.247 0.643 0.698 27 00 578 343 0.081 0.430 0-553 3700 509 320 0.250 0.647 0.700 10 577 343 0.084 0-433 0-55S 10 507 320 0.253 0.650' 0.703 20 576 343 0.087 0.437 0-S57 20 506 320 0.255 0-654' 0.706 30 S7S 342 0.090 0.441 0-559 30 505 319 0.258 0.657 0.709 40 574 342 0.093 0.445 0.562 40 504 319 0.260 o.66i 0.712 SO 573 342 0.096 0.448 0-564 50 503 318 0.263 0.665 0.715 2800 57 1 341 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 S69 341 0.105 0.460 0.570 20 499 3'7 0.271 0.67s 0.723 30 S68 340 0.108 0.463 0-573 30 498 317 0.273 0.679 0.726 40 567 340 O.III 0.467 0-575 40 496 316 0.276 0.683 0.729 SO 566 340 0.114 0-471 0-S77 SO 495 316 0.278 0.686 D.732 2900 565 339 0.117 0.474 0.579 3900 494 315 0.281 0.690 0-735 0-738 10 564 339 0.120 0.478 0.582 10 492 315 0.284 0.693 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 °-5!^ 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 S6o 337 0.132 0.493 0.591 SO 487 313 0.294 0.708 0.750 3000 SS8 337 0-135 0.496 0-593 40 00 486 313 0.296 0.7 1 1 0-733 SjnjTHScniAM 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.] ai il=ci a2 i2 C2 «i h—n 02 h (2 40°oo' 7.99486 7-99313 0.296 0.71 1 0.752 So°oo' 7.99409 7.99287 0.448 0-939 0-955 0.958 10 48s 312 0.299 0.715 0-7SS 10 408 287 0.450 0.944 20 484 312 0.301 0.719 0-759 20 407 287 0-453 0.948 0.962 3° 482 312 0.304 0.722 0.762 30 406 286 0.45s 0.458 0.460 0.952 0.966 40 481 311 0.307 0.726 0.76? 0.768 40 404 286 0.956 0.970 50 480 3" 0.309 0.730 50 403 285 0.960 0-974 41 00 479 310 0.312 0-733 0.771 51 00 402 285 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 3° 475 309 0.319 0.744 0.780 30 398 284 0.471 0.976 0.989 40 473 309 0.322 0.748 0.783 40 397 283 0.473 0,981 0-993 50 472 308 0.324 0.751 0.786 50 396 283 0.476 0.985 0-997 42 00 471 308 0.327 0.755 0.789 52 00 394 282 0.478 0.989 1. 001 10 4^2 307 0.329 0-759 0.792 10 393 282 0.481 0-993 0.998 1.005 20 468 307 0.332 0.762 0.796 20 392 281 0.484 1.009 30 467 306 0-334 0.766 0.799 30 391 281 0.486 1.002 1.013 40 466 306 0.337 0.770 0.802 40 389 281 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 3f7 280 0.494 1.015 1.025 10 462 305 0.344 0.781 0.812 10 386 279 0.497 1.019 1.030 20 461 304 0-347 0-785 0.815 20 384 279 0.499 1.023 1.034 3° 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 5° 457 303 0-354 0.796 0.824 50 381 278 0.507 1.036 1.046 4400 455 303 0-3S7 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 1.055 20 453 302 0.362 0.807 0.834 20 377 277 0.515 1.049 1.059 30 452 301 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 SO 449 300 0-370 0.818 0.844 50 373 275 0.523 1.063 1.072 4.500 448 300 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 1.081 1.089 40 443 298 0.382 0.837 0.861 40 367 273 0.537 1.085 1.093 1.098 SO 441 298 0-385 0.841 0.865 50 366 273 0.539 1.090 4600 440 297 0-387 0.845 0.868 56 00 36s 273 0.542 1.094 1. 102 10 439 297 0.390 0.849 0.872 10 364 272 0.545 1.099 i.io6 20 437 -297 0.392 0.853 0.87 s 20 363 272 0.547 1. 104 I. Ill 30 436 296 0.39s 0.856 0.878 30 361 271 0.550 1. 108 1. 115 40 435 296 0.397 0.860 0.882 40 360 271 0-553 0.556 1. 113 1. 120 50 434 295 0.400 0.864 0.885 50 359 271 1. 118 I.I 24 4700 432 295 0.402 0.868 0.889 57.00 358 270 0.558 1. 122 1. 129 to 431 294 0.405 0.872 0.892 10 357 270 0.561 0.564 1. 127 1-134 20 430 294 0.407 0.876 0.896 20 356 269 1. 132 1.138 30 428 294 0.410 0.880 0.900 30 354 269 0.567 1-137 I. '43 40 427 426 293 0.412 0.884 0.903 40 353 269 0.569 1.141 1.147 50 293 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 351 268 0-575 0.578 0.581 1.151 1.157 10 423 292 0.420 0.89s 0.914 10 350 267 1.156 1. 162 20 422 291 0.422 0.899 0.918 20 349 267 1. 161 1.166 30 421 291 0.425 0.903 0.921 30 347 267 0.583 1.166 1.171 40 420 418 291 0.427 0.907 0-925 40 346 266 0.586 1. 170 1.176 50 290 0.430 0.9H 0.929 50 345 266 0.589 1. 175 1. 181 4900 416 414 290 0.432 0.915 0.932 S9 00 344 266 0.592 1. 180 1.185 1.190 1.19s 10 20 289 0.435 0.438 0.919 0.923 0.936 0.940 10 20 343 342 26s 265 0-595 0.598 1.185 1. 190 30 413 289 288 288 0.440 0.927 0.943 30 341 264 0.600 1.19s 1.200 40 412 0.443 0.931 0-947 40 339 264 0.603 1.200 1.205 50 411 0.445 0-935 0-951 SO 338 264 0.606 1.205 1.210 50 00 409 287 0.448 0-939 0-955 6000 337 263 0.609 1. 210 1.215 Smithsonian Tables. 72 Table 15. LOCA>»rTHMS 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.] * "1 6i = ci 32 h C2 ^ fll il = ci 02 h f2 6o°oo' 7-99337 7.99263 0.609 1. 210 [.215 70°oo' 7.99278 7.99244 0.809 1-575 1.576 10 336 263 0.612 1.216 [.220 10 277 243 0.813 1-583 1-584 20 33S 263 0.615 1. 221 1.225 20 277 243 0.817 1-590 1. 591 30 334 262 0.618 1.226 [.230 30 276 243 0.821 1.598 1-599 40 333 262 0.621 1.231 1-235 40 275 242 a.825 1.605 1. 606 50 332 261 0.624 1.236 1.240 50 274 242 0.829 1.613 1.614 61 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 50 325 259 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 GO 269 240 0.858 1.669 1.669 10 323 259 0.648 1.279 1.282 10 268 240 0.862 1.677 ^■^11 20 322 258 0.651 1.284 1.288 20 267 240 0.866 1.685 1.686 3° 321 2S8 0.654 1.290 1.293 30 266 240 0.871 1.694 1.694 40 320 2S7 0.657 1.295 1.298 40 266 239 °iP 1.702 1.702 s° 319 257 0.660 1.301 1.304 50 265 239 0.880 1.710 1.711 63 00 318 2S7 0.663 1.306 1-309 7300 264 239 0.884 1.719 1.720 10 317 256 0.666 1.312 1-315 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 30 31S 256 0.672 1-323 1.326 30 262 238 0.898 1-745 1.746 40 314 2SS 0.676 1.329 1-332 40 261 238 0.903 1-754 I -7 55 50 313 25s 0.679 1-335 1-337 50 261 238 0.907 1-763 1.764 6400 312 2SS 0.682 1-341 1-343 7400 260 238 0.912 1-772 1-773 10 3" 2S4 0.685 1.346 1-349 10 259 237 0.917 1.782 1.782 20 310 254 0.688 1-352 I-35S 20 259 237 0.922 1.791 1. 791 30 309 254 0.692 1-358 1.360 30 258 237 0.927 1.800 1. 801 40 308 2S3 0.695 1-363 1.366 40 257 237 0.931 1.810 i.8io s° 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 10 305 252 0.705 0.708 1.382 1.384 10 25s 236 0.946 1.839 1.839 20 304 252 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 0.718 1.400 r.402 40 254 235 0.962 1.869 1.869 so 301 251 1.406 1.408 SO 253 23s 0.967 1.879 1.880 6600 300 251 0.721 1-413' 1.414 7600 252 235 0.973 1.890 1.890 10 250 0.725 1-419 1.421 10 252 235 0.978 i.goo 1. 901 20 298 250 0.728 1.425 1.427 20 251 235 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 0-735 1.438 1.440 40 250 234 0-995 1-933 1-933 5° 29s 249 0-739 1-444 1.446 50 249 234 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 I -457 1.459 10 248 234 1.012 1.966 1.966 20 292 248 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 so 289 247 0.760 1.484 1.485 SO 246 233 1.036 2.013 2.013 6800 289 247 0.763 1.491 1.492 7800 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 50 243 232 1-074 2.088 6900 283 245 0.786 1-532 I-S33 7900 242 232 1.081 1.087 2.101 2.101 10 282 245 0.789 I-S39 1.540 10 242 232 2.114 2.128 2.114 2.128 20 282 245 0-793 1.546 1-547 20 242 231 1.094 30 281 244 0.797 1-553 1-554 30 241 231 1. 101 2.142 2.156 2.142 2.156 40 280 244 0.801 1.561 1.562 40 241 231 1.108 so 279 244 0.805 1.568 1.569 SO 240 231 1.116 2.170 2.170 70 00 278 244 0.809 1-575 1.576 8000 240 231 1.123 2.184 2.184 Smithsonian Tables. 13 Table t6. LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI- TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANCULATION. UNIT = THE METRE. [Derivation and use o£ table explained on p. Ix.] ^ fli h=n "2 ii ^2 ^ «l *i=fi «! h (1 o°oo' 8.51268 8.50973 00 — 00 1.404 io°oo' 8.51254 8.50968 0.653 0.958 1.430 lO 268 973 8.871 9.169 1.404 10 254 9g 0.660 0.965 1-431 20 268 973 9.172 9.470 1.404 20 253 968 0.668 0-973 1432 30 26S 973 9-348 9.646 1.404 30 253 ^0 0.675 0.980 1-433 40 268 973 9-473 H'^l 1.404 40 253 968 0.682 0.987 1-434 50 268 973 9-570 9.868 1.404 50 252 967 0.689 0-995 I-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 40 250 967 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 9.985 0.248 1.405 12 00 1$ 966 0.734 1.042 1-441 10 267 972 0.283 1.405 10 966 0.740 1.048 1.442 20 267 972 0.017 0.31 5 1.405 20 248 966 0.746 I.0S5 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 50 266 972 0.1 01 0.400 1.406 SO 246 965 0.764 I-073 1-447 300 266 972 0.126 0.425 r.406 1300 245 96s 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 3° 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 5° 266 972 0-233 0.532 1.408 SO 242 964 0.798 1.109 1-454 4 00 26s 972 0.252 0.551 1.408 14 00 242 964 0.803 1.115 1-455 10 265 972 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 I.I 26 1-457 30 26s 972 0-303 0.602 1.409 30 240 963 0.819 I.I32 1.458 40 26s 972 0.319 0.618 I.4IO 40 239 963 0.824 I -1 37 1.460 50 264 972 0-334 0.634 I.4IO 50 239 963 0.830 "•143 1.461 5 00 264 972 0-349 0.649 1. 41 1 15 00 238 963 0.835 1. 148 1.462 ID 264 97" 0-363 0.663 1. 41 1 10 237 963 0.840 1-153 1.463 20 264 971 0-377 0.677 I.4II 20 237 962 0.845 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 50 263 971 0.416 0.717 1-413 50 235 962 0.859 1.174 1.469 600 263 971 0.428 0.729 I-4I3 1600 234 961 0.864 1.179 1.470 10 263 971 0.440 0.741 1.414 10 233 961 0.869 1.18s 1-471 20 262 971 0.452 0-753 1-415 20 233 961 0.873 1.190 1-473 3P 262 971 0.464 0.764 1-415 30 232 961 0.878 1-195 1.474 4Q 262 971 0.475 0.776 I.416 40 231 961 0.883 1.200 1-475 50. 261 971 0.485 0.787 I.416 50 231 960 0.887 1.205 1477 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 1.480 20 260 970 0.516 0.818 I.418 20 228 960 0.901 1.219 1.481 3° 260 970 0.526 0.828 1-419 30 228 959 0.905 1.224 1.482 40 260 970 0.536 °M I.4I9 40 227 959 0.910 1.229 1-484 50 259 970 0-545 0.848 1.420 50 226 959 0.914 1.234 1.485 800 259 970 0-555 °i^l 1. 42 1 1800 225 959 0.918 1-238 1.457 lO 250 970 0.564 0.866 1.421 10 225 958 0.922 1.243 1.489 20 258 970 0-573 0.875 1.422 20 224 958 0.927 1.248 1.490 30 '5f 969 0.581 0.884 1-423 30 223 958 0.931 1.252 1-491 40 258 969 0.590 0.893 1.424 40 223 958 0-935 1.257 1-493 50 257 969 0.598 0.902 1.424 50 222 957 0-939 1-495 900 257 969 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. 27 1 1.498 20 256 969 0.623 0.927 1.427 20 219 957 0.951 1.275 1-499 30 256 969 0.630 0-935 1.428 30 218 956 0-955 1.501 40 2SS 969 0.638 0.942 1.428 40 218 956 0.959 1.284 1-502 50 255 968 0.646 0.950 1.429 50 217 956 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.] * 01 ii=ci at *2 C2

"1 h=n fla h '^2 « Ol *i = a az h (2 40°oo' 8.5x085 8.50912 1.328 1-743 1-784 5o°oo' 8.51008 8.50886 1.480 1.971 1.987 10 084 911 I-33I 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 1.994 30 081 911 1.336 1-754 1-793 30 005 f|5 1.487 1.984 1.998 40 080 910 1-338 1.758 1-797 40 003 885 1.490 1.988 2.002 S° 079 910 1-341 1.762 1.800 50 002 884 1.492 1.992 2.006 41 00 078 909 1-344 1-765 1.803 51 00 001 884 1.49 c 1.498 1.996 2.010 10 076 1.346 1-769 1.806 10 000 883 2.000 2.014 20 075 908 1-349 1.772 1.809 20 8.50998 883 1.500 2.004 2.017 30 074 908 I-3SI 1.776 1.812 30 997 882 1-503 2.008 2.021 40 072 908 1-354 1.780 1.815 1.818 40 996 882 m 2.013 2.025 5° 071 907 1-356 1-783 SO 994 882 2.017 2.029 42 00 070 907 1-359 1-787 1.821 52 00 993 881 1. 510 2.021 2.033 10 .069 906 1.361 1.791 1.824 10 992 881 1-513 2.025 2.037 20 067 906 1.364 1.794 1.828 20 991 880 1.516 2.030 2.041 30 066 905 1.366 1.798 1.831 30 '^ 880 1.518 2.034 2.045 40 065 90s 1.369 1.802 1.834 40 880 1.521 2.038 2.049 SO 063 90s I-37I 1.805 I-837- SO 987 879 1-523 2.042 2.053 4300 062 904 1-374 1.809 1.840 S3 00 986 §7g 1.526 2.047 2.057 10 061 904 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-S3I 2.055 2.066 30 058 903 1.381 1.820 1.850 30 982 877 1-534 2.060 2.070 40 0S7 902 1.384 1.824 1-853 40 981 877 I -537 2.064 2.074 50 056 902 1.386 1.828 1.856 SO 980 877 1-539 2.068 2.078 44 00 054 902 1.389 1.832 1.860 5400 978 876 1.542 2.073 2.082 10 053 901 »-39i 1-835 1-863 10 977 876 I-S44 2.077 2.086 20 052 901 1-394 1.839 1.866 20 976 87s 1-547 2.081 2.091 30 051 900 1-396 1.843 1.870 30 975 87s 1-550 2.086 2.095 40 049 900 1-399 1.847 'in 40 973 87s 1-552 2.090 2.099 SO 048 899 1. 40 1 1.850 1.876 50 972 874 1-555 2.095 2.104 4500 047 899 1.404 '•^54 1. 880 55 00 971 874 1-558 2.099 2.108 10 045 ^^i 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 2.116 .30 043 89S 1.412 1.865 1.890 30 967 873 1.566 2.113 2.121 40 042 897 1-414 1.869 1-893 40 966 87i 1.568 2.117 2.125 SO 040 897 1-417 1-873 1-897 SO 965 872 1-571 2.122 2.130 46 00 039 ^96 1.419 'ir 1.900 5600 964 871 1-574 2.126 2.134 ro 038 M 1.422 1. 88 1 1.903 10 963 871 1-577 2.131 2.138 30, 036 896 1.424 1.885 1.907 20 961 871 1-579 2.136 2.143 30 03s 895 1.427 1.888 1.910 30 960 870 1-582 2.140 2.147 40 034 895 1.429 1.892 1-914 40 959 870 1.585 2.145 2.152 SO 033 894 1.432 1.896 1-917 SO 958 869 1.588 2.150 2.156 .4700 031 894 1-434 1.900 1. 92 1 57 00 957 869 1.590 2.154 2.161 10 030 893 1-437 1.904 1.924 10 956 869 1-593 2.159 2.166 20 029 893 1-439 1.908 1.928 20 954 868 1.596 2.164 2.170 30 027 026 893 1.442 1.912 1.932 30 953 868 1-599 2.169 2.17s 40 892 1.444 1.916 1-935 40 952 867 1.601 2.173 2.178 2.179 SO 025 892 1.447 1.920 1-939 SO 951 867 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 2.19^ 2.198 20 021 890 1.454 1-931 1.950 20 947 866 1.613 2.193 30, 020 890 1-457 1-935 1-953 30 946 866 1.615 2.197 2.202 2.203 40 019 890 1-459 1-939 1-957 40 945 865 1.618 50 017 889 1.462 1-943 1.961 SO 944 865 1.621 2.207 2.213 4900 10 016 015 889 888 888 1.464 1.467 1-947 1-951 1.964 1.968 59 00 10 943 942 864 864 1.624 1.627 2.212 2.217 2.217 2.222 20 013 1.469 1-955 1-972 20 941 864 1.630 2.222 2.227 30 40 SO 5000 012 on 010 008 888 887 887 886 1.472 1-475 1-477 1.480 1-959 1.963 1.967 1-971 1-975 1-979 1.983 1.987 30 40 SO 60 00 939 938 937 936 863 863 863 862 1.632 1-638 1.641 2.227 2.232 2-237 2.242 2.232 2.237 2.242 2.247 Smithsonian Tables. 76 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.] « a\ b\ = cx a2 bi <:■!. ^ a\ *i=a "1 *2 C-i 6o°oo' 8.50936 8.50862 1.641 1.2\Z 2.247 7o°oo' 8.50877 5.50842 1.841 2.607 2.608 10 935 862 1.644 2.247 2.252 10 876 842 1.84s 2.615 2.616 20 934 86i 1.647 2-253 2.257 20 87s 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 4° 932 861 1-653 2.263 2.267 40 874 841 1.857 2.637 2.638 SO 931 860 1.656 2.268 2.272 50 873 841 1.861 2.645 2.646 6i 00 860 1.659 2.273 2.277 71 00 872 841 1.86s 2.653 2.653 10 928 860 1.66I 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 30 870 840 ^■W 2.676 2-677 4° 925 858 1.671 2.295 2.298 40 869 840 1.881 2.684 2.685 5° 924 858 1.674 2.300 2.303 SO 868 840 1.886 2.692 2.693 6200 923 858 1.677 2.305 2.309 72 00 868 839 1.890 2.701 2.701 10 922 8S7 1.680 2.311 2.314 10 867 839 '■^94 2.709 2.709 20 921 857 1.683 2.316 2.320 20 866 839 1.898 2.717 2.718 30 920 857 1.686 2.322 2.325 30 865 In 1.903 2.725 2.726 40 919 856 1.689 2.327 2.330 40 ?^s l^l 1.907 2.734 2.734 S° 918 856 1.692 2.333 2.336 SO 864 838 I 912 2.742 2.742 6300 917 856 ;S 2-338 2.341 7300 f^ l^. 1.916 2.751 2.751 10 916 855 2-344 2.347 10 862 838 1. 92 1 2.760 2.760 20 91 S 8SS 1.701 2.350 2-352 20 862 837 1-925 2.769 2.769 30 913 855 1.704 2-355 2.358 30 861 ^37 1.930 ^■^^^ 2.778 40 912 854 1.708 2.361 2.364 40 860 837 1-935 2.786 2.787 50 911 854 1.711 2.367 2.369 SO 860 837 1-939 2.795 2.796 6400 910 854 1.714 2-373 2-375 7400 f^i ¥i 1.944 2.804 2.805 10 909 853 1.717 2.378 2.381 10 ^5? ¥i 1.949 2.814 2.814 20 908 853 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 \% f36 1.958 2.832 2.833 40 906 85i 1.727 2.396 2.398 40 ¥i 836 1.963 2.842 2.842 SO 90s 852 1-730 2.402 2.404 50 856 835 1.968 2.851 2.852 65 00 904 852 1-733 2.408 2.410 7500 ^55 ^35 1-973 2.861 2.861 10 903 851 1-737 2.414 2.416 10 854 ^35 1.978 'ir '•«^' 20 902 851 1.740 2.420 2.422 20 854 83s 1.984 2.881 2.881 30 901 851 1-743 2.426 2.428 30 f" 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 SO 900 850 1.750 2.438 2.440 50 852 834 1.999 2.911 2.gi2 6600 899 850 1-753 2.445 2.446 76 00 f5' 834 2.005 2.922 2.922 10 898 849 1-757 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 40 895 848 1.767 2.470 2472 40 \n 833 2.027 2.965 2.965 s° 894 848 1.771 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 1.788 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-°33 3033 50 888 846 1.792 2.516 2.517 50 845 832 2.068 3-045 3-045 ' 6S00 887 846 1-795 2.522 2.524 7800 844 832 2.074 3-057 3-057 10 20 887 886 846 845 1.799 1-803 2.529 2.536 2-531 2-537 10 20 844 843 ^3' 831 2.080 2.086 3.069 3.082 3.069 3.082 30 88s 845 1.806 2-543 2.544 30 843 i3' 2.093 3-094 3-094 40 884 845 1.810 2.550 2.551 40 842 ^3' 2.099 3-107 3-107 SO 883 844 1.814 2-557 2.558 5° 842 831 2.106 3.120 3.120 6900 10 882 844 i.8r8 2.564 2.565 7900 841 ?3' 2.113 3-133 3-133 881 844 1.821 2.571 2.572 10 841 830 2.119 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 t^A 3-'74 3.188 40 879 843 1-833 2-593 2.594 40 839 830 2.140 Sf 87I 843 1-837 2.601 SO 839 830 2.148 3.202 3.202 70 60 877 842 1.841 2.607 2.608 80 00 839 830 2.155 3.216 3.216 tm^KtStum 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 1007.67 1007.68 1007.71 20 Z015.31 2015.3Z 2015.34 2015.37 2015.41 30 302Z.97 3022.98 3023.01 3023,06 3023.12 40 4030.63 4030.64 4030.68 4030.74 4030.83 50 S038.Z8 5038.30 5038.35 5038.42 5038.54 60 6045.94 6045.96 6046.02 6046.11 6046.24 ic/ 60459.4 60459.6 60460,2 60461. 1 60462.4 zo 1Z0918.S 1Z0919.Z 120920,4 12092Z.2 120924.8 30 181378.3 181378.8 181380.6 181383.3 181387.3 40 Z41837.7 241838.4 241840.8 241844.4 241849.7 5° 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 Z015.71 Z015.S2 30 3023.20. 3023.31 3023.43 30Z3.56 3023.72 40 4030.94 4031.08 4031.24 4031.42 4031.63 1° 5038.67 5038.84 503904 5039.28 5039.54 60 6046.41 6046.61 6046.85 6047.13 6047.4s 10' 60464.1 60466.1 60468.5 60471.3 60474.5 20 120928.2 J20932.3 120937-1 120942.6 120949.0 30 181392.3 181398.4 181405.6 181413.9 181423.4 40 Z41856.4 Z41864.6 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° n° 12° 13° 14° 10" 1007.97 1008.03 1008,10 1008.18 1008.26 20 2015.93 2016.06 2016,20 2016.35 Z016.51 30 3023.90 3024.09 3024.30 3024.52 3024.77 40 4031.86 403Z. 12 4032.40 4032.70 4033.02 1° 5039.83 5040.15 5040.50 5040.88 5041.28 60 6047.80 6048.18 6048.60 6049.05 6049.54 10' 60478.0 60481.8 60486.0 60490.5 120981.0 60495.4 20 120955.9 120963.6 120972.0 120990.7 30 181433.9 181445.4 181458.0 181471.5 181486.1 40 241911.8 241927.Z 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 Z016.69 2016.87 2017.06 2017.27 Z017.48 30 30Z5.03 3025.30 3025.60 3025.90 3026.23 40 4033.37 4033.74 4034.13 403454 4034.97 1° 5041.7Z 5042.18 5042.66 5043. 18 5043.71 60 6050.06 6050.61 6051.19 6051.81 6052.45 10' 60500,6 60506.1 60511.9 605 18. I 60524.5 20 I2IOOI.Z 121012.2 121023.8 121036.2 121049.0 30 1815OI.7 181518.3 181535.8 181554.3 181573.6 40 Z4200Z.3 242024.4 242047.7 242072.4 Z42098.1 SO 30250Z.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 1009.10 2018.10 3027.28 1009.22 1009.35 2018.70 zo Z017.71 2017.95 2018.44 30 3026.56 3026.92 3027.66 3028.06 40 4035.42 4035.89 4036.38 4036.88 4037.41 f 5044.28 5044.86 5045.48 5046. 10 5046^76 60 6053.13 6053.84 6054.57 6055.33 6056. 1 1 lo/ 60531.3 60538.4 60545.7 60553-3 60561.1 20 121062.6 121076.8 121091.4 I2II06.S 121122.2 30 181593.9 181615.1 181637.1 181659.8 181683.4 40 242125.2 242153.5 242182.8 242213.0 Z42244.5 so 302656.5 302691,9 302728.5 302766.3 302805.6 60 363187.8 363330.3 363274.2 363319-6 363366.7 Smithsonian Tables. 78 Table 17. LENGTHS OF TERRESTRtAL 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 2019.83 1010.07 20 2018.97 2019.25 2019.54 2020.13 30 302S.46 J028.88 3029.31 3029.75 3030.20 40 4037.95 4038.51 4039.08 4039.67 4040.27 5° 5047.44 5048-13 5048.85 5049.58 5050.33 60 6056.92 6057.76 6058.62 6059.50 6060.40 ly 60569.2 60577.6 60586.2 60595.0 60604.0 20 121138.5 121155.2 121172.3 Z21190.0 121208.0 30 181707.7 181732.7 181758.5 181785.0 181812.0 40 242276.9 242310.3 242344.7 242379.9 242416.0 50 302846.1 302887.9 302930.9 302974.9 303019.9 60 363415-4 363465.5 363517-1 363569.9 363623.9 30° 31° 320 33° 34° 10" 1010.22 1010.38 1010.54 1010.70 1010.86 20 2020.44 2020.7s 2021.07 2021.40 2021.73 30 3030.66 3031.13 303 I -61 3032.10 3032.59 40 4040.88 4041.51 4042-15 4042.80 4043.46 so 5051.10 5051.89 5052.68 5053.50 5054.32 60 6061.32 6062.26 6063^2 6064.20 6065.19 10' 60613.2 60622.6 60632.2 60642.0 60651.9 20 121226.4 121245.3 12 1 264. 4 121283.9 121303.8 30 181839.7 181867.9 :8i896.6 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 3639i'-3 35° 36° 37° 38° 39° 10" 1011.03 I0II.20 1011.37 1011.55 IOII.72 20 2022.06 2022.40 2022.75 2023.09 2023.44 30 3033.10 3033.61 3034->2 3034.64 3035-17- 40 4044.13 4044.81 4045.50 4046.19 4046.89 SO 5055.16 5056.01 5056.87 5057-74 5058.61 6a 6066.19 6067.21 6068.24 6069.29 6070.34 10' 60661.9 60672.1 60682.4 60692.9 60703.4 20 121323.9 I2I344-3 121364.9 121385.7 121406,7 30 181985.8 182016.4 182047.3 182078.6 182IIO.I 40 242647.8 242688.S 242729.7 242771-4 242813.4 so 303309-7 303360.6 303412.2 303464.3 303516.8 60 363971.7 364032.8 364094.6 364157. 1 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 2024.87 2025.23 30 3035-70 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 6072.46 6073.53 6074.61 6075.69 lo/ 60713.9 60724.6 60735.3 60746.1 60756.9 20 121427.9 121449.2 121470.6 121492.2 121513-7 30 182141.8 182173.8 182206.0 182238.2 182270.6 40 242855.S 242898.4 242941.3 242984.3 243027.4 50 303569.7 303623.0 303676.6 303730.4 303784-3 60 364283.7 364347.6 364411.9 364476.5 48° 364541-2 45° 46° 47° 49° 10" 1012.79 1012.97 1013.15 1013.33 1013.51 20 2025.59 2025.95 2026.31 2026.67 2027.02 30 3038.38 3038.92 3039.46 3040.00 3040.54 40 4051.18 4051.90 4052.62 4°53-34 4054.05 5° 5063.97 5064.87 5065.77 5066.67 5067.56 60 6076.77 6077.85 ^ 6078.93 6080.00 6081.08 lO* 60767.7 60778.5 60789.3 60800.0 60810.8 20 I2I53S-3 121556.9 12157S.5 121600.1 121621.5 30 182303.0 182335.4 182367.8 182400.1 182432.3 40 243070.6 243"3-9 243157.0 243200.1 243243.0 50 303838.3 303892.4 303946.3 304000.1 304053.8 60 364606.0 364670.8 364735.S 364800.2 364864.5 Smithsonian Tables. 79 Table 1 7. LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN. [Derivation ot table explained on p. xlvi.] Latitude Latitude. Latitude. Latitude. Latitude. Latitude. Latitude. Interval. 50° 51° 52° 53° 54° 55° Feet. Feet. Feet. Feet. Feet. Feet. lo" 1013.69 1013.87 1014.04 1014.22 1014.39 1014.56 20 2027.38 2027.74 2028.09 2028.44 2028,78 2029.12 30 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 5° 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 10/ 60821.5 60832.1 60842.6 60853.1 60863.5 60S73.7 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° 58° 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 2031.07 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 5073.6s 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 60932.0 20 121767.6 121787.5 121807.2 121826.5 182739.8 121845.5 121864.1 30 182651.4 182681.3 182710.8 18276S.2 182796.1 40 243535.2 243575.0 243614.4 243653.0 243691.0 243728.2 50 304419.0 304468.8 304518.0 304566.3 304613.7 304660.2 60 365302.8 365362.6 365421.6 365479.6 365536.4 365592.2 62° 63° 64° 6s° 66° 67° 10" 1015.69 1015.83 1015.98 1016.12 1016,26 1016.39 20 2031.37 2031.67 2031.96 2032.24 2032,51 2032.78 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 10' 60941.2 60950.0 60958.7 60967.1 60975.4 60983.3 20 121882.3 121900.1 121917.5 121934.3 121950.7 121966.6 30 182823.5 182850.1 182876.2 182901.4 1S2926.1 182949.8 40 243764.6 243800.2 243835.0 243868.6 243901.4 243933.1 SO 304705.8 304750.2 304793.7 304835.7 304876.8 304916.4 60 365647.0 365700.2 365752.4 365802.8 3658|;2,2 365899.7 68° 69° 70° 71° 72° 73° 10" 1016.52 1016.64 1016.76 1016.87 1016,98 1017.09 20 2033.03 2033.28 2033.52 2033-75 2033,96 2034.17 30 3049.55 3049.92 3050.28 3050.62 30.50.95 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 5085.43 60 6099. 10 6199.84 6100.55 6101.24 6101.89 6102.52 10' 60991.0 61998.4 61005.5 61012.4 61018.9 61025.2 20 121982.0 121996.8 I22011.1 122024.8 122037.8 122050.3 30 182973.1 182995.2 183016.6 183037.1 183056.8 1 83075. 5 40 243964.1 243993.6 244022.2 244049.5 244075.7 244100.6 50 30495s. I 304992.0 305027.7 305061.9 305094,6 305125.8 60 365946.1 365990.4 366033.2 366074.3 366113,5 366151.0 74° 75° 76° 77° . 78° 79° 10" 1017.18 1017.28 1017.37 1017.45 '017-53 1017.60 20 2034.37 2034.56 2034.73 2034.90 2035,05 2035.19 30 3051.56 3051.84 3052.10 3052.3s 3052,58 3052.79 40 4068.74 4069.12 4069.46 4069.80 4070.10 4070.38 1° 5085.92 5086.40 5086.83 S087.24 50S7.63 5087,98 60 6103.11 6103.67 6104.20 6104.69 6105.16 6105.58 10' 61031. I 61036.7 61042.0 61046.9 61051.6 61055.8 20 122062.2 122073.5 122083.9 122093.9 122103,1 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 1° ^IV^Wl 305183.7 305209.8 305234.7 305257.8 305278.8 60 366186.6 366220.4 366251.8 366281.6 366309.4 366334.6 Smithsonian Tables. 80 Table 18. LENGTHS OF TERRESTRIAL ARCS OF PARALLEL. [Derivation of table explained ou p. xlix.] Longitude Latitude. Latitude. Latitude. Latitude. Latitude. Interval. 0° 1° 2° 3° 4° Feet. Feet. Feet. Feet. Feet. lo" 1014.52 1014.37 1013.91 1013.14 1012.07 20 202g.o5 2028.74 2027.82 2026.29 2024. 14 30 3043.57 3043- I' 3041.73 3039-43 3036.21 40 4058.10 4057.48 4055-64 405257 4048.28 5° 5072.62 5071.86 5069.5s 5065.72 5060.35 60 6087.14 6086.23 6083.46 6078.86 6072.42 10' 60871.4 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 5° 304357-2 304311.3 304173-0 303942-9 303621.0 6a 365228.6 365173-6 365007.6 364731-5 364345.2 5° 6° 7° 8° 9° lO" 1010.69 1009.00 1007.01 1004.72 1002.12 20 2021.38 2018.01 2014.03 2009.43 2004.23 30 3032.07 3027.01 3021.04 3014.15 3006.3s 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 10' 60641.4 60540.2 60420.8 60283.0 60127.0 20 121282.S 1210S0.5 120841.6 120566.0 120254.0 30 181924.2 181620.7 181262.3 180849.1 18038 1. I 40 242565.6 242161.0 241683.1 241132.1 240508. 1 50 303207.0 302701.2 302103.9 301415.1 300635.1 60 363848.4 363241-4 362524.7 361698. 1 360762.1 10° 11° 12° 13° 14° jo" 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 3954-76 3938-34 so 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 59550.0 59321.5 59075-0 20 H9905.6 119520.8 119100.0 118642.9 118150.1 30 179858.3 179281.3 178650.0 177964.4 177225.1 40 23981 I. I 239041.7 238200.0 237285.8 236300.2 50 299763.9 298802.1 297750.0 296607.3 295375-2 60 359716.7 358562.5 357300.0 355928.S 3544S0.2 15° 16° 17° 18° 19° 10" 980.18 975-47 970.48 965.18 959-60 20 1960.3s 1950.95 1940.95 1930.36 1919.19 30 2940.53 2926.42 2911.42 2895.55 ^S'S-^? 40 3920.71 3901.90 3881.90 3860.73 3838.38 50 4900.88 4877-37 4852.38 4825.91 4797-98 60 5881.06 5852.84 5822.85 5791.09 5757-58 10' 58810.6 58528.4 58228.5 57910.9 57575-8 20 117621.2 117056.9 116457.0 115821.8 iiS'5i-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" 953-72 947-55 941.10 S'l'* 927-33 20 1907.44 1895.10 1882.19 1868.71 1854.67 30 2861,15 2842.66 2823.29 2803.07 2782.00 40 3814-87 4768,59 3790,21 3764-38 3737-43 3709-33 50 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 113706.2 112931.3 112122.8 111280.0 30 171669.2 170559-4 169397-3 168184.3 166919.9 40 228892.3 227412.5 225863.0 224245.7 222559.9 50 286115.4 284265.6 282328.8 280307. 1 278199-9 60 343338.S 34i"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. 25° 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 3S85-76 3552.12 so 4600.14 4562.21 4522.89 4482.20 4440.15 5328.18 60 SS20.I7 5474-65 5427-47 5378.64 10' 55201.7 54746.5 54274-7 53786.4 53281.8 20 110403.3 109493.0 108549-5 107572.9 106563. 5 30 165605.0 164239-5 162824.2 161359-3 159845-3 40 220806.6 218986. 1 217099.0 215145-7 ';i"?-j SO 276008.3 273732.6 271373-7 268932.2 266408.8 60 331209.9 328479.1 325648.4 322718.6 319690.6 30° 31° 32° 33° 34° to" 879-35 1758.70 870.40 861.18 851.71 841.97 20 1 740. So 1722.37 1703.41 1683.94 30 2638.04 2611.20 2583.55 2555-12 =s?5.9; 40 3517-39 3481.59 3444-74 3406.83 3367-88 50 4396-74 4351.99 4305-92 4258.53 4209.8s 60 5276-09 5222.39 5167.10 5110.24 5051.82 10' 52760.9 52223.9 S1671.0 51102.4 50518.2 20 105521.8 104447.8 103342.1 102204.8 101036.4 30 158282.6 156671.8 155013-1 153307-3 151554-6 40 21 1043.5 208895.7 206684.2 204409-7 202072.8 SO 263804.4 261119.6 258355-2 255512.1 252591.0 60 316565.3 313343.5 310026.3 306614.5 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 2401.4s 2368.48 40 3327-9I 3286.91 3244.92 3201.93 3157-97, SO 4159-88 4108.64 4056.15 4002.42 3947-46 60 4991.86 4930.37 4867.38 4802.90 4736.9s 10' 49918.6 49303.7 48673.8 48029.0 47369-5 20 99837.2 98607.4 97347-6 96058.0 94739-1 30 "49755-8 147911.2 1 4602 1. 4 1440S7.0 142108.6 40 199674.3 197214.9 194695.2 192116.0 189478.2 SO 249592-9 246518.6 243369.0 240145.0 236847.7 60 2J95I1-5 295822.3 292042.8 288174.0 284217.2 40° 41° 42° 43° 44° 10" 778.26 766.79 75 =.08 743-iS 730.98 20 1556-52 1533.58 1510.17 1486.29 1461.90 30 2334-78 2300.37 2265.25 2229.44 2192.95 40 3l'3-o4 3067.16 3020.33 2972.59 2923.93 SO 3891.30 3833.94 3775-42 3715.73 3654.91 60 4669.56 4600.73 4530.50 4458.88 4385.89 10' 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 1359'50 133766.4 131576.8 40 186782.3 184029.3 181220.0 178355.2 175435-8 SO 233477-9 230036.7 226525.0 222944.0 219294.7 6a 280173.S 276044.0 271830.1 267532.8 263153.6 45° 46° 47° 48° 49° 10" 718.59 705-99 693.16 680.12 666.87 20 I437-I9 1411.97 1386.32 1360.24 1333.75 30 2155.78 2117.96 2079.48 2040.36 2000.62 40 2874.38 2823.94 2772.64 2720.49 2667.50 SO 3592.97 3529.93 3465-80 3400.61 3334-37 60 4311-56 4235-91 4158.96 4080.73 4001.25 jo/ 43115-6 42359-1 84718.2 41589-6 40807.3 40012.5 20 86231-3 83179.2 81614.6 80024.9 30 129346.9 127077.3 124768.7 122421.9 120037.4 40 172462.5 169436.S 166358.3 163229.2 160049,9 50 215578.2 2II795-6 207947.9 204036.4 200062.3 6a 258693.8 254'54.7 249537-5 244843.7 240074.8 Smithsonian Tables. 82 Table i8. LENGTHS OF TERRESTRIAL ARCS OF PARALLEL. [Derivation of table explained on p. xlix.] Longitude Latitude. Latitude. Latitude. Latitude. Latitude. Latitude. Interval. 50° SI° 52° 53° 54° 55° J'eet. Feet. Feet. Feet. Feet. Feet. lO" 653.42 639.77 625.92 611.88 597-65 583-23 20 1306.85 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 2503.68 2447-51 2390,59 2332.93 50 3267.12 3198-85 3129.60 3°59-39 2988.28 2916.16 6a 3920.54 3838.62 3755-52 3671.27 3585-89 3499-40 10' 39205.4 38386.2 37555-2 36712.7 35858.9 349940 20 78410.8 76772.4 75110.4 73425-4 71717.8 69988.0 30 117616.1 11515S.6 112665.6 110138.0 107576.6 104981.9 40 156821.5 153544-8 150220.8 146850.7 ■43435-5 ■39975-9 1° 196026.9 19193 1 -0 187776.0 ■83563-4 179294.4 174969.9 60 235232.3 230317.2 225331.2 220276. 1 2I5I53.3 209963.9 S6° 57° 58° 59° 60° 61° 10" 568.64 553-87 538.93 523.82 508.55 493-13 20 1137-28 1 107.74 1077.86 1047.65 1017.11 986.26 30 1703.92 1661.61 1616.79 ■571-47 1525.66 ■479-38 40 2274.56 2215.48 2155-72 2095.29 2034.22 1972-52 50 2843.20 2769-35 2694.64 2619.12 2542-77 2465.64 60 34"-83 3323.22 3233-57 3142.94 3051-33 2958-77 icy 341 18.3 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 9428S.1 91539-9 88763.2 40 ■36473.4 132928.8 129343.0 125717.5 122053.2 118351.0 50 170591.7 166161.0 161678.7 157146.9 ■52566.5 ■47938-7 60 204710.0 199393-2 194014,4 188576.3 183079.8 ■77526.4 62° 63° 64° 65° 429-95 66° 67° 10" 477-55 461.83 445-96 413-82 397-55 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.S4 1719.81 1655.26 1590.19 50 2387.76 2309.14 2229.80 2149.76 2069.08 1987,74 60 2865.31 2770.96 2675.75 2579.72 2482.89 2385-29 lo/ 28653.1 27709.6 26757-5 25797.2 24828.9 23852,9 20 57306.2 55419-2 53515-1 51594.4 49657.8 47705,8 30 85959.4 83128.9 80272.6 77391-S 74486.7 71558-6 40 114612.5 110838.5 107030.2 103188.7 99315.6 95411.5 50 143265.6 138548.1 133787-7 128985.9 ■24144-5 119264.4 60 17>9«8.7 166257.7 160545.2 154783.1 1489734 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 1392.12 1325.20 1257.88 1190.16 SO 1905.79 1823.25 1740. 14 1656.50 1572-34 ■487 70 60 2286.95 2187.90 208S.17 1987.81 1886.81 ■785-23 10' 22869.5 21879.0 20881.7 19878.1 18868,1 17852.3 20 45739-0 43758.0 41763-S 39756-1 37736.3 35704-7 30 686084 65637.0 62645.2 59634.2 56604.4 53557-0 40 91477.9 87516.0 83527-0 795^2-2 75472.6 71409,4 so 114347-4 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° 10" 280.52 263.41 246.22 228.96 211.62 194,22 20 561.04 526.82 492.44 4S7-9' 423-24 388,43 30 841.56 790.23 738.66 686.86 634-85 582.64 40 fl22.08 1053.64 984.88 915.82 846,47 776.86 50 X4O2.60 1317-06 1231.10 1144.78 1058.09 971.08 6a 1683.11 1580,47 1477-33 ■373-73 1269.71 1165.29 zol 1683I.I 15804.7 ■4773-3 13737-3 12697.1 11652.9 20 33662.3 31609.3 29546.5 27474.6 25394-2 23305-8 30 50493.4 47414.0 44319.8 412 1 1.9 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.S 60 100986.8 94828.0 88639.6 82423.8 76182.5 69917.4 Smithsonian Tables. «3 Table 1 9. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE teAttt- [Derivation of table explained on pp. liii — Ivi.] P Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR- 15' longitude. yj longitude. 45' longitude. 1° longitude. X y X y X y X y Inches. Inches, Inches. Inches. Inches. Inches. Inches. Itiches. Inches. o°oo' IS 3° 45 13-059 4-383 4-383 4-383 4.382 .000 .000 .000 .000 8.766 8.766 8.76s 8.76s .000 .000 .000 .001 13-148 13-148 13.148 13.147 .000 .000 .001 .001 17.531 17.531 17-530 17-530 .000 .001 .001 .002 i 00 IS 30 45 17.412 4.382 4-382 4.381 4.381 .000 .000 .000 .000 8.764 8.764 8.763 8.762 .001 .001 .001 .001 13.146 13.145 13-144 13.142 .001 .002 .002 .003 17.528 17.527 17-525 17.523 .003 .003 .004 .005 13-059 2 00 IS 3° 4S 17.412 4.380 4-379 4-379 4-378 .000 .000 .000 .000 8.760 8.759 8.757 8.755 .001 .001 .001 .002 13.141 13.138 13.136 13.133 •003 .003 .004 .004 17.521 17.518 17.514 17.511 .005 .006 .007 .007 4-353 8.706 13-059 3 00 IS 30 45 17-413 4-377 4-376 4-375 4-373 .001 .001 .001 .001 8.753 8.751 8.749 8.747 .002 .002 .002 .002 13.130 13.127 13.124 13.120 .004 .005 .005 .006 17.507 17.503 17.498 17.494 .008 .008 .009 .009 8.706 13.060 400 IS 30 45 17-413 4-372 4-371 'A .001 .001 .001 .001 8.744 8.742 8.739 8.736 .003 .003 .003 ■003 I3.I16 13.II2 13.108 13.104 .006 .006 .007 .007 17.488 17-483 17.478 17.472 .010 .011 .012 .013 4-3S3 8.707 13.060 5 00 IS 30 45 17-413 4-366 4-364 4-363 4-361 .001 .001 .001 .001 8.732 8.729 8.725 8.722 .003 .003 .004 .004 13-099 13.082 .007 .008 .008 .008 17.465 17.458 17.451 17.443 .013 .014 .014 .015 4-353 8.707 13.060 6 00 IS 30 45 17-414 4-359 4-3S7 4-355 4-353 .001 .001 .OOI .001 8.718 8.714 8.710 8.70s .004 .004 .004 .004 13.076 13.071 13064 13.058 .009 .009 .010 .010 17.435 17.428 17.419 17.410 .016 .017 .017 .018 4-3 54 8.707 13.061 7 00 IS 30 45 17.414 4-350 4-348 4-346 4.343 .001 .001 .001 .001 8.701 8.696 8.691 8.686 .005 .005 .005 .005 13.051 13-044 13-036 13.029 .010 .011 .011 .011 17.4^- 17.392 17.382 17.372 .019 .019 .020 .020 4-354 8.707 13.061 800 IS 30 45 17-415 4-340 4-338 4-335 4-332 .001 .001 .001 .002 8.681 8.675 8.670 8.664 .005 .006 .006 13.021 13-013 13-00S 12.996 .012 .012 .013 .013 17.362 17-351 17-340 17.328 .021 .022 .022 .023 ^•354 8.708 13.062 900 IS 30 45 17.416 4-329 4-326 4-323 4.320 .002 .002 .002 .002 8.658 8.652 8.646 8.640 .006 .006 .006 .006 12.987 12.979 12.969 12,960 .013 .014 .014 .014 17,316 17.305 17.292 17.280 .024 .024 .026 4-3S4 8.708 13.062 10 00 17.417 4-317 .002 8.633 .006 12.950 .015 17,266 .026. Smith soNiAr 1 TABtes. 84 Table 19. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iiiUi - [Derivation of table explained on pp. liii— Ivi.j ■s . ■J a 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 Inchts. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. io°oo' IS 30 4S 4-354 8.709 13-063 4-317 4-313 4.310 4-306 .002 .002 .002 .002 8-633 8.626 8.620 8.613 .006 .007 .007 .007 12.950 1 2.940 12.930 12.919 .015 •015 .015 .016 17.266 17-253 17.240 17.226 .026 .027 .027 .028 II 00 IS 30 4S 17.418 4-303 4.299 4.29s 4.292 .002 .002 .002 .002 8.606 8.S98 I:lt3 .007 .G07 .007 .008 12.908 i2!886 12.87s .016 .016 .017 .017 17.211 17.196 17.182 17.166 .029 .029 .030 .031 4-355 8.709 13.064 12 00 IS 30 4S 17.419 4.288 4.284 4.280 4.27s .002 .002 .002 .002 8-575 8-567 8.SS9 8-551 .008 .008 .008 .008 12.863 12.851 12.839 12.826 .017 .018 .018 .019 17.150 17-134 17.118 17.102 .031 .032 .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-S34 8.52s 8.516 .008 .009 .009 .009 12.813 12.800 12.787 12.774 .019 .019 .030 .020 17.084 17.067 17.050 17.032 •034 -034 ■035 •035 4-3SS 8.7 II 13.066 1400 IS 30 4S 17.421 4-2S3 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 .02 1 .021 .021 17.013 16.99s 16.976 16.957 .036 .036 io38 4-356 8.7 1 1 13.067 1500 15 30 45 17-423 4-234 4.229 4.224 4.219 .002 .002 .002 .002 8.469 8.4S9 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 4S 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 .on 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 3° 4S 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 .on 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 1800 IS 30 45 17.427 4.170 4.164 4-158 4.152 .003 .003 -003 .003 8-339 8.327 8.316 8-3°3 .011 .011 .012 .012 12.509 12.491 12-473 J 2-455 .025 .026 .026 .026 16.679 '^■^" 16.631 16.606 -045 -045 .046 .046 4.357 8-715 13.072 1900 15 30 4S 17.429 4.14s 4-139 4-133 4.127 .003 .003 .003 .003 8.291 8.278 8.266 8.253 .012 .012 .012 .012 12.436 12.418 12.399 12.380 .036 .027 .027 .027 16.582 16-557 16.506 .047 .048 , -048 .048 4-358 8.716 13-073 2000 17-43' 4.120 .003 8.240 .012 12.360 .028 16,480 .049 Smithsonian Tables. 8,"! Table 1 9. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE tt^ts- [Derivation of table explained on pp. liil-lvi.] 20"00 IS 3° 45 21 00 15 3° 45 15 3° 45 23 00 15 30 45 24 00 15 30 45 25 00 IS 30 45 2600 IS 30 45 27 00 15 30 45 28 00 15 30 45 2900 15 30 45 .2 ""o^ ? " S S Inches, 4-358 8.717 13-075 17-433 ^■359 8.718 i3-°76 17-435 4-359 8.719 13.078 17-437 4.360 8.720 13.080 17-439 4.360 8.721 13.081 17.442 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 15' longitude. 4.361 8.722 13-083 17-444 4.362 8.723 13.085 17.446 4.362 8.724 13-087 17-449 4-363 8.726 13.088 17.451 4-363 8.727 13.091 3000 17.454 Inches. 4.120 4.I14 4.107 4.100 4.094 4.087 4.080 4-073 4.066 4.058 4.051 4.044 4.036 4.029 4.021 4.014 4.006 3-998 3-990 3.982 3-974 3.966 3-958 3-950 3-942 3-933 3-925 3.916 3.908 3-899 3-881 3-873 3.863 3-854 3-845 3-836 3.827 3-817 3.808 3-799 Inches. .003 .003 .003 .003 •003 .003 -003 .003 .003 .003 .003 .003 .003 .003 .003 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 so' longitude. Inches. 8.240 8.227 8.214 8.200 8.187 8-173 8.159 8-145 8.131 8.117 8.102 8.088 8.073 8.058 8.043 8.028 8.012 7-997 7.981 7.965 7-949 7-933 7.916 7.900 7-883 7.866 7-849 7-833 7-8i6 7-798 7.780 7-763 7-745 7-727 7.709 7.691 7-673 7.654 7-635 7-616 7.598 Inches. .012 .012 .013 •013 •013 .013 .013 .013 ■013 .013 .014 .014 .014 .014 .014 .014 .014 .014 .014 .015 .015 .015 .015 .015 .015 .015 .015 .015 .015 .016 .016 .016 .016 .016 .016 .016 .016 .016 .016 .016 .017 45' longitude. Inches. 12.360 12.340 12.321 12.301 12.280 12.260 12.239 12.218 12.197 12.175 12.154 12.132 12.109 12.087 12.064 12.041 12.018 "•995 11.971 11.948 11.923 11.899 11.874 11.850 11.825 n.800 11.774 11.749 11.723 11.697 11.671 11.644 11.618 11.591 11.563 11.536 11.509 11.481 "•453 11.425 11.396 Inches. .028 .028 .023 .029 .029 .029 .029 .030 •030 .030 •030 .031 .031 .031 .031 -032 .032 .032 .032 ■033 ■033 -033 -033 •034 -034 -034 -034 •035 •035 •03s •035 .036 -036 .036 .036 ■036 .036 •037 •037 •037 ■037 x" longitude. Inches. 16.480 16.454 16.428 16.401 16.374 16.346 16.318 16.291 16.262 16.234 16.205 16.176 16.146 16.116 16.086 16.055 16.024 15-993 1 5.962 15-930 15.898 15.865 15-832 1 5.800 15.767 15-733 15.699 15-665 15.631 15-596 15.561 15.526 15.490 15-454 15.418 15.382 15-345 15-308 I 5.270 15-233 15-195 Inches, .049 .050 .050 .051 .051 .052 -052 •053 ■053 ■054 -054 •055 -055 .055 .056 -056 •057 -057 .058 .058 -059 -059 •059 .060 .060 .061 .061 .061 .062 .062 .063 -063 .064 .064 .064 .065 .065 .065 .066 .066 .066 Smithsonian Tables. 86 Table 19. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Wtinr- [Derivation of table explained on pp. liii-lvi.] •3 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 Inches. Inches. Inches. Inches, Inches. Inches. Inches. Inches. Inches. 30°oo' 30 4S 4364 8.728 13.092 3-799 3-789 3-779 3-770 .004 .004 .004 .004 7-S98 7-578 7-SS9 7-540 .017 .017 .017 .017 11.396 11-367 11-338 11.309 •037 ■038 .038 15-195 15.156 15.I18 15.079 .066 .067 .067 .067 31 00 IS 30 45 17457 3.760 3-7 SO 3-740 3-730 .004 .004 .004 .004 7.520 7.500 7.480 7.460 .017 .017 .017 .017 11.280 11.250 II. 221 II. 191 -038 •038 .038 .038 15.040 15.001 14.961 14.921 .068 .068 .068 .068 4-365 8.730 13095 3200 IS 30 4S 17.460 3.720 3-710 3-700- 3.690 .004 .004 .004 .004 7.441 7-420 7.400 7-379 .017 .017 .017 .017 II.l6l 11-130 1 1. 100 11.069 -039 -039 •039 -039 14.881 14.840 14-799 14.758 .069 .069 .069 .070 4.366 8731 13-097 3300 IS 30 4S 17.462 3-679 3.648 .004 .004 .004 .004 7-359 7-338 7-317 7.296 .017 .018 .018 .018 11.038 11.007 10.975 10.943 -039 -039 .040 .040 14.718 14.676 14-633 14-591 .070 .070 .070 .071 4-366 8-733 13.099 3400 IS 30 4S 17-465 3-637 3.626 3-6i6 3.605 .004 .004 .004 .004 7-275 7-253 7-231 7.210 .018 .018 .oi8 .018 10.912 10.879 10.847 10.815 .040 .040 .040 .040 14.549 14.506 14.463 14.420 .071 .071 .071 .072 4-367 8-734 13.101 3S00 IS 30 45 17.468 3S94 3-583 3-572 3-561 .004 .004 .004 -005 7.188 7.166 7.144 7.122 .018 .018 .018 .018 10.782 10.749 10.716 10.683 .040 .041 .041 .041 14-376 I4.'288 14.244 .072 .072 .072 •073 4-368 8-735 13-103 3600 IS 30 45 17.471 3-550 3-539 3-527 3-516 .005 .005 .COS .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 IS - 30 45 17-473 3-504 3-493 3.481 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 13.108 3800 IS 30 45 17-477 3-458 3446 3-434 3.422 .005 .005 .005 .005 6.916 6.892 6.869 6.84s .019 .019 .019 .019 10.374 10-339 10.303 10.267 .042 .042 .042 .042 13-832 13-785 13-737 13.690 .074 .074 -07 s .075 4-370 8.740 13-110 3900 IS 30 45 17.480 3-411 3-398 3-386 3-374 .005 .005 .005 .005 6.821 6.797 6.773 6.748 .019 .019 .019 .019 10.232 I0.I9S 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 Smithsonia N Tables c Table 1 9. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE 7i;ijW> [Derivation of table explained on p. liii-lvi.j •S3 3« 4o°oo' IS 30 45 41 00 IS 3° 4S 42 00 IS 30 45 4300 15 30 45 4400 15 30 4S 45 00 15 30 45 46 00 IS 30 45 4700 15 30 45 48 00 15 30 45 4900 IS 30 45 50 00 sag .S m-a^ Inches, 4-371 8743 13-114 17.486 4-372 8.744 13-117 17.489 ^■373 8.746 13.119 17.492 4-374 8.747 13-121 17-495 .4-375 8.749 13.124 17.498 4-375 8.751 13-126 17-501 4-376 8.752 13-128 17-504 4-377 8.754 '3-131 17.508 4-378 8-755 13-133 17-511 4-378 8-757 13-13S 17.514 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 15' longitude. Inches. 3-362 3-35° 3-337 3-325 3-312 3-300 3-287 3-27S 3.262 3-249 3-230 3-223 3.210 3-197 3-184 3-170 3-158 3-144 3-131 3-1 18 3-104 3-091 3-077 3-063 3-050 3-036 3.022 3.008 2.994 2.980 2.966 2-952 2.938 2.924 2.909 2-895 2.881 2.866 2.852 2.837 2.823 so' longitude. Inches. .005 .005 .005 .005 .005 .005 •005 .005 .005 .005 -005 -005 -005 .005 •005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 ■005 .005 .005 .005 .005 .005 .005 .005 .005 .005 Inches, 6.724 6.699 6.650 6.625 6.600 6-575 6.549 6.524 6.498 6.472 6.447 6.421 6.394 6.368 6.342 6-316 6.289 6.262 6.235 6.209 6.i8i 6.154 6.127 6.100 6.072 6.044 6.017 5-989 5.961 5-933 5.904 5.876 5-848 5.819 5-790 5.762 5-733 5.704 5-675 5.646 Inches. .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 .019 45' longitude. Inches. 10.086 10.049 10.012 9-975 9-937 9.900 9.862 9.824 9.786 9-747 9-709 9.670 9.631 9.592 9-552 9-513 9-473 9-433 9-393 9-353 9-313 9.272 9-231 9.190 9.150 9.108 9.067 9-025 8.983 §■?*' 8.899 8.857 8.814 8.771 8.728 8.686 8.643 8.599 8-555 8.512 8.468 Inches. .042 .042 •043 •043 ■043 •043 -043 ■043 -043 ■043 -043 •043 •043 •043 -043 -043 •043 •043 •043 •043 -043 -043 •043 •043 •043 -043 -043 •043 •043 •043 •043 -043 •043 -043 -043 •043 ■043 •043 -043 .042 .042 1° longitude. Inches. 3-448 3-399 3-349 3-300 3-250 3-200 3-149 3-098 3.048 2.996 2-945 2-893 2.842 2.789 2.736 2.684 2.631 2.578 2.524 2.471 2-417 2-363 2.308 2.254 2.200 2.144 2.089 2.033 1-978 i.§6s 1.809 1-752 1.69^ 1.638 1-581 1.524 1.465 1.407 1-349 11.291 Inches. •075 .075 .076 .076 .076 .076 .076 .076 .076 .076 .076 .076 .076 .076 .076 .076 .077 .077 .077 •077 •077 •077 .077 .077 .077 •077 ,077 .077 .076 .076 .076 .076 .076 .076 .076 ,076 .076 .076 .076 .076 .076 Smithsonian Tables. 88 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on p. liii-lvi.} Table 19. scale ts-^utts- Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR — is' longitude. 30* longitude. 4S' longitude. 1° longitude. 1 X y X y X y X y Inches. Itukes. Inches. Inches. Inches. Inches. Inches, Inches. Inches. So°oo' 15 30 4S 4-379 8.758 13-137 2.823 2.808 2.793 2.779 .005 .005 .005 .005 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.174 II. 114 .076 -075 -075 -075 51 00 IS 30 4S 17-S17 2.764 2.749 2-734 2.719 .005 .005 .005 .005 5-528 5.498 5.468 5-438 .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 13.140 52 00 >S 30 4S 17.520 2.704 2.689 2.674 2.659 .005 .005 .005 .005 5.408 5-378 5-347 S-317 .019 .019 .019 .0l8 8.II2 8.067 8.021 7.976 .042 .042 .041 .041 10.816 10.756 10.695 10.634 .074 .074 -074 .074 ^■.71; 13.142 S3 00 IS 3° 4S 17-523 2.643 2.628 2.613 2.597 .005 .005 .005 .005 5.287 5.256 5.225 S-195 .018 .0x8 .018 .018 7-93° 7.884 7.838 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 3° 45 17.526 2.582 2.566 2.551 2.S35 .005 .005 .005 .005 5.164 S-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 8.764 13-147 S5o° IS 30 45 17.529 2.520 2.504 2.488 2.472 .005 .004 .004 .004 5-039 5.008 4.976 4-945 .018 .018 .018 .018 7-5S9 7-512 7.465 7-417 .041 .040 .040 .040 10.078 10.016 9-953 9.890 .072 .072 .072 .071 8.766 13-149 5600 IS 30 4S 17-532 2.456 2.441 2.425 2.409 .004 .004 .004 .004 4-913 4.881 4.849 4.817 .oi8 .018 .018 .018 7-370 7.322 7-274 7.226 .040 .040 .040 .040 9.826 9-763 9.699 9-635 .071 .071 .071 .070 4-384 8.767 13-151 5700 15 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 9-571 9.507 9-442 9-378 .070 .070 .070 .069 4-384 8.769 13-153 5800 17-537 2.328 .004 4.656 .017 6.985 ■039 9-313 .069 IS 3° 4S 4-385 8.770 13-155 2.312 2.296 2.279 .004 .004 .004 4.624 4.591 4-559 .017 .017 .017 6.838 •039 .038 9.248 9.183 9.117 .069 .068 .068 5900 IS 3° 4S 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 ^■^° 6.690 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 BMITHSONrAN TABLES. 8q Table 1 9. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jrniW- [Derivation of table explained on pp. liii-lvi.] 6o°oo' 15 30 45 61 00 IS 30 45 62 00 15 30 45 6300 IS 30 45 64 00 15 3° 45 6500 IS 30 45 6600 IS 30 45 67 00 15 30 45 6800 IS 30 45 69 00 IS 30 45 70 00 .S "i-oii as B-g N B U = 4.386 8-773 i3-'59 17.546 4-387 8.774 13.161 17.548 4.388 8.776 13-163 17-551 4-388 8.777 13.165 17-554 4-389 8.778 13.167 17-556 4-390 8.779 13.169 17-559 CO-ORDINATES OF DEVELOPED PARALLEL FOR — 15' longitude. 4-390 8.780 13-171 17.561 4-391 8.782 13.172 17-563 4-391 8.783 13-174 17-565 4-392 8.784 13.176 17-568 Ittches. 2.197 2.180 2.164 2.147 2.130 2.II4 2.097 2.080 2.063 2.046 2.029 2.012 1-995 1.978 1.961 1.944 1.926 1.909 1.892 1.875 1.857 1.840 1.823 1.805 1.770 1-753 1-735 1.717 1.700 1.682 1.664 1.647 1.629 i.6n I -593 1-575 I-5S7 1.540 1.522 1.504 Inches. .004 .004 •P04 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .003 .003 .003 .003 -003 .003 .003 .003 .003 •003 .003 .003 .003 .003 so' longitude. Inches- A-3 s g jS 0) p. ABSCISSAS OF DEVELOPED PARALLEL. 5' longitude. lo' longitude. IS longitude. 20' longitude. 25' longitude. 30' longitude. ORDINATES OF DEVELOPED PARALLEL. 8°00' lO 20 3° 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 IS 00 10 20 30 40 SO 1600 5.805 H.610 17.416 23.221 29.026 5.806 11.611 17.417 23.222 29.028 5.806 11.612 17.417 23.223 29.029 5.806 11.612 17.419 23.225 29.031 5.807 11-613 17.420 23.226 29.033 5.807 H.614 17.421 23.228 29-°3S 5.808 11-615 17.422 23.230 29.038 5.808 ii.6i6 17.424 23.232 29.040 Inches. 2.894 2.892 2.891 2.890 2.888 2.887 2.886 2.885 2.883 2.882 2.88i 2.879 2.878 2.876 2.875 2.S73 2.872 2.870 2.869 2.867 2.865 2.864 2.862 2.860 2.858 2.857 2-8SS 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 2.8x8 2.816 2.814 2.8l2 2.809 Inches. S-787 5.784 5.782 S-779 S-777 S-77S 5.772 5.769 S-767 5.764 5.761 S-7S8 S-7SS S-7S2 S-749 5.746 S-743 5.740 S-737 5-734 5-730 5.727 5.724 5.720 5-717 5-713 5-709 5.706 5.702 5-698 5-695 5.691 5.687 5.683 5.679 S-675 5.671 5.667 5.663 5.658 5.654 5.650 5.646 5.641 5-637 S-^32 5.628 5.623 5.619 Inches. 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 ^•585 8.580 8.S7S 8.570 8.564 8-559 ^■553 8.548 8.542 8.536 8.530 8.524 8.519 8-513 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 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 Inches, 14.468 14.461 14-455 14.448 14.442 14.436 14.430 14.424 14.416 I4410 14.402 14.396 14.388 14.380 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 I4.II4 14.103 14.092 14.080 14.069 14.058 14.046 Inches. 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 Inches. 0.000 .001 -003 .005 .007 .010 0.000 .001 .003 .006 .009 .013 0.000 .002 .004 .007 .Oil .016 14" s 0.000 10 .002 IS 20 :o^^ 25 30 .012 .018 16° O.OOI .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 iTTTnnr- [Derivation oJ table explained on pp. liil-lvi.] ■SG 3°- i6°oo' 10 20 3° 40 SO 17 00 10 20 3° 40 50 18 00 10 20 30 40 50 1900 10 20 3° 40 50 20 00 10 20 30 40 5° 21 00 10 20 30 40 50 22 00 10 20 30 40 50 2300 10 20 30 40 5° 24 00 _ C OJ Inches. 5.809 U.617 17.426 23-234 29.043 5.809 H.618 17.427 23.236 29.046 5.810 11.619 17.429 23239 29.049 5.810 11.621 17431 23.242 29.052 5.8 1 1 11.622 17433 23.244 29.055 5.812 n.623 17435 23.247 29.058 5.812 11.625 17437 23.250 29.062 11.626 17439 23.252 29.066 s' longitude. ABSCISSAS OF DEVELOPED PARALLEL. Indus. 2.809 2.807 2.804 2.802 2.800 2.797 2.795 2.792 2.790 2.787 2.785 2.782 2.780 2.777 2-774 2.772 2.769 2.766 2.764 2.761 2.758 2-75S 2.752 2.750 2.747 2-743 2.741 2.738 2-735 2.732 2.729 2.726 2.723 2.720 2.717 2.714 2.710 2.707 2.704 2.701 2.697 2.694 2.691 2.688 2.684 2.681 2.677 2.674 2.671 10' longitude. Inches. 5.619 5.614 5.609 5.604 5-599 5-595 5-590 5-585 5-580 5-575 5-570 5-564 5-559 5-554 5-549 5-543 5-538 S-533 5-527 5-522 5-516 5.510 5-505 5-499 5-493 5-487 5.482 5-476 5-470 5.464 5-458 5.452 5-445 5-439 5-433 5.427 5.421 5.414 5-408 5.401 5-395 5-388 5-382 5-362 5-355 5-348 5-341 15 longitude. Inches. 8.428 8.421 8.414 8.406 8-399 8.392 8.385 8.377 8.369 8.362 8-354 8.347 8-339 8-331 8-323 8.315 8.307 8.299 8.291 8.282 8.274 8.266 8.257 8.249 8.240 8.231 8.222 8.213 8.204 8.196 8.187 8.177 8.168 8.159 8.150 8.141 8.131 8.122 8.112 8.102 8.092 8.083 8.073 8.063 8.053 8.042 8.032 8.022 longitude. Inches. 1 1.237 11.228 II. 218 11.208 II.199 II. 189 1 1. 180 1 1. 170 II. 159 II.I49 II. 139 II. 129 II. 119 II. 108 11.097 11.087 11.076 n.065 11.054 11.043 11.032 II.02I 11.009 10.998 10.987 10.975 10.963 10.951 10.939 10.928 10.916 10.903 10.891 10.878 J0.866 10.854 10.842 10.829 io.8t6 10.802 10.790 10.777 10.764 10.750 10.737 10.723 10.710 10.696 8.012 10.683 25' longitude. Inches. 14.046 14.034 14.022 14.010 13-998 13.986 13-974 13.962 13-949 13-936 13-924 13.91 1 13.898 13-885 13.872 13-859 13-845 13-832 13.818 13.804 13-790 13.776 13.762 13-748 J 3-734 13-719 13-704 13.689 13-674 13.660 13-645 13.629 13.614 13-598 13-583 13.568 13-552 13-536 13.520 13-503 13-487 13-471 13455 13-438 13.421 13.404 13-387 13-371 13-354 30' longitude. Inches. 16.856 16.841 16.827 16.813 16.798 16.784 16.769 16.754 16-739 16.724 16.709 16.693 16.678 16.662 16.646 16.630 16.614 16.598 16.582 16.565 16.548 16.531 16.514 16.497 16.480 i6..i62 16.445 16.427 16.409 16.391 16373 16.336 16.318 16.300 16.281 16.262 16.243 16.223 16.204 16.184 16.165 16.145 16.125 16.105 16.085 16.064 16.045 16.024 ORDINATES OF DEVELOPED PARALLEL. a g •QS 1-1 •" 16° Inches. 0.00 1 .002 .005 .009 .014 .020 0.001 .002 .006 .010 .016 .022 .003 .006 .Oil .017 .025 0.00 1 .003 .007 .012 .018 .027 24" 0.001 .003 .007 .013 .020 .028 17° InchtSi 0.00 1 .002 .005 .010 .015 .021 19° .003 .000 .010 .016 .024 0.001 •003 .006 .oil .018 .026 23° O.OOI .003 .007 .012 .019 .028 Smithsonian Tables. 94 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation ^£ table explained on pp. liii^-lvi.] Table 20- SCALE ttbW??- •g lip ABSCISSAS OF DEVELOPED PARALLEL. ^^ T-* X-^ T HT jl T^ ORDINATiLo \jv ■g" nil S' 10' IS' 20' 25' 30' DEVELOPED PARALLEL. Jl* longitude. longitude. longitude. longitude. longitude. longitude. Inches Inches. Inches. Inches. Inches. Inches. Inches. 24°00' 2.671 S-34I 8.012 10.683 13-354 16.024 24° 25° lO ■■'s-Sh 2.667 S-334 8.002 10.669 13-336 16.003 P 20 3° 40 11.628 17.442 23.256 2.664 2.660 2.657 S-327 5-320 S-3I3 7.991 7.981 7.970 10.655 10.641 10.627 13-319 13-301 13.284 15.982 15.961 15.940 Inches. Inches. S° 29.069 2.653 5.306 7.960 10.613 13.266 15.919 5' 10 0.001 .003 0.001 .003 2500 2.650 5.299 7.949 10.599 13-249 15.898 IS .007 .007 10 ■■■5.8is' 2.646 5.292 7938 10.584 13-231 15.877 20 .013 -013 20 11.629 2.642 5.2§^ 7.927 10.570 13.212 15-854 25 .020 .028 .020 30 17.444 2.639 5.278 7.916 10-555 13-194 15-833 30 .029 40 23.259 2.635 5.270 7.905 10.540 13.176 15.811 50 29.074 2.631 2.628 s-263 5.256 7.883 10.526 10.511 13-157 13-139 15.788 15.767 2600 26° 27° 10 5.816' 2.624 5.248 7.872 10.496 13.120 15-744 20 11.631 2.620 5.240 7.861 10.481 13.101 15.721 5 10 0.001 0.001 30 17.446 2.616 s-233 7.849 10.466 13.082 15.698 .008 .008 40 23.262 2.613 5.225 5.218 7.838 10.451 13-063 15.676 IS 20 SO 29.077 2.609 7-827 10.436 13-045 15.654 .013 .014 27 00 2.605 5.210 7.816 10.421 13.026 15.631 25 30 .021 ■030 .022 -031 10 s'.sie' 2.601 5.203 7.804 10.405 13.006 15.608 20 30 "■633 17.449 2-597 2.593 S-igS 5.187 7.792 7.780 10.390 10.374 12.987 12.967 15-584 - 15.560 40 23.265 2.589 S-I79 7.768 10.358 12.947 15-537 28° 29° SO 28 00 29.082 2.586 2.582 5.171 S-163 7-7S7 7-74S 10.342 10.327 12.928 12.909 15-514 15.490 s O.OOi 0.001 10 "s-Si?" 2.578 S-'SS 7-733 10.311 12.889 15.466 10 .004 .004 20 11.634 2.574 S-H7 7.721 10.294 12.868 15.442 15 .008 .008 30 17.451 2.570 S-I39 7.709 10.278 12.848 15.418 20 .014 .014 40 23.268 2.566 S-131 7.697 10.262 12.828 15-394 25 .022 .023 SO 29.086 2.562 S-I23 7.685 10.246 12.808 15-369 30 .032 -032 29 00- 10 2.558 2.SS3 5.115 5.107 7.660 10.230 10.213 12.788 15.345 '"s-sis 12.767 15.320 ^_o .itO 20 11.636 2.549 5.098 7.648 10.197 12.746 15.295 30 31 30 17.454 2.54s 5.090 7.63s 10.180 12.725 15.270 40 23.272 2.541 5.082 7.622 10.163 12.704 15-245 5 10 O.OOI 0.001 SO 29.090 2.537 S-073 7.610 10.146 12.683 15.220 .004 .004 3000 2.533 5.065 7-598 10.130 12.662 15.195 '5 20 .008 .015 .023 •033 .008 .015 .023 ■034 10 s'sig' 2.528 5.056 7-585 10.113 12.641 15.169 25 30 20 11.638 2.524 5.048 7.572 10.096 12.620 iS-143 30 17-457 2.520 S-039 7-559 10.078 12.598 15.118 40 SO 23.276 29.094 2.515 2.511 S-031 5.022 7-546 7-533 10.061 10.044 12.577 12.555 15.066 t-iO 31 00 10 '5.820' 2.507 2.502 5.014 S.oo| 7.520 7-507 10.027 10.009 12.534 12.512 15.040 15.014 32 20 11.640 2.498 7-494 9-992 12.490 14.987 5 0.001 30 17.460 2.493 4.987 7.480 9-974 12.467 14.960 10 .004 40 23.280 2.489 4.978 7.467 9.956 12.445 14-934 15 .009 SO 29.100 2.485 4.969 7-454 9-938 12.423 14.908 20 25 .015 .024 32 00 2.480 4.960 7.441 9.921 12.401 14.881 30 •034 Smithsonian Tables. 9S Table 20. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] SCALE iT^innr' ■is" ABSCISSAS OF DEVELOPED PARALLEL. •M ORDINATES OF DEVELOPED •S3 5' 10' IS' 20' as' 30' PARALLEL. ^^ |ISa longitude. longitude. longitude. longitude. longitude. longitude. Inches. Inches. Inches. Inches. Inches. Inches, Inches. H 32°00' 2.480 4.960 7.441 9.921 12.401 14.881 ■l| 32° 33° 10 20 5.821 ' 11.642 2.476 2.471 4-9SI 4.942 7.427 7413 9-S°3 9.884 12.379 12-355 14.854 14.827 ►3-^ 30 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 O.OOI 5° 29.104 2.458 4-915 7-373 9.830 12.288 I4-74S .004 .004 33 00 10 20 5.822 11.643 2.448 2.444 tg 7-359 7-345 7-331 9.812 9-793 9-774 12.265 12.241 12.218 14.717 14.689 14.661 15 20 25 30 .009 .015 .024 -034 .009 .016 .024 ■035 30 17.465 2-439 4'^J 7-316 9-7 SS 12.194 14-633 40 SO 23.287 29.109 2-434 2.429 4.868 4.859 7.288 9-736 9-718 12.171 14.605 14.576 12.147 3400 "S-Szs' 2.425 4.850 4.840 7.274 9-699 12.124 14.549 34° 35° 10 2.420 7.260 9.680 12.100 14.520 20 11.645 17.468 2.41S 4-830 7.246 9.661 12.076 14.491 s 0.001 O.OOI 30 2.410 4.821 7-231 9.642 12.052 14.462 10 .004 .004 40 23.291 2.406 4.811 7.217 9.622 12.028 14-434 '5 .009 .009 SO 29.113 2.401 4.802 7-203 9.604 12.004 14.405 20 .016 .016 3S00 2.396 4.792 7.188 9-584 11.980 14-376 25 30 .036 ■036 10 ■■5'824' 2.391 4.782 7-174 9-565 11.956 14-347 20 11.647 2.386 2.381 4-773 4-763 7-159 9-545 11.932 '"^-^si 30 17.471 7.144 9.526 11.907 14.288 40 23.294 2-377 4-753 7.130 9.506 11.883 11.858 11-833 14.259 36° 37° SO 3600 29.118 2.372 2.367 4-743 4-733 7.115 7.099 9.4S6 9.466 14.230 14.200 5 0.001 0.001 10 'V-824' 2.362 4-723 7.085 9.446 11.868 14-170 10 .004 .004 20 11.649 2-3S7 4-713 7.070 9.426 11.783 "4-139 IS .009 .009 30 17473 2-3SI 4-703 7-055 9.406 11-757 14.109 20 .016 .016 40 23.297 2.346 4-693 7-039 9.386 11.732 14.078 25 .025 .026 SO 29.122 2.341 4.683 7.024 9.366 11.707 14.048 30 ■036 -037 3700 2-336 4-673 7.009 9-345 11.682 14.018 10 20 '5-826' 11.651 2-331 2.326 4.662 4.652 '^,\ 9325 9-304 11.656 11.630 13-987 13-956 38° 39° 30 17477 2.321 4.642 6.963 6.947 9.284 9.263 11.605 11.579 13-925 13.894 40 23.302 2.316 4-631 5 10 SO 29.128 2.311 4.621 6.932 9.242 "-SS3 13-864 0.001 .004 O.OOI .004 38 00 2-30S 4.611 6.916 9.222 11.527 13-832 15 .009 .009 10 ■ "5.827' 2.300 4.600 5-§s° 9.200 11.501 13.801 20 25 .017 .026 .017 .026 20 11-653 17.480 2.295 4.590 6.884 9.179 11.474 13-769 30 2.290 4-S79 6.869 9.158 11.448 '3-737 30 •037 •037 40 SO 23.306 29-133 2.279 4.568 4.558 6.853 6.837 9-137 9.116 11.421 13-705 13673 11-395 3900 " '5.828' 2.274 2.268 4.548 6.821 6.805 9-095 11.369 13-642 40° 10 4-537 9-073 11-342 13.610 20 11.655 17-483 2.263 4.526 6.789 9.052 "•31S '3-577 5 0.001 30 2.258 4-515 6-773 9.030 11.288 '3-545 10 .004 40 23.310 2.252 4-504 6.756 9.008 11.261 '3-S'3 15 .009 SO 29.138 2.247 4-493 6.740 8.987 11.234 13.480 20 .017 .026 .038 4000 2.241 4-483 6.724 8.965 11.207 13.448 25 30 Smithsonian Tables. 96 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] Table 20. SCALE irsWir- "s nal dis- from igree ls> ABSCISSAS OF DEVELOPED PARALLEL. ^%T*T'\TTfcT An HIM ORDINATiia ur |3 .2 w'aJi DEVELOPED ■U •c i §1 S' 10' 15' 20' 25' 30' PARALLEL. 2°- gaSe, longitude. longitude. longitude. longitude. longitude. longitude. Inches. Inches, Inches. Inches. Inches. Inches. Inches. I| 4o°oo' 2.241 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 8.921 II. 152 13-382 3° 17.486 2.225 4-450 ^.■^i 8.899 II. 124 13-349 Inches. Inches. 40 23-314 2.219 4-439 6.658 8.877 11.097 13-316 5' 10 0>OOI O.OOI so 29-'43 2.214 4-428 6.641 8.855 11.069 13-283 .004 .004 41 00 2.208 4.417 6.625 6.608 !-?34 11.042 13-250 15 20 .009 .026 .009 .017 .026 10 ■'s'sp' 2.203 4.406 8.81 1 H.O14 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 ^■57| 8.766 10.958 13-149 40 23-319 2.186 4-372 6-558 8.744 10.929 13-I15 5° 29.149 2.180 4.360 6.541 8.721 10.901 13.081 42 00 '"5-831' 2.175 2.169 4-349 4-338 6.524 6.507 8.698 8.676 10.873 10.844 13.048 42° 43° 10 13-013 20 II.66I 2.163 4.326 6.490 8.653 10.816 12.979 5 0.001 O.OOI 30 17.492 2.157 4-31 S 6.472 8.630 10.787 12.94s 10 .004 .004 40 23-323 2.152 4-303 6.455 f-^7 10.759 12.910 15 .010 .010 S° 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 f-5^J 10.702 12.842 30 .038 -038 10 ""5-832' 2-135 4.269 6.403 8.538 10.672 1 2.807 20 H.663 2.129 4.257 6.386 8.514 10.643 12.772 30 17-495 2.123 4.246 6-368 8.491 10.614 12.737 40 23-327 2.II7 4-234 6-351 8.468 10.585 12.701 44° 45° 5° 4400 29.159 2.III 2.105 4.222 4.210 6-333 6.316 8.444 8.421 10.556 10.526 12.667 12.631 5 O.OOI O.OOI 10 '"'s-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 15 .010 .010 30 17.498 2.087 4-175 6.262 8.350 10.437 12.524 20 .017 .017 40 23-331 2.081 4.163 6.244 8.326 10.407 12.489 25 .027 .027 S° 29.164 2.076 4.151 6.227 8.302 10.378 12-453 30 .038 -038 45 00 10 2.070 2.064 4-139 4.127 6.209 6.191 8.278 10.348 10.317 12.417 12.381 ""s-834" 8.254 46° .„o 20 30 11.668 17.501 2.057 2.051 4-1 1 5 4.103 6.172 6.154 8.230 8.206 10.288 10.257 12.345 12.308 47 40 23-335 2.045 4.091 6.136 8!i8i 10.226 12.272 S 10 0.001 O.OOI SO 29.169 2.039 4.079 6. 1 18 8.157 10.197 12.236 .004 .004 4600 2-033 4.067 6.100 l;^ 10.166 12.199 IS .010 .017 .027 -038 .010 .017 .027 -038 10 '"'5-835' 2.027 4-054 6.081 10.136 12.163 25 30 20 11.670 2.021 4.042 6.063 8.084 10.104 12.125 30 17.504 2.015 4.030 6.044 8.059 10.074 12.089 40 23-339 2.009 4.017 6.026 6.008 8.034 8.010 10.043 12.052 50 29.174 2.003 4.005 10.013 12.015 48° 4700 10 '5-836' 1.996 1.990 3-992 5.989 S-970 7.985 7.960 9.981 9.951 11.978 1 1. 941 20 11.672 1.984 3.968 S-9SI 7-935 9§'9 11.903 5 0.001 30 17.508 1.978 3-955 5-933 7-§'° 9.888 11.866 10 .004 40 23-344 I.97I 3-943 5-914 7.885 9-f57 11.828 15 .010 SO 29.180 1.965 3-930 5-895 7.860 9.826 11.791 20 25 .017 .026 4800 1-959 3-9«7 5.876 7-835 9-794 11.752 30 ■038 Smithsonian Tables, 97 Table 20. , CO-ORDINATES FOR PROJECTION OF MAPS. SCALE irtVinr' [Derivation of table explained on pp. liu-lvi.] 48°oo' 10 20 3° 40 SO 4900 10 20 3° 40 5° 50 00 10 20 30 40 50 51 00 10 20 30 40 50 52 00 10 20 3° 40 5° 53°° 10 20 30 40 SO 54 00 10 20 30 40 50 55 00 10 20 30 40 50 5600 ■°as is Saf C c 0) S ABSCISSAS OF DEVELOPED PARALLEL. Inches, 5-837 11.674 17.511 23-348 29.185 H.676 17.514 23-352 29.190 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 S.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 longitude, Inchts. 1.959 .1.952 1.946 1.940 1-933 1.927 1.921 1.914 1.908 1. 90 1 1.894 1.888 1.882 1.875 1.869 1.862 1.856 1.849 1.^2 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 1-735 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 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 3-750 3-737 3-724 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 3-5" 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 Indus. 5-876 5-f57 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 5-587 5-567 5-547 5-528 S-59Z 5.488 5.468 5.448 5.428 5.408 5.388 5-367 5-347 S-327 5-307 5.287 5.266 5.246 5-225 5.205 5.184 5.164 5-143 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 6.635 6.607 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 9-31 1 9.278 9-245 9.212 9.179 9.146 9-"3 9.080 9.046 9.013 8.980 ■ 8.946 8.912 8.878 8.844 8.8 II 8-777 8.742 8.708 8.674 8.640 8.606 8.572 8-537 8.502 8.468 8-433 8.398 8.328 8.294 8.258 8.224 8.188 30' longitude. Inches. 1.752 I.714 1.677 1.638 1.600 1.562 1-523 1.485 1.446 1.408 1-369 1-330 I.291 1. 251 1. 212 I-I73 1-134 1.094 1-055 I.0I5 0-975 0.936 0.895 0.855 0.816 0.775 0.734 0.694 0.654 0.613 0-573 0-532 0.491 0.450 0.409 0.368 0.327 0.286 0.244 0.202 0.1 61 0.120 0.078 0.036 9-994 9-952 9.910 9.868 9.826 ORDINATES OF DEVELOPED PARALLEL. Inches. 5 0.00 1 10 .004 15 .oto 20 .017 2'! .026 30 .038 48° 50° 0.00 1 .004 .009 .017 .026 .038 52° 0.00 1 .004 .009 .017 .026 -037 54° 5 0.00 1 10 .004 15 .009 20 .016 2S .025 30 .036 56° .004 .009 .0x6 .025 .036 49° Inches. 0.00 1 .004 .010 .017 .026 .038 51° O.OOI .004 .009 017 .026 -037 53° O.OOI .004 .009 .016 .026 -037 55° 0.001 .004 .009 .016 .025 .036 Smithsonian Tables. 98 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TTTirw [Derivation of table explained on pp. liii-lvi.] Table 20. •s nal dis- from egree Is. ABSCISSAS OF DEVELOPED PARALLEL. /TD T\ T XT A T 4) .-J UKJJIJN Al jLa \jr ■1^ .S»-ojJ DEVELOPED 11 •si SI S' 10' IS' 20' 25' 30' PARALLEL. 2^ gaSft longitude. longitude. longitude. longitude. longitude. longitude. • IncAes. Inches. Inches, Inches. Inches. Inches. Inches. li S6°oo' 1.638 3-27 s 4-913 6-551 8.188 9.826 •|£ 56° 57° 10 ■■■5.845' 1.631 3.261 4.892 6.522 IWI 9.784 3-2 20 11.690 I7-S3S 1.624 1.616 3-247 3-233 4.870 4-849 6.494 6.466 9.741 9.698 3° 8.082 Inches. Inches. 40 23.3»o 1.609 3.219 4.828 6-437 8.046 9.656 5' 10 0.00 1 0.001 SO 29.224 1.602 3-204 4.807 6.409 8.011 9.613 .004 .004 5700 \$l 3.190 4-785 6.380 7.976 9-571 15 20 .009 .016 .016 10 ' 5.846' 3.176 4.764 6.352 7.940 9.527 25 30 -036 .024 -035 20 30 11.692 I7-S37 1-581 1-574 3.162 3-147 4.742 4.721 6-323 6.294 9.485 9.442 40 23-3»3 1.566 3-133 4.699 6.266 7-832 9-398 SO 29.229 I-S59 3-119 4.678 6.237 7.796 9-356 5800 "s"847' I-SS2 3-104 4.656 4-634 6.208 6.179 7.760 9-3!3 9.269 58° 59° 10 1-545 3.090 7-724 20 11.694 1-538 3-07S 4.613 6.150 7.688 9.226 s 0.001 0.001 30 17.540 1-530 3-061 4-591 6.122 7.652 9.182 10 .004 .004 40 23-387 1-523 3.046 4-569 6.092 7.616 9.139 15 .009 .008 SO 29.234 1.516 3-032 4-547 6.063 7-579 9.095 20 25 .015 ,024 .015 .024 5900 1.509 3-017 4.526 6-034 7-543 9.052 30 •034 -034 10 ■■'s-sis' I'Soi 3-003 4.504 6.005 7.506 9.008 20 11.695 1.494 2,98§ 4.482 S.976 7.470 8.963 8.920 30 17-543 1-487 2.973 4.460 5.946 7-433 40 23-391 1.479 2-959 4-438 5-917 7-396 8.876 60° 6i° SO 6000 29.238 1.472 1.465 2.944 2.929 4.416 4-394 5.888 5.858 7.360 7-323 8.831 8.788 5 0.001 0.001 10 5.849 1-4S7 2.914 - 4-372 ■5.829 7.286 8.743 10 .004 .004 20 11.697 1.450 2.900 4-349 S-799 7-249 8.699 15 .008 .008 30 17.546 1.442 2.885 4-327 5.770 7.212 8.654 20 .015 .014 40 23-394 I-43S 2.870 4-305 5.740 7-I7S 8.610 25 .023 .023 SO 29.243 1.428 2.855 4.283 5.710 7-138 8.566 30 -033 -033 61 00 1.420 2.840 4.261 5.681 7.101 8.521 10 5.850 1-413 2.825 4.238 5-651 7.064 8.476 62° 63° 20 11.699 1.405 2.810 4.216 5.621 7.026 8.431 30 17-549 1-398 2-795 4-193 5-591 6.988 8.386" 40 23-398 1.390 2.781 4-171 5.561 6.952 8.342 5 10 0.001 0.001 SO 29.248 1-383 2.766 4.148 S-S3I 6.914 8.297 .004 .003 62 00 1-375 2.751 4.126 5- SOI 6-877 8.252 15 20 .008 '.014 .022 .008 .014 .022 10 ■5-850' 2.736 4.103 S-471 6.839 8.207 25 30 20 11.701 1.360 2.720 4.081 5.441 6.801 8.i6i .032 -031 30 17-551 1-353 2.705 4.058 5.410 6.763 8.116 40 23.402 ;-^^i 2.690 2.675 4-035 S-380 - 6.726 6.688 8.071 8.026 SO 29.252 1-338 4.013 5-350 64° 6300 10 ■ ■5.851 ■ 1-330 1.322 2.660 2.645 3-990 3-967 S-320 5.290 6.650 6.612 7.980 7.934 20 11.702 1-315 2.630 3.944 5.259 6.574 7.889 5 0.001 30. 17-554 1-307 2.614 3.921 5.228 6.536 7.843 10 ■003 40 23-405 1-300 2.599 3-899 5.198 6.498 7.797 15 .008 SO 29.256 1.292 2-584 3-876 5.168 6.460 7.751 20 25 .013 .021 64 00 1.284 2.569 3-853 S-137 6.422 7.706 30 .030 Smithsonian Tables. 99 Table 20. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi*] SCALE TTsVlTT' i... ABSCISSAS OF DEVELOPED PARALLEL. "3 Meridional c tances froir even degret parallels. ORDINATES OF DEVELOPED M'e s' 10' IS' 20' 25' 30' PARALLEL. 32- longitude. longitude. • longitude. longitude. longitude. longitude. IncAes. Inches. Inches. Itches. Inches. Inches. Inches. 1-= 64°oo' 1.284 2.569 3-853 S-I37 6.422 7.706 •|| 64° 6s° 10 '"ih'sV 1.277 2-5S3 3-830 5.106 6.383 7.660 j.l 20 11.704 1.269 2-538 3-807 5.076 6-345 7-614 30 I7-SS6 1. 261 2.523 3-784 5.045 6.307 7-568 Inches. Inches. 4° 23.408 1.254 2.507 3-761 5.014 6.268 7.522 5' 10 0.00 1 0.001 50 29.260 1.246 2.492 3-738 4.984 6.230 7.476 -003 -003 6s 00 1.238 2.477 3-715 4-953 6.192 7-430 7-384 IS 20 .008 -013 .021 .007 .013 .020 10 5.853' I.231 2.461 3.692 4-922 6-153 25 30 20 11.706 1.223 2.446 3.668 4.891 6.1 14 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 5° 29.264 1.200 2-399 3-599 4.798 5-998 7.198 66 00 "V-Sw 1. 192 I.184 '^A 3-575 4.767 5-959 7.151 66° 67° 10 3-552 4-736 5'9zo 7.104 20 11.707 '•'?o 2.352 3-529 4.705 5-881 7-057 5 0.001 O.OOI 30 17.561 I.I68 2-337 3-505 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-I53 2-305 3-458 4.611 5.764 6.916 20 25 .013 .020 .012 .010 .028 67 00 1.145 2.290 3-43S 4.580 5-725 6.869 3° .029 10 "s-SsV I-I37 2.274 3-4" 4.548 5.685 6.822 20 11.709 17.563 1. 129 2.258 3-388 3-364 4.517 4.485 5.646 5.607 U'd 30 1. 121 2.243 40 23.418 1.113 2.227 3-340 4-454 5-567 6.680 68° 69° 5° 6800 29.272 1. 106 1.098 2.21 1 2.I9S 3-317 3-293 4.422 4-391 5.528 5.489 6.634 6.586 5 0.001 O.OOI 10 "V-Sss' 1.090 2.180 3.269 4-359 5-449 6-539 10 .003 .003 20 II.7I0 1.082 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 .011 40 23.420 1.066 2.132 3.198 4.264 5-330 6.396 25 .019 .018 50 29.276 1.058 2.1 16 3-174 4.232 5.291 6.349 30 .027 .026 6900 10 "'s.Ss6 1.050 1.042 2.100 2.084 3-iSi 3-127 4.201 4.169 5.251 5.211 6.301 6-253 20 II.7I2 1.034 2.068 3-103 4-137 5.171 6.205 70° 71° 30 17.567 23-423 1.026 2.052 2.037 3-079 3-055 4.105 4-073 5-131 5-092 6.157 6.110 40 1. 01 8 5 10 so 29.279 I.OIO 2.021 3-031 4.041 5.052 6.062 0.00 1 .003 O.OOI .003 70 00 1.002 2.005 3-007 4.009 5.012 6.014 IS .006 .006 10 "'Vase' -994 1.989 2.983 3-977 4-972 5.966 20 25 30 .Oil .017 .024 .010 .016 .024 20 30 "■713 17.570 «78 1.972 1.956 2.959 2-935 3-945 3-913 4-931 4.891 1^ 40 23.426 .970 1.940 2.9H 2.886 3.881 4.851 5-821 50 29.282 .962 1.924 3-848 4.811 5-773 71 00 ■■■5.857' -954 .946 1.892 2.862 2.838 3.816 4.771 5-725 72° 10 3-784 4-730 5.676 20 11.714 -938 1.876 2.814 3-752 4.690 5.628 5 0.001 30 17-572 •930 1.860 2.790 3.720 4.650 5-S79 10 .003 40 23.429 .922 ■•§44 2.765 3.687 4.609 5-531 IS .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. 100 Table 20. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE tjsW- [Derivation of table explained on pp. liii-lvi.] 1 ABSCISSAS OF DEVELOPED PARALLEL. "3 llli ORDI NATES OF DEVELOPED ■ss 9*3 ■11 si S' 10' 15' 20' 25' 30' PARALLEL. ^S. gflSa longitude. longitude. longitude. longitude. longitude. longitude. , IncheS' Inches. Inches, Inches. Incites. Inches. Inches. ll 72*'oo' iSgS I.8II 2.717 3-623 4- 529 5-434 72° 73° 10 ■'V-Ssf' n.716 '•795 'ieei 3-590 3558 4.488 5-386 5-33f J -2 20 1-779 4-447 3° I7-S73 .881 1.763 2.644 3-525 4.407 5.288 Inches. Inches. 40 23-431 •^P 1.746 2.620 3-493 4.366 5-239 5' 10 O.OOI O.OOI 5° 29.289 .865 1-730 2-595 3-460 4-325 5.190 -003 .002 7300 .857 1-714 2.571 3-428 4.285 5-141 IS 20 .006 .010 .005 .010 10 ■■'s-sss' .849 1.697 2.546 3-395 4-244 5.092 25 .016 .015 20 11.717 .841 1.681 2.522 3-362 4-203 5-044 30 -023 .021 30 17-575 .832 1.665 1.648 2.497 3-330 4.162 4-994 40 23434 .824 2.473 2.448 3-297 4.121 4-945 SO 29.292 .8i6 1.632 3-264 4.081 4-897 7400 10 "V-Ssg' .808 .800 1.616 1.599 2.424 2-399 3-232 3-199 4.040 3-999 4.847 4.798 20 11.718 •791 1-583 2-374 3.160 3-957 4.748 74° 75° 30 40 17-577 23-436 -783 -775 1.566 i-SSo 2-350 2-325 3-133 3.100 3-916 3-875 4-699 4.650 5° 29.295 .767 1-534 2.300 3.067 3-834 4.601 s 10 0.001 .002 O.OOI .002 7500 •759 1.517 2.276 3-034 3-793 4.552 IS .005 -005 10 5.860 -750 1. 501 2.251 3.002 3-752 4.502 20 .009 .009 20 H.719 .742 1.484 2.226 2.968 3-711 4-453 25 .014 .013 30 17.578 -734 1.468 2.201 2-93S 4-403 30 .020 .019 40 23-438 .726 1.451 2.177 2.902 3:62! 4-354 SO 29.298 .717 1-435 2.152 2.870 3-587 4-304 7600 5.8&)' .709 1.418 2.127 2.836 2.803 3^546 4-255 10 .701 1.402 2.102 3^504 4-205 20 11.720 1-352 2.078 2.770 3-463 4-155 76° 77° 30 40 17.580 23.440 .676 2.053 2.028 2-737 2.704 3.421 3380 4.105 4-056 SO 29.300 .668 I -335 2.003 2.671 3-339 4.006 5 10 0.001 .002 0.000 .002 7700 .659 1-319 1.978 2.638 3-297 3-956 15 .005 .008 .004 to 5.860' .651 1.302 1-953 2.604 3.256 3-907 20 .007 20 11.721 -643 1.285 1.928 2.571 3.214 3.856 25 .013 .012 30 17.582 •634 1.269 \^ 2-538 3-172 3.806 30 .018 .017 40 23-442 .626 1.252 2.504 3-131 3-089 3-757 SO 29.302 .618 1-235 1-853 2.471 3.706 7800 .609 1.219 1.828 2.438 3-047 3-656 10 " s'sei' 11.722 .601 1.202 1.803 1-778 2.404 2-371 3.005 2.964 3.606 3-556 20 ■593 1.185 78° 30 17-583 .576 1. 169 \VI 2-338 2.922 2.880 3.506 3.456 79 40 23-444 1. 152 1.725 2.304 SO 29.304 .568 1-135 1-703 2.270 2.838 3.406 5 10 0.000 .002 0.000 .002 7900 •SS9 1. 119 i.67'8 2-237 2-797 3.356 15 .004 .004 10 "5.861" •SSI 1. 102 1.653 2.204 2-755 3.305 20 .007 .006 20 11.723 .542 1.085 1.628 2.170 2-713 3.255 25 30 .011 .010 30 17-S84 ■534 1.068 1.602 2.136 2.671 3.205 .016 .014 40 23-445 .526 1.052 1-577 2-103 2.629 3.155 SO 29.306 •517 1-035 I-SS2 2.070 2.587 3.104 8000 .509 1.018 1.527 2.036 2-545 3-054 Smithsonian Tables. loi Table 21. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE nsWir- [Derivation of table explained on pp. liii-lvi.] •3 •|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. Inches. Inches. Inches. Inches. o°oo' 30 45 "'8.588' 17.176 25.764 8.647 8.646 8.646 8.646 .000 .000 .000 .ono 17.293 17-293 17.292 J7.29I .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 IS 30 4S 34-352 8.645 8.644 8.643 8.642 .000 .000 .000 .001 17.291 17.289 17.287 17.285 .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 .007 .008 .009 8.588 17.176 25.764 2 00 IS 30 4S 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 3° 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 .011 34-539 34-530 34-521 34-512 .016 .018 .019 .020 8.588 17-177 25.765 400 15 30 45 34-353 8.626 8.623 8.620 8.617 .001 .001 .001 .002 17.251 17.245 17.240 17-234 .005 .no6 .006 .006 25-877 25.868 25-859 25.850 .012 .012 .013 .014 34-502 34-491 34-479 34-467 .021 .023 .024 .025 8.589 17-177 25.766 500 15 30 45 34-354 8.614 8.610 8.607 8.603 .002 .002 .002 .002 17.228 17.221 17.213 17.206 .007 .007 25.842 25-831 25.820 25.809 .015 .016 .016 .017 34-456 34-441 34-427 34.412 .026 .028 .029 .030 8.589 25.766 600 15 30 45 34-355 8.600 8- 595 8.591 8.587 .002 .002 .002 .002 17-199 I7-I9I :7.i82 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 25.767 700 IS 3° 45 34-356 8-583 8.578 IP .002 .002 .003 .003 17-165 17-1SS 17-145 17.136 .009 .009 .009 .010 25.748 25-733 25.718 25.704 .021 .022 .022 .023 34-330 34-310 34.291 34-272 -037 .038 .040 .041 8-589 17.179 25.768 800 15 30 45 34-358 8-563 8.558 8.552 8.546 .003 .003 .003 .003 17.126 17-115 17.104 17-093 .010 .010 .011 .011 25.689 25-673 25-656 25-639 .023 .0^4 .024 .025 34-252 34-230 34-208 34.186 .042 ,044 .046 17.180 25.769 900 IS 3° 45 34-359 8.541 8.52^ 8.522 .003 .003 .003 .003 17.082 17.069 17.057 17.045 .012 .012 .012 .013 25.622 25.604 25-585 25-567 .026 .027 .027 .028 34-163 34-138 34114 34.089 •047 .048 .050 .051 8.590 17.180 25-771 1000 34.361 8.516 .003 17.032 -013 25-548 .029 34.064 .052 Smith soNiA N Tablci ). Table 21. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE WriET- [Derivation of table explained on pp. liii-lvi.] ■s ■S'3 11 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 Inches, Ittc/us. Inches. Inches. Inches. Inches. Inches. Inches. Inches. io°oo' IS 3° 45 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-°37 34.010 33.982 .052 .054 .055 .056 II 00 IS 3° 45 34-363 8.489 8.481 1% .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 •°33 •033 ■°34 33-955 33-925 33-895 33-865 .057 .058 .059 8.591 17.183 25-774 12 OO IS 3° 4S 34-365 8-459 8.451 8-443 8-434 .C04 .004 .004 .004 16.918 16.901 16.885 16.869 .015 .^016 .016 .016 25-376 25-352 25-328 25.304 -03s .036 .036 33-835 33-8°3 33-77° 33-738 .061 •063 .064 .065 8.592 17.184 25.776 1300 IS 30 4S 34-368 8.426 8.418 8.409 8.400 .004 .004 .004 .004 16.853 16.835 16.818 16.800 .017 .017 .017 .018 25.279 25-253 25.227 25.201 •037 .038 •039 .040 33-7°6 33-636 33-601 .066 .067 .069 .070 8.592 17.185 25.778 1400 IS 30 45 34-37° 8-391 8.382 8-373 8-363 .004 .005 .005 .005 16.783 16.764 .018 .018 .018 .019 25-174 25.146 25.118 25.090 .040 .041 .041 .042 33-566 33-528 33-49° 33-453 .071 .072 -073 .074 8-593 17.186 25.780 15 00 IS 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 25.782 1600 IS 3° 45 34-376 8.314 8-303 8.292 8.282 .005 .005 .005 .005 16.627 16.606 16.585 16.564 .020 .020 .020 .021 24.941 24.909 24.877 24.845 •°45 -°4S .046 .046 33-255 33-212 33-17° 33-127 .080 .081 .082 -083 8.595 17.190 25.784 17 00 15 3° 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 .087 .088 8.596 17.191 25-787 18 00 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 15 30 45 34-386 8.178 8.166 8.153 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 2000 34-39°" 8.128 .006 16.257 .024 24.385 -OSS 32-513 .097 Smithsonian Tables. 103 Table 21. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TT^WiT- [Derivaiion of table explained on pp. Uii-lvi.] Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR — is' longitude. lof longitude. 45' longitude. 1° longitude. X y X y X y X y Inches, Inches, Inches. Inches, Inches, Inches, Inches. Inches, Inches. , 20°00' IS 3° 45 17.197 25795 8.128 8.IIS 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 .055 .056 .056 •057 32.513 32.461 32.408 32.356 .099 .100 21 00 15 45 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 32.248 32.193 32.138 .101 .102 .103 .104 8.599 17.199 25.798 22 00 15 30 45 34-398 8.021 8.006 7.992 7-978 .007 .007 .007 .007 16.042 16.013 15.984 IS-9SS .026 .026 .027 .027 24.062 24.019 23.976 23-933 •059 .060 .060 .o6i 32.083 32.026 31.968 31-91 1 .105 .io6 8.600 17.201 25.801 2300 15 30 45 34-402 7-963 7-948 7-933 7.918 .007 .007 .007 .007 15.927 15-897 15.867 15-837 .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 .III 8.602 17-203 25.804 2400 IS 30 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 25!8o8 25 00 15 30 45 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 ioel .067 31-365 31.300 31-235 31-170 .116 -117 ::5i 8.604 17.207 25.8H 2600 IS 3° 45 34-415 7.776 7.760 7-743 7.726 .007 .007 .008 .008 15-553 15.519 1 5.486 15.452 .030 .030 .030 -030 23-329 23.279 23.229 23-179 .067 .067 .068 .068 31.106 31-039 30-972 30-905 .119 .120 .121 .121 8.605 17.210 25.8:4 27 00 IS 3° 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 2800 IS 3° 45 34.424 7.640 7.622 .008 .008 .008 .008 15.280 15-244 15.208 1S-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 34-430 7.568 7-SSo 7-S3I 7513 .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 .128 .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. SCALE xidrw [Derivation of table explained on pp. liii-lvi.] •s n Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR- 15' longitude. 30' lo igitude. 45' longitude. 1° longitude. X y X y X y - y Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. IncJtes. 3o°oo' IS 30 4S "8.610 17.220 25.830 7-494 7-475 7-456 7-437 no8 .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 3100 IS 30 4S 34.440 7.418 7-398 7-379 7-359 .008 .008 .008 .008 14.836 14-797 14-758 14.718 •033 -033 -034 -034 22.254 22.195 22.137 22.078 -075 .075 .076 .076 29.672 29-594 29-515 29-437 •133 -134 •13s -135 8.61 1 17.213 25-834 3200 "S 30 4S 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 21.958 21.898 21.837 .076 -077 .077 .077 29.358 29.278 29.197 29.116 -136 .136 -137 •137 8.613 17.225 25-838 3300 IS 30 4S 34-451 7-259 7.238 7.217 7-197 .009 .009 .009 .009 14.518 14.476 14-435 14-393 -034 -03s -03s -03s 21-777 21.714 21.652 21.590 .078 .078 .078 .078 29.036 ^^■§53 28.869 28.786 .138 .138 -139 -139 8.614 17.228 25.842 3400 IS 30 45 34-456 7.176 7-154 7-133 7. 112 .009 .009 .009 .009 14-352 14.309 14.266 14.224 •03s -03s ■03s -035 21.527 21.464 21.400 21.336 -079 •079 .079 .080 28.703 28.618 28-533 28.448 .140 .141 .141 .142 8.615 17.231 25.846 3S00 IS 30 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 IS 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 30 45 34.480 6.822 6.799 6.775 6.752 .009 .009 .009 .009 13-645 13-598 13-SSi 13-504 -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 IS 3° 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 ■0&3 -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 TableSi loS Table 21 . CO- ORDINATES FOR PROJECTrON OF MAPS. SCALE yWrinr- [Derivation of table explained on pp. liii-lvi.] 3 a MehdioDal dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED 1 PARALLEL FOR — 15' longitude. 3c/ longitude. 4S' longitude. 1° longitude. X y X y X y X y Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. 4o°oo' IS 30 45 8.624 17.249 25-873 6:584 6.560 .009 .009 .009 .009 13.266 13.217 13.168 13-II9 ■037 ■037 •037 •037 19.899 19.825 19-752 19.679 .084 .084 .084 .084 26-532 26-434 26-336 26-238 .149 •149 .149 .149 41 00 IS 30 45 34-497 6-535 6.510 6.485 6.460 .009 .009 .009 .009 13.070 13.020 12.970 12.920 037 •037 •037 -037 19.605 19-530 19.456 19381 .084 .084 .084 .084 26.140 26-041 25-941 25.841 .150 .150 .150 .150 8.&2S 17.250 25.875 42 OQ »S 30 45 34-500 6-435 6.410 6.385 6-359 .009 .009 .009 .009 12.871 12.820 12.769 12.718 •037 -038 .038 19-306 19-230 19.154 19.077 .085 .085 .085 .085 25.741 25.640 25-538 25-436 .150 -150 .151 .151 8.627 25.882 4300 IS 30 45 34-510 6-334 6.308 6.282 6.256 .009 .009 .009 .009 12.667 12.615 12.563 12.512 .038 .038 .038 -038 19.001 18.845 18.767 .085 .085 .085 .085 25-335 25.231 25.127 25.023 .151 -151 -151 .151 8.629 ^^5:li 4400 »5 3a 4S 34-515 6.230 6.203 6.177 6.151 .009 .009 .009 .009 12.460 12.407 12.354 12.301 ■038 -038 .038 -038 18.689 18.610 18.531 18.452 .085 .085 .085 .085 24.919 24.814 24.708 24.603 -151 -151 •151 .151 8.630 17.261 25.891 4500 IS 30 45 34-522 6.124 6.097 6.071 6.044 .009 .009 .009 .009 12.249 12.195 12.141 12.088 .038 .038 -038 -038 18.373 18.292 18.212 18.131 .085 .085 .085 .085 24.497 24.390 24-283 24.175 .151 -151 .151 .151 8.632 17.264 25.896 4600 IS 30 45 ' 34-528 6.017 5-935 .009 .009 .009 .009 12.034 11-979 11.925 11-870 -038 -038 .038 •038 18-051 17-969 17-887 17.805 .085 .085 .085 .085 24.068 23-959 23-849 23-740 .151 .151 .151 •151 8-633 17-267 25.901 4700 IS 30 45 34-534 5.908 5-880 5.852 5-824 .009 .009 .009 .009 11-815 11-760 11.704 11-648 .038 .038 .038 .038 17-723 17.640 17556 17-473 .085 .085 .085 .085 23-631 23.520 23-408 23-297 -151 .151 .151 .151 8-635 17.270 25.905 48 00 IS 30 4S 34-540 5-796 5.768 5-740 5-7 1 2 .009 .009 .009 .009 11-593 11-536 11.480 11.424 .038 .038 .038 -037 17-389 17-30S 17.220 17-135 .085 .085 .084 .084 23-186 23-073 22-960 22.847 .150 .150 .150 .150 8.637 17-273 25-910 4900 IS 30 45 34-546 5.684 5.626 5-598 .009 .009 .009 .009 11-367 11-310 "•253 11-195 •037 -037 -037 •037 17.051 16.965 16.879 16.793 .084 .084 .084 .084 22.734 22.620 22.505 22.391 .150 -150 .150 .150 8.638 17.276 25914 5000 34-552 5.569 .009 11.138 ■037 16.707 .084 22.276 .150 SrvirTHsoNiAN Tables. 106 Table 21, CO-ORDINATES FOR PROJECTION OF MAPS. SCALE n^m - [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 iMies. /ttcAes. IficAes, Inches. Inches. Inches. Inches. Inches. Inches* 45 8.640 17.279 25.919 5.569 S-S40 5.511 5.482 .009 .009 .009 .009 II.138 11.080 11.022 10.963 •037 -037 •037 -037 16.707 16.620 16.532 16.445 .084 .084 .084 .083 22.276 22.160 22.043 21.927 .150 .149 .149 .149 51 00 IS 30 4S 34-558 5-453 5-423 S-394 5-364 .009 .009 .009 .009 10.905 10.846 10.787 10.728 •037 •037 .037 16.358 16.269 16.181 16.092 .083 .083 .083 -083 21.810 21.692 21.574 21.456 .148 .148 .148 .147 8.641 17.282 25.924 5200 30 45 34-565 S-334 S-305 S-275 5-245 .009 .009 .009 .009 10.669 10.609 10.549 10490 -037 ■036 •036 •036 16.004 15.914 15.824 15-734 •083 .082 .082 .082 21.338 21.218 21.099 2a979 .147 .146 .146 .145 8.643 17-285 25.928 S3 00 IS 30 45 34-571 5-215 5-185 5-154 5.124 .009 .009 .009 .009 10.430 10.369 10.309 10.248 .036 .036 ■036 •036 15.645 15-554 15-463 15-372 .082 .082 .081 .081 20.860 20.738 20.617 20.496 •145 .145 .144 .144 8.644 17.288 25-932 54 00 15 30 45 34-576 5.094 5-063 5-032 5-002 .009 .009 .009 .009 10.187 10.126 10.064 10.003 .036 .036 .036 .036 "15.281 15.189 15.097 15.004 .081 .081 .080 .080 20.374 20.252 20.129 20.006 .144 -143 -143 .142 8.646 17.291 25-937 SS 00 IS 30 45 34-582 4.971 4.940 .009 .009 .009 .009 9.942 9.879 9.817 9-7SS .036 •035 •035 -03s 14.912 14.819 14.726 14-633 .080 .080 -079 .079 19.883 19-759 19.634 19.510 .142 .141 .141 .140 8.647 17.294 25.941 5600 IS 30 45 34-588 4.846 4.815 4.784 4-752 .009 .009 .009 .009 9-693 9.630 9-567 9.504 •03s •035 •03s •035 14-539 14.445 14-351 14.256 -079 .079 .078 .078 19.386 19.260 19-134 19.008 .140 .140 •139 -139 8.648 17.297 25.946 57 00 IS 30 45 34-594 4.720 4.689 4.657 4.625 .009 .009 .009 .009 9.441 9-377 9-314 9.250 •035 •03s -034 -034 14.162 14.066 13-970 13-875 .078 .077 .077 .077 18.882 18.754 18.627 18.500 .138 .138 ■137 -137 8.650 17.300 25.950 5800 IS 30 4S 34.600 4-593 4.561 4-529 4-497 .009 .008 .008 .008 9.186 9.122 9.058 8-993 -034 -034 -034 •034 13779 13-683 13586 13.490 .076 .076 .076 ■075 18.372 18.244 18.115 17.986 -136 -135 •13s -134 8.651 17-303 25-954 5900 IS 30 45 34.605 4.464 4-432 4-399 4-367 .008 .008 .008 .008 8.929 8.864 8.799 8-734 •033 •033 •033 •033 13393 13.296 13.198 13.100 r075 -075 .075 .074 17.858 17.728 17-597 17.467 -134 •133 •133 .132 8.653 17-305 25.958 6000 34.611 4-334 .008 8.669 ■033 13003 -074 17-337 ■131 Smithsonian Tables. 107 Table 21. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation o£ table explained on pp. liii-lvi.] SCALE i^?Vm- ■s Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR - 15' longitude. 30^ longitude. 45' longitude. 1° longitude. 1 X y X y X y X y Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. eo^oo' 15 30 45 8.654' 17.308 25.962 4-334 4.301 4.269 4.236 .008 .008 .008 .008 8.669 8.603 8-537 8.471 •033 .032 •032 .032 13-003 ;X6 12.707 .074 .074 •073 .073 ■7-337 17.206 17.074 16.943 •131 ■131 .130 .129 61 00 '5 30 45 34.616 4-203 4.170 4.136 4.103 .008 .008 .008 .008 ^.406 8.339 8-273 8.207 .032 ..032 .032 -031 12.608 12.509 12.410 12.310 .072 .072 .072 .071 16.81 1 16,679 16,546 16,413 .128 .128 .126 8.655 25.966 62 00 IS 30 45 34.621 4.070 4.036 4-003 3-970 .008 .008 .008 .008 8.140 8.006 7-939 .031 .031 .031 .031 12.210 12.110 12.009 11.909 .071 .071 .070 .070 16,280 16.146 16.012 15.878 .125 .125 .124 .123 8.657 17-313 25-970 6300 15 30 45 34.626 3-936 1^8 3-835 .008 .008 .007 .007 7.872 7.804 .031 .030 .030 .030 11.808 11,707 11.605 11,504 .069 !o68 15.744 15.609 15-474 15-338 .122 .122 .121 .120 8.658 17.316 25.974 6400 IS 30 45 34-632 3.801 3-767 .007' .007 .007 .007 7.602 7.533 7.465 7-397 .030 .029 .029 .029 11,402 11,300 1 1. 198 11.096 .067 .067 .066 .066 15-203 15.067 14.930 14,794 .119 .119 .118 .117 8,659 17.318 25-977 6500 15 30 45 34-636 3.664 3630 3.596 3-561 .007 .007 .007 .007 7-329 7.260 7-191 7-123 .029 .028 .028 .028 10.993 10.890 10.787 10.684 .065 .065 .064 .064 14.658 14-520 14-383 14.245 .116 ,115 ■114 .113 8.660 17.321 25.981 6600 I.S ; 30 1 •45 ; 34.641 3-527 3-492 3.458 3-423 .007 .007 .007 .007 7.054 6.984 6.915 6.846 .028 .028 .027 .027 10.581 10-477 10-373 10.269 .063 .063 .062 .062 14.108 13-969 13-830 13,692 .112 .III .III .110 8.661 17-323 25.984 67 00 : 15 30 -45 34.646 3-388 3-353 3-318 3-283 .007 .007 .007 .C07 6.776 6.706 6-637 6-567 .027 .027 .026 .026 10.165 10.060 9-955 9-850 .061 .061 .060 .060 13-553 13-413 13-273 13-134 .109 .108 ,107 ,106 8.663 25:988 6800 ; -15 1 30 45 34-650 3.248 3-213 3-178 3-143 .007 •007 .006 .006 6-497 ^^"•^ 6.356 6.286 .026 .026 .025 .025 9,746 9.640 9-535 9.429 .059 .059 .058 .058 12.994 12.854 12.713 12.572 .105 .104 .103 .102 8.664 17-327 25.991 6900 IS 30 45 34-655 3.108 3.072 3-037 3.002 .006 .006 .006 .006 6.2x6 6.145 6.074 6.003 .025 .025 .024 .024 9-323 9,217 9,111 9.005 -057 -057 .056 .056 12.431 12.290 12,148 12,006 .101 .100 8.665 17-329 25.994 7000 34-659 2.966 .006 S-932 .024 8.899 -055 11,865 .097 Smithsonian Tables. 108 Table 21 . CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] SCALE issVao- 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. IncAes, Inches. Inches. Inches. Inches. Inches. Inches. Inches. 70°oo' IS 3° 45 'im 17-331 25.997 2.966 2.930 2.895 2.859 .006 .006 .006 .006 S-932 5.861 5.790 5-718 .024 .024 •023 .023 8.899 8.792 8.68s 8.578 ■055 •055 .054 -053 11.865 11.722 11.580 11-437 .097 .096 •095 .094 71 00 IS 3° 45 34-663 2.824 2.788 2.752 2.716 .006 .006 .006 .006 5.647 5-576 5- 504 S-432 .023 .023 .022 .022 8.471 8-363 8.256 8.148 .052 .052 .051 .051 11.294 11.151 11.008 10.864 -093 .092 .091 .090 8.667 17-333 26.000 72 00 IS 30 45 34.667 2.680 2.572 .006 .006 .005 ■005 5.360 5.288 5.216 5.144 .022 .022 .021 .021 8.040 7-932 7.824 7-716 .050 .050 .049 .C49 10.720 10.576 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 7.281 .048 .048 .047 .046 10.144 9.998 9.854 9.708 .085 .084 .083 .081 8.668 17-337 26.006 7400 15 30 45 34-674 2.391 2.354 2.318 2.281 .005 .005 .005 .005 4.782 4.709 4.636 4-563 .020 .020 .019 .019 7.172 7.063 .045 -044 .044 -043 9-563 9.417 9.272 9.126 .080 .079 .078 .077 8.669 26.008 7500 IS 3° 45 34-677 2.245 2.208 2.172 2-135 .005 .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 8.834 8.687 8.540 .076 .074 -073 .072 8.670 17-340 26.010 7600 IS 3° 45 34.680 2.098 2.062 2.025 i.98§ .004 .004 .004 .004 4.197 4123 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, 15 30 45 34.684 1. 95 1 1-914 1.877 1.840 .004 .004 .004 .004 3-903 3.829 nil .017 .017 .016 .016 S-8S4 - 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 IS 30 45 34.686 1.804 1.766 1.729 1.692 .004 .004 .004 .004 3.607 3-533 3-459 3-385 -015 -015 •015 .014 5.41 1 5-3°° 5.188 5-077 •03s -034 -034 -033 7.214 7.066 6.918 6.769 .062 .060 •059 .058 8.672 17-344 26.017 7900 15 30 45 34.689 1-655 i.6i8 1.581 1-544 .004 .C03 .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 ■055 •054 ■053 8.673 17-346 26.018 8000 34-691 1.506 .003 3-013 .013 4-519 .029 6.026 -052 Smithsonian Tables. 109 Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ttW [Derivation of table explained on pp. liii-lvi.] Meridional dis> tances from even degree parallels. ABSCISSAS OF DEVELOPED PARALLEL. ORDINATES OF DEVELOPED PARALLEL. 5' longitude. 10' longitude. 15' longitude. 20' longitude. 25' longitude. 30' longitude. o°oo' 10 20 30 40 50 1 00 10 20 30 40 50 2 00 10 20 30 40 50 300 10 20 30 40 50 400 10 20 30 40 50 500 10 20 30 40 50 600 10 20 30 40 50 700 Itiches. Inches. 5.764 5-764 5-764 5-764 5-764 5.764 5-764 5-763 5-763 5-762 5-762 5.761 S-761 5.760 5-759 5-759 5-758 5-757 5-756 5-756 5-754 5-753 5-752 5-751 5-750 5-749 5-748 5-746 5-745 5-744 5-743 S-741 5-739 5-738 5-736 5-735 5-733 5-731 5-729 5-727 ■ 5.726 5-724 5.722 Inches. 11.529 11.528 11.528 11.528 11.528 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 "-513 11.511 11.509 11.507 11.505 11.503 11.501 11.498 .11.496 "•493 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 "-443 Inches. '7-293 17-293 17.292 17.292 17.291 17.291 17.291 17.289 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 17.188 17.182 17.177 17.171 17.165 Inches. 23.058 23-057 23-056 23.056 23-055 23.054 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 28.786 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 •s_- Is 5_H 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° 11.451 22.902 34-353 45-804 57-254 68.705 5 10 15 20 25 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° f 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 34.355 45.806 57.258 68.710 6° 7° 5 10 '5 20 25 30 0.000 .002 .004 .007 .011 .016 0.000 .002 .013 .018 11.452 22.904 3*-356 45.808 57.260 68.7 1 2 Smithsok IAN TaBI ES. ■■"■" Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE 77}^. [Derivation of table explained on pp. liii-lvi.] ■-3 g u .2 K-Si! S „ = S S ABSCISSAS OF DEVELOPED PARALLEL. S' longitude. 10' longitude. IS' longitude. longitude. 25' longitude. 30' longitude. ORDINATES OF DEVELOPED PARALLEL. 7°oo' 10 20 30 40 SO 800 10 20 3° 40 so 900 10 20 30 40 so 10 20 30 40 so 10 20 30 40 so 10 20 30 40 so 1300 10 20 30 40 so 14 00 Inches. 68.712 11.452 22.905 34-35» 45.810 57.262 68.715 "■4S3 22.906 34-3S9 45.812 57.265 68.718 "•454 22.907 33-361 45.814 57.268 68.722 11.454 22.909 34-263 45-817 57.272 68.726 11.455 22.910 34-365 45.820 S7-27S 68.730 11.456 22.912 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 S-717 S-71S S-7I3 S-7II S-709 5.706 S-704 S-701 5.696 S-694 5.691 5.688 5.686 5.683 5.680 S-677 S-674 5-671 5.668 ^■^^ 5.662 5.659 5.656 5.652 5.649 5.646 5.642 S-639 5.636 S-632 5.628 5.625 5.621 5.618 5.614 5.610 5.606 5.602 S-S98 S-S94 Inches. "-443 "-439 "-43S 11.430 11.426 11.422 11.417 11.412 11.407 11.403 11.398 "-393 11.388 n.382 "•377 "•371 11.366 11.360 "•355 "•349 "■343 "•337 "•331 11.324 11.318 11.312 11.305 11.298 11.292 n.285 11.278 II. 271 11.264 11.257 11.250 11.242 "•235 11.227 11.220 11.212 11.204 II. 196 11.188 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 22.455 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 28.088 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 33-659 33-635 33-612 33-589 33-565 3-s Inches. 0.000 .002 .005 .008 .013 .018 .003 .006 .010 .016 .023 O.OOI -003 .007 .013 .020 .028 13° .004 .008 .015 .023 -033 Inches. 0.001 .002 .005 .009 .014 .021 .003 .006 .Oil .018 .026 O.OOI .003 .008 .014 .021 .031 14° O.OOI .004 .009 .016 .025 -035 Smithsomian Tables. Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ssW [Derivation of table explained on pp. liii-lvi.] II i4°oo' 10 20 30 40 5° 15 00 10 20 30 40 5° BO .2 »-aJJ Inches. 68.740 11.458 22.915 34-373 45.830 57.2S8 68.746 11.459 22.917 34-376 45-834 57-293 ABSCISSAS OF DKVELOPED PARALLEL. S' longitude. 1600 68.752 10 11.460 20 22.919 .30 34-379 40 45-838 50 57.298 17 00 68.758 10 11.461 20 22.921 30 34-382 40 45-843 50 57-304 1800 68.764 10 11.462 20 22.924 30 45;848 40 5° 57-310 19.00 68.771 10 11.463 20 22.926 30 34-390 40 45-853 SO 57-316 2000 68.779 :o 11.464 20 22.929 30 34-394 40 45-858 50 57.322 21 00 68.787 Inches* 5-594 5-590 5.586 5.582 5-578 5-573 5.569 S-565 5.560 5-556 S-551 5-547 5.542 5-538 5-533 5-528 5-524 5-519 5-514 5.509 5-504 5-499 5-494 5.489 5-484 5-479 5"^73 5.468 5-463 5.458 5-452 5-447 5-441 5-436 5-430 5.424 5-419 5-413 5-407 5.401 5-396 5-390 5-384 10' longitude. Indus. II.I88 n.iSo II. 172 II. 163 "-155 II. 147 II. 138 II. 130 II. 121 II. [12 11. 103 11.094 11.085 1 1.076 11.066 11.057 11.047 11.038 11.028 ii.oiS 11.008 10.998 10.988 10.978 10.968 10.957 10.947 10.936 10.926 10.915 10.905 10.893 10.882 10.871 10.860 10.849 10.838 10.826 10.814 10.803 10.791 10.779 10.768 15' longitude. Inches. 16.783 16.770 16.758 16.745 16.733 16.720 16.708 16.694 16.681 16.667 16.654 16.641 16.628 16.613 16.599 16.585 16.556 16.542 16.527 16.512 16.497 16.482 16.467 16.452 16.436 16.420 16.404 16.389 16.373 16.357 16.340 16.324 16.307 16.290 16.274 16.257 16.239 16.222 16.204 16.187 16.169 16.151 20' longitude Inches. 22.377 22.360 22.344 22.327 22.310 22.294 22.277 22.259 22.241 22.223 22.206 22.188 22.170 22.151 22.132 22.113 22.094 22.075 22.056 22.036 22.016 21.996 21.976 21.956 21-936 21.915 21.894 21.872 21.852 21.830 21.809 21.787 21.765 21.742 21.720 21.698 21.676 21.652 21.629 21.605 21.582 21.558 21-535 25' longitude. 30' longitude. Inches. 27.971 27.950 27-930 27.909 27.888 27.867 27.846 27.824 27.802 27.779 27-757 27-735 27-713 27.689 27.665 27.642 27.618 27-594 27.571 27.546 27-521 27-495 27.470 27.445 27.420 27-394 27-367 27-341 27-3I5 27.288 27.262 27-234 27.206 27.178 27.150 27.123 27.095 27.065 27.036 27.007 26.978 26.948 26.919 Inches, 33-565 33-540 33-515 33-490 33-465 33-440 33-415 33-389 33-362 33-335 33-308 33.282 33-255 33-227 33-198 33-170 33-142 33-113 33-085 33-055 33-025 32-994 32.964 32-934 32-904 32.872 32.840 32.809 32-777 32.746 32-714 32.680 32-647 32.614 32.580 32-547 32-513 32-478 32-443 32.408 32-373 32-338 32-303 ORDINATES OF DEVELOPED PARALLEL. 'EbS 14° Inches. 0.00 1 .004 .009 .016 .025 •035 16= 0.00 1 .004 .010 .018 .028 .040 18° 0.00 1 .005 .oil .020 .031 .044 O.OOI .005 .012 .022 -034 •049 15° Inches. O.OOI .004 .009 .017 .026 .038 17° O.OOI .005 .oil .019 .029 .042 19° O.OOI .005 .012 .021 .032 .046 O.OOI .006 .013 .022 •03s .051 Smithsonian Tables. 112 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] Table 22. SCALE ^jJsT- •g . Meridional dis- tances from even degree parallels. ABSCISSAS OF DEVELOPED PARALLEL. ORDINATES OF DEVELOPED PARALLEL. s' longitude. 10' longitude. IS' longitude. 20' longitude. 25' longitude. 30' longitude. 2I°00' 10 20 30 40 SO 22 00 10 20 30 40 so 2300 10 20 30 40 50 24 00 10 20 30 40 SO 25 00 10 20 30 40 SO 2600 10 20 30 40 SO 27 00 10 20 30 40 SO 2800 Inches. 68.787 Inches. S-384 S-378 5.366 5-359 5-353 5-347 5-341 5-334 5-328 5-322 5-315 S-309 5-302 5.296 5.289 5.282 5.276 5.269 5.263 5-256 5.249 5.242 5-235 5-227 5.220 5-213 5-206 5-199 5.191 5.184 5-177 5.169 5.162 5-1 54 S-147 5.140 5-132 5.124 5.116 5.109 5.101 S-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 10.218 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 '¥ 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 '5-349 15-326 15-303 15.280 Inches, 2I-53S 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 20.99s 20.967 20.938 20.910 20.881 20.852 20.824 20.795 20.766 20.737 20.708 20.678 20.648 20.6 r 8 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 3I-S35 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 30.882 30.838 30-791 30-745 30-699 30-653 30.607 30.560 S' 10 IS 20 25 30 21° 22° 11.466 22.932 34-397 45-863 57-329 68.79s Inches, 0.001 .006 ■013 .022 -035 .051 Inches, 0.001 .006 .013 •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 5 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 .014 .026 .040 .058 0.002 .007 .026 .041 -059 ir.472 22.943 3+415 • 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. 113 Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] SCALE ?jjinr- .i ABSCISSAS OF DEVELOPED PARALLEL. vh ORDINATES OF DEVELOPED .§1 -1^£« 5' 10' 15' 20' 25' 30' PARALLEL. r- ^Bss. longitude. longitude. longitude. longitude. longitude. longitude. Inclus. Inches. Inches. Inches. Inches. Inches, Inches. 28°00' 10 68.849 5-093 5.085 10.187 IO.17I 15.280 15-256 20.374 20.342 25.467 25.427 30.560 30-SI3 S.5 28° 29° 11.476 20 22.953 s-077 10.155 15-232 20.310 25-387 30-465 Inches. Inches. 3° 34-43° 5.069 10.139 15.208 20.278 25-347 30-417 30-369 5' 0.002 0.002 40 45.906 5.061 10.123 .5.185 20.246 25.308 10 .007 .007 5° 57-383 S-0S4 10.107 I5.161 20.214 25.268 30.321 15 20 .016 .028 .016 .028 2900 68.859 S-046 10.091 'S-I37 20.182 25.228 30.274 25 30 ■Z -044 .064 10 U.478 5-037 10.075 10.058 15.112 20.150 25.187 30.224 20 22.957 5.029 15.087 20.117 25.146 30-175 30.126 3° 34-435 5.021 10.042 15.063 20.084 25.105 40 45-913 S-013 10.025 15.038 20.051 25.064 30.076 50 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.Ot8 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 19.917 24.896 29.876 5 0.002 0.002 3° 34.440 4.971 9.942 14.912 19.883 24.854 29.825 10 .007 .007 40 45.920 4.962 9-925 9.908 14.887 19.849 24.812 29.774 15 .016 .017 5° 57.400 4-954 14.862 19.815 24.769 29.723 20 .029 .030 25 •045 .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 34.446 4.928 9.856 14.784 19.712 24.640 29.568 . 30 4.919 9.838 14.758 19.677 24.596 29-515 40 5° 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° 3200 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 5 0.002 0.002 .008 11.484 20 22.967 4-875 9.750 14.625 14.598 19.500 24.376 29.251 10 .007 30 34-451 4.866 9-732 19.465 24-331 29.197 15 .017 .017 40 45-934 4.857 9.714 14-572 19.429 24.286 29-143 20 .030 .031 .048 .069 5° 57.418 4.848 9.696 14-545 19-393 24.241 29.089 25 30 t^ 33 °o 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 '4-435 14.407 19.246 19.210 24.058 24.012 28.870 28.814 1A° 35° 5° 3400 57-427 68.913 4-793 4.784 9-586 9-568 14-379 14-352 19-173 19.136 23.966 23.920 28.759 28.704 34 5 0.002 0.002 10 11.487 4-774 9-549 14-323 19.098 23-872 28.647 10 15 20 .008 .017 .031 -049 .070 .008 .018 20 22.975 4.765 9-530 14-295 19.000 23.825 23-778 28.590 .031 •049 .071 30 34.462 4-755 9.511 14.267 19.022 28.533 25 30 40 45.949 4.746 9-492 14.238 18.984 23:6^3 28.476 SO 57-437 4-737 9-473 14.210 18.946 28.420 3500 68.924 4.727 9-454 14.181 18.908 23.636 28.363 Bmithsonian Tables. 114 Table 22i CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^^hs- [Derivation of table explained on pp. liii-lvi.] ABSCISSAS OF DEVELOPED PARALLEL. "S .2 M"^^ ORDINATES OF DEVELOPED PARALLEL. ■5 s S' 10' IS' 20' 25' 30' 2'^ gSss. longitude. longitude. longitude. longitude. longitude. longitude. Inches. Inches. Inches. Inches. Inches. Inches, Inches. 35°oo' 10 68.924 4.727 4^717 9-454 9-435 14.181 14.152 18.908 18.870 23.636 23-587 28.363 28.305 35° 36° 11.489 20 22.978 4.708 9.415 14.123 18.831 23-539 '?-^46 Inches. Inches. 30 34.468 4.698 9-396 14.094 18.792 23.490 28.188 5' 10 0.002 0.002 40 45-957 4.688 9-377 14.065 18.753 23-442 28.130 .008 .C08 50 57.446 4^679 9-357 14.036 18.714 23-393 28,.072 15 .018 .018 3600 10 68.935 4.669 4.659 9-338 9.318 14.007 13-977 18.676 18.636 23-345 23-295 28.014 27-954 27-894 20 25 30 •031 •049 .071 .032 .050 .072 II.491 20 22.983 4.649 9.298 13-947 18.596 23.245 30 34-474 4^639 9.278 '3-917 18.556 23-195 27-835 40 45.965 4^629 9.258 '3-f7 18.517 23.146 27.775 5° 3700 10 57457 68.948 4^619 4^609 4^599 9.238 9.219 9.198 13-858 13.828 i3'-797 18.477 18.437 18.396 23.096 23.046 22.995 27.715 27.656 27-594 37° 38° "•493 20 22.986 4-589 9.178 13-767 18.356 22.944 27-533 5 0.002 0.002 30 34.480 4-579 9-1 57 13-736 18.315 22.894 27.472 10 .008 .008 40 45-973 4.568 9-137 13-706 18.274 22.843 27.411 15 .018 .018 50 57.466 4-558 9.117 13-675 18.234 22.792 27-350 20 25 .032 .050 •033 .051 3800 10 68.959 4-548 4-538 9.096 9.076 13-645 J 3-613 18.193 18.151 22.741 22.689 27.289 27.227 30 -073 -073 11.495 20 22.990 4-527 9.055 13.582 '?•'?? 22.637 27.164 30 34-485 4-517 9-034 13-551 18.068 22.585 27.102 40 50 45.980 57-475 4.506 4.496 9.013 8.992 13.520 13-488 18.026 17.984 22.533 22.481 27-039 26.977 39° 40° 3900 10 68.970 4.486 4-475 8.971 8.950 13-457 13-425 17-943 17.900 22.429 22.375 26.914 26.851 26.787 5 0.002 0.002 11.497 20 22.994 4.464 8.929 13-393 17.858 22.322 10 .008 .008 30 34-491 4-454 8.908 13-361 17.815 22.269 26.723 15 .018 .019 40 45.988 4-443 8.886 13-330 17-773 22.216 26.659 20 •033 -033 50 57-485 4-433 8.865 13.298 17-730 22.163 26.595 25 30 .051 .074 .052 .074 4000 10 68.982 4.422 4-4" S.844 8.822 13.266 13-233 17.688 17-644 22.110 22.055 26.532 26.466 "•499 20 22.998 4.400 8.800 13.201 17.601 22.001 26.401 30 34-497 4-389 8.779 13.168 '7-557 21.947 26.336 40 45.996 4-378 8-757 13-135 17-514 21.892 26.271 ^tO A'yO so 41 00 57^495 68.994 4-368 4-357 8-735 8-713 13-103 13-070 17.470 17.427 21.838 21.784 26.206 26.140 41 42 S 0.002 0.002 10 .008 .008 10 1 1. 501 4-346 ?-^i' 13-037 17-383 21.728 26.074 15 .019 -033 .052 -075 .019 20 23.002 4-335 8.669 13.004 17-338 21.673 26.007 20 -033 30 34^503 4-324 8.647 12.971 17.294 2i.6i8 25.941 25 •052 .075 40 46.004 4.312 8.62s 12.937 17.250 21.562 \$ol 30 SO 57^So6 4.301 8.603 12.904 17.205 21.507 42 00 69.007 4.290 8.581 12.871 17.161 21.451 25-742 .^^ Smithsonian Tables. "S Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] SCALE vsWs' o -si 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 42°00' 10 20 30 40 50 4300 10 20 30 40 50 4400 10 20 30 40 50 4500 10 20 30 40 50 4600 10 20 30 40 50 4700 10 20 30 40 SO 48 00 10 20 30 40 50 4900 Inches. 69.007 Inches. 4.290 4.256 4-245 4-234 4.222 4.211 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 4.011 3-975 3-963 3-95' 3-938 3.926 3-9'4 1.8^9 3-877 3-864 3.852 3-839 3.827 3.814 3.802 3-789 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 8.21 -t 8.189 8.166 8.142 8.118 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 ^•^^ 7.603 7-578 Inches. 12.871 12.837 12.803 12.769 '2.735 12.701 12.667 12.633 12.598 12.564 12.529 12.494 12.460 12.425 12.390 12.354 12.319 12.284 12.249 12.213 12.177 12.141 12.105 12.070 12.034 11.997 11.961 11.925 11.888 11.852 11.815 11.778 11.741 11.704 11.667 11.630 "-593 "-5S5 11.518 11.480 11.442 11.405 11.367 Inches. 17.161 17.116 17.071 17.025 16.980 '6.935 16.890 16.844 16.798 16.751 16.659 16.613 16.566 16.519 16.473 16.426 '6.379 16.332 16.284 16.236 16.188 16.141 16.093 16.045 '5-997 15.948 15.899 15.851 15.802 '5-754 15.704 15.606 15.556 15-507 « 5-457 '5-407 '5-357 'S-307 '5-257 15.206 15.156 Inches. 21.451 21.39s 21.318 21.282 21.225 21.169 21.112 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 '9-974 19.813 '9-753 19.692 19.630 19.569 19-507 19.445 '9-383 '9-322, 19.259 19.196 i9-'34 19.071 19.008 18.945 Inches. 25.742 l\^ 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 23.260 23.186 23.1 11 23-035 22.960 22.885 22.810 22.734 o.g 42° 43° "-503 23.006 34-5'0 46.013 57-516 69.019 5' 10 '5 20 25 30 Inches. 0.002 .008 .019 -033 .052 ■075 Inches. 0.002 .008 .019 •033 .052 •075 11.505 23.010 34-515 46.020 57-525 69.030 44° 45° 11.507 23.014 34.522 46.029 57-536 69.043 5 lO '5 20 25 30 5 10 '5 20 25 30 0.002 .008 .019 •034 .052 -075 0.002 .008 .019 -034 -053 .076 11.509 23.018 34-528 46.037 57-546 69.055 46° 47° 11.511 23.023 34-534 46.045 57-557 69.068 0.002 .008 .019 -034 -053 .076 0.002 .008 .019 -034 .052 .075 "-513 23.027 34-540 46.053 57-567 69.080 II. 516 23.031 34-546 46.062 57-577 69.093 48° 49° 5 10 '5 20 25 30 0.002 .008 .019 -033 .052 -075 0.002 .008 .019 -033 .052 -075 6MITH90N PIN T«BU 9. ■~~" 116 Table 22, CO-ORDINATES FOR PROJECTION OF MAPS. SCALE 7;W [Derivation of table explained on pp. liii-Ivi.] •s u. ABSCISSAS OF DEVELOPED PARALLEL. ^^ T\ ir\ T XT Jt Tl f^-! cJsgfjS ORDINATta ur •S3 2 w'Sji DEVELOPED II l = ll S' . 10' 15' 20' 25' 30' PARALLEL, J3- |i2So. longitude. longitude. longitude. longitude. longitude. longitude. Inches. Inches. iTtches, Ittches. Inches. Inches. Inches. 49°oo' 10 69.093 3-789 3-776 7-578 7-553 11.367 11.329 15.156 15.105 18.945 18.882 22.734 22.658 ■as 49° 50° II.SI7 20 23-035 3-764 7-527 11.291 15.054 18.818 22.581 Inches. Inches. 3° 34-SS2 3-751 7.502 11.253 15.003 18.754 22.505 5' 10 0.002 0.002 40 46.070 3-738 7.476 II.214 14.952 18.690 22.429 !oo8 !oo8 5° S7-S87 3-725 7-451 II. 176 14.901 18.627 22.352 15 .019 .019 SO 00 69.10S 3-713 7-425 II.138 14.850 18.563 22.276 20 25 •033 .052 •033 .052 lO II.S20 3.700 7-399 11.099 14-799 18.499 22.198 30 .075 .075 20 23-039 3.687 7-374 11.000 14-747 18.434 22.121 3° 34-5SS 3-674 7-348 II.02I 14.695 18.369 22.043 40 46.078 3.661 7.322 10.983 14.644 18.305 21.965 5° SI 00 10 S7-S98 69.117 3.648 3-635 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.IIO 21.888 21.811 21.732 51° 52° II. 521 20 23-043 3-609 7.218 10.827 14.436 18.045 21.653 5 0.002 0.002 3° 34-564 3-596 7.191 10.787 14-383 17-979 21.574 10 .008 .008 40 46.086 3-583 7.165 10.748 14-330 17-913 21.496 15 .019 .018 50 57-607 3-570 7-139 10.709 14.27S 17.848 21.417 20 -033 -033 S2 00 10 69.128 3-556 3-543 7-"3 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. 1 19 17-649 21.179 3° 34-570 3516 7-033 10.550 14.066 17-583 21.099 40 so 46.094 57.6x7 3-503 3-490 7.006 6.980 10.510 10.470 14.013 13.960 17.516 17.450 21.019 20.939 53° 54° S3 00 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 S 0.002 0.002 11.525 20 23.051 3-450 6.899 10.349 13-798 17.248 20.698 10 .008 .008 3° 34-576 3-436 6.872 10.309 •3-745 17.181 20.617 15 .018 .018 40 46.102 3-423 6.845 6.8i8 10.268 13.691 17.114 20.536 20 .032 -032 5° 57.627 3-409 10.228 13-637 17.046 20.455 25 .050 .050 30 -073 .072 54 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 34-582 46. log 3368 6-737 ■ 10.105 13-474 16.842 20.210 30 40 3-355 3-341 6.709 6.682 10.064 10.023 13-419 13-364 16.774 16.706 20.128 20.047 rpO 56° SO SSoo 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 55 5 0.002 0.002 10 .008 .008 10 11.529 3-300 6.600 9.900 13.200 16.500 19.800 15 20 .018 .018 20 23.059 34.588 3.286 6.572 9.859 13-145 16.431 19.717 .032 -031 30 3.272 6.545 9.817 13.089 16.362 19.634 25 30 -049 .049 40 46,117 3-258 6.517 9.776 13-034 16.293 '^•5|« .071 .070 5° 57-646 3-245 6.489 9-734 12.979 16.224 19.468 S6oo 69.176 3-231 6.462 9-693 12.924 16.155 19-385 Smithsonjan Tables. 117 Table 22. . CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^■stTn- [Derivation of table explained on pp. liii-lvi>] ABSCISSAS OF DEVELOPED PARALLEL. ■83 =3 ORDINATES OF DEVELOPED ■^i till S' 10' •5' 20' 25' 30' PARALLEL. u s > d g £1 u p< longitude. longitude. longitude. longitude. longitude. longitude. Inches. ■ Inches. Inches. Inches. Inches. Inches. Inches. ^.S 56-00' 10 69.176 3-231 3.217 6.462 6-434 9-693 9-651 12.924 12.868 16.15s 16.085 19-385 19.301 56° 57" "•S31 20- 23.063 3-203 6.406 9.609 12.812 16.015 19.217 Inches. Inches. 30 34-S94 3.189 6.378 9-567 12.756 '5-945 15-875 '9-134 5' 0.002 0.002 40 46.125 3-'7S 6.350 9-525 12.700 'i'°i2 10 .008 .008 5° 57.656 3.161 6.322 9-483 12.644 15-805 18.966 '5 .018 .017 57 00 69.188 3-147 6.294 9.441 12.588 '5-735 18.882 20 25 .031 -049 .031 10 "■533 3-133 6.266 9-398 '2-531 15.664 18.797 30 .070 20 23.066 3-i'9 6.237 9-356 12-475 12.418 '5-594 18.712 30 34-599 3.104 6.209 9-314 '5-523 18.627 40 46.132 3.090 6.181 9.271 12.362 15-452 15.381 iS-3" 15-239 15.168 18.542 so 5800 10 57.666 69.199 3.076 3.062 3.048 6.152 6.124 6.096 9.229 9.186 9-143 12.305 12.248 12.191 18.457 '8.373 18.287 58° 59° "•535 20 23.070 3-034 6.067 9.IOI 12.134 18.201 S 0.002 0.002 30 34.605 3.019 6.038 9-058 12.077 15.096 18.115 10 .008 .007 40 46.140 3.005 6.010 9.015 12.020 15.025 18.029 '5 .017 .017 5° S7'67S 2.991 5.981 8.972 11.962 '4-953 17.944 20 25 -030 -047 .030 .046 59 00 10 69.210 2.976 2.962 S-9S3 S-924 8.929 11.905 11.847 14.882 14.809 17-858 17.771 30 .068 .067 "•537 20 23-074 2.947 5-895 8.842 H.790 14-737 17.684 30 34-6 10 2.933 5.866 8-799 11.732 14.665 '7-597 40 SO 46.147 57.684 2.918 2.904 $^ 8-755 8.712 11.674 11.616 14.592 14.520 17.510 17.424 60° 6i» 6000 10 69.221 2.890 2.875 5-779 5-750 8.669 8.625 U.558 11.500 14.448 '4-375 '7-337 17.249 5 0.002 0.002 "•S39 20 23-077 2.860 5-721 8.581 1 1. 441 14.302 17.162 10 .007 .007 30 34-616 2.846 5.691 8-537 11-383 14.229 17-074 '5 .016 .016 40 46.154 2.831 5.662 8-493 11.324 14.156 14-083 16.987 16.899 20 .029 .029 SO 57-693 2.816 5-633 8.450 11.266 25 .045 •045 30 .065 .064 6i 00 10 69.232 2.802 2.787 5.604 S-S74 8.406 8.361 H.208 II. 148 14.010 13-936 16.811 16.723 11.540 20 23.081 2.772 5-545 8.317 11.090 13.862 16.634 30 40 46.162 2-758 2-743 2.728 2-713 2.699 5.115 8.273 11.030 13-788 16. w6 i48l 8.229 10.972 '3-715 16.457 fi-yO 63° SO 62 00 10 57.702 69.242 5.456 5.427 5-397 8.184 8.140 8.096 10.912 10.854 10.794 13.641 13-567 13-493 13.418 16.369 16.280 16.191 02^ 5 10 '5 20 0.002 .016 0.002 .007 .015 .027 11.542 20 23.084 2.684 5-367 8.051 10.734 16.102 .02S .044 -063 30 34.626 2.669 5-337 8.006 10.675 '3-344 16.012 25 30 40 SO 46.168 57.710 2.654 2.639 5-308 5.278 7.961 7.917 10.61 5 10.556 13.269 '3-'9S '5-923 'S-833 :^? 6300 69-253 2.624 5.248 7.872 10.496 13.120 '5.744 1 1 Smithsonian Tabues. 118 Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE zvhs- [Derivation of table explained on pp. liii-lvi.] .10. 63°oo' 10 20 30 40 SO 6400 10 20 30 40 50 6500 10 20 30 40 50 66 00 10 20 3° 40 SO 67 00 10 20 30 40 SO 6800 10 20 30 40 50 69 00 10 20 30 40 SO 7000 ■a S S s 3 2'S:S S 5 cd > S Inches. 69-2S3 11.544 23.087 34-631 46.175 57.718 69.262 "•S4S 23.091 34-635 46.182 S7727 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 n.552 23.103 34-654 46.206 S7-7S8 69.309 "-S53 23.106 34-659 46.212 57-764 69-317 ABSCISSAS OF DEVELOPED PARALLEL. S longitude. Inches. 2.624 2.609 2.594 2.579 2.564 2.549 2-S34 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 2.1 19 2.103 2.088 2.072 2.056 2.040 2.025 2.009 1-993 1.977 10 longitude. Inches. 5-248 5.218 5.188 5.158 5.128 5.098 5.068 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 4-45S 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 3-955 15' longitude. 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 6.309 6.263 6.216 6.169 6.121 6.074 6.027 5.980 S-932 20' longitude. 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 9-035 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 25' longitude. 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 9-7" 12-139 ^:5^8 12.062 11.986 9.527 11.909 9.466 "-833 9-405 11.756 9-343 11.679 9.282 11.602 9.220 ir.525 9-158 11.448 9-097 "-371 H.294 II. 217 II. 139 1 1. 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 30' longitude. 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 H-383 14.291 14.199 14.107 14.015 13.922 13-830 13-738 13-645 13-553 13.460 13.366 13-273 13.180 13.087 12.994 12.900 12.806 12.712 12.619 12.525 12.431 12.337 12.242 12.148 12.054 11.959 1 1.865 ORDINATES OF DEVELOPED PARALLEL. ■at Inches s' 0.002 10 .007 «5 .015 20 .027 25 -043 30 .061 63° 65° 0.002 .006 .014 .026 .040 .058 67° O.OOI .006 .014 .024 .038 .054 69° O.OOI .006 .013 .022 ■035 .051 64= Inches. 0.002 .007 .015 .026 .041 .060 66° 0.002 .006 .014 .025 -039 .056 (&° O.OOI .006 .013 .023 •036 -053 70" O.OOI .005 .012 .022 -034 •049 Smithsonian Tables. 119 Table 22. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE siftZ- [Derivation o£ table explained on p, liii-lvi.] ■a J! is 70°oo 10 20 30 40 5° 71 00 10 20 30 40 SO 72 00 10 20 30 40 50 7300 10 20 30 40 50 7400 10 20 30 40 5° 7500 10 20 30 40 50 7600 10 20 30 40 5° 7700 Bo; Itickes. 69-317 "•554 23.109 34-663 46.217 57.772 69.326 11.556 23.1 1 1 34.667 46.222 57-778 69-334 11-557 23.114 34.670 46.227 57-784 69.341 11.558 23.116 34-674 46.232 57.790 69.348 "-559 23.118 34-677 46.236 57-796 69-355 11.560 23.120 34.681 46.241 57.801 69.361 1 1. 561 23.122 34-683 46.244 57-806 69-367 ABSCISSAS OF DEVELOPED PARALLEL. 5' longitude. Inches. 1.977 .962 .946 -930 .914 .898 .882 .866 .850 .835 .819 .803 .787 -771 -755 -739 -723 .707 .691 .674 .658 .642 .626 .610 -594 .578 .562 ■545 .529 -513 •497 .480 .464 .448 •432 -415 -399 .366 •35° •334 •317 .301 Smithsonian Tables. 10 longitude. Inches, 3-955 3-923 3.892 3.860 3.828 3-796 3-765 3-733 3.701 3.669 3-637 3.605 3^574 3542 3^5°9 3-477 3-445 3-413 3-381 3-349 3-3 '7 3.284 3-252 3.220 3.188 3-' 55 3-123 3.091 3.058 3.026 2-993 2.961 2.928 2.896 2.863 2.831 2.798 2.765 2-733 2.700 2.667 2.634 2.602 15' longitude. Iftches. 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 S-312 5.264 5.216 5.168 5.120 5.072 5.024 4-975 4.927 4.878 4.830 4.782 4-6^5 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 3903 20' longitude. Inches. 7.910 7.846 7-783 7-720 7.656 7^593 7^530 7.466 7.402 7^338 7^275 7.211 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 5-726 5.661 5-596 5-530 5-465 5.400 5-334 5.269 5.204 25' longitude. 9.888 9.808 9.729 9.650 9-571 9.491 9.412 9-333 9-253 9-173 9.094 9.014 8.934 8.854 8.774 8.694 8.614 8-533 8-453 8-373 8.292 8.211 8.131 8.050 7.970 7.889 7.808 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 30' longitude. Inches. 11.865 1 1.770 11.675 11.579 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 9-757 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 ORDINATES OF DEVELOPED PARALLEL. 'Ebu Inches. < 0.00 1 10 .005 15 .012 20 .022 25 •034 30 .049 70° 72° O.OOI .005 .011 .020 ■031 •044 74° O.OOI .004 .010 .018 .028 .040 76° O.OOI .004 .009 .016 .025 .036 71° Inches. O.OOI .005 .012 .021 .032 .047 73° O.OOI .005 .011 .019 .029 .042 75° .004 .009 .017 .026 .038 77" 0.001 .004 .008 .015 -023 -033 120 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on p. liii^lvi.] Table 22. SCALE isTihTi- •s 3^ Meridional dis- tances from even degree parallels. ABSCISSAS OF DEVELOPED PARALEL. ORDINATES OF DEVELOPED PARALLEL. 5' longitude. 10' longitude. 15' longitude. 20' longitude. 25' longitude. 30' longitude. 77°oo' 10 20 30 40 50 7800 10 20 30 40 SO 7900 10 20 30 40 SO 8000 Inches. 69.367 Ittches. 1.301 1.284 1.268 1.252 I-23S 1.219 1.202 1.186 1.169 I-I53 1.136 1.120 1.104 1.087 1.070 1.054 1-037 1.021 1.004 Inches. 2.602 2.569 2.536 2.503 2.470 2.438 2.405 2.372 2-339 2.306 2.273 2.240 2.207 2.174 2.141 2.108 2.075 2.042 2.009 Inches. 3-903 3-854 3.804 3-755 3.706 3-656 3-607 3-558 3.508 3-459 3.410 3-360 3-3" 3-261 3-2" 3.162 3.112 3.062 3-013 Inches. 5.204 S-I38 5.072 5.006 4.941 4-875 4.810 4-744 4-678 4.612 4-546 4.480 4.414 4-348 4.282 4.216 4.150 4-083 4.017 Incites. 6.505 6.423 6.341 6.258 6.176 6.094 6.012 s-930 5-847 5-765 s-683 5.600 S-S18 5-435 5-352 5.270 5.187 5.104 5.022 Inches. 7-805 7.707 7.609 7-510 7-4" 7-313 7.214 7.115 7.016 6.918 6.819 6.720 6.621 6.522 6.422 6-323 6.224 6.125 6.026 77° 78° 11.562 23.124 34.686 46.248 57.810 69-373 S' 10 15 20, 25 30 Inches. O.OOI .004 .008 .015 -023 •°33 Inches. O.OOI ■n .014 .021 .031 11.563 23.126 34-689 46-252 57.814 69-377 5 10 15 20 25 30 79° 80° 0.001 .003 .007 .013 .020 .028 0.001 .006 .oil .018 .026 11.564 23.127 34.691 46.25s 57.818 69.382 Smithsonian Tables. 121 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Wirw [Derivation of table explained on pp. liii-lvi.] dional dis- les from 1 degree illels. co-ordinat£s of DEVELOPED PARALLEL FOR— lo' longitude. 20/ longitude 30' longitude. 40/ longitude so' longitude 1° longitude. "C H 5 B 2a V d > S X y X y X y X y X y X y fnjtt. mm. mm. Tnm. m-m. mm. mm. mm. mm. mm. mm. mm. mm. b°oo' 92.8 .0 185.5 .0 278.3 .0 37I-I .0 463.8 .0 556.6 .0 10 92.1 92.8 .0 185.5 .0 278.3 .0 37 1 -I .0 463.8 .0 556.6 .0 20 184.3 92.8 .0 185-5 .0 278.3 .0 37 1 -I .0 463-8 .0 556-6 .0 30 ^l^i 92.8 .0 185.5 .0 278.3 .0 371-0 .0 463.8 .0 556-6 .0 40 368.6 92.8 .0 18S-S .0 278.3 .0 371-0 .0 463-8 .0 556.6 .0 5° 460.7 92.8 .0 185-5 .0 278.3 .0 371-0 .0 463-7 .0 556-5 .1 I 00 92.8 .0 185-5 .0 278.3 .0 371-0 .0 463.7 556-5 .1 10 92.1 92.7 .0 i8S-S .0 278.2 .0 371-0 .0 463-7 556-4 .1 20 184.3 92.7 .0 185.5 .0 278.2 .0 371-0 .0 463-7 556-4 .1 3° ^ZB 92.7 .0 185.5 .0 278.2 .0 370-9 .0 463-7 556-4 .1 40 368.6 92.7 .0 185.4 .0 278.2 .0 370-9 .0 463-6 556-3 .1 SO 460.7 92.7 .0 185.4 .0 278.2 .0 370-9 .1 463.6 556-3 .2 2 00 92.7 .0 185.4 .0 278.1 .0 370.8 .1 463.6 556-3 .2 10 92.1 92.7 .0 185-4 .0 278.1 .0 370.8 .1 463.5 556.2 .2 20 184.3 92.7 .0 185.4 .0 278.1 .0 370.8 .1 463.4 556-1 .2 3° 276.4 92.7 .0 185-3 .0 278.0 .0 370.7 .1 463.4 556-0 .2 40 368.6 92.7 .0 185-3 .0 278.0 .0 370.6 .1 463.3 .2 556.0 .2 50 460.7 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 5SS-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 .2 555-5 .3 40 368.6 92.6 .0 185.1 .0 277.7 .1 370.3 .1 462.8 .2 555-4 .3 SO 460.7 92.6 .0 185.1 .0 277.7 .1 370.2 .1 462.8 ,i 555-4 •3 400 92. s .0 185.1 .0 277.6 .1 370.2 .2 462.7 .2 555-2 •3 10 92.1 92-S .0 185.0 .0 277.6 .1 370.1 .2 462.6 .2 S55-I .3 20 184.3 92.5 .0 185.0 .0 277.5 .1 370.0 .2 462.5 .2 555-0 .3 30 276.4 92.5 .0 185.0 .0 277-4 .1 369-9 .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 •3 554-6 •4 500 92.4 .0 184.8 .0 277-3 .1 3697 .2 462.1 .3 554-5 .4 10 '5^ 92.4 .0 184.8 .1 277.2 .1 369-6 .2 462.0 .3 554-3 •4 20 92.4 .0 184.7 .1 277.1 .1 369-5 .2 461.8 .3 554.2 •4 3° 276.4 923 .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 -3 553-9 • 5 SO 460.7 92-3 .0 184.6 .1 276.9 .1 369-2 .2 461.4 -3 553-7 -5 6 00 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 .1 368.0 368.1 .2 461.2 .4 553-4 .5 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 .1 368.7 .2 460.8 •4 553.0 :l 40 368.6 92.1 .0 184.3 .1 276.4 .1 368.6 .2 460.7 .4 552.8 50 460.7 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 •4 552.1 .6 3° 275.4 368.6 460.7 92.0 .0 184.0 .1 275-9 .1 367.9 -3 459.9 .4 55'.9 .6 40 91.9 .0 'f3-§ .1 275.8 .1 367.8 -3 459-7 .4 55'-6 .6 SO 91.9 .0 183.8 .1 275-7 .1 367.6 -3 4S9-S •5 .551-4 .7 800 91.9 .u ■83.7 .1 275.6 .z 367-S -3 459-4 .5 551-2 .7 Smithsonian Tables. "~~- Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE aooiaa - [Derivation of table explained on pp. liii-lvi.] "S . "■sss CO-ORDINATES OF DEVELOPED PARALLEL FOR— 10' longitude. ao' longitude. 30/ longitude. 40/ longitude. 50^ longitude. 1° longitude. fl X y X y X y X y X y A y mm. mm. tnm. mm. mm. tnm. mm. mm. mm. mm. mm. mm. mm. 8°oo' 10 20 3° 40 50 184.3 276-5 368.6 460.8 91.9 91.8 91.8 91.8 91.7 91.7 .0 .0 .0 .0 .0 .0 183.7 183.6 183.5 183.4 183-3 .1 275.6 275-5 275.4 275.2 275.1 275-0 .2 .2 .2 .2 .2 .2 367-5 367-3 367.2 367.0 366.8 366.7 •3 •3 -3 •3 •3 -3 459-4 459.2 til 458-6 458.4 •5 -5 -5 -5 -5 •5 551-2 551.0 550-7 550-5 550-3 550.0 -7 ■7 -7 ■7 -7 -7 9 oo 10 20 3° 40 SO 92.2 184.3 276.5 368.6 460.8 91.6 91.6 91.5 91.5 9I-S 91.4 .0 .0 .0 .0 .0 .0 183.3 183.2 183.1 183.0 182.9 182.8 .1 274.9 274.8 274.6 274-5 274.4 274.2 .2 .2 .2 .2 .2 .2 366.4 366.2 366.0 365.6 •3 -3 ■3 ■3 •4 -4 458.2 458.0 457-7 457-5 457-3 457.0 ■5 •5 •5 .6 549-8 549-5 549.2 548.5 .8 .8 .8 .8 .8 .8 10 00 10 20 30 40 50 92.2 184.3 276.5 46o!8 91.4 91.3 9«-3 91.2 91.2 91.1 .0 .0 .0 .0 .0 .0 182.7 182.6 182.5 182.4 182.3 182.2 274.1 274.0 273-8 273-7 273-5 273-4 .z .z .2 .2 .2 .2 365.5 365-3 365-1 364-9 364-7 364-5 -4 -4 •4 •4 •4 -4 456.6 456.4 456.1 455-9 455-6 .6 .6 .6 .6 .6 .6 548.2 547.9 547-6 547-3 547.0 546.7 .8 .8 -9 -9 -9 -9 II 00 10 20 30 40 5° 92.2 184.3 276.5 460.8 91.1 91.0 91.0 90.9 90.9 90.8 .0 .0 .0 .0 .0 .0 182.1 182.0 181.9 181.8 181.7 181.6 273.2 273-1 272.9 272.7 272.6 272.4 .z .2 .2 .2 .2 .2 364-3 364.1 363-6 3634 363-2 -4 •4 -4 -4 -4 -4 455-4 455- ■ 454.8 454.6 454-3 454-0 .6 .6 .6 546.4 546.1 545-8 545-5 545-2 544.8 -9 -9 -9 -9 1.0 I.O 12 00 10 20 30 40 SO 184.4 276.5 368.7 460.9 90.8 90.7 90.6 90.6 90.5 90.5 .0 .0 .0 .0 .0 .0 181. 5 181.4 181.3 181. 1 181.0 180.9 272.2 272.1 271.9 271.7 271.6 271.4 .2 .2 .2 •3 -3 -3 363-0 362.8 3625 362.3 362.1 361.8 -4 •4 453-8 453-4 453-2 452-8 452.6 452-3 544-5 544.1 543-8 543-4 543-1 542.8 1.0 1.0 1.0 1.0 1.0 i.i 1300 10 20 30 40 50 92.2 Si 368.8 461.0 90.4 90.3 90.3 90.2 90.2 90.1 .0 .0 .0 .0 .0 .0 180.8 180.7 180.6 180.4 180.3 180.2 271.2 271.0 270.8 270.6 270.4 270.3 •3 .3 .3 .3 •3 •3 361.6 361.4 361.1 360.8 360.6 360.4 452.0 451-7 451-4 451.0 450.8 450.4 -7 .8 .8 .8 542.4 542.0 541.7 541.3 540.9 540.5 i.i i.i 1.1 I.I 1.1 1.1 14 00 10 20 30 40 SO 92.2 184.4 276.6 368.8 461.0 90.0 t9°9 89.8 89.8 89.7 .0 .0 .0 .0 .0 .0 180.1 179.9 179.8 179.7 179.5 179.4 270.1 269.9 269.7 269.5 269.3 269.1 •3 •3 •3 .3 -3 ■3 360.1 359-8 359-6 359-3 359-° 358.8 450.2 449.8 449-5 44S.5 .8 .8 .8 :i .8 540.2 539-8 539-4 539-0 538-6 538-2 i.i 1.2 1.2 1.2 1.2 1.2 15 00 10 20 30 40 SO 92.2 184.4 276.6 368.8 461.0 89.6 89.6 89.5 89.4 89.3 89-3 .0 .0 .0 .0 .0 .0 179.3 179.1 178.8 178.7 178.5 •I 268.9 268.7 268.5 268.3 268.0 267.8 -3 •3 -3 -3 -3 -3 358-5 358-2 358-0 357-7 357-4 357-1 -S '.6 .6 .6 448.2 447-8 447.4 447.1 446.7 446.4 .8 .8 .8 -9 -9 -9 537-8 537-4 536-9 536.5 536.0 S3 5-6 1.2 1.2 1.2 1.2 1-3 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 ^tuVtHT' [Derivation of table explained on pp. liii-lvi.] in CO-ORDINATES OF DEVELOPED PARALLEL FOR — "3 ^i^js 10/ longitude. 20^ longitude. 30' longitude. 40^ longitude. 50/ longitude. 1° longitude. •S3 .2«-o^ 3'ti |is^ la |fssa X y X 1 X mm. X ■ y X y mm. X y mm mm. mm. mtn. mm. mm. inm. mm. mm. mm. mm. le-'oo' 89.2 .0 178.4 267.6 •3 356.8 .6 446.0 -9 535-2 1-3 10 92.2 89.1 .0 178.2 267.4 •3 3S6-S .6 445-6 •9 534-7 '-3 20 184.4 89.0 .0 I78.I .1 267.2 -3 356-2 .6 445.2 -9 534-3 533-8 1-3 3° 276.6 8g.o .0 177.9 266.9 •3 3SS-9 .6 444.8 ■9 1-3 40 368.8 88.9 .0 177.8 266.7 •3 35S-6 .6 444-4 -9 533-3 1-3 SO' 461.0 88.8 .0 177.6 266.5 •3 3SS-3 .6 444.1 •9 532-9 1.4 17 00 88.7 .0 I77-S .2 266.2 •3 3SS-0 .6 443-7 •9 532-4 1-4 10 92.2 88.7 .0 177-3 .2 266.0 •3 354-6 .6 443-3 •9 532.0 1.4 20 184.4 88.6 .0 177.2 .2 265.7 -3 354-3 .6 442.9 I.O 531-5 1-4 30 276.7 88.5 .0 177.0 .2 265.5 •3 354-0 .6 442.5 1.0 531-0 1-4 40 368.9 88.4 .0 176.8 .2 265.2 -4 353-6 .6 442.0 1.0 530-S 1-4 SO 461. 1 88.3 .0 176.7 .2 265.0 -4 353-3 .6 44:. 6 1.0 530.0 1-4 1800 !!-3 .0 176.5 .2 264.8 -4 353-0 .6 441.2 1.0 529-5 1-4 10 92.2 88.2 .0 176.3 .2 264.5 -4 352-6 .6 440.8 1.0 529.0 1.4 20 184.5 88.1 .0 176.2 .2 264.2 -4 352-3 .6 440-4 1.0 528.5 i-S 30 276.7 88.0 .0 176.0 .2 264.0 •4 352-0 .6 440.0 1.0 528.0 i-S 40 368.9 87.9 .0 '75-8 .2 263.7 -4 351-6 .6 439-6 1.0 527-5 i-S SO 461.2 87.8 .0 175.6 .2 263.5 -4 3S'-3 ■7 439-1 1.0 526.9 i-S 19 00 87.7 .0 I7S-S .2 263.2 -4 351-0 -7 438-7 1.0 526.4 i-S 10 92.2 87.6 .0 175-3 .2 263.0 -4 350.6 -7 438.2 1.0 525-9 I-S 20 184.5 87.6 .0 175.1 .2 262.7 -4 350.2 -7 437-8 1.0 525-4 '-5 30 276.7 87.5 .0 174-9 .2 262.4 •4 349-9 •7 437-4 I.I 524.8 1-5 40 369.0 87.4 .0 174.8 .2 262.1 -4 349-5 •7 436-9 I.I 524-3 50 461.2 87-3 .0 174.6 .2 261.9 ■4 349-2 •7 436-4 I.I 523-7 1.6 20 00 87.2 .0 174-4 .2 261.6 •4 348-8 -7 436.0 I.I 523.2 1.6 10 92.2 87.. .0 174.2 .2 261.3 -4 348.4 •7 435-6 I.I 522.7 1.6 20 184.^ 87.0 .0 174.0 .2 261.0 -4 348.0 •7 43S-0 I.I 522.1 1.6 30 276.8 86.9 .0 173-8 .2 260.8 •4 347-7 •7 434-6 I.I 521.5 1.6 40 369.0 86.8 .0 173-7 .2 260.5 •4 347-3 •7 434-2 I.I 521.0 1.6 SO 461.2 86.7 .0 173-S .2 260.2 -4 346-9 -7 433-6 I.I 520.4 1.6 21 00 86.6 .0 173-3 .2 259.9 •4 346.6 -7 433-2 I.I 519.8 1.6 10 92-3 86.5 .0 173-1 .2 259.6 -4 346.2 •7 432-7 I.I 519.2 1.6 20 184.5 86.4 .0 172.9 .2 259-3 -4 345-8 -7 432.2 I.I 518.6 1.6 30 276.8 86.3 .0 172.7 .2 259.0 -4 345-4 -7 43 '-7 1.2 518.0 1-7 40 369.0 86.2 .0 172.5 .2 258.8 -4 345-0 •7 43'-2 1.2 517-5 1-7 SO 461.3 86.1 .0 172-3 .2 258.4 -4 344-6 •7 430.8 1.2 516.9 '-7 22 00 86.0 .0 172.1 .2 258.2 -4 344-2 .7 430.2 1.2 S16.3 1-7 10 92.3' l^-l .0 I7I-9 .2 257.8 -4 343-8 .8 429.8 1.2 515-7 1-7 20 184.5 276.8 85.8 .0 171-7 .2 257.6 •4 343-4 .8 429.2 428.8 1.2 S'S-' 1-7 30 f5i .0 171-5 .2 257.2 -4 343-0 .8 1.2 S14-S S'3-8 1-7 40 369.1 85.6 .0 171-3 .2 256.9 •4 342.6 .8 428.2 1.2 ••7 SO 461.4 85.5 .0 171. 1 .2 256.6 ■4 342-2 .8 427-7 1.2 5 '3-2 1-7 23 00 85.4 .0 170.9 .2 256-3 -4 341.8 .8 427.2 1.2 512.6 1-7 10 92-3 85.3 .0 170.7 .2 256.0 -4 341-3 .8 426.6 1.2 512.0 1.8 20 184.6 85.2 .0 170.4 .2 25S-7 -4 340-9 .8 426.1 1.2 5"-3 1.8 30 276.8 85.1 .0 170.2 .2 25S-3 -4 340.4 .8 425.6 1.2 510.7 1.8 40 369.1 85.0 .0 170.0 .2 25S-0 ■4 340.0 .8 425.0 1.2 510 I 1.8 SO 461.4 84.9 .0 169.8 .2 254-7 -4 339-6 .8 424-5 1.2 509.4 1.8 24 00 84.8 .0 169.6 .2 254.4 •4 339-2 .8 424.0 '-3 508.7 1.8 Smithsonian Tables. 124 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TsWsTnr- [Derivation of table explained on pp. liii-lvi.] •s •S3 lis CO-ORDINATES OF DEVELOPED PARALLEL FOR— 10/ longitude. 2q/ longitude. 30' longitude. 40^ longitude. so' longitude. 1° longitude. 3^ *3 s ^ i si 3- a-"^ X y X tnm. y X y X y X y mm. X y ntm. mm* mtn. mm. mm. mm. mm. ■mm. mm. mm. mm. 24°00' 84.8 .0 169.6 .2 254.4 -4 339-2 .8 424.0 1-3 508.7 1.8 lO 92-3 84.7 .0 169.4 .2 254.0 -5 338-7 .8 4234 1-3 508.1 1.8 20 184.6 84.6 .0 169.I .2 253-7 •5 338-3 .8 422.8 1-3 507-4 1.8 3° 276.9 84-S .0 168.9 .2 253-4 -5 337-8 .8 422.3 1-3 506.8 1.8 40 369.2 84.4 .0 168.7 .2 253.0 -5 337-4 .8 421.8 1-3 506.1 1.8 SO 461.5 84.2 .0 168.5 ,2 252.7 -5 337-0 .8 421.2 1-3 505-4 1-9 25 00 84.1 168.3 .2 252.4 •S 336-5 .8 420.6 1-3 504.8 1-9 10 92-3 S4.0 168.0 ^2 252.0 -5 336-0 .8 420.0 1-3 504.1 1-9 20 184.6 83-9 167.8 .2 251.7 -5 335-6 .8 419.5 1-3 503-4 1-9 30 276.9 83.8 167.6 .2 251-3 •5 335-1 .8 418.9 1-3 502.7 1-9 40 369.2 837 167-3 .2 251.0 -5 334-6 .8 418.3 1-3 502.0 1-9 SO 461.6 83.6 167.I .2 250.6 -5 334-2 .8 417.8 1-3 501-3 1-9 2600 834 166.9 .2 250-3 -5 333-7 -9 417-2 1-3 500.6 1-9 10 92-3 83-3 166.6 .2 249-9 ■5 333-2 -9 416.6 1-3 499-9 1.9 20 184.6 83.2 166.4 ,2 249.6 -5 332-8 -9 416.0 1-3 499.1 1-9 30 277.0 83.1 166.I .2 249.2 -S 332-3 -9 415.4 1-3 498.4 1-9 40 369-3 82.9 165.9 .2 248.8 -5 331-8 -9 414.8 1.4 497-7 2.0 5° 461.6 82.8 165.7 .2 248.5 -5 331-3 -9 414.2 1.4 497.0 2.0 27 00 82.7 .1 165.4 .2 248.1 -5 330-8 -9 413-6 14 496.3 2.0 10 1847 82.6 165.2 .2 247-8 -5 330-4 ■9 415.0 14 495-5 2.0 20 82.S 164.9 .2 247.4 -S 329-8 -9 412-3 1.4 494-8 2.0 30 277.0 82.3 164.7 .2 247-0 ■5 329-4 -9 411.7 14 494.0 2.0 40 3693 82.2 164.4 .2 246.7 ■5 328.9 -9 411. 1 14 493-3 2.0 SO 461.6 82.1 164.2 .2 246-3 -5 328.4 ,-9 410.4 1.4 492.5 2.0 2800 82.0 163.9 .2 245.9 •5 327-9 -9 409.8 14 491.8 2.0 10 92.4 81.8 163-7 .2 245-5 •5 327-4 -9 409.2 1.4 491.0 2.0 20 184.7 81.7 163.4 .2 245-1 -5 326.8 -9 408.6 1.4 490-3 2.0 30 277.0 81.6 163.2 .2 244-7 -5 326.3 -9 407.9 14 a 2.0 40 3694 81.S 162.9 .2 244-4 -5 325-8 -9 407-3 1.4 2.0 SO 461.8 81.3 162.7 _2 244.0 -S 325-3 -9 406.6 1.4 488.0 2.1 2900 81.2 162.4 ^2 243.6 ■5 324.8 -9 406.0 1.4 487.2 2.1 10 92.4 184.7 81. 1 162.1 .2 243.2 -5 324-3 ■9 405-4 14 486.4 2.1 20 80.9 161.9 .2 242.8 -5 323-8 -9 404-7 1.4 4!5-^ 2.1 30 277.1 80.8 i6i.6 .2 242.4 -5 323-2 -9 404.0 14 484.8 2.1 40 3694 80.7 1 61. 3 .2 242.0 -5 322.7 -9 403-4 14 484.0 2.1 SO 461.8 80.S 161. 1 .2 241.6 -5 322.2 -9 402.7 1-5 483.2 2.1 3000 80.4 160.8 .2 241.2 •5 321-6 -9 402.0 1-5 482.5 2.1 10 92.4 80.3 160.5 .2 240.8 -5 321-1 •9 401.4 1-5 481.6 2.1 20 184.8 80.1 160.3 ,2 240.4 -5 320.6 -9 400.7 1-5 480.8 2.1 30 277.1 80.0 160.0 n 240.0 •5 320.0 -9 400.0 1-5 480.0 2.1 40 369s 79-9 159-7 ^2 239.6 -5 3194 -9 399-3 1-5 479-2 2.1 SO 461.9 79-7 IS9-S .2 239.2 -5 318.9 -9 398.6 1-5 478.4 2.1 31 00 79.6 159.2 .2 238.8 -5 318.4 1.0 398-0 1-5 477-5 2.1 10 924 794 158.9 .2 238.4 •5 317-8 I.O 397-2 1-5 476.7 2.1 20 184.8 79-3 158.6 .2 237-9 -5 317.2 I.O 396.6 1-5 475-9 2.2 30 277.2 79.2 158-3 .2 237-5 -5 316-7 1.0 395-8 1-5 475.0 2.2 40 369.6 79.0 158.. .2 237-1 -5 316.1 1.0 395-2 1-5 474.2 2.2 SO 462.0 78.9 157.8 .2 236.7 -5 315-6 I.O 3944 -5 473-3 2.2 3200 78.8 .1 157-5 _2 236.2 •5 31S-0 1.0 393-8 1-5 472-5 2.2 Smithsonian Tables. "5 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^mrW- [Derivation of table explained on pp. liii-lvi.] •s 1^ Meridional dis- tances from even degree parallels. CO-ORDINATES OF DEVELOPED PARALLEL FOR— . It/ longitude. k/ longitude. 30^ longitude. 40^ longitude. so' longitude. I*' longitude. X y X y mm. X y X y X y X y mm- mitt. mm. mm. mm. mm. mm. mm. mm. mm. Tlim. mm. 32°00' 7f? •S7-S .2 236.2 ■5 315-0 I.o 393-8 1-5 472-5 2.2 10 i^-i 78.6 157.2 .2 235.8 -5 314-4 I.O 3930 i-S 471.6 2.2 20 184.8 78.5 156.9 .2 235-4 -S 313-8 1.0 392-3 1-5 470.8 2.2 3° 277.2 78-3 156.6 .2 235.0 •S 3'3-3 1.0 391.6 i-S 469-9 2.2 40 3696 78.2 IS6-3 .2 234-S •s 312-7 1.0 390-8 I -5 469.0 2.2 50 462.0 78.0 156.0 .2 234-1 -5 312.1 1.0 390.1 1-5 468.1 2.2 3300 77-9 155.8 .2 233-6 .6 311-S 1.0 3i§-4 i-S 467-3 2.2 10 92.4 77-7 iSS-S .2 233-2 .6 310.9 1.0 388.6 1-5 466.4 2.2 20 184.8 77.6 155.2 .2 232.7 .6 310.3 1.0 387-9 1-5 465-S 2.2 30 277-3 77-4 154.9 .2 2323 .6 309-7 1.0 387.2 1.6 464.6 2.2 40 369-7 77-3 154.6 .2 231.9 .6 309.2 1.0 386.4 1.6 463-7 2.2 so 462.1 77-1 IS4-3 .2 231-4 .6 308.6 1.0 385-7 1.6 462.8 2.2 3400 7^-S 154.0 -3 231.0 .6 308.0 1.0 384-9 1.6 461.9 2-3 10 92.4 76.8 1S3-7 -3 230.5 .6 307-4 1.0 384.2 1.6 461.0 2-3 20 184.9 76.7 IS3-4 •3 230.0 .6 306.7 1.0 383-4 1.6 460.1 2-3 30 277-3 76.5 IS3-I -3 229.6 .6 306.1 1.0 382.6 1.6 459-2 2-3 40 369-7 76.4 ■ I 152.8 ■3 229.1 .6 30S-S 1.0 381.9 1.6 458-3 2-3 SO 462.1 76.2 152.4 -3 228.7 .6 304-9 1.0 381.1 1.6 457-3 2-3 35 00 76.1 .1 152.1 -3 228.2 .6 304-3 1.0 380-4 1.6 456.4 2-3 10 92.4 7S-9 151.8 -3 227.8 .6 303-7 1.0 379-6 1.6 4SS-S 2-3 20 184.9 75.8 iSi-S -3 227.3 .6 303-0 I.o 378.8 1.6 454.6 2-3 30 277.4 75-6 I5I.2 -3 226.8 .6 302-4 1.0 378.0 1.6 453-6 2-3 40 369.8 7S-4 150.9 -3 226.4. .6 301.8 1.0 377-2 1.6 452-7 2-3 5° 462.2 7S-3 •I 150.6 ■3 225.9 .6 301.2 1.0 376.5 1.6 451.8 2-3 3600 7S-I •So-3 -3 225.4 .6 300.6 1.0 375-7 1.6 450.8 2-3 10 92.5 75-0 150.0 -3 224.9 .6 299-9 1.0 374-9 1.6 449-9 2-3 20 184.9 74.8 149.6 -3 224.5 .6 1.0 374-1 1.6 448.9 2-3 30 277-4 74-7 149-3 •3 224.0 .6 298.6 1.0 373-3 1.6 448.0 2-3 40 369.8 74-S 149.0 -3 223.5 .6 298.0 1.0 372-S 1.6 447-0 2-3 so 462.3 74-3 148.7 •3 223.0 .6 297-4 1.0 371-7 1.6 446.0 2-3 37 00 74-2 148.4 -3 222.5 .6 296.7 1.0 370-9 1.6 445.1 2-3 lO 92.5 74-0 .1 148.0 •3 222.1 .6 296.1 1.0 370.1 1.6 444-1 2-3 20 185.0 73-8 147-7 •3 221.6 .6 295.4 1.0 369.2 1.6 443-1 2-3 30 277-4 73-7 147.4 -3 221. 1 .6 294.8 1.0 1.6 442.1 2.3 40 369-9 73-S .1 147.1 -3 220.6 .6 294.1 1.0 367-6 1.6 441.2 2.4 so 462.4 73-4 146.7 ■3 220.1 .6 293-4 1.0 366.8 1.6 440.2 2.4 3800 73-2 146.4 -3 219.6 .6 292.8 1.0 366.0 1.6 439-2 2-4 10 92-s' 730 1 46. 1 •3 219.1 .6 292.1 1.0 365-' 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 30 277.5 72-7 145.4 •3 2l8.I .6 290.8 I.I g:l 1.6 436-2 2.4 40 370.0 72.5 145.1 •3 217.6 .6 290.1 2S9.4 I.I 1.6 435-2 2.4 SO 462.5 72.4 144.7 ■3 2I7.I .6 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 92.5 72.0 144-0 -3 2I6.I .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-S 71.7 H3-4 ■3 215.0 .6 286.7 I.I 358-4 '-7 430.1 2.4 40 370.0 462.6 71-S •I 143.0 -3 214.5 .6 286.0 I.I 357-5 1-7 429.0 2.4 SO 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 3SS-8 1-7 427.0 2.4 Smithsonian Tables. ^^^ 126 Table 23. \t\J- unuira/' k ■ Ba rwr [ rnu oa\d 1 lun \ir lYlMf- 9. i >OMUI - iiHOOi- [Derivation of table explained on pp. liii-lvi.] idnal dis- i from degree els. CO-ORDINATES OF DEVELOPED PARALLEL FOR — lo' longitude. 20' longitude. 30' longitude. 40' longitude. so' longitude. 1° longitude. p Meridi tance! even i parail 1 t-i X y X y X y X y X y X y mm. mm mm. mm. mm. frtjn. mm. mm. 9nm. mm. tnm. mm. mm. 40°oo' 71.2 142.3 -3 213-5 .6 284.6 l.I 355-8 1-7 427-0 2-4 10 92.5 71.0 142.0 •3 212.9 .6 283.9 1.1 354-9 1-7 425.9 2.4 20 185.1 70.8 .1 141.6 -3 212.4 .6 283.2 I.l 354-0 1-7 424.9 2-4 3° 277.6 70.6 .1 141-3 ■3 211.9 .6 282.6 1.1 353-2 1-7 423.8 2.4 40 370.1 70.5 .1 140.9 ■3 211.4 .6 281.8 1.1 352-3 1-7 422.S 2.4 5° 462.6 70-3 140.6 -3 210.8 .6 281.1 I.I 351-4 1-7 421.7 2.4 41 00 70.1 140.2 ■3 210.3 .6 280.4 l.I 350.6 1-7 420.7 2.4 10 "■ 92.5 69.9 139-9 -3 209.8 .6 279.7 l.I 349-6 1-7 419.6 2.4 20 185.1 69.8 139-S •3 209.2 .6 279.0 1.1 348.8 1-7 418.5 2.4 3° 277.6 69.6 139.2 •3 208.7 .6 278.3 l.I 347-9 1-7 417.5 2-4 40 370.2 69.4 138.8 -3 208.2 .6 277.6 1.1 347-0 1-7 416.4 2.4 5° 462.7 69.2 •I 138.4 •3 207.7 .6 276.9 l.I 346.1 1-7 415.3 2.4 42 00 69.0 138.1 •3 207.1 .6 276.2 1.1- 345-2 1-7 414.2 2.4 10 92.6 68.9 137-7 -3 206.6 .6 2754 I.l 344-3 1-7 413.2 2.4 20 185.1 ^•7 137-4 ■3 206.0 .6 274.7 l.I 343-4 1-7 412.1 2.4 30 277.7 68.5 137-0 -3 205.5 .6 274.0 l.I 342.4 1-7 410.9 2.4 40 370.2 68.3 136.6 -3 204.9 .6 273.2 I.l 341-5 1.7 4°g-9 2.4 5° 462.8 68.1 136-3 •3 204.4 .6 272.5 l.I 340.6 1.7 408.8 2.4 4300 68.0 .1 135-9 -3 203.8 .6 271.8 l.I 339-8 1.7 407.7 2.4 10 '9i.6 ' 67.8 I3S-S •3 203.3 .6 271.0 l.I 338-8 1.7 406.6 2.4 20 . 185.2 67.6 135-2 •3 202.7 .6 270.3 1.1 - 337-9 1-7 405.5 2.4 30 277.7 67.4 134.8 •3 202.2 .6 269.6 l.I 337-0 1.7 404.4 2.4 40 370.3 67.2 134-4 •3 201.6 .6 268.8 1.1 336-0 1-7 403.3 2.4 50 462.9 67.0 134.0 -3 201.1 .6 268.1 1.1 335-1 1-7 402.1 2.4 4400 66.8 133-7 •3 200.5 .6 267.4 l.I 334-2 1-7 401.0 2.4 10 ""92.6" 66.6 133-3 -3 200.0 .6 266.6 1.1 333-2 1-7 399-9 2.4 20 185.2 66.5 132.9 ■3 199.4 .6 265.8 I.l 332-3 1-7 398.8 2.4 30 277.8 66.3 132.6 •3 198.8 .6 265.1 l.I 331-4 1-7 397.7 2.4 40 3704 66.1 132.2 -3 198.3 .6 264.4 I.l 330-4 1-7 396.5 2.4 5° 463.0 65.9 131.8 -3 197-7 .6 263.6 l.I 329-5 1-7 395.4 2.4 4500 65.7 131-4 -3 197.1 .6 262.8 1.1 328.6 1-7 394.3 2.4 10 92.6 65-5 131.0 ■3 196.6 .6 262.1 l.I 327.6 1-7 393.1 2.4 20 185.2 65-3 130.6 •3 196.0 .6 261.3 l.I 326.6 1-7 391.9 2.4 30 277.8 65.1 130-3 •3 195-4 .6 260.5 l.I 325.6 1-7 390.8 2.4 40 370.4 . 64.9 129.9 -3 194.8 .6 259.8 l.I 324.7 1-7 ^si-^ 2.4 so 463.0 64.7 129.5 ■3 194.2 .6 259.0 1.1 323.7 1-7 388.4 2.4 4600 64.6 129.1 •3 193.6 .6 258.2 I.l 322.8 1-7 3tJ-3 2.4 10 "92.6" 64.4 128.7 -3 193-1 .6 257-4 1.1 321.8 1-7 386.2 2.4 20 185.3 64.2 128.3 -3 192.5 .6 256.6 l.I 320.8 1-7 ^l^-l 2-4 3° 277.9 64.0 127.9 •3 191.9 .6 255-9 1.1 319.8 1-7 383-8 2.4 40 370.5 63.8 127.6 -3 191-3 .6 255-1 l.I 318.9 1-7 382.7 2.4 5° 463.1 63.6 127.2 •3 190.7 .6 254-3 I.l 317-9 1-7 381-5 2.4 47 0° 63-4 126.8 -3 190.1 .6 253-5 1.1 316.9 1-7 380.3 2.4 10 92.6 63.2 126.4 •3 189.5 .6 252.7 l.I 315-9 1-7 379-1 2.4 20 185-3 63.0 126.0 -3 188.9 .6 251.9 l.I 3'4.9 1-7 377-9 2.4 3° 277.9 62.8 125.6 ■3 188.3 .6 251.1 1.1 313.9 1-7 376.7 2.4 40 370.6 62.6 125.2 •3 187.8 .6 250.4 l.I 313-0 1-7 375-5 2.4 50 463.2 62.4 124.8 •3 187.2 .6 249.6 l.I 312.0 1-7 374-3 2.4 48 00 62.2 .1 124.4 -3 186.6 .6 248.8 l.I 311.0 1-7 373-1 2.4 Smithsonian Tables. 127 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi,] SCALE ^T^TS' 11 lo 20 30 40 5° 4900 10 20 30 40 5° 50 00 10 20 30 40 50 51 00 10 20 30 40 5° 5200 10 20 30 40 5° S3 00 10 20 30 40 50 5400 10 20 30 40 50 10 20 30 40 50 5600 w S >■ rt 92.7 278.0 370.6 463-3 CO-ORDINATES OF DEVELOPED PARALLEL FOR — ■y longitude. 92.7 185.4 278.0 370.7 4634 92.7 185.4 278.1 370.8 4634 92.7 185.4 278.1 370.8 463.6 92.7 1854 278.2 370-9 463.6 62.2 62.0 61.8 61.6 61.4 61.2 61.0 60.8 60.6 60.4 60.2 60.0 59.8 59-5 59-3 S9.I 58.9 58.7 58-5 58-3 58.1 57-9 57-6 57-4 57-2 57-0 56.8 56.6 56-4 56.2 56.0 92.7 55-7 185.5 278.2 55-5 55-3 371-0 S5-I 463-7 54-9 54-6 92.8 54-4 ^7^:.^ 54-2 54-0 371-0 51.8 463.8 53-6 53-3 92.8 53-1 185.5 278.3 52-9 52-7 371-1 52-4 463.8 52.2 52.0 20' longitude. 124.4 124.0 123.6 123.2 122.8 122.4 122.0 I2I.6 121. 1 120.7 120.3 1 1 9.9 "9-5 119.1 1 18.7 1 18.2 117.8 1 17.4 1 17.0 1 16.6 1 16.2 115.7 "5-3 "4-9 "4-5 1 14.0 "3-6 1 13.2 1 1 2.8 1 1 2.3 1 1 1.9 111. 5 III.O 1 10.6 1 10.2 109.7 109.3 108.9 108.4 108.0 107.5 107. 1 106.7 106.2 105.8 105-3 104.9 104.4 104.0 30^ longitude, mm. mm. 186.6 186.0 1854 184.7 184.I 183-5 182.9 182.3 181.7 181.I 180.5 179.9 179.2 178.6 178.0 1774 176.8 176.1 I75-S 174.9 174.2 173-6 173-0 172-3 I71.7 171. 1 170.4 169.8 169. 168.5 167.9 167.2 166.6 165-9 165.2 164.6 164.0 162.6 162.0 161.3 160.6 160.0 159-3 158.7 158.0 157-3 156.7 156.0 40^ longitude. 248.8 248.0 247.2 246.3 245.5 244.7 243-9 243.1 242.3 241.4 240.6 239-8 239-0 238.2 237-3 236.5 235-7 234-8 234.0 233-2 232-3 231-5 230.6 229.8 228.9 228.1 227.2 226.4 225-5 224.6 223.8 222.9 222.1 221.2 220.3 219-5 218.6 217.7 216.8 216.0 215.1 214.2 213-3 212.4 211.6 210.7 209.8 208.9 208.0 l.I 1.1 I.l l.I I.I 1.1 I.l I.l l.I l.I 1.1 I.l 1.1 l.I 1.1 l.I l.I I.l l.I 1.1 l.I l.I 1.1 I.l I.O I.O 1.0 I.O I.O I.O I.O I.O I.O I.O 1.0 I.O I.O I.O I.O 1.0 I.O I.O I.O 1.0 1.0 I.O I.O I.O 50^ longitude. 311.0 310.0 309.0 307-9 306.9 305-9 304-9 303-9 302.8 301.8 3015.8 299.8 298.8 297.7 2966 295.6 294.6 293.6 292.5 291.4 290.4 289.4 288.2 287.2 286.2 285.1 284.0 283.0 281.9 280.8 279.8 278.6 277.6 276.5 2754 274.4 273.2 272.1 271.0 269.9 268.8 267.7 266.6 265.6 264.4 263.4 262.2 261.1 260.0 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1-7 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1° longitude. 373-1 371-9 370-7 369-5 368-3 367-1 365-9 364-7 363-4 362.2 361.0 359-8 358-5 357-2 356.0 354-7 353-5 352-3 351-0 349-7 348-5 347-2 345-9 344-6 343-4 342-1 340-8 339-5 338-3 337-0 335-7 334-4 333-1 331-8 3305 329-2 3279 326.6 325-3 323-9 322.6 321.3 320.0 3'8.7 317-3 316.0 314.6 3 '3-3 312.0 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2-4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2-3 2-3 2-3 • 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 Smithsonian Tables. I2S Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE WffiyTy [Derivation of table explained on pp. liii-lvi,] naldis- from egree Is. CO-ORDINATES OF DEVELOPED PARALLEL FOR— to' longitude. 20^ longitude. 30' longitude. 40^ longitude. 50^ longitude. 1° longitude. •Si! 11 Meridio tances even d paralle X y X y X y X y X y X y mm. mm. mm. mm. mm. mm. mm. mm. mm. 7«m. 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 '"92.8" Si-8 103.6 .2 ISS-3 .6 207.1 1.0 258.9 1.6 310.7 2-3 20 185.6 27S.4 51.6 103. 1 .2 154.6 .6 206.2 1.0 257.8 1.6 3°9-3 2.2 3° 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 IS3-3 .6 204.4 1.0 255-5 1-5 306.6 2.2 SO 464.0 50.9 I0I.8 .2 152.6 .6 203.5 1.0 254.4 '•5 30S-3 2.2 S7 00 50.6 IOI.3 .2 152.0 .6 202.6 1.0 253.2 i-S 303-9 2.2 lO '"'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 50.2 100.4 .2 150.6 .6 200.8 I.O 251.0 ■•5 301.1 2.2 3° 278.4 50.0 99-9 ^2 149.9 .6 199.8 1.0 249.8 1-5 299.8 2.2 4° 371-2 49-7 99-5 .2 149.2 .6 199.0 1.0 248.7 I-S 298.4 2.2 5° 464.0 49-S 99-0 .2 148.5 •S 198.0 1.0 247.6 I-S 297.1 2.2 S8oo 49-3 98.6 .2 147-8 197.1 1.0 246.4 I-S 295-7 2.2 10 ""'92.8" 49.0 98.1 .2 147.2 196.2 1.0 245.2 1-5 294-3 2.2 20 185.6 278.5 48.8 97.6 .2 146.5 145-8 195-3 1.0 244.1 I-S 292.9 2.2 3° 48.6 97-2 .2 194.4 1.0 243.0 I-S 291.5 2.2 40 371-3 48.4 96.7 .2 145.1 193.4 1.0 241.8 I-S 290.2 2.2 5° 464.1 48.1 96-3 .2 144.4 192.5 1.0 240.6 1-5 288.8 2.1 S9 00 47-9 95.8 .2 143-7 191.6 1.0 239-5 I-S 287.4 2.1 10 92.8 47-7 9S-3 .2 143.0 190.7 1.0 238.4 1-5 286.0 2.1 20 185.7 278.5 47-4 94-9 .2 142.3 189.7 1.0 237.2 ■1-5 284.6 2.1 3° 47.2 94.4 .2 141.6 188.8 1.0 236.0 I-S ^^3-i 2.1 40 371-3 47.0 93-9 .2 140.9 187.9 -9 234.8 I-S 281.8 2.1 5° 464.2 46.7 93-S .2 140.2 186.9 -9 233-6 I-S 280.4 2.1 6ooo 46.5 93-0 .2 \m 186.0 -9 232.5 I-S 279.0 2.1 10 "92.8 ' 46-3 92.5 .2 185.0 -9 231-3 I-S 277.6 2.1 20 185.7 46.0 92.1 .2 138.1 184.1 9 230.2 1-4 276.2 2.1 3° 278.6 45.8 91.6 .2 137-4 183.2 •9 229.0 1-4 274.8 2.1 40 371-4 45.6 91.1 .2 136-7 182.2 -9 227.8 1-4 273-4 2.1 5° 464.2 4S-3 90.6 .2 136.0 181.3 -9 226.6 1-4 271.9 2.1 6i 00 4S.I 90.2 .2 135-3 180.4 ■9 225.4 1-4 270.5 2.1 10 92-9 44.8 89.7 .2 134.6 179.4 -9 224.2 1.4 269.1 2.1 20 278.6 44.6 89.2 .2 133-9 178.5 -9 223.1 1-4 267.7 2.1 3° 44.4 88.8 .2 133-1 I77-S -9 221.9 1.4 266.3 2.0 40 371-4 44-1 88.3 .2 132-4 176.6 -9 220.7 1.4 264.8 2.0 SO 464-3 43-9 87.8 .2 131-7 175.6 -9 219.6 1-4 263.5 2.0 62 00 43-7 87-3 .2 131.0 174-7 -9 218.4 1-4 262.0 2.0 10 92.9 43-4 86.9 .2 130-3 173-7 -9 217.2 1-4 260.6 2.0 20 278.6 43-2 86.4 .2 129.6 172.8 -9 216.0 1.4 259.1 2.0 30 43-0 •I 85.9 .2 128.8 171.8 ■9 214.8 1.4 257-7 2.0 40 37I-S 42.7 .1 85.4 .2 128.1 170.8 -9 213.6 1-4 256.3 2.0 SO 464-4 42.5 84-9 .2 127.4 169.9 •9 212.4 1.4 254.8 2.0 6300 42.2 -^ 84-S .2 126.7 168.9 •9 211.2 1.4 2S3-4 2.0 10 92.9 42.0 84.0 .2 126.0 168.0 -9 210.0 1.4 251.9 2.0 20 185.8 41.7 83-S .2 125.2 167.0 -9 208.8 1.4 250.5 2.0 30 278.7 41-S 83-0 .2 124.5 166.0 -9 207.5 1-3 249.0 1-9 40 371.6 41-3 82.S .2 123.8 165.0 -9 206.3 1-3 247.6 1-9 SO 464.4 41.0 82.0 .2 123.1 164.1 -9 205.1 1-3 246.1 1-9 6400 40.8 .1 81.6 .2 122.3 •5 163.1 •9 203.9 1-3 244-7 1.9 Smithsonian Tables. 129 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE its^tsts- [Derivation o£ table explained on pp. liii.-lviii.] ■S3 Lonaldis- s from degree [els. CO-ORDINATES OF DEVELOPED PARALLEL FOR — to' longitude. zof longitude iof longitude. 40/ longitude. so' longitude. 1° longitude. ■S a nil JJft g 5 u p. X y X y X y X y JL y X y mtft' mm. mm. mm. mm. mm. mm. ?«7«. mm. WW. mm. mm. tnm. 64°oo' 40.8 81.6 .2 122.3 •5 163. 1 i 203.9 1-3 ZiA.1 1-9 10 92.9 40.5 81. 1 .2 I2I.6 -5 162.2 .8 202.7 1-3 243.2 1.9 20 185.8 278.7 40-3 80.6 .2 120.9 •5 I6I.2 .8 201.4 1-3 241.7 1.9 30 40.0 80.1 .2 120.1 -S 160.2 .8 200.2 1-3 240.2 1.9 40 371-6 39-8 79.6 .2 "9-4 -5 159.2 .8 199.0 1-3 238.8 1-9 SO 464.5 39-6 79.1 .2 1 18.7 -5 158.2 .8 197.8 1-3 237-4 1-9 65 00 39-3. 78.6 .2 1 17.9 -5 157.2 .8 196.6 1-3 235-9 1-9 10 92.9 39-i 78.1 .2 117.2 •5 156.2 .8 195-3 1-3 234-4 1.9 20 185.8 278.7 3f-? 77.6 .2 116.5 -5 I5S-3 .8 I94.I 1-3 232.9 1.8 30 38.6 77.2 .2 1 1 5.7 •5 154-3 .8 192.9 1-3 231-5 1.8 40 371-6 38-3 76.7 .2 H5.0 -5 "53-3 .8 191.6 1-3 230.0 1.8 50 464.6 38-1 76.2 .2 1 14.2 -5 152.3 .8 190.4 1-3 228.5 1.8 6600 37-8 .1 75-7 .2 II3-S 1 1 2.8 -5 151.4 .8 189.2 1-3 227.0 1.8 10 "92-V 37-6 .0 75.2 .2 ■4 150.4 .8 188.0 1-3 225-5 1.8 20 lUt 37-3 .0 74-7 .2 11 2.0 -4 149.4 .8 186.7 1.2 224.0 1.8 30 ^li .0 74-2 .2 111.3 •4 148.4 .8 185.4 1.2 222.5 1.8 40 371-7 36.8 .0 73-7 .2 H0.6 -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 02.9 36.1 .0 72.2 .2 108.3 -4 144.4 .8 180.S 1.2 «6.6 1.7 20 3S-8 .0 71-7 .2 '°^S •4 «43.4 .8 179.2 1.2 215.1 1.7 30 27^.8 35-6 .0 71.2 .2 106.8 -4 142.4 .8 178.0 1.2 213.6 1.7 40 371-8 3S-4 .0 70.7 .2 io6jd -4 141.4 .8 176.8 1.2 212.1 1-7 50 464.7 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 '39-4 .8 174.2 1 2 209.1 1-7 10 93-0 34-6 .0 69.2 .2 103.8 -4 138.4 •7 173-0 1.2 207.6 1-7 20 278.8 34-4 .0 ^■^ .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 1.1 204.5 1-7 40 Vo 33-8 .0 67.7 .2 101.5 •4 135.4 .7 169.2 168.0 1.1 203.0 1-7 50 464.8 33-6 .0 67.2 .2. 100.8 •4 134.4 -7 I.I 201.5 1.6 69 00 33-3 .0 66.7 .2 1 00.0 •4 133-4 •7 166.7 1.1 200.0 1.6 10 93-0 33-1 .0 66.2 .2 99-3 •4 132.4 •7 165.4 i.i 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 1.1 195.5 1.6 40 ^I'i 32.3 .0 64-7 .2 97.0 .4 129.3 .7 161.6 1.1 194.0 1.6 5° 464.8 32.1 .0 64.1 .i 96.2 •4 128.3 •7 160.4 1.1 192.4 1.6 70 00 3,.8 .0 63.6 .2 95-5 -4 127.3 ■7 159.1 1.1 190.9 1.6 10 93-0 185.9 278.9 31.6 .0 Pi .2 94-7 -4 126.2 •7 157.8 1.1 189.4 1.6 20 3'-3 .0 62.6 .2 93-9 -4 125.2 .7 156.6 1.1 187.9 1.6 30 3'-J .0 62.1 .2 93-2 •4 124.2 .7 155-3 1.1 186.4 1.5 40 371-9 30.8 .0 61.6 .2 92-4 -4 123.2 •7 154.0 1.1 184.8 1-5 5° 464.9 30-5 .0 61.1 .2 91.6 •4 122.2 -7 152.7 1.0 183.2 1-5 71 00 30-3 .0 60.6 .2 90.9 -4 121.2 .7 151.4 I.O 181.7 1-5 10 93.0 30.0 .0 60.1 .2 -4 120.2 .7 150.2 1.0 180.2 1.5 20 186.0 278.9 29.8 .0 59.6 .2 89.3 •4 1 19. 1 .7 148.9 1.0 178.7 1-5 30 29.5 .0 59-0 .2 88.6 -4 ii8.i .7 147.6 1.0 177.1 1-5 1-5 1.4 40 371-9 29-3 .0 58.S .2 87.8 -4 117.1 .6 146.4 1.0 175-6 174.1 5° 464.9 29.0 .0 58.0 .£. 87.1 •4 116.1 .6 145.1 1.0 72 00 28.8 .u 57-5 .2 86.3 •4 1 1 5.0 .6 143-8 1.0 172.6 1-4 Smithsonian Tables. ■^~" — ^ 130 Table 23. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^TiiW- [Derivation of table explained on pp. liii-lvi.] 11 dional dis- es from 1 degree Uels. CO-ORDINATES OF DEVELOPED PARALLEL FOR — 10^ longitude. 20' longitude. 30^ longitude. 40^ longitude. so' longitude. I** longitude. .s 2 ■C S B E r- iSsk X y X y X y X y X y X y mfft. mm. mm. mm. mm. mm. mm. mm. mm. tnm. m7n. Tnm. mvt. 72°oo' 28.8 .0 S7-S .2 86.3 ■4 1 1 5.0 .6 143-8 1.0 \T2..i, 1-4 10 93.0 28.5 .0 57.0 .2 85.S ■4 1 1 4.0 .6 142.5 I.O 171.0 1-4 20 186.0 28.2 .0 56.5 .2 84.7 •3 113-0 .6 141.2 1.0 169.4 1.4 3° 279.0 28.0 .0 56.0 .2 83-9 •3 1 1 1.9 .6 139-9 1.0 167.9 1.4 40 372.0 27.7 .0 SS-S .2 83.2 •3 1 10.9 .6 138.6 1.0 166.4 1-4 SO 465.0 27.5 .0 54.9 .2 82.4 •3 109.9 .6 '37-4 1.0 164.8 1.4 7300 27.2 .0 54-4 .2 81.6 •3 108.8 .6 136.0 -9 163-3 1-4 10 93-0 27.0 .0 S3-9 .1 80.8 •3 107.8 .6 134.8 ■9 161.7 1-4 20 186.0 26.7 .0 53-4 .1 80.1 •3 106.8 .6 133-4 -9 160.1 1-3 3° 279.0 26.4 .0 52.9 .1 79-3 •3 105.7 .6 132.2 ■9 158.6 1-3 40 .372.0 26.2 .0 52.3 .1 78.5 •3 104.7 .6 130.8 -9 157.0 1-3 5° 465.0 25.9 .0 51.8 .1 77-7 ■3 103.6 .6 129.6 -9 IS5-5 1-3 7400 25.6 .0 5'-3 .1 77.0 •3 102.6 .6 128.2 -9 153-9 1-3 10 93-0 25.4 .0 50.8 .1 76.2 •3 101.6 .6 127.0 -9 152-3 1-3 20 186.0 25.1 .0 S°-3 .1 75-4 ■3 100.5 .6 125.6 ■9 150.8 1-3 30 279.0 24.9 .0 49-7 .1 74.6 ■3 99- S .6 124.4 -9 149.2 1-3 40 372.0 24.6 .0 49-2 .1 73-8 •3 98.4 .6 123.0 -9 147-7 1.2 SO 465.0 24.4 .0 48.7 .1 73-° •3 97-4 -S 121.8 -9 146.1 1.2 75 00 24.1 .0 48.2 .1 72.3 •3 96.4 -5 120.4 .8 144.5 1.2 10 93.0 23.8 .0 47-7 .1 7I-S •3 95-3 ■5 119.2 .8 143.0 1.2 20 186.0 23.6 .0 47.1 .1 70.7 •3 94-2 -5 117.8 .8 141.4 1.2 30 279.1 23-3 .0 46.6 .1 69.9 •3 93-2 -5 116.5 .8 139.8 1.2 40 372.1 23.0 .0 46.1 .1 6g.i •3 92.2 -5 115.2 .8 138.2 1.2 SO 465.1 22.8 .0 4S-S .1 68.3 •3 91.1 -5 113.8 .8 136.6 i.i 7600 22.5 .0 45.0 .1 66!8 •3 90.0 •S 112.6 .8 135-1 1.1 10 930 22.2 .0 44-S .1 •3 89.0 •s 111.2 .8 133-S i.i 20 186.1 22.0 .0 44.0 .1 65.9 •3 87-9 -s 109.9 .8 131-9 i.i 30 279.1 21.7 .0 43-4 .1 65.2 ■3 86.9 -5 108.6 .8 130-3 1.1 40 3721 21.5 .0 42.9 .1 64.4 •3 -5 107.3 .8 128.8 1.1 SO 465.1 21.2 .0 42.4 .1 63.6 ■3 84.8 -S io5.o -7 127.1 1.1 7700 20.9 .0 41.9 .1 62.8 •3 83-7 •5 104.6 -7 125.6 1.1 10 93-0 20.7 .0 41-3 .1 62.0 •3 82.7 -S 103.4 -7 124.0 1.1 20 1 86. 1 20.4 .0 40.8 .1 61.2 •3 81.6 -5 102.0 -7 122.4 1.0 30 279.1 20.1 .0 40-3 39-8 .1 60.4 •3 80.6 -5 100.7 -7 120.8 1.0 40 372.2 19.9 .0 .1 59.6 •3 79-5 •4 99.4 -7 "9-3 1.0 SO 465.2 19.6 .0 39-2 .1 58.8 •3 78.4 -4 98.0 -7 117.7 1.0 7800 19.4 .0 38.7 .1 58.0 .2 77-4 -4 96.8 •7 116.1 1.0 10 93-0 I9.I .0 38.2 .1 57-2 .2 76.3 -4 95-4 •7 114.5 1.0 20 186.1 18.8 .0 37-6 .1 56- S .2 7S-3 -4 94.1 -7 1 1 2.9 1.0 30 279.1 18.6 .0 37-1 .1 SS-7 .2 74.2 -4 92.8 -7 111.4 1.0 40 372.2 18.3 .0 36.6 .1 S4-9 .2 73-2 •4 91.4 .6 109.7 -9 SO 465-2 i8.o .0 36.0 .1 54.1 .2 72.1 -4 90.1 .6 108.1 •9 7900 17.8 .0 3S-S .1 S3-3 .2 71.0 -4 88.8 .6 106.6 -9 10 93-0 I7-S .0 3S-0 .1 52.5 .2 l°^° -4 87.4 .6 104.9 -9 20 1 86. 1 17.2 .0 34-5 .1 S'-7 .2 68.9 -4 86.2 .6 103.4 -9 30 279.2 17.0 .0 33-9 .1 50.9 .2 67.8 •4 84.8 .6 101.8 ■I 40 372.2 16.7 .0 334 .1 50.1 .2 66.8 -4 83-4 .6 • lOO.I so 465.2 16.4 .0 32.9 .1 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 sTshT!- [Derivation o£ table explained on pp. liii-lvi.] 0"00' 10 20 30 40 SO 1 00 10 20 30 40 50 2 00 10 20 30 40 SO 300 10 20 30 40 SO 400 10 20 30 40 SO 500 10 20 30 40 SO 600 10 20 30 40 SO 7 00 10 20 30 40 SO 800 230.4 460.7 691.0 921.4 1151.8 230.4 460.7 691.0 921.4 1151.8 230.4 460.7 691.0 921.4 1151.8 230.4 460.7 691. 1 921.4 1151.8 230.4 460.7 691. 1 921.4 :i5i.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 92.1.5 1151.9 ABSCISSAS OF DEVELOPED PARALLEL. 5' longitude. 16.0 16.0 16.0 16.0 16.0 15.9 IS-9 15.9 IS-9 15.9 IS-9 15.9 IS9 15.9 IS-9 .5.8 ,5.8 15.8 15.8 15-8 15.8 1 5-7 iS-7 15-7 iS-7 15.6 15.6 15.6 15.6 ■s-s iS-S ■S-S 15.4 1 5-4 15-4 15-3 iS-3 15.2 15.2 15-2 1 5.1 15-1 1 5-1 15.0 15.0 14.9 14.9 1 14.8 10 longitude. 231.9 231.9 231.9 231.9 231.9 231.9 231.9 231.9 231.8 231.8 231.8 231.8 231.8 231-8 231.7 231-7 231.7 231.6 231.6 231.6 231.S 231.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 347-5 347.5 347-4 347-3 347-3 347-2 347-2 347-1 231.4 347-0 231-3 347.0 231-3 346-9 231.2 346.8 231. 1 346.7 23I.I 346.6 231.0 346.6 231.0 346.5 230.9 346.4 230.8 346.3 230.8 346.2 230.7 346.1 230.7 346.0 230.6 345-9 230.5 345-8 230.4 345-7 230.4 345- S 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. 463-8 463-8 463.8 463.8 463.8 463.8 463-8 463-7 463.6 463.6 463.6 463.6 463.5 463.4 463.4 4633 463-3 463.2 463.1 463-0 463.0 462.9 462.8 462.7 462.6 462.5 462.4 462.3 462.2 462.1 462.0 461.8 461.7 461.6 461.4 461.3 461.2 461.0 460.9 460.7 460.6 460.4 460.2 460.0 459-9 459-7 459-5 459-4 25' longitude. 579-8 579-8 579-8 579-8 579-8 579-7 579-7 579-6 579-6 579-6 579.6 579-5 579-4 579-4 579-3 579.2 579-2 579- 1 579.0 578.8 578.7 578.6 578.5 578.4 578.2 578.2 578.0 577-8 577.8 577-6 577.4 577-3 577.1 577.0 576.8 576.6 576.4 576.2 576.1 575-9 575-7 57 S- 5 575-3 57S-0 574.8 574.6 574-4 574-2 30 longitude. ORDINATES OF DEVELOPED PARALLEL. 695.8 695.8 695-7 695.7 695-6 695.6 69S-S 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 •g>s mm. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 O.I O.I 0.0 0.0 O.I O.I O.I 0.2 6° 0.0 0.0 O.I O.I 0.2 0.3 0.0 0.0 O.I 0.2 0-3 0.4 mvt, 0.0 0.0 0.0 0.0 0.0 O.I 0.0 0.0 0.0 O.I O.I 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. 132 Table 24. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TirW- [Derivation of table explained on pp. liii-lvi.] sasa ABSCISSAS OF DEVELOPED PARALLEL. S' longitude. 10' longitude. longitude. 20' longitude. 25' longitude. 30' longitude. ORDINATES OF DEVELOPED PARALLEL. 8°00' 10 20 3° 40 SO goo lO 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 1500 10 20 30 40 SO 1600 230.4 460.8 691.2 921.6 1152.0 230.4 460.8 691.2 921.6 1 1 52.0 230.4 460.8 691.3 921.7 II 52.1 230.4 460.9 691.3 921.8 1 1 52.2 230.4 460.9 691.2 921.8 1152.2 230.5 460.9 691.4 921.9 1152.4 230.5 461.0 691.5 922.0 1152.4 230.5 461.0 691.5 922.0 1 152.6 14.8 14.8 147 14.7 14.6 14.6 14.5 I4-S 14.4 14.4 14-3 14-3 14.2 14.2 14.1 14.0 14.0 139 13.8 13-8 13.6 13.6 I3-S '3-4 134 13-3 13.2 13.2 I3-I 13.0 12.9 12.8 12.8 12.7 12.6 I2-S 12.5 12.4 12.3 12.2 I2.I 12.0 1 1.9 1 1.8 1 1.8 II.7 1 1.6 111.5 229.7 229.6 229.5 229.4 229.3 229.2 229.1 229.0 228.9 228.7 228.6 228.5 228.4 228.3 228.2 228.0 227.9 227.8 227.7 227.5 227.4 227.3 227.1 227.0 226.9 226.7 226.6 226.4 226.3 226.2 226.0 225.9 225.7 225.6 225.4 225.2 225.1 224.9 224.7 224.6 224.4 224.2 224.1 223.9 223.7 223.5 223.3 223.2 344- S 344-4 344-2 344-1 343-9 343-8 343-6 343-4 343-3 343-1 3430 342.8 342.6 342-4 342-3 342-1 341-9 341-7 341-5 341-3 341 -I 340-9 340.7 340-5 340.3 340-1 339-9 339-7 339-4 339-2 339-0 338-8 338-6 338-3 338-1 337-9 337-6 337-4 337-1 336-6 336-4 336-1 335-8 335-6 335-3 335-0 334-7 223-0 334-5 459-4 459-2 459-0 458.8 458.6 458.4 458.2 457-9 457-7 457-5 457-3 457.0 456.8 456.6 456.4 456.1 455-8 455-6 4SS-4 45S-I 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-S 448.1 447-8 447-4 447.0 446.7 446.3 446.0 574-2 574.0 573-7 573-4 573-2 573-0 572.7 572.4 572-2 571.8 571-6 571-3 571.0 570-8 570.4 570.1 569.8 569.5 569.2 568.8 568.6 568.2 567-6 567-2 566.8 566.1 565.8 565.4 564.6 564.2 563-9 563-S 563-1 562.7 562-3 561.8 561.4 561.0 560.6 560.2 559-7 559-2 558-8 558-4 557-9 557-4 689.0 688.7 688.4 688.1 687.8 687.5 687.2 686.9 686.6 686.2 685.9 685.6 685-3 684.9 684.5 684.1 683.8 683.4 683.0 682.6 682.3 681.8 681.4 681. 1 68D.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 ^■2 8° 0.0 0.0 0.1 0.2 03 0.4 0.0 0.1 0.1 0.2 0.4 0-5 12° 0.0 0.1 0.2 0.3 0.4 0.6 14' 5 0.0 10 0.1 15 0.2 20 0.3 25 0.5 30 0.7 16° 0.0 O.I 0.2 0.4 0.6 0.8 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 0.1 0.2 0-3 0.5 0.7 15° 0.0 O.I 0.2 0.3 0.5 0.8 Smithsonian Tables, 133 Table 24. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ^irW- [Derivation of table explained on pp. liii-lvi.] r 10 20 3° 40 SO 17 00 10 20 30 40 SO iS 00 10 20 30 40 SO ig 00 10 20 30 40 5° 20 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 .2 »>'d2 230.5 461.1 691.6 922.1 1152.6 ABSCISSAS OF DEVELOPED PARALLEL. S' longitude. 230.6 461.1 691.6 922.2 1152.8 230.6 461.1 691.7 922.3 1152.8 230.6 461.2 691.8 922.4 1153.0 230.6 461.2 691.9 922.5 "S3-1 230.6 692.0 922.6 1153.2 230.7 461.4 692.0 922.7 '1 534 230.7 461.4 692.1 922.8 "53-6 111.5 111.4 111.3 111.2 iii.i III.O 110.9 110.8 H0.7 110.6 110.5 1 10.4 110.3 110.2 IIO.l II 0.0 109.9 109.8 109.7 109,6 109.5 109.4 109.2 1 09. 1 109.0 108.9 108.8 108.7 108.5 108.4 108.3 108.2 108.1 107.9 107.8 107.7 107.6 107.4 107-3 107.2 107.1 106.9 106.8 106.7 106.5 106.4 106.3 106.1 106.0 longitude. 223.0 222.8 222.6 222.4 222.2 222.0 221.8 221.6 221.4 221.2 221.0 220.8 220.6 220.4 220.2 220.0 219.8 219.6 219.4 2ig.i 218.9 218.7 218.5 218.2 218.0 217.8 217.5 217-3 217.1 216.8 216.6 216.4 216.1 215.9 215.6 215.4 215.1 214.9 214.6 214.4 214.1 213.9 213.6 213-3 213.1 212.8 212.5 212.3 IS' longitude. 334-S 334-2 333-9 333-6 333-3 333-1 332-8 332- S 332-2 331-9 331-6 331-3 33'-o 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 32S-3 324-9 324-S 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 20' longitude. 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-S 436.0 43S-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 25' longitude. SS7-4 SS7-0 5S6-5 556.0 SSS-6 SSS-i 554.6 SS4-I SS3-6 S53-I 552.6 SS2-1 SSi-6 551.0 550.6 550.0 S49-4 S49-0 S48.4 S47-8 546.8 546.1 S4S-6 S4S-0 S44-4 S43-8 543-3 542.7 542.1 S4I-S 540.9 540-3 539-6 539-0 S3»-4 537-8 537-2 536.6 536.0 535-3 534-6 534-0 533-3 532-6 532.0 53 '-3 530.6 530-0 30' longitude. 668.9 668.3 667.8 667.2 666.7 666.1 664.9 663.7 663.1 662.5 661.9 661.3 660.7 660.0 659-3 658.7 658.1 656.8 656.1 6SS-4 654.7 654.1 till 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 ORDINATES OF DEVELOPED PARALLEL. .2| 16° 0.0 0.1 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.4 0.6 0.9 0.0 0.1 0.2 0.4 0.7 1.0 5 0.0 10 0.1 15 0-3 20 0.5 25 0.7 30 1.1 24" 0.0 O.I 0-3 li i.i 17° mm. 0.0 0.1 0.2 0.4 0.6 0.8 19" 0.0 O.l 0.2 0.4 0.6 0.9 0.0 0.1 0-3 0.5 0.7 I.O 23° 0.0 O.I 0-3 0.5 0.8 i.i Smithsonian Tables. 134 CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] Table 24, SCALE iTsW- i ABSCISSAS OF DEVELOPED PARALLEL. ■s . 1.g»^ ORDINATES OF I§ Meridio tancea even di paralle! S' 10' 'S' 20' 25' 30' DEVELOPED PARALLEL. s°- longitude. longitude. longitude. longitude. longitude. longitude. mm. mm. mm. mm.' mm. mm. mm. -o-i p n n 24°00' 106.0 212.0 318.0 424.0 530.0 636.0 24° 25° 10 230.7 105.9 211.7 317.6 423-4 529-3 635-2 hJ'" 20 461.S 105.7 211.4 317-2 422.9 528.6 634-3 3° 692.2 105.6 211.2 316.7 422.3 527-9 633-5 mtn. mm. 40 923.0 105.4 210.9 316-3 421.8 527-2 632.6 5' 10 0.0 0.0 50 "S37 ioS-3 210.6 31S-9 421.2 526.5 631.8 0.1 0.1 25 00 105.2 210.3 31S-S 420.6 525-8 631.0 '5 20 0-3 0-3 o.e, 10 230.8 105.0 210.0 3IS-0 420.0 525.0 630.1 25 30 ok 20 461.S 104.9 209.7 314.6 41 9- 5 524.4 629.2 i.i 1.2 30 692.3 104.7 209.4 314.2 418.9 523.6 628.3 40 SO 923.1 1153.8 104.6 104.4 . 209.2 208.9 313-7 418.3 522.9 627.5 626.6 3^3-3 417-7 522.2 2600 10 t1 104.3 1 04. 1 208.6 208.3 312.9 312.4 417.2 416.6 521.4 520.7 625.7 624.8 26° 27° 20 461.6 104.0 208.0 312.0 416.0 520.0 623.9 5 0.0 0.0 30 692.4 103.8 207.7 311-S 415.4 519.2 623.0 10 O.I 0.1 40 923.2 103.7 207.4 311.1 414-8 518.4 622.1 15 0-3 0-3 SO 1154.0 103-S 207.1 310.6 414.2 S17-7 621.2 20 0.5 0.8 0.5 25 0.8 27 00 103.4 206.8 310.2 413.6 517-0 620.3 30 1.2 1.2 10 230.8 103.2 206.5 3°9-7 413.0 516.2 619.4 20 30 461.7 1 03. 1 102.9 206.2 205.8 1^1 412.3 411.7 515-4 514.6 618.5 617.5 40 923-3 102.8 205.5 308.3 411. 1 513-8 616.6 28° 29° 50 1 1 54.2 102.6 205.2 307-9 410.5 513-1 615.7 28 00 102.5 204.9 307-4 409.8 512.3 614.8 5 0.0 0.0 10 230.9 102.3 204.6 306.9 409.2 S"-5 613.8 10 0.1 0.1 20 461.7 1 02. 1 204.3 306.4 408.6 510.7 612.8 15 0-3 0.6 0.3 0.6 30 102.0 204.0 305-9 407.9 509-9 611.9 20 40 923-S 101.8 203.6 305-5 407-3 509.1 610.9 25 0.9 0.9 SO 1154.4 101.7 203-3 305.0 406.6 S08.3 6ro.o 30 1-3 1-3 2900 101.5 203.0 304-5 406.0 507-5 609.0 10 230.9 101.3 202.7 304.0 405-4 506.7 608.0 30° 3t° 20 461.8 692.7 101.2 202.3 303-S 404.7 505.8 607.0 606.0 30 101.0 202.0 303-0 404.0 505.0 40 923.6 100.8 201.7 302.5 403-4 504.2 605.0 5 0.0 0.0 SO "S4-S 100.7 201.4 302.0 402.7 503-4 604.1 10 0.1 0.1 3000 100.5 201.0 301-S 402.0 502.6 603.1 15 20 0-3 o.a 0-3 0.6 10 230.9 100.3 200.7 ^ 301.0 401.4 501.7 602.0 25 0.9 0.9 20 461.9 100.2 200.3 300.5 400.7 500.8 601.0 30 1-3 1-3 30 692.8 100.0 200.0 300.0 400.0 500.0 599-9 40 SO 923.8 1 1 54.7 99.8 99-6 199.6 199-3 299.5 299.0 iii 499.1 498.2 598.9 597-9 --0 3100 10 231.0 99S 99-3 199.0 198.6 298.4 297.9 397-9 397-2 497-4 496.5 596-9 595.8 32 20 461.9 99.1 198.3 297.4 396-S 495-6 594-8 5 0.0 30 692.9 197.9 296.9 395-8 494.8 593-8 10 0.2 40 923-9 98.8 197.6 296.3 395-1 493-9 592-7 IS 0.3 SO 1 154.8 98.6 197.2 295.8 394-4 493-0 591.6 20 25 0.6 0.9 3200 98.4 196.9 29S-3 393-7 492-2 590.6 30 1-4 Smithsonian Tables. ^ZS Table 24. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE sD^i)- [Derivation of table explained on pp. liii-lvi.] •g ional dis- 5 from degree lels. ABSCISSAS OF DEVELOPED PARALLEL. Oil r-o r»w •S2 DEVELOPED S' 10' IS' 20' 2S' 30' PARALLEL. 2^ ^JSSo. longitude. longitude. longitude . longitude. longitude- longitude. mm. fntft. mm. mm. mtfi. mm. mm. it 32°00' 98.4 196.9 29S-3 393-7 492.2 g ■II 32° 33" 10 231.0 462.0 98.2 98.1 196.5 1 96. 1 294.8 393-0 491.2 490-4 1-1 ■" 20 294.2 392-3 3° 693.0 97-9 195.8 293-7 391.6 489-4 kp mm. mm. 40 924.0 97-7 195.4 293.1 390-8 488.6 586.3 5' 10 0.0 0.0 SO 1155.0 97-s 195.I 292.6 390.1 487.6 585.2 0.2 0.2 33 00 97-4 194.7 292.1 389-4 486.8 584.1 '5 20 ol 0.6 10 231.0 97.2 «94-3 291.5 388.6 485.8 583-0 25 30 0.9 1-4 1.0 20 462.1 97.0 194.0 290.9 387-9 484.9 ^^'■i 1.4 30 693.2 96.8 193.6 387-2 484.0 580.8 40 SO 924.2 U55.2 96.6 96.4 193.2 192.8 2^918 289.3 386.4 385-7 483.0 482.1 S79-7 578.5 3400 96.2 192.5 288.7 385.0 481.2 S77-4 34° 35° 10 231.I 96.0 192.1 288.2 384-2 480.2 576-3 20 462.2 9S-9 191.7 287.6 383-4 479-3 575-2 s 0.0 0.0 30 693.2 95-7 1 91 -3 287.0 382.6 478-3 574.0 10 0.2 0.2 40 924-3 9S-5 190.9 286.4 381.9 477-4 572.8 IS 0.4 0.4 SO 1155.4 9S-3 190.6 285.8 381. 1 476-4 571-7 20 25 0.6 I.O 0.6 1.0 3500 95.1 190.2 285.3 380.4 475-4 S70-S 30 1-4 1-4 10 231.I 94.9 189.8 284.7 379-6 474-S 569.4 20 462.2 94-7 189.4 284.1 378.8 473-5 568.2 30 6934 94-5 189.0 283-S 378.0 472.5 567-0 40 924.5 94-3 188.6 282.9 377-2 471.6 565-9 36° 37" SO 3600 1 1 55.6 94-1 93-9 i88.2 187.8 282.4 281.8 376.S 37S-7 470.6 564.7 563-5 S 0.0 0.0 10 231.2 93-7 187.4 281.2 374-9 562.3 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 30 693-5 924.6 93-3 186.6 280.0 373-3 466.6 559-9 20 0.6 0.6 40 93-1 186.2 279-4 372-5 465.6 SS»-7 25 1.0 1.0 SO 1 1 55.8 92.9 185.8 278.8 371-7 464.6 S57-5 30 1-4 I-S 37 00 , ... 92.7 185.4 278.2 370-9 370-1 462.6 556-3 SSS-i 10 231.2 92.5 185.0 277.6 38° — «0 20 462.4 92-3 184.6 276.9 369-2 461.6 553-9 39 30 40 693-6 924.8 92.1 91.9 184.2 183.8 276.3 275-7 367:6 460.5 4S9-S 552.6 551-4 s 10 0.0 0.2 0.0 SO 1 1 56.0 91.7 183.4 275.1 366.8 458-5 550.2 0.2 3800 91.5 183.0 274.C 366.0 457-4 548-9 15 20 0.4 0.7 1.0 0.4 0-7 1.0 10 231.2 462.5 91-3 182.6 273-8 365-1 456-4 547-7 25 30 20 91.1 182.1 273.2 364-3 455-4 546.4 i-S I-S 30 693-7 90-9 181.7 272.6 363-S 454-4 545-2 40 SO 925.0 1 1 56.2 90.7 90-4 181.3 180.9 272.0 362-6 361.8 453-3 544.0 271.4 452.2 542.7 3900 90.2 180.5 180.1 270.7 361.0 451.2 541-4 40° 10 23'-3 90.0 270.1 360.1 450.2 540.2 20 462.6 89.8 179-6 269.4 268.8 3S9-2 449.0 538.9 s 0.0 30 693.8 89.6 '^g-o 358.4 448.0 537-6 10 0.2 40 925.1 1 1 56.4 89.4 178.8 268.2 357-6 447-0 S36.3 15 0-4 SO 89.2 178.3 267.5 356-7 445.8 535-0 20 0.7 25 1.0 4000 89.0 177-9 266.9 355-8 444.8 533-8 30 i-S SMITHSOr SCALE iihi - 3^ 40"oo' 10 20 30 40 50 41 00 10 20 30 40 SO 42 00 10 20 30 40 so 4300 10 20 30 40 so 4400 10 20 30 40 so 4500 10 20 30 40 so 46 00 10 20 30 40 so 4700 10 20 30 40 SO 4800 ■Sag 231-3 462.6 694.0 1156.6 231.4 462.7 694.1 925.4 1156.8 231.4 462.8 694.2 925.6 1 1 57.0 231.4 462.9 694-3 925.8 1157.2 231-S 463.0 694.4 925.9 1157.4 23I-S 463.1 694.6 926.1 1 1 57.6 231.6 463.1 694.7 926.3 1 1 57.8 231.6 463.2 694.8 926.4 1 1 58.0 ABSCISSAS OF DEVELOPED PARALLEL. 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,-S 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 17S-7 I7S-3 174-8 174-4 173-9 173-S 173-0 172.6 172.1 171.7 171.2 170.8 170.3 169.9 169.4 169.0 168.5 168.0 167.6 167.1 166.6 166.2 165.7 165.2 164.7 164.3 i63.§ 163.3 162.8 162.3 161.9 161.4 160.9 160.4 159.9 IS9-4 158.9 158.5 158.0 IS7-S 157.0 156-5 156.0 iSS-S IS' 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 25S-5 254.8 254.1 253-4 252.8 252.0 251-3 250.6 249.9 249.2 248.5 247.8 247.1 246.4 245.7 245.0 244.2 243-5 242.8 242.1 241.4 240.6 239-9 239.2 238.4 237-7 236.9 236.2 235-5 234-7 234.0 233-2 20' longitude. 355-8 355-0 354-1 353-2 352-3 351-4 350.6 348.8 347-9 347-0 346-1 345-2 344-3 343-4 342- 5 341.6 340-7 339-8 338-8 337-9 337-0 336-0 335-1 334-2 333-2 332-3 331-4 330-4 329-5 328.5 327.6 326.6 325-6 324-7 323-7 322.8 321.8 320.8 319.8 318-9 317-9 316.9 315-9 314-9 314.0 313-0 312.0 31 1.0 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-S 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 402.2 401.0 399-8 398.6 397-4 396.2 394-9 393-6 392-4 391.2 390.0 388.7 30' longitude. S33-8 532-4 531-1 529.8 528-S 527.2 525.8 524-5 523-1 521.8 520.5 519-1 517-8 516.4 515-1 513-7 512-3 511-0 509.6 508.3 506.9 505-5 504-1 502.7 501-3 499-9 498.5 497.0 495-6 494-2 492.8 491-3 489.9 488.5 487.0 485.6 484.1 482.7 481.2 479.8 478-3 476.8 475-4 473-9 472.4 470.9 469.4 467-9 466.4 ORDINATES OF DEVELOPED PARALLEL. 40° tnm. 0.0 0.2 0.4 0.7 I.O 1-5 42" 0.0 0.2 0.4 0.7 1.0 1-5 44" 0.0 0.2 0.4 0-7 I.I 1-5 46° 0.0 0.2 0.4 0-7 I.I i-S 0.0 0.2 0.4 0.7 1.0 1-5 41° mjn. 0.0 0.2 0.4 0-7 1.0 1-5 43° 0.0 0.2 0.4 0.7 I.I 1-5 45° 0.0 0.2 0.4 0-7 I.I 1-5 47° 0.0 0.2 0.4 0.7 i.i 1-5 Smithsonian Tables. 137 Table 24. CO-ORDINATES FOR PROJECTION OF MAPS. [Derivation of table explained on pp. liii-lvi.] SCALE Tir^T- ■| = » ABSCISSAS OF DEVELOPED PARALLEL. •s •o a s nil OPTITKATTTQ CYW II •S3 DEVELOPED •11 SI 5' lO' «5' 20' 25' 30' PARALLEL. s^ |ssa longitude. longitude. longitude. longitude. longitude. longitude. mm. mm. mm. mtn. mm. mm. mm. 48°oo' 77-7 155-5 233-2 311-0 388.7 466.4 Jl 48° 49° lO " V3V.6' 463-3 77-5 1 55-0 232-5 231-7 310.0 308.9 387-4 386.2 464.9 4634 20 77.2 154-5 30 695.0 77.0 154-0 230.9 307-9 384-9 461.9 mm . mm. 40 926.6 76.7 153-5 230.2 306.9 383-6 460.4 5' 10 0.0 0.0 5° 1158.2 76.5 152.9 229.4 3OS-9 382.4 458.8 0.2 0.2 4900 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 23J-7 76.0 151.9 227.9 303-8 379-8 25 .30 I.O 1.0 20 4634 75-7 151.4 227.1 302.8 378.6 454-3 1.5 1-5 30 ^^■l 75-4 150.9 226.4 301.8 377-2 452-7 40 SO 926.8 1158.4 75-2 150.4 225.6 224.8 300.8 299.8 376.0 374-7 45I-I 449.6 74-9 149.9 50 00 10 231-7 74-7 74-4 Itt 224.0 223-3 298.7 297.7 373-4 372-1 448.1 446.5 50° 51° 20 463-5 74-2 148.3 147.8 222.5 296.6 370.8 445.0 5 0.0 0.0 30 695.2 926.9 73-9 221.7 295.6 368.2 443-4 10 0.2 0.2 40 73-6 '47-3 220.9 294.6 441.8 15 0.4 0.4 5° 1 1 58.6 73-4 146.8 220.1 293-5 366.9 440-3 20 25 0-7 1.0 0.7 1.0 51 00 73-1 146.2 219.4 292.5 365-6 438-7 30 1-5 1-5 10 231.8 72-9 145-7 218.6 291.4 364-3 437-2 20 t?-^ 72.6 145-2 217.8 363-0 435-5 30 695-3 72-3 144.7 217.0 289.3 361.6 434-0* 40 ^^li 72.1 144.1 216.2 288.3 360.4 432-4 52° 53° SO 52 00 1158.8 71.8 7J-S 143-6 143-1 215-4 214.6 287.2 286.2 359-0 357-7 430.8 429.2 5 0.0 0.0 10 ^i'i 71-3 142.5 213.8 285.1 356-4 427-6 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 3° 695.4 70.7 141-5 212.2 283.0 353-7 424.4 20 0.7 0.6 40 . 927-2 70-5 140.9 2II.4 281.9 352-4 422.8 25 1.0 1.0 50 1159.0 70.2 140.4 210.6 280.8 351-0 421-3 30 1-5 'S 5300 10 231.8 463-7 69.9 69.7 139-9 209.8 209.0 279-8 278.7 349-7 348-4 419.6 418.0 20 69.4 138.8 208.2 277.6 347-0 416.4 54° 55° 30 695-6 927.4 68.8 138-3 137-7 207.4 206.6 276-5 275.4 345-6 344-2 414.8 413-1 40 0.0 0.2 0.0 0.2 50 1159.2 68.6 137-2 205-7 274-3 342-9 411-5 5 10 54 00 68.3 136.6 204.9 273.2 341-6 409.9 15 li o°:l 10 231.9 68.0 136-1 204.1 272.2 340-2 408.2 20 20 463.8 67.8 135-5 203-3 271.0 338-8 406.6 25 1.0 1.0 30 695-7 67.5 135-0 202.4 269.9 337-4 404.9 30 1.4 14 40 5° 927.6 "59-4 67.2 66.9 134-4 133-9 201.6 200.8 26718 336-0 403-3 401.6 334-7 SSoo 66.7 66.4 133-3 132.8 200.0 266.6 333-3 400.0 56° 10 231.9 199-1 265-S 331-9 398-3 396.6 20 463-9 695.8 66.1 132.2 198-3 264.4 330.5 5 0.0 30 Pi 131-7 197-5 196.6 263.3 329.2 395-0 10 0.2 40 927-7 65.6 131.1 262.2 327.8 393-3 15 0.4 SO 1159.6 65-3 130-S 195.8 261.1 326.4 391.6 20 0.6 5600 65.0 130.0 195.0 260.0 325.0 389.9 25 30 1.0 1.4 Smithsonian Tables. 138 Table 24. CO-ORDINATES FOR PROJECTION OF MAPS. SCALE rihu- [Derivation of table explained on pp. liii-lvi.] •ss .2 "-Sii S c s s ABSCISSAS OF DEVELOPED PARALLEL. longitude. 10' longitude. longitude. 20' longitude. 25' longitude. 30' longitude. ORDINATES OF DEVELOPED PARALLEL. S6°oo' 10 20 3° 40 5° 5700 10 20 30 40 SO 5800 10 20 30 40 SO S9 0O 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 232.0 463-9 695.9 927.9 1159.8 232.0 464.0 696.0 928.0 1 1 60.0 232.0 464.1 696.1 928.2 II60.2 232.1 464.2 696.2 928.3 1 1 60.4 232.1 464.2 696.4 1 160.6 232.2 464-3 696.4 928.6 II60.8 232.2 464-4 696.6 928.8 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 6i.6 61.3 61.0 60.7 60.4 60.2 59-9 S9-6 S9-3 S9-0 58.7 58.4 58.1 S7-8 S7-S 57.2 57.0 56.7 56.4 56.1 55.8 SS-S 55.2 S4-9 54.6 S4-3 54-0 S3-7 S3-4 S3- 1 52.8 52.5 52.2 51.6 Si-3 51.0 130.0 129.4 128.9 128.3 127.7 127.2 126.6 126.0 125.5 124.9 124-3 123.8 123.2 122.6 122.0 121.5 120.9 120.3 "9-7 1 19.2 1 18.6 1 18.0 "7-4 1 16.8 101.9 195.0 194-1 193-3 192.4 191.6 190.8 189.9 189.1 188.2 187.4 186.5 185.6 184.8 183.9 183.1 182.2 181.4 180.5 179.6 178.7 177-9 177.0 176.1 I7S-3 116.3 174.4 115.7 173-S 115.1 172.6 1 14.5 171.7 "3-9 170.8 "3-3 170.0 112.7 169.1 112.1 168.2 111.5 167-3 110.9 166.4 1 10.3 109.8 i6s-S 164.6 109.2 163.7 108.6 162.8 108.0 161.9 107.4 161.0 106.8 160.1 106.2 159.2 105.5 158.3 105.0 IS7-4 104.4 I.S6-5 103.8 155.6 103.1 1 54-7 102.5 153-8 152.9 260.0 258.8 257.7 256.6 255-5 254.4 253-2 252.1 251.0 249.8 248.7 247.5 246.4 245-2 244.1 242-9 241.8 240.6 239-5 238-3 237.2 236.0 234.8 233-7 232- 5 231.4 230.2 229.0 227.8 226.6 225.4 224.2 223.1 221.9 220.7 219.5 218.3 217.1 215.9 214-7 213-5 212.3 211. 1 209.9 208.7 207-5 206.3 205.1 203.9 325.0 323-6 322.2 320.8 319-4 318.0 316.6 31 5-1 313-7 312-3 310.8 309-4 308.0 306.6 30S-1 303-6 302.2 300.8 299.4 297.9 296.4 295.0 293.6 292.1 290.6 289.2 287.7 286.2 284.8 283-3 281.8 280.3 278.8 277-4 275-8 274.4 272.9 271.4 269.9 268.4 266.9 265.4 263.9 262.4 260.9 259-4 257.8 256.4 254.8 3f9-9 388.3 386.6 384-9 383-2 381-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 3S4-0 352-3 350-5 348-8 347-0 345-2 343-4 341-7 340-0 338-2 336.4 334-6 332-8 331-0 329-3 327-S 325-7 323-9 322.1 320.3 318.5 316.7 3 '4-9 313-1 3"-3 309-4 307.6 305.8 J-5 56° mm, 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 0.6 0.9 »-3 64° 0.0 0.1 0-3 ti 1.2 57° mt». 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 1-3 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 fnhv- [Derivation of uble explained on pp. liii-lvi«] I .2 "'OJ! 3s PS iSSi ABSCISSAS OF DEVELOPED PARALLEL. 5' longitude. longitude. IS' longitude. longitude. 2S' longitude. 30' longitude. ORDINATES OF DEVELOPED PARALLEL. 64°00' 10 20 30 40 SO 6500 10 20 30 40 SO 6600 10 20 30 40 SO 67 00 10 20 30 40 SO 6800 10 20 30 40 SO 6900 10 20 30 40 SO 70 00 10 20 30 40 SO 71 00 10 20 30 40 SO 72 00 232.2 464.5 696.8 929.0 Il6l.2 464.6 696.9 929.1 II6I.4 232-3 464.6 697.0 929-3 II6I.6 232.4 464.7 697.0 929.4 1I6I.8 232.4 464.8 697.1 929.5 1161.9 232.4 464.8 697.2 929.6 1 1 62.0 232-4 464.9 697-3, 929.7 1 162.2 232.5 464.9 697.4 929.8 1162.3 51.0 50.7 50-4 50.1 49.8 49-4 49-1 48.8 48.5 48.2 47-9 47-6 47-3 47-0 46.7 46.4 46.1 45-8 4S-4 45.1 44-8 44-5 44-2 43-9 43-6 43-2 42.9 42.6 42-3 42.0 41.7 41.4 41.0 40.7 40.4 40.1 39-8 39-S 3§i 38.8 38-S 38.2 37-9 37-6 37-2 36.6 36-3 3S-9 101.9 101.3 100.7 1 00. 1 99-S 98.9 98-3 97-7 97.1 96-4 95-8 9S-2 94.6 94-0 93-4 92-7 92.1 91-S 90.9 90-3 89.6 89.0 88.4 87-7 87.1 86.5 85-9 85.2 84.6 84.0 83-4 82.7 82.1 8i.q 80.8 80.2 79-6 78-9 78-3 77-6 77.0 76-4 7S-7 7S-I 74-5 73-8 73-2 72.5 71.9 152.9 152.0 151.1 150.2 149.2 148.3 147.4 146.5 145.6 144.7 143-7 142.8 141.9 141.0 140.0 139-1 138.2 137-2 136-3 "35-4 134-4 133-S 132.6 131.6 130-7 129.8 128.8 127.9 126.9 126.0 125.0 124.1 123.2 122.2 121.2 120.3 "9-3 1 1 8.4 117.4 116.5 115-5 114.6 113.6 112.6 1H.7 110.7 109.7 108.8 107.8 203.9 202.6 201.4 200.2 199.0 197.8 196.6 195-3 194.1 192.9 191.6 190.4 189.2 188.0 186.7 185.5 184.2 183.0 181.8 180.5 179.2 178.0 176.8 175-5 174.2 173-0 171.7 170-5 169.2 168.0 166.7 165.4 164.2 162.9 161.6 160.4 159.1 157.8 156.6 I5S-3 154.0 152.8 151-5 150.2 148.9 147.6 146.3 HS-o 143.8 254.8 253-3 251.8 250.3 248.8 247.2 245.7 244.2 242.6 241.1 239-6 238.0 236.5 235.0 233-4 231.8 230-3 228.8 227.2 225.6 224.0 222.5 221.0 219.4 217.8 216.2 214.6 213.1 211.6 210.0 208.4 206.8 205.2 203.6 202.0 200.5 198.9 197-3 195-7 194-1 192.6 191.0 189.4 187.8 186.2 184.5 182.9 181.3 179-7 305.8 304.0 302.2 300.4 296.6 294.8 293.0 291.2 289-3 ,287.5 285-7 283.8 281.9 280.1 278.2 276.4 274- 5 272.6 270.8 268.9 267.0 265.1 263.2 261.4 259-S 257.6 255-7 253-9 251.9 250.1 248.2 246.3 244-4 242.5 240.6 238-7 236.8 234.8 232.9 231. 1 229.1 227.2 225-3 223.4 221.4 219-5 217.6 215.6 64° mm. 0.0 0.1 0-3 0.5 0.8 1.2 66° 0.0 0.1 0-3 i.i 68° 0.0 0.1 0-3 0-5 0.7 1.1 70° s 0.0 10 0.1 15 0.2 20 0.4 25 0.7 30 1.0 72° 0.0 0.1 0.2 0.4 0.6 0.9 6S° tnvt. 0.0 O.I 0-3 67° 0.0 0.1 0-3 i.i 69° 0.0 0.1 0-3 o-S 0.7 i.o 71° 0.0 O.I 0.2 0.4 0.7 0-9 Smithsonian Tables. 140 Table 24 CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Tjshm- [Derivation of table explained on pp. liii-lvi.] Igj, ABSCISSAS OF DEVELOPED PARALLEL. "g ORDINATES OF DEVELOPED Is s-g liii s' 10' IS- 20' 25' 30' PARALLEL. S^ ^SS^p. longitude. longitude. longitude. longitude. longitude. longitude. mm. mm. mm. mm. mm. mm. mm. lis 72°00' 3S-9 71.9 107.8 143.8 179-7 215.6 §1 72° 73° 10 232.5 3S-6 71.2 106.9 142.5 178.1 213-7 J- 20 465.0 3S-3 70.6 105.9 141.2 176.5 211.8 30 697.4 35° 70.0 104.9 1399 174.9 209.9 fnTn. mtn. 4° 929.9 34.6 69.3 104.0 138.6 173-2 207.9 5' 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 340 68.0 102.0 136.0 170.0 204.1 IS 20 0.2 0.2 0.4 0.6 10 232.5 33-7 67.4 lOI.O 134-7 168.4 202.1 25 30 20 465.0 334 66.7 100.1 •334 166.8 200.2 0.9 0.9 30 697-S 330 66.1 99.1 132.2 165.2 198.2 40 H°t 32-7 ^5-4 98.1 130.8 163.6 196.3 50 1 162.6 324 64.8 97.1 129.5 161.9 194-3 74° 75° 7400 10 232.5 32.1 31-7 64.1 63-5 96.2 95.2 128.2 127.0 160.3 158.7 192.4 '§2-4 20 465.1 314 62.8 94.2 125.6 157.0 188.5 S 0.0 0.0 30 697.6 311 62.2 93-2 124.3 iSS-4 186.5 10 0.1 0.1 40 930.1 30.8 61.5 923 123.0 1 53-8 184.6 15 0.2 0.2 SO 1 1 62.6 304 60.9 913 1 21. 8 152.2 182.6 20 2S l^ 0-3 0.8 7500 30.1 60.2 90-3 120.4 150.6 180.7 30 0.8 10 '"iile 29.8 59.6 89-3 1 19. 1 148.9 178.7 20 697.6 29.4 58.9 S8-3 88.4 874 117.8 1 16.5 147.2 145.6 176.7 30 174.8 76= .t.tO 40 930L2 28.8 57.6 86.4 1 1 5.2 144.0 172.8 77 SO 1 162.8 28.5 56.9 854 "3-9 142.4 170.8 7600 28.1 S6-3 84.4 112.6 140.7 168.8 5 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-S 55.0 82.4 109.9 137-4 164.9 0-3 0.5 0-3 o-S 0.7 30 697.7 HI S4-3 81.4 108.6 I3S-8 162.9 25 40 930-3 26.8 S3-7 80.5 107.3 134.2 1 61.0 30 0.7 SO 1 1 62.8 26.5 26.2 S30 79S 78.5 io6.o I32-S 130.8 159.0 7700 52-3 104.7 157.0 10 232.6 25.8 Si-7 77-5 103.4 129.2 iSS-o 78° 79° 20 30 465.2 697.8 25-S 25.2 51.0 504 76.5 7S-| 74.6 102.0 100.7 127.6 125.9 1 53- 1 151.1 40 9304 24.8 49-7 99.4 124.2 149.1 5 0.0 0.0 SO 1163.0 24-S 49.0 73-6 98.1 122.6 147.1 io IS 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 10 232.6 239 47-7 71.6 95-4 "9-3 143-2 25 0.4 0.4 20 465.2 23s 47.1 70.6 94.1 117.6 141.2 30 0.6 0.6 30 697.8 23.2 46.4 69.6 92.8 1 1 6.0 139-2 40 SO 9304 1 163.0 22.9 22.5 45-7 45.1 67I6 91.4 90.1 1 14-3 112.6 137-2 I3S-2 80° 7900 10 232.6 22.2 21.9 444 43-7 66.6 65.6 88.8 III.O 109.4 133-2 131.2 20 465.2 21.5 43-1 64.6 86.1 107.6 129.2 S 0.0 30 697.9 21.2 42.4 63.6 84.8 106.0 127.2 10 0.1 40 930-S 20.9 41.7 62.6 83-5 104.4 125.2 15 0.1 SO 1 163.1 20.5 41.1 61.6 82.1 102.6 123.2 20 25 0.2 0.4 8000 20.2 40.4 60.6 80.8 lOl.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. 1-lii.] Middle Latitude of Quadrilateral. Area in Square Miles. 0° 474653 5 47289s 10 467631 '5 458891 20 446728 2S 431213 30 412442 3S 390533 40 365627 4S 337890 5° 307514 55 274714 6o 239730 65 202823 70 164279 7S 124400 80 83504 85 41924 Smithsonian Tables. 142 Table 26. AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1° EXTENT 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' 30 1 00 I 30 47S2-33 47S2.I6 4751-63 47 50-7 S 26° 00' 26 30 27 00 27 30 4282.50 4264.51 4246.20 4227.56 52° 00' 52 30 53 00 53 30 2950.58 2917.85 2884.88 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 4695-38 4688.89 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 n 00 11 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 3?i3-i4 3687.18 64 00 64 30 65 00 65 30 2106.26 2068.68 2030.94 1993.04 14 00 14 3° 15 °o 15 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 1878.37 1839.84 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- '7 3525-54 3497-62 3469-44 68 00 68 30 69 00 69 30 1801.16 1762.33 1723.36 1684.24 18 00 18 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 2S 30 4351-21 4334-52 4317-51 4300.17 50 00 50 30 51 00 51 30 3079-15 3047-37 2983.08 76 00 76 30 77 00 77 30 1164.49 "23.75 1082.91 1041.99 Smithsonian Tables, 144 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1< LATITUDE AND LONGITUDE. [Derivation of table explained on pp. l-lii,j Table 26. EXTENT IN Middle latitude of quadrilateral. 78° 00' 78 30 79 00 79 30 80 00 80 30 81 00 81 30 Area in square miles. 1000.99 959-90 918.73 877.49 836.18 794-79 753-34 711-83 Middle latitude of quadrilateral. 82° 00' 82 30 83 00 83 30 84 00 84 30 85 00 85 30 Area in square miles. 670.27 628.64 586.97 545-24 461.66 419.81 377-93 Middle latitude of quadrilateral. 86° 00' 86 30 87 00 87 30 88 00 88 30 89 00 89 30 Area in square miles. 336,02 294.08 252.11 210.12 168.12 126.10 84.07 42.04 Smithsonian Tables. 145 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. o°oo' IS 30 45 1 188.10 1188.08 1188.05 1188.00 13° 00' 13 IS 13 30 13 45 1158.44 1157-29 1156.12 "54-93 26° 00' 26 15 26 30 26 4S 1070.64 1068.40 1066.14 1063.86 I 00 I IS I 30 I 45 1187.92 1187.82 1187.70 1187.56 14 00 14 15 14 30 14 45 1153.72 1152.48 1151-23 1149.95 27 00 27 IS 27 30 27 45 1061.56 1059.24 1056.90 1054.54 2 00 2 IS 2 30 2 45 1187.39 1187.20 1 180.99 1186.76 15 00 IS 15 IS 30 15 45 1148.65 1147-33 1145-99 1144.63 28 00 28 15 28 30 28 45 1052.16 1049.76 1047-34 1044.90 3 00 3 15 3 3° 3 45 1 186.51 1186.24 1185.95 1185.62 16 00 16 15 16 30 16 45 1143.25 1 141.84 1 140.41 1138.96 29 00 29 15 29 30 29 45 1042.44 1039.97 1037-47 1034-95 - 4 00 4 15 4 30 4 45 1185.28 1184.92 1184.53 1184.13 17 00 17 15 17 30 17 45 1137-50 1136.00 1134.49 1132.96 30 00 30 IS 30 30 3° 45 1032.41 1029.85 1027.27 1024.68 5 00 5 IS 5 30 5 45 1183.70 1183.24 1182.77 1182.28 18 00 18 IS 18 30 18 45 1131.41 1129.83 1128.24 1126.62 31 00 31 15 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 15 19 30 19 45 1124.98 1123.32 1121.64 1 1 19-93 32 00 32 15 32 30 32 45 1011.40 1008.69 1005.96 1003.20 7 00 7 15 7 30 7 45 1179.48 1178.85 1178.20 1177-53 20 00 20 IS 20 30 20 45 1118.21 1 1 16.47 1 1 14.7 1 1 1 12.92 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 1175-39 1174.63 21 00 21 IS 21 30 21 45 nil. II H09.28 1107.44 1105.57 34 00 34 IS 34 30 34 45 989.16 986.29 983.41 980.50 9 00 9 IS 9 30 9 45 1173.86 1173.06 1172.23 1171-39 22 00 22 15 22 30 22 45 1103.68 1101.77 1099.84 1097.88 35 00 35 15 35 30 35 45 977-58 974.64 ' 97I-68 968.70 10 00 10 IS 10 30 ID 4S 1170.52 1169.63 1168.73 1167.80 23 00 23 IS 23 30 23 45 1095-91 1093.92 1 09 1. go 1089.87 36 00 36 15 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 15 24 30 24 45 1087.81 1085.74 1083.64 1081.52 37 00 37 15 37 30 37 45 953-52 950-43 947-32 944.21 12 00 12 IS 12 30 12 45 1 162.81 1161.75 1160.67 1159-56 25 00 25 IS 25 30 25 45 1079.39 1077.23 1075-05 1072.85 38 00 38 IS 38 30 38 45 941.05 937-88 934-71 931-51 Smithsonian Tabl ES. 146 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. 39 00 39 IS 39 30 39 45 40 00 40 IS 40 30 40 4S 41 00 41 IS 41 30 41 4S 42 00 42 IS 42 .30 42 45 43 00 43 15 43 30 43 45 44 00 44 15 44 30 44 45 45 00 45 IS 45 30 45 45 46 00 46 IS 46 30 46 4S 47 00 47 15 47 30 47 45 48 00 48 IS 48 30 48 45 49 00 49 15 49 30 49 45 50 00 SO IS 50 30 SO 45 SI 51 00 IS SI 30 51 45 Area in square miles. 928.29 925.06 921.80 91853 915-25 911.94 908.61 905.27 901.91 898.54 895.14 891-73 888.30 884.85 881.39 877.91 874.41 870.90 867.37 863.82 860.25 856.67 853-07 849.46 845.82 842.18 838.51 834.83 831.13 827.42 823.68 819.94 816.18 812.40 808.60 804.79 800.97 797-13 793-27 789.39 785-50 781.60 777.68 773-74 769.79 765-83 761.85 757-85 753-84 749.82 745-78 741.72 Middle latitude of quadrilateral. 52° 00' 52 IS 52 30 52 45 53 00 S3 15 S3 30 53 4S .54 00 54 15 .54 30 54 45 55 00 55 IS 55 30 55 45 56 00 S6 15 5b 30 56 45 57 00 57 IS 57 30 57 45 58 00 58 15 ■^S ,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 IS 63 30 63 45 64 00 64 IS 64 30 64 45 Area in square miles. 737.65 733-57 729.47 725-36 721.23 717.08 712.93 708.76 704-57 700.38 696.16 691.94 687.70 683.44 679.17 674.89 670.60 666.29 661.97 657-64 653-29 648.93 644.5s 640.17 635-77 631.36 626.93 622.49 618.05 613.59 609.11 604.62 600.13 595.62 591.09 586.56 582.01 577-45 572.88 568.30 563.71 S59-II 554-49 549.86 545-23 540.58 535-92 531.25 526.57 521.88 517-17 512.46 Middle latitude of quadrilateral. 65° 00' 65 IS 65 30 65 45 66 00 66 5 66 30 66 4S 67 00 67 IS 67 30 67 45 68 00 68 15 68 30 68 45 69 00 69 IS 69 30 69 45 70 00 70 15 70 30 70 45 71 00 71 IS 71 30 71 45 72 00 72 15 72 30 72 45 73 00 73 15 73 30 73 45 74 00 74 15 74 30 74 45 75 00 75 IS 75 30 75 45 76 00 76 15 76 30 76 45 77 00 77 15 77 30 77 45 Area in square miles. 507-74 503.01 498.26 493.51 488.75 483.97 479.19 474.40 469.60 464.78 459.96 455-13 450.29 445-45 440.59 435-72 430.84 425.96 421.06 416.16 411.25 406.34 401.41 396.47 386.58 381.62 376.65 371.68 366.70 361.71 356.71 346.69 341.68 33665 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. 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° 00' 78 IS 78 30 78 45 250.25 245.12 239.98 234-83 82° 00' 82 IS 82 30 82 45 167.57 162.37 157.16 151.95 86° 00' 86 IS 86 30 86 45 84.01 78.76 79 00 79 IS 79 30 79 45 229.68 224-53 219-37 214.21 83 00 83 IS 83 30 83 45 146.74 141-53 136-31 131.09 87 00 87 15 87 30 87 45 63.03 57-78 52-53 47.28 80 00 80 IS 80 30 80 45 209.05 203.88 198.70 193-52 84 00 84 IS 84 30 84 45 125.87 120.64 115.42 110.18 88 00 88 IS 88 30 88 45 31-53 26.27 81 00 81 15 81 30 81 45 188.34 183.IS 177.96 172.77 8s 00 8s IS 8s 30 85 45 104-95 99-72 94.48 89.2s 89 00 89 15 89 30 89 45 21.02 15.76 10.51 5.26 Smithsonian Tables. 148 Table 28. AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 15' EXTENT IN LATITUDE AND LONGITUDE. [DeiivatioD ol ubla explalntd 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. o" 07' 30" 15 00 22 30 30 00 297.02 297.02 297.02 297,01 f 37'30" 6 45 00 6 52 30 7 00 00 295.09 295.02 294.9s 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 IS 00 7 22 30 7 30 CO 294.79 294.71 294.63 294-55 13 37 30 13 45 00 13 52 30 14 00 00 288.88 288.73 288.58 288.43 I 07 30 I 15 00 I 22 30 1 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 IS 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 00 296.91 296.89 296.87 296.85 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 00 287.65 287.49 287-33 287.17 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 15 07 30 15 15 00 IS 22 30 IS 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 15 37 30 15 45 00 15 52 30 16 00 00 286.33 286.16 SI 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 15 00 16 22 30 i6 30 00 285.64 285.46 285.28 285.10 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 IS 00 4 22 30 4 30 00 296.28 296.23 296.18 296.13 «o 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 II 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 00 283-43 283.24 283.05 282.86 5 07 30 5 IS 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 5 37 30 5 45 00 5 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 00 295-37 295-3« 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 Tab LES. 150 Table 28. AREAS OF QUADRILATERALS OF EARTH'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. 19° 37' 30" 19 45 0° 19 52 30 20 00 00 280.20 279.99 279.77 279-SS 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 15 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 3° 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 27778 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 15 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 °° 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 °o 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 15 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 3° 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 IS 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 3° 24 4S 00 24 S2 3° 25 00 00 270.6s 270.38 270.11 269.85 31 07 3° 31 15 00 31 22 30 31 30 00 254-53 254.19 37 37 30 37 45 °° 37 52 30 38 00 00 236.44 236.05 235.66 235.26 25 07 30 25 15 00 25 22 30 25 30 00 269.58 269.31 31 37 30 31 45 °o 31 52 30 32 00 00 253.86 253-53 253-19 252-85 38 07 30 38 15 00 38 22 30 38 30 00 234-87 234-47 234-07 233-68 2S 37 30 25 4S 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 3° 39 00 00 233.28 232.88 232.48 232.07 Smithsonian Tables. 151 Table 28. AREAS OF QUADRtLATERALS OF EARTH'S SURFACE OF 16' EXTENT IN LATITUDE AND LONGITUDE. [Derivation of table explained on pp. 1-lii.] Middle latitude of quadrilateral. 39° 07 3° 39 'S 00 39 22 30 39 30 00 39 37 30 39 45 00 39 52 30 40 00 00 40 07 30 40 15 00 40 22 30 40 30 00 40 37 30 40 45 00 40 52 30 41 00 00 41 07 30 41 15 00 41 22 30 41 30 00 41 37 30 41 45 00 41 52 30 42 00 00 42 07 30 42 15 00 42 22 30 42 30 00 42 37 30 42 45 00 42 52 30 43 00 00 43 07 30 43 IS 00 43 22 30 43 30 00 43 37 30 43 45 00 43 52 30 44 00 00 44 07 30 44 15 00 44 22 30 44 30 00 44 37 30 44 45 00 44 52 30 45 00 00 45 07 30 45 15 00 45 22 30 45 30 00 Area in square miles 231.67 231.27 230.86 230.45 230.04 229.63 229.22 228.81 228.40 227.99 227.57 227.15 226.73 226.32 225.90 225.48 225.06 224.64 224.21 223.79 223.36 222.93 222.50 222.08 221.65 221.21 220.78 220.35 219.91 219.48 210.04 218.60 218.16 217.73 217.28 216.84 216.40 215.96 215.51 215.06 214.61 214.17 213.72 213.27 212.82 212.37 211.91 211.46 211.00 210.55 210.09 209.63 Middle latitude of quadrilateral. 45° 37 30 45 45 00 45 52 30 46 00 00 46 07 30 46 15 00 46 22 30 46 30 00 46 37 30 46 45 00 46 52 30 47 00 00 47 07 30 47 15 00 47 22 30 47 30 00 47 37 30 47 45 00 47 52 30 48 00 00 48 07 30 48 15 00 48 22 30 48 30 00 48 37 30 48 45 00 48 52 30 49 00 00 49 07 30 49 15 00 49 22 30 49 30 00 49 37 30 49 45 00 49 52 30 50 00 00 SO 07 30 50 15 00 50 22 30 50 30 00 SO 37 30 50 45 00 50 52 30 51 00 00 S> 07 30 5' '5 00 SI 22 .30 SI 30 00 SI 37 30 SI 45 00 SI 52 30 52 00 00 Area in square miles. 209.17 208.71 208.25 207.78 207.32 206.86 206.39 205.92 205.4s 204.99 204.52 204.05 203-57 203.1a 202.63 202.15 201.67 201.20 200.72 200.24 199.76 199.28 198.80 198.32 197-83 197-35 196.86 196-38 195.89 195-40 194-9' 194-42 193-93 193-44 192-94 192.45 191.95 191.46 190.96 190.46 189.96 189.46 188.96 188.46 187.96 187.46 186.95 186.45 185.94 185-43 184.92 184.41 Middle latitude of quadrilateral. 52° 07' 30" 52 15 00 52 22 30 52 30 00 52 37 30 52 45 00 52 52 30 53 00 00 S3 07 30 S3 15 00 S3 22 30 53 30 00 S3 37 30 S3 45 00 53 52 30 54 00 00 54 07 30 54 15 00 54 22 30 54 30 00 54 37 30 54 45 00 54 52 30 55 00 00 55 07 30 55 15 00 55 22 30 55 30 00 55 37 30 55 45 00 55 52 30 56 00 00 56 07 30 56 15 00 56 22 30 56 30 00 56 37 30 56 .45 00 56 52 30 57 00 00 57 07 30 57 IS 00 57 22 30 57 30 00 57 37 30 57 45 00 57 52 30 58 00 00 58 07 30 58 15 00 58 22 30 58 30 00 Area in square miles. 183.90 182.88 182.37 181.85 181.34 180.82 180.31 179.79 179-27 178-75 178.23 177-71 176.67 176.14 175.62 175-10 '74-57 174.04 '73-51 172.99 172.46 i7'-93 '7'-39 170.86 170.33 169.79 169.26 168.72 168.19 167.65 167.11 166.57 166.03 165.49 164.95 164.41 163.87 163-32 162.78 162.23 161.6S 161.14 160.59 160.04 159-49 158.94 158-39 157-84 157-29 156-73 Smithsonian Tabus. 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. 5f 37'30" 58 45 00 58 52 30 59 00 00 156.18 155.62 155-07 154.51 65007' 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 15 °o 59 22 30 59 30 00 153-96 153-40 152.84 152.28 65 37 30 65 45 00 65 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 IS 00 60 22 30 60 30 00 149-47 148.91 148.34 »47-77 66 37 30 66 45 00 66 52 30 67 00 00 119.20 1 18.60 118.00 117.40 73 07 30 73 >5 00 73 22 30 73 30 00 86!i7 86.05 85.42 60 37 30 60 45 00 60 52 30 6i 00 00 146.64 146.07 145.50 67 07 30 67 15 00 67 22 30 67 30 00 116.80 116.20 "5-59 114.99 73 37 30 73 45 00 73 52 30 74 00 00 84.79 84.16 83-53 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 "4-39 113.78 113.18 112-57 74 07 30 74 IS 00 74 22 30 74 30 00 82.28 81.65 81.01 80.38 61 37 30 61 45 00 61 52 30 62 00 00 142.65 142.08 141.50 140.93 68 07 30 68 IS 00 68 22 30 68 30 00 "1-97 II 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 15 00 75 22 30 75 30 00 77.22 76.59 75-95 75-32 62 37 30 62 4S 00 62 52 30 63 00 00 138.04 137-47 136-89 136-31 69 07 30 69 15 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 00 135-73 135-15 134-56 133-98 69 37 30 69 45 00 69 52 30 70 00 00 104.6s- 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 6S.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 9?:^8 77 07 30 77 15 00 77 22 30 77 30 00 67.04 66.41 6577 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 15 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 tMITHSONIAN TABLCe. IS3 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" 61.92 7« 1=; 00 61.28 78 22 ^o 60.64 7» 30 00 60.00 78 .17 .30 59-35 78 4^ 00 5871 78 52 30 58.06 79 00 00 57.42 79 07 .30 56.78 79 IS 00 56-13 79 22 30 55-49 79 30 00 54-84 79 37 3° 79 45 °o 79 52 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 8i 00 00 81 07 30 81 15 00 81 22 30 81 30 00 81 37 30 81 45 00 81 52 30 82 00 00 Area in square miles. 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. 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 83 15 00 83 22 30 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 2493 24.27 23.62 22.97 22.31 21.06 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 89 15 00 89 22 30 89 30 00 89 37 30 89 45 00 89 52 30 Area in square miles. 20.35 19.69 19.04 18.38 17.72 17.07 16.41 'S-76 15.10 14.44 '3-79 13-13 12.48 11.82 II. 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 I -31 0.66 Smithsonian Tables. IS4 Table 29. AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN LATITUDE AND LONGITUDE. [Derivation of table explained on pp. Hii.] 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 2S 35 132.01 13Z.01 132.01 132.00 8°45' 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 4S 55 1 OS 132.00 131-99 131-99 131.98 9 25 9 35 9 45 9 55 130.28 130.22 130-15 130.09 18 OS 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 13 J -94 10 OS 10 15 10 25 10 35 130.02 129.96 129.82 18 45 18 55 19 05 19 15 125.18 125.06 124.94 124.81 2 OS 2 IS 2 2S 2 35 131-93 131.91 10 45 10 55 11 05 II 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 OS 3 «5 131.86 131-84 131.82 131.80 II 25 11 35 II 45 II 55 129-47 129-39 129.32 129.24 20 OS 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 15 12 25 12 35 129.16 129.08 129.00 128.92 20 45 20 55 21 05 21 15 123.66 123.52 123-39 123.25 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 45 21 55 123.12 122.98 122.84 12270 4 45 4 55 5 05 5 IS 131-57 131-54 131-50 131-47 13 25 13 35 13 45 13 55 128.50 128.41 128.33 128.24 22 05 22 15 22 25 22 35 122.56 122.42 122.28 122.13 5 25 5 35 5 45- 5 55 131-44 131-40 131-36 131-33 14 05 14 15 14 25 14 35 128.14 128.05 127.96 127.87 22 45 22 55 23 OS 23 '5 121.99 121.84 121.69 121-55 6 OS 6 25 e 35 131.29 131.25 131-21 131-16 14 45 14 55 15 05 15 15 127.77 127.67 127.58 127.48 23 25 23 35 23 45 23 55 121.40 121.25 121.10 120.94 6 45 6 55 7 OS 7 15 131-12 131-07 131-03 130-98 15 2S 15 35 15 45 IS 55 127.38 127.28 127.18 127.08 24 05 24 15 24 25 24 35 120.79 120.64 120.48 120.33 7 25 7 35 7 45 7 55 130-93 130.88 130.84 130-79 16 05 16 15 16 25 16 35 126.98 126.87 126.77 126.66 24 45 24 55 25 OS 25 IS 120.17 120.01 119.85 119.69 8 OS f '5 8 25 8 35 130.68 130-63 130-57 16 45 16 55 17 05 17 15 126.SS 126.44 126.33 126.22 25 25 2S 35 25 45 25 55 119.53 119.37 1 19.21 119.04 Bmithsonian Tabl E8. ^56 Table 29 AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10' EXTENT IN LATITUDE AND LONGITUDE. i^^ 1 1« i m [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. 26005' 26 15 26 25 26 35 118.87 118.71 118.54 118.37 34° 45' 34 55 35 05 35 IS 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 15 118.21 118.04 117.87 117.69 35 25 35 35 35 45 35 55 108.07 107.85 107.63 107.41 44 05 44 15 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 OS 36 15 36 25 36 35 10696 106.74 106.51 44 45 44 55 45 05 45 IS 94-38 94.11 93-84 93-58 28 05 28 IS 28 25 28 35 116.82 116.64 116.46 116.28 36 45 36 55 37 OS 37 IS 106.29 106.06 105.83 105.60 45 25 45 35 45 45 4S 55 93-30 93-03 92.76 92.48 28 45 28 55 29 OS 29 IS 1 16.10 115.92 115-73 "5-55 37 25 37 35 37 45 37 55 105-37 105.14 104.91 104.68 46 05 46 15 46 25 46 35 92.21 91.38 29 25 29 35 29 45 29 55 "5-37 115.18 114.99 ■ 114.81 38 05 38 15 38 25 38 35 104.44 104.21 103.97 103.74 46 45 46 55 47 OS 47 15 91.10 90.82 90-SS 90.27 30 05 30 IS 30 25 30 35 114.62 114-43 114.24 114.04 38 45 38 55 39 05 39 IS 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 30 55 31 OS 31 15 113-47 113.27 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 55 112.68 112.48 40 05 40 15 40 25 40 35 101.57 101.33 101.08 100.83 48 4S 48 55 49 05 49 15 87.71 87.42 87-13 86.84 32 OS 32 IS 32 25 32 35 112.28 112.08 111.87 111.67 40 45 40 55 41 OS 41 15 100.59 100.34 100.09 99.84 49 25 49 35 49 45 49 55 86.55 86.26 ^5-97 85.68 32 45 32 55 33 OS 33 15 111.47 111.26 111.06 110.85 41 25 41 35 41 45 41 55 99-59 98.83 50 05 50 IS 50 25 50 35 8S-39 85.09 84.80 84.50 33 25 33 35 33 45 33 55 1 10.64 110.43 110.22 110.01 42 05 42 IS 42 25 42 35 98.57 98.32 98.06 97.80 SO 45 50 55 51 OS 51 IS 84.21 83.91 83.61 83-31 34 05 34 15 34 25 34 35 ■109.80 109.59 109-37 109.16 42 45 42 55 43 05 43 15 97-55 97.29 97-03 96.77 51 25 51 35 51 45 51 55 83.01 82.71 82.41 82.11 Smitm^omian 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° OS' 52 15 52 25 52 35 8i.8i 81.51 81.20 80.90 60° 45' 60 55 61 05 61 15 65.17 64.84 64.16 69° 25' 69 35 69 4S 69 55 a s 52 45 52 55 53 05 53 15 80.60 80.29 61 25 61 35 61 4S 61 55 63.82 63-48 63.14 62.80 70 OS 70 IS 70 25 70 35 45.51 45-iS 44.78 44.42 53 25 53 35 53 45 53 55 79-37 79.06 7875 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 15 54 25 54 35 78.13 77.82 77-51 77-19 62 45 62 55 63 05 63 15 61.10 60.75 60.41 60.06 71 25 71 3S 71 4S 71 55 42.58 42.22 41.85 41.48 54 45 54 55 55 05 55 IS 76.88 76.57 76.25 75-94 63 25 63 35 63 45 63 55 59-72 59-37 72 05 72 15 72 2S 72 3S 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 IS 64 25 64 35 58-33 57-99 57-64 57-29 72 45 72 55 ■ 73 05 73 15 39.63 38.52 56 OS 56 15 56 25 56 35 74-35 74-03 73-71 73-39 64 45 64 55 65 OS 65 IS 56-94 56-59 56.24 55-89 73 25 73 35 73 45 73 55 38.15 37.78 37-41 37.03 56 45 56 55 57 OS 57 15 73-07 72.75 7243 72.10 65 25 65 35 65 4S 65 55 55-54 55-19 54.83 54.48 74 05 74 IS 74 BS 74 3S 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 OS 66 IS 66 25 66 35 S4.I3 53-78 53-42 53-06 74 45 74 SS 75 05 75 IS 35-17 34-79 34-42 34.04 58 OS 58 15 58 25 58 35 70.48 70-15 69.82 69.49 66 55 67 OS 67 IS 52.71 52.3s 52.00 51.64 75 25 75 35 75 45 75 SS 33-66 33-29 32.91 32.53 58 45 58 55 59 OS 59 15 ■ 68.84 68.51 68.18 67 2S 67 35 67 4S 67 55 S1.28 50-93 50-57 50.21 76 05 76 IS 76 2S 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 05 68 IS 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 4S 68 55 69 OS 69 IS 48.41 48.05 47.69 47-33 77 25 77 35 77 45 77 SS 28.37 27.99 Smithsonian Tab LCS. 158 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. 78° 05' 78 IS 78 2S 78 35 27.62 27.24 26.8s 26.47 82° 05' 82 15 82 25 82 35 18.43 18.04 17.65 17.27 86° 05' 86 15 86 25 86 35 B 7-97 78 45 78 55 79 05 79 IS 26.09 25.71 25-33 24.9s 82 45 82 55 83 OS 83 15 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 SS 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 OS 84 IS 84 25 84 35 13-79 13.40 13.02 12.63 88 OS 88 IS 88 25 88 35 4-47 4.09 3-70 331 80 45 80 SS 81 05 81 15 21.50 21.12 20.73 20.35 84 45 84 SS 85 OS 85 15 12.24 11.86 11.47 n.08 11 45 88 55 89 05 89 15 2.92 2-53 2.14 1-75 8i 25 81 35 81 45 81 55 19.97 19.58 19.20 i8.8i 85 25 85 35 85 45 8S 55 10.69 10.30 9-92 9-53 89 25 89 35 89 45 89 55 1-36 0.97 0.58 0.19 Smithsonian Tables. 159 Table 30. DETERMINATION OF HEIGHTS BY THE BAROMETER. Formula of Babinet. y f-i B^ B Bo + B C(in feet) = 52494 fi + ^o + ^~^4 "] _ English Measures. L 900 J r 2 it A- i\~\ C (in metres) = 16000 i + ^^^ — — Metric Measures. In which 2';= Difference of height of two stations in feet or metres. ^o. ■5 = Barometric readings at the lower and upper stations respectively, corrected for all sources of instrumental error. ta, t^ Air temperatures at the lower and upper stations respectively. Values of C. ENGLISH MEASURES. METRIC MEASURES. H'o+')- F. 10° IS 20 30 35 40 45 5° 55 60 65 70 75 80 85 90 95 100 logC. 4.69834 •70339 .70837 •71330 .71818 4.72300 •72777 .73248 •73715 •74177 4^74633 •75085 •75532 •75975 •76413 4.76847 .77276 .77702 .78123 C. Feet. 49928 50511 51094 51677 52261 52844 53428 5401 1 54595 55178 55761 56344 56927 575" 58094 58677 59260 59844 60427 i('o+'). C. —10° — 8 — 6 — 4 — 2 +2 4 6 10 12 14 16 18 20 22 24 26 28 30 32 36 logC. 1.18639 .t9ooo •19357 .19712 .20063 4.2041 2 .20758 .21101 .21442 .21780 4.22II5 .22448 •22778 .23106 ■23431 4-23754 •24075 •24393 .24709 .25022 4^25334 •25643 .25950 .26255 Metres. '5360 15488 15616 15744 15872 16000 1 61 28 16256 16384 1-6512 16640 16768 16896 17024 17152 17280 17408 17536 17664 17792 17920 18048 18176 18304 Smithsonian Tables. 160 o o 10 20 30 40 I o 2 O 10 20 30 40 SO 3 ° 4 o 10 20 3° 40 5° 5 ° 10 20 30 40 5° 60 10 20 30 40 50 7 o Table 31. MEAN REFRACTION. Refraction. 34 54-1 32 49.2 3° 52-3 29 3-S 27 22.7 25 49-8 24 24.6 23 6-7 21 55.6 20 50.9 't 51-9 18 58.0 18 8.6 17 23.0 1640.7 16 0.9 1523-4 14 47-8 14 14.6 1343-7 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 3-3 946-5 930-9 916.0 848.4 835-6 823.3 811.6 8 0.3 7 49-5 7 39-2 7 29.2 719-7 124.9 116.9 108.8 ioa.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 35-6 33-2 30.9 28.7 26.7 24.6 23.0 21.8 20.6 19.7 19.0 18.4 17.9 16.8 15.6 14.9 14. 1 '3-5 12.8 12.3 II. 7 1 1.3 10.8 10.3 10.0 9-5 C u a 3 < " o ' 7 o 10 20 30 40 5° 8 o 10 20 30 40 5° 9 o 10 20 30 40 5° 10 20 30 40 5° 10 20 30 40 50 10 20 30 40 so 13 o 10 20 30 40 50 14 o Refraction 719-7 710.5 7 1-7 653-3 645-1 637-2 629.6 622.3 615.2 6 8.4 6 1.8 SSS-4 5 49-3 5 43-3 5 37-6 S32-0 526.5 521-3 516.2 5 II. 2 5 6.4 5 1-7 457.2 452-8 448.5 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 4 4-9 4 1.8 358-8 3 55-9 3 53-0 350-2 347-4 14 o 20 40 15 o 20 40 16 o 20 40 17 o 20 40 18 O 20 40 19 O 20 40 20 40 20 40 20 40 23 O 20 40 24 O 20 40 25 o 20 40 26 o 20 40 27 o 20 40 28 o Refraction. 3 47-4 342.1 3 37-0 332-1 327-4 322.9 318-6 314-5 310-5 3 6.6 3 2.9 259-3 255-8 252-5 249-3 246.1 243.1 2 40.2 237-3 2 34-5 231-9 229.3 226.8 224.3 2 21.9 2 19.6 217-4 2 15.2 213.0 2 10.9 8.9 2 3-2 2 1.4 159-6 157-8 I 56.1 I 54-4 I 52.8 I 51.2 149-7 148.2 Refraction. 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 o 20 40 36 o 20 40 3Z_ 20 40 38 o 39 o 20 40 40 o 20 40 41 o 20 40 42 o 48.2 46.7 45-3 43-8 42.4 41.0 397 38-4 37-1 35-8 34-5 33-3 32.1 30-9 29.8 28.7 27.6 26.5 25-4 24-3 23-3 22-3 21.3 20-3 '9-3 18.3 17-4 ^6^5 15.6 14-7 iiS 12.9 12.0 10.3 9-5 M 7-9 7-1 Al 5-5 4-7 4.0 132 44 46 i2. 54 I 60 62 63 22. 71 72 73 77 78 79 80 sr 82 86 90 Refraction. 64.0 61.8 59-7 57-7 53-8 51-9 50.2 "4ST 46.7 45-1 43-5 41.9 40.4 38-9 36.1 J±7_ 33-3 32.0 30-7 29-4 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 4.1 0.0 2.0 2.0 1.9 1.9 '.7 1.8 »-7 1.6 1.6 1.6 "•5 1-5 1.4 1.4 1-4 1.4 1.3 «.3 1.3 1.2 1-3 1.2 1.2 I.I I.I I.I 1.0 i.o i.i i.o 4.0 4-1 Smithsonian Tables. 161 Table 32. FOR CONVERSION OF ARC INTO TIME. o I 2 3 5^ 6 I 9 10 II 12 13 15^ i6 \l 19 20 h. m. 1 h. m. h. m. h. m. h. m. h. m. / m. s. // s. 4 8 12 16 020 024 028 032 036 60 61 62 64 65 66 67 68 6q 4 4 i 4 8 412 416 420 424 428 432 436 120 121 122 123 124 125 126 \% 129 8 8 4 8 8 812 816 820 824 828 832 836 180 181 182 183 ll5^ 186 187 188 189 190 12 12 4 12 8 12 12 12 16 1220 12 24 1228 1232 12 36 240 241 242 243 244 245 246 247 248 249 16 16 4 16 8 16 12 16 16 1620 1624 1628 1632 1636 300 301 302 303 306 309 20 20 4 20 8 2012 2016 20 20 2024 2028 2032 2036 I 2 3 4 5 6 7 8 9 10 o°8^ 12 016 020 024 028 032 036 I 2 3 6 I 9 0.000 0.067 0-133 0.200 0.267 0-333 0.400 0.467 0-533 040 70 440 130 840 1240 250 1640 310 2040 040 10 0.667 044 048 052 056 I I 4 I 8 I 12 116 71 72 73 ^5 76 77 78 7P 444 448 4S2 456 S 5 i 5 8 ■516 13' 132 133 l'3^5^ 136 \'^ ISP 852 856 9 ° It 9 12 916 191 192 193 I'ls^ 196 199 1244 1248 12 52 12 56 13 lit 13 12 13 16 251 252 2S3 255 256 257 258 2S9 1644 1648 1652 1656 17 17 4 17 8 17 12 17 16 311 312 313 3l'5^ 316 318 319 2044 2048 2052 2056 21 21 4 21 8 21 12 21 16 II 12 13 14 15 16 17 18 19 20 044 048 052 056 I I 4 I 8 I 12 116 II 12 13 14 15 16 ;i 19 20 0-733 0.800 0.867 0-933 1.000 1.067 1-133 1.200 1.267 I 20 80 5 20 140 9 20 200 1320 260 17 20 320 21 20 I 20 1-333 21 22 23 24 25 26 27 28 29 30 136 I 40 144 148 i 33338 3 39 45 9 39 53 9 45 59 1546 8 15 52 14 21 52 22 21 58 28 0.3s 0.36 2 12 216 2 19 223 226 5 II 515 39 34551 351 57 358 3 952 5 95812 10 4 18 15 58 20 16 4 26 16 10 33 22 4 35 22 10 41 22 16 47 0.37 0.38 0.39 5 19 5 22 526 40 4 4 10 10 10 24 16 16 39 22 22 53 0.40 o.go 530 41 42 43 44 % 4 10 16 10 16 30 16 22 45 22 29 0.41 230 0.91 5 33 4 16 22 10 22 37 16 28 51 2235 6 0.42 234 5 37 422 28 10 28 43 16 34 57 22 41 12 0.43 237 0-93 5 41 4283s 43441 44047 44653 4 53 10 34 49 10 40 55 1047 2 10 S3 8 10 59 14 16 41 4 1647 10 16 S3 16 16 59 22 17 529 22 47 18 22 S3 24 22 59 31 23 5 37 23 11 43 0.44 0.45 0.46 0.47 0.48 241 248 252 256 0.94 0-95 0.96 0.97 0.98 0.99 548 5 52 5 55 5 59 6 3 40 4 eg 6 II 5 20 17 II 35 23 17 49 0.49 2 59 50 5 5 12 II II 27 17 17 41 23 23 56 0.50 3 3.,. 1. 00 6 6 51 5 II 18 II 17 33 17 23 47 2330 2 52 51725 II 2339 17 29 54 2336 8 Example : Gi ven is'c "o". S3 52331 II 2945 1736 1742 6 23 42 14 23 48 21 The table giv es ,54 52937 113552 first for 14' 57° l8" 2» 27* 55 5 35 43 II 41 58 17 48 12 23 54 27 then for 2 42 0.44 56 59 5 41 so 54756 5 54 2 g 8 II 48 4 11 54 10 12 17 12 623 17 54 19 18 025 18 631 18 12 37 24 033 24 639 24 12 46 24 r8 52 15 The differenc IS'O^O' — 2'°27=. 2 e 44 = 14' Tiean time 27.44 57"32'-S6 60 6 615 12 12 29 18 18 44 24 24 58 Smithsonian Tables. i6s Table 36. LENGTH OF ONE DECREE OF THE MERIDIAN AT DIFFERENT LATITUDES. [Derivation of table explained on pp. xlvi-xlviii.] Statute MUes. Geographic Statute Geographic Latitude. Metres. Miles. i' of the Eq. Latitude. Metres. Miles. Miles. I' of the Eq. 0° 1 10568.5 68.703 59-594 45° IH132.1 69.054 59-898 I 1 10568.8 68.704 59-594 46 111151-9 69.067 59.908 2 1 10569.8 68.705 59-595 ^l 111171.6 69.079 59.919 3 110571.5 68.706 59-596 48 111191,3 69.091 59.929 4 1 10573.9 68.707 59-597 49 1 11210.9 69.103 59.940 5 110577.0 68.709 59-598 50 111230.5 69.115 59-95^ 6 1 10580.7 68.711 59.600 51 111249.9 69.127 59.961 7 110585.1 68.714 59.603 52 111269.2 69.139 59-972 8 110590.2 68.717 59.606 S3 11 1288.3 69.151 59.982 9 110595.9 68.721 59-609 54 1 1 1307.3 69.163 59.992 10 1 10602.3 68.725 59.612 55 1 11326.0 69.175 60.002 II 1 10609.3 68.729 59-616 56 1 11344.5 69.186 60.012 12 110617.0 68734 59.620 1 1 1362.7 69.198 60.022 13 1 10625.3 68.739 59.625 58 111380.7 69.209 60.032 14 1 10634.2 68.745 59.629 59 1 1 1398.4 6g.22o 60.041 13 1 10643.7 68.751 59-634 60 111415.7 69.230 60.051 16 1 10653.8 68.757 59.640 61 11 1432.7 69.241 60.060 17 110664.5 68.763 59.646 62 111449.4 69.251 60.069 18 110675.7 68.770 59.652 63 11 1465.7 69.261 60.077 19 1 10687.5 68.778 59-658 64 111481.5 69.271 60.086 20 110699.9 68.786 59.665 65 1 1 1497.0 69.281 60.094 21 110712.8 68.794 59.672 66 111512.0 69.290 60.102 22 1 10726.2 68.802 59-679 67 n 1526.5 g-29^ 60.110 23 1 107 40.1 68.810 59.686 68 11 1540.5 60.118 24 110754.4 68.819 59.694 69 II 1554.1 69.316 60.125 25 110769.2 68.829 59.702 70 111567.1 69.324 60.132 26 110784.5 68.838 59.710 71 II 1579.7 69-332 60.139 ^2 1 10800.2 68.848 59-719 72 111591.6 69.340 60.145 28 110816.3 68.858 59-727 73 1 1 1603.0 69-347 60.151 29 110832.8 68.868 59-736 74 111613.9 69-354 60.157 30 110849.7 68.879 59-745 75 111624.1 69.360 60.163 60.168 31 1 10866.9 68.889 59-755 76 1 1 1633-8 69.366 32 1 10884.4 68.900 59764 77 1 1 1642.8 69.372 60.173 33 1 10902.3 68.911 59-774 78 111651.2 69-377 60.177 34 110920.4 68.923 59-784 79 1 1 1659.0 69-382 60.182 35 1 10938.8 68.934 59-794 80 1 1 1666.2 69.386 60.186 36 1 10957.4 68.946 59.804 81 11 1672.6 69.390 60.189 37 1 10976.3 68.957 59.814 82 1 1 1678.5 69.394 60.192 38 110995.3 68.969 59.824 83 1 1 1683.6 69397 60.195 39 11 1014.5 68.981 59-834 84 1 1 1688.1 69.400 60.197 40 1 1 1033.9 68.993 59-845 85 111691.9 69.402 60.199 41 11 1053.4 69.005 59-855 86 111695.0 69.404 60.201 42 IH073.0 69.017 59.866 87 111697.4 69.405 60.202 43 11 1092.6 69.029 59.876 88 111699.2 69.407 60.203 44 111112.4 69.042 59.887 89 1 11700.2 69.407 60.204 « 111132.1 69.054 59-898 90 1 1 1700.6 69.407 60.204 Smithsonian Tables, 166 Table 37, LENGTH OF ONE DECREE OF THE PARALLEL AT DIFFERENT LATITUDES. [Derivation of table explained on p. xHx.] Latitude. Metres. Statute Miles. Geographic Miles. i' of the Eq. Latitude. Metres. Statute Miles. Geographic Miles. 1' of the Eq. 0° 111321.9 69.171 60.000 45° 78850.0 48.995 42.498 I 111305.2 69.162 59.991 46 77466.5 48.13s 41-753 2 1 1 1254.6 69.130 59-964 '^l 76059.2 47-261 40.994 3 111170.4 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 H0901.2 68.911 59-773 50 71698.9 44-552 38.644 6 110716.2 68.796 59-673 S' 70200.8 43.621 37-837 7 1 10497.7 68.660 59-556 52 68681. 1 42.676 37.018 8 II 0245.8 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 14 108036.6 67.131 58.229 59 57478-1 35-715 30-979 15 107553.1 66.830 57.969 60 55802.8 34-674 30.076 i6 107037.0 106488.5 66.510 57.690 61 541 10.2 33-622 29.164 17 66.169 57-395 62 52400.9 32.560 28.243 18 105907.7 65.808 57.082 63 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 1 01 7 54.6 63.227 54-843 69 40012.4 24.862 21.566 25 100953.0 62.729 54-4" 70 38188.6 23.729 20.583 26 1 00 1 20.6 62.212 53-963 53-498 71 36353-0 22.589 19-593 27 99257.8 61.676 72 34506.2 21.441 18.598 28 98364.8 61. 121 53.016 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 95S°7-3 59-345 51-476 76 27017.4 16.788 14.562 32 94496.2 58-717 50-931 77 25122.8 ■ 15.611 13-541 33 93456-3 58.071 50-371 78 23220.4 14.428 "■^kI 34 92387.9 57-407 49-795 79 21310.8 13.242 11.486 35 91291.3 56.726 49.204 80 19394.6 12.051 10.453 36 90166.8 56.027 48.598 81 17472.4 10.857 P'l 37 89014.8 55-3" 47-977 82 15544-7 9.659 8.378 38 87835.6 54-578 47-341 83 13612.2 8.458 7-337 39 86629.6 53.829 46.691 84 "675.5 7-255 6.293 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 urn 3-632 3-151 43 81544.2 50.669 43-950 88 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. I nautical mile* ^6080.27 feet. Nautical Miles. Statute Miles. Statute Miles. Nautical Miles. 1 2 3 4 5 6 I 9 1.1516 2.3031 3-4547 4.6062 5-7578 6.9093 8.0609 9.2124 10.3640 1 2 3 4 5 6 9 0.8684 1.7368 2.6052 34736 4-3420 1.0788 6.9472 7.8155 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 * ^ J vara *Klafter, Wiener (Vienna) *Line, old French = -j-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 *Ruthe, Prussian, Norwegian Sagene, Russian *Toise, old French ^6 feet *Vara, Spanish Mexican Werst, or versta, Russian =: 500 sagene Metric Equivalent. metre. I 1.7814 0.31608 ' 0.32484 ' 0.30480 ' 0.31385 ' 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 " 0-8359 " 0.8380 " t.o668 km. metre. 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 " t 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 + 0.05221 sin* ^ 1=9.80599 — 0.02610 cos 2^ metres.* r= geographical latitude. <(> g log^ log- e,^g logV?j- Metres. Metres. 0° 9.7798 0-99033 8.70864-10 0.64568 0.99090 S .7803 035 862 569 095 10 .7814 040 857 572 106 >s .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 207 690 655 488 65 .8227 223 674 663 525 70 .8261 238 659 670 559 75 .8286 249 64S 676 584 80 .8306 258 639 680 60s 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 Office, xSgx. t This 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 ^ SUver .... Steel, hard . . Steel, soft . . . Tin Zinc Expansion per degree C. 20 19 17 9 IS II 12 28 9 8.7 19 12 II '9 29 Expansion per degree F. 10.S 9.4 6.1 6.7 iS-S 4.8 I0.S 6.7 6.1 10.5 16.1 Smithsonian Tables. 1 Of International Prototype Metres. Table 42. FRACTIONAL CHANCE IN A NUMBER CORRESPONDING TO A CHANCE IN ITS LOGARITHM. Computed from the formula, AAf _ A log AT N ~ |U. ' fi = modulus of common logarithms ^ 0.43429448. For A log .A'' = I unit in A^ N For A log N_ = 4 units in AAT N (in round numbers) 4th place 5th " 6th " 7th « 494l44i. 4th place Sth " 6th " 7th " 10 44 10406 146'446 164^444 Smithsonian Tables. 170 APPENDIX. CONSTANTS. Numerical Constants. Number. Logarithm. Base of natural (Napierian) logarithms, = ,• = 2.7182818 0.434294s Log e, modulus of common logarithms, = M = 0.4342945 9.6377843- 10 Circumference of circle in degrees, 360 2.5563025 " " in minutes. = 21600 4-3344538 " " in seconds, = 1296000 6.1126050 Circumference of circle, diameter unity. = •■ = 3.14159265 0.4971499 Number. Logarithm. 2ir = 6.2831853 0.7981799 l/ir^ = 0.1013212 9.0057003- ■10 — = 1. 047 1 976 0.0200286 Vx = 17724539 0.2485749 3 -^=0.5641896 9.7514251 - ■10 i-= 0.3183099 9.5028501 — 10 X V 7= 1.4142136 0.1 505 1 50 ifi = 9.8696044 0.9942997 V 3= 1.7320508 0.2385607 The arc of a circle equal to its radius is in degrees, p° = i8o/ir 57.29578° I. 7 58 I 226 in minutes, p = 60 p° = 3437.7468' 3-5362739 in seconds, f" = 60 p' = 206264.8" 5.3144251 For a circle of unit radius, the arc of 1° = i/p° = 0.0174533 8.2418774- -10 arc of i' = i/p' = 0.0002909 6.4637261 — -10 arc (or sine) of i"= i/p" = 0.00000485 4.6855749- -10 Geodetical Constants. Dimensions of the earth (Clarke's spheroid, 1866) and derived quantities. Equatorial semi-axis in feet. = a = 20926062. 7.3206875 in miles. = a = 3963.3 3.5980536 Polar semi-axis in feet. = *= 208551 21. 7.3192127 in miles. = * = 3949-8 3-5965788 (Eccentricity)^ = " ~ «2 = 0.00676866 7.8305030- ■10 Flattening = ^^:^^ =/= 1/294-9784 7.5302098- -10 Perimeter of meridian ellipse, = 24859.76 miles. Circumference of equator. = 24901.96 * Area of earth's surface. = 196940400 square miles. Mean density of the earth (Harkness) = 5.576± 0.016. = 2.56 ± 0.16. Surface density " Acceleration of gravity (Harkness) : g (cm. per second) = 980.60 (i —0.002662 cos 2^) for latitude ip 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) / = 39.012540 -f 0.208268 sin'' ^ inches = 0.990910 -f 0.005290 sit 2 ^ metres. Smithsonian Tables. 171 APPENDIX. CONSTANTS. -Continued. Astronomical Constants (Hakkhess). Sidereal year = 365.256 357 8 mean solar days. Sidereal day = 23* 56»« 4.1100 mean solar time. Mean solar day == 24^ 3»< 56.'i546 sidereal time. Mean distance of the earth from the sun — 92 800 000 miles. Physical Constants. Velocity of light (Harkness) = 186 337 miles per second = 299 878 km. per second. Velocity of sound through dry air = 1090 v'l + 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. 4° C. 62° F. 4° C. I cubic inch (determination of 1890) 252.286 252.568 16.3479 16.3662 I cubic centimetre (1890) IS-39S3 ^SAT-^S 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 its weight of carbonic acid) : I cubic centimetre at temperature 32° F. and pressure 760 mm. and under the standard value of gravity weighs o.ooi 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.) : 0.003 670. Coe6Scient of expansion of mercury for Centigrade temperatures (Broch) : A = Aj, (i — 0.000 i8i 792 t — 0.000000000 175 <^ — .000000000035 "6 fi). Coefficient of linear expansion of brass for 1° C, p = 0.0000174 to 0.0000190. Coefficient of cubical expansion of glass for 1° C, y = 0.000 021 to 0.000 028. Ordinary glass (Recknagel) : at 10° C, y = 0.000 0255 ; at 100°, y = 0.000 0276. Specific heat of dry air compared with an equal weight of water : at constant pressure, £}> = 0.2374 (from 0° to 100° C, Regnault). at constant volume, Ji^v — 0.1689. Ratio of the two specific heats of air (Rontgen) : A}) /jt» = 1.4053. Thermal conductivity of air (Graetz) : i — 0.000 04S4 (i A- 0.00 1 %t,f, C ) ^''"'"'°' [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 niass degrees, C. Latent heat of vaporization of water = 606.5 — 0.695 '° C. Absolute zero of temperature (Thomson, Heat, Encyc. Brit.) : — 273.°o C. = 4S9.°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. I calorie or kilogramme-degree, C. = 3087 foot-pounds = 426.8 kilogram- metres = 4187 joules (for ^ = 981 cm.). Smithsonian Tables. * Based on Prof. Rowland's determinations, (Proc, A m, Acad. Arts and Set* t88o.) 171 APPENDIX. SYNOPTIC CONVERSION OF ENGLISH AND METRIC UNITS. English to Metric. Units of length. 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. Units of volume. I cubic inch. I cubic foot. I cubic yard. Metric equivalents, 2.54000 centimetres. 0.304801 metre. 0.914402 " 1.60935 kilometres. 6.45163 929.034 0.836131 0.404687 2.59000 259.000 16.3872 0.028317 0.764559 Units of capacity i I gallon (U. S.) = 231 cubic inches. I quart (U. S.). I Imperial gallon (BritisW. 277.463 cubic inches (1890). I bushel (U. S.) = 2150.42 cubic inches. I bushel (British). Units of mass. I grain. I pound avoirdupois. I ounce avoirdupois. I ounce troy. 1 ton (2240 Ibs.^. 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 ^'^- Per hr. 64.7990 0-453593 28.3496 31-1035 1. 01 605 0.907186 square centimetres. II (I 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. 0.404 835 9.484016 — 10 9.961 137 — 10 0.206 650 0.809 669 2.968 032 9.922 274 — 10 9.607 120 — 10 0.413 300 2.413 300 1.214504 8.452 047 — 10 9.883 41 1 — 10 0.578 116 9.976056 — 10 0.657 709 1.547027 1.560477 1.811 568 9.656 666 — 10 1.452 546 1.492 810 0.006914 9.957 696 — 10 Units of force. I poundal. Weight of I grain (for^ = 981 cm.). Weight of I pound av. (for ^= 981 cm.). 13825.5 dynes. 63.57 dynes. 4.45 X 10' dynes. 4.140682 1.803 237 5-648 335 1.846997 0.688 634 5.624 698 Units of stress— In gravitation measure. I pound per square inch = 70.307 grammes per sq. centimetre. I pound per square foot = 4-8824 kUogrammes per sq. metre. Units of woric — in absolute measure. I foot-poundal. 421 403 ergs. — 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). 1 foot-pound per minute (fori-= 981 cm.) = 0.022605 watts. I horse-power (33 000 foot-pounds per mm.) = 746 wa s = 1.01387 force de cheval. Units of heat. I pound-degree, I''. I pound-degree, C. = 252 small calories or gramme-degrees, C. = 1.8 pound-degrees, F. SmTHSONiAM Tables. 173 APPENDIX. SYNOPTIC CONVERSION OF ENGLISH AND METRIC UNITS. Metric to English. English equivalents. Logarithms. Units of length. I metre (lo' microns). 39.3700 inches. 1.59516s li 3.28083 feet. 0.515984 u 1. 09361 yards. 0.038863 I kilometre. 0.62137 miles. 9-793350—10 Units of area. I square centimetre. 0.15500 square inches. 9.190 331 — 10 I square metre. 10.7639 square feet. 1. 03 1 968 (( ti 1-19599 square yards. 0.077 726 I hectare. 2.47104 acres. 0.392 880 I square kilometre. 0.38610 square miles. 9.586701 — 10 Units of volume. I cubic centimetre. 0.0610234 cubic inches. 8.785496—10 I cubic metre or st6re. 35-3145 cubic feet. I -547 953 (( .1 (1 1-30794 cubic yards. 0.116589 Units of capacity. I litre (61.023 cubic inches). 0.26417 gallons (U. S.). 9.421884— 10 (. 1.05668 quarts (U. S.). 0.023944 u 0.21993 Imp. gallons (British). 9.342 291 — 10 || I hectolitre. 2.83774 bushels (U. S.). 0.452973 it 2.75121 bushels (British). 0-439523 Units of mass. I gramme. 15.4324 grains. 1.188433 I kilogramme. 2.20462 pounds avoirdupois 0-343334 u 35-2739 ounces avoirdupois. 1-547 454 tl 32.1507 ounces troy. 1.507 190 I tonne. 0.98421 tons (2240 lbs.). 9.993086—10 tf 1.10231 tons (2000 lbs.). 0.042304 Units of velocity. I metre per second. 3.2808 feet per second. 0.515984 (( tl tl 2.2369 miles per hour. 0-349653 I km. per hr. (0.2778 m. per sec). 0.62137 miles per hour. 9-793350-10 Units of force. I dyne (weight of (981)-^ grammes, for^=98i cm.) = 7.2330 X 10- -6 poundals. Units of stress — In gravitation measure. I gramme per square centimetre. 014223 pounds per sq. inch. I kilogramme per square metre. 0.204817 pounds per sq. foot. I standard atmosphere. 14.7 pounds per sq. inch. (See def. p. 172.) Units of work — In absolute measuri . I erg. 2.3730 X lo"' foot poundals. I megalerg = 10^ ergs ; i joule = 10' ergs. — In gravitation measure. I kilogramme-metre (for ^= 981 cm .) = 981 X 106 ergs = 7.2330 foot-pounds. | Units of activity (rate of doing work). I watt = I joule per sec. (= 44.2385 foot-pound s per minute, for g = 981 cm.) = 0.10194 kilogramme-metre per sec, for J- =981 cm I force de cheval = 75 kilogramme-metres per sec. = 735I watts = .98632 horse-power. Units of heat. I calorie or kilogramme-degree = 3 968 pound-degrees, F. — 2.2046 pound-degrees, C. 11 I small calorie or therm, or gramme-degree = 0.001 calorie gr kilogramme-degree. SuiTHSONiAN 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. [L^] Moment of Inertia. [M] Force. [M L""S] Stress (per unit area). [LT-i] Work or Energy. [LT~2] Rate of Working {Power). [L^UT^] [o] Heat. [L2 M T-2] [1—1] Thermal Conductivity. [L~i M T~i] Dimensions. [L M T-i] [ML2] IL M T-2] [L-i M T-^] [L2 M T-2] In Electrostatics. Quantity of Electricity. Surface Density: quantity per unit area. Difference of P o t e n t i a 1 : quantity of work required to move a quantity of electricity ; (work done) -^ (quan- tity moved). Electric Force, or Electro-motive Intensity: (quantity) -^ (distance^). Capacity of an accumulator : e -i- £. Specific Inductive Capacity. In Magnetics. Quantity 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 ; (work done) -i- (quantity moved). Magnetic Inductive Capacity. Symbol. e a E Dimensions in electrostatic system. [L^ M* T^i] [L-* M* T-i] [Li mJ t-1] F [L-» Ml T-i] C or q k [L] [0] m S Dimensions In electro-magnetic system. [L^ M* T-i] [L-^ M* T-i] ml r [L-*M*T 1] [l5 M* T-1] [L-^ M* T-1] In Electro-magnetics. Symbol. Intensity of Current. i Quantity of Electricity conveyed by current : e (intensity) X (time). Potential, or difference of potential: (work E done) -f- (quantity of electricity upon which work is done). Electric Force; the mechanical force act- IE [L* M* T"^] ing on electro-magnetic unit of quantity; (mechanical force) -f- (quantity). Resistance of a conductor : E-^i. R \L. T" *] Capacity: quantity of electricity stored up q \yr^ T^] per unit potential-difference produced by it. Specific Conductivity: the intensity of [L"^ T] current passing across unit area under the action of unit electric force. Specific Resistance: the reciprocal of r [L^ T~i] specific conductivity. Vox a [L*M*T-i] M [o] Dimensions In Name of electro-magnetic practical unit system. [L* MJ T-1] AmpSre. [L* M*] Coulomb. [L» M* T-2] Volt. Ohm. Farad. Smithsonian Tables. 175 INDEX. PAGB Acceleration, dimensions of 175 of gravity, formula for 171 table of values of 169 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 Ivi Aluminum, linear expansion of 170 Ampere, dimensions of 175 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 171-175 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 Ixvu latitude xliv time Ixxii Astronomy Ixvii-lxxxii references to works on Ixxxii Atmosphere, mass of earth's Ixvi standard pressure of 172 vpeight 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 PAGB Babinet, barometric formula of , . . 160 Barometer, heights by 160 Binomial series xiv Brass, linear expansion of 170 Briinnovir, 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 xli 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 , Ixvi Coefiicient, 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 Iviii of mean and probable errors xcv Conductivity, thermal, of air 172 Cone, surface of xxxi volume of xxxii Constants, astronomical 172 geodetical 17I numerical 17I 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-Ivi table of, scale 1/250000 84-91 table of, scale 1/125000 92-101 £78 INDEX. Co-ordinates {continued). table of, scale 1/126720 102-109 table of, scale 1/63360 1 10-121 table of, scale 1/200000 122-131 table of, scale 1/80000 132-141 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 30> 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 160 by trigonometric leveling Ixi Differences, of latitude, longitude, and azi- muth, on earth's spheroid Iviii table for computation of 70-77 Differential 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 Ixvi 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 .Hi 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 168 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 xv' 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.lviii-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 vaporization of water 172 mechanical equivalent of 172 Hectare xli Heights, average, of continents Ixv determination of, by 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 English and Metric units 2,3, 173, 174 of sidereal and solar time Ixxiii tables for 164, 165 Iron, linear expansion of 170 Joule, value of I74 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 '68 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 Ivil 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 i6g Leveling, trigonometric Ixi Line (French), value of 168 Lines, lengths of xxviii on a spheroid Ivi Linear measures, British xxxvii Metric xli tables for interconversion 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 xxxvij 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 analogies xx Natural logarithms xiv Nautical mile, table of equivalents in statute miles 168 Numerical constants 171 Ohm, dimensions of 175 Palm, length of, English 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 i erimeter, 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 xzx 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 x] 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 S7-6S 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 Ratio, 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 168 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. l8l 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) . . . . 1 viii 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 Stire 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 'xv of sphere and spheroid xxxi Surveyor's chain, length of xxxvni Table for conversion of arc into time 162 conversion of mean into sidereal time . . 164 conversion of sidereal into mean time. .165 conversion of time into arc io3 determination of heights by barometer. . 160 interconversion of British and Metric units 2, 3, 173. 174 interconversion of nautical and statute 168 miles Table of acceleration of gravity and derived quantities 169 Table of {continued). antilogarithms, 4-place 26, 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 1 72 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 84-91 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, .xix 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 i65 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, 6g 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 of {continued). British, of length xxxviii British, of surface xxxviii British, of weight xxxix Metric xli tables for interconversion of A 3. 173. "74 natural cosines 28, 29 natural tangents 3°) 3' radii of curvature, logarithms of, for meridian section 48-55 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 (i — 2m) and log (1 — m) used in trigonometric leveling . . .Ixil weights and measures (see Taile 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-xxvli 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 2, 3, 173, 174 Werst, Russian 168 Work, dimensions of 175 Wright, Prof. T. W., treatise cited xcviii Yard, imperial xxxiv ratio of, to metre xxxvi, xxxvii Zachariae, G., treatise cited xlvi Zenith distances, use of, in trigonometric leveling Ixi Zenith telescope, use of Ixxix Zero, of absolute temperature 172 Zinc, linear expansion of 170 Zones, of earth's surface, area of 1