.Division Range Shelf Received 1S7// No. 12. PROFESSIONAL PAPERS OF THE CORPS OF ENGINEERS OF THE UNITED STATES ARMY. PUBLISHED BY AUTHORITY OF THE SECRETARY OF WAR. HEADQUARTERS CORPS OF ENGINEERS. 1873. TABLES AND FORMULAE SURVEYING, GEODESY, PRACTICAL ASTRONOMY, ELEMENTS FOR THE PROJECTION OF MAPS, INSTRUCTIONS FOR FIELD MAGNETIC OBSERVATIONS. v -t r ; ^.Cv^cv Third Edition, Revised and Enlarged. WASHINGTON: GOVERNMENT PRINTING OFFICE. 1873. OFFICE OF THE CHIEF OF ENGINEERS, Washington, February 24, 1873. GENERAL : In preparing this third edition of a volume com piled in 1849 for tne use of the Corps of Topographical Engineers, when an officer in that corps, I have made such additions and corrections as experience has suggested and the requirements of the service seemed to demand. Intended more especially for field-use by officers engaged in surveys or explorations, and aspiring to no further merit than that of accuracy, utility, and convenience, it is submitted with the hope that it may continue to be favorably received by those who may have occasion to refer to it. The revision was undertaken at the suggestion of others, and not without reluctance. I beg indulgence for its defects. Very respectfully, THOS. J. LEE. Brigadier-General A. A. HUMPHREYS, Chief of Engineers, United States Army. In the additions and corrections introduced into this volume, besides the authorities named in the text, I have availed myself, in the article on the Ganging of Rivers, of notes by MAJOR ABBOT, Corps of Engineers, on the practical gauging of rivers, printed in the proceedings of the Essay ons Club at Willefs Point. The article on Trigonometrical Leveling is taken principally from Appendix No. 7 of the Coast-Surjey report for 1868, by ASSISTANT R. D. CUTTS, United States Coast Survey. I have also made use of Appendices Nos. 9, io, and II of Coast-Survey report of 1866, 07 ASSISTANT C. A. SCHOTT, United States Coast Survey, in the revision of the portions relating to the use of the transit-instrument, of the zenith-telescope, and the determination of astronomical azimuths. The article on Longitude by Lunar Culminations was prepared in 1858 /I?/ the Topographical Bureau, by PROFESSOR BARTLETT, United States Military Academy. I am indebted to CAPTAIN ERNST and LIEUTENANT MERCUR, Corps of En gineers, and to ASSISTANTS WOODWARD and WRIGHT, Survey of the Lakes, for suggestions and corrections ; & LIEUTENANT MERCUR for the article on Probable Errors, &c.; and to CAPTAIN RAYMOND, Corps of Engineers, for the valuable contribution on Magnetic Field Observations, -which forms the Appendix; and I have to regret that LiEUTENANT-CoLONEL WILLIAMSON was prevented by absence from the country from preparing a suitable set of hvpsometrical tables, which he had consented to do. T. J. L, CONTENTS. PART I. MISCELLANEOUS. Page. Trigonometry, equivalent expressions 3 Solution of plane triangles 5 Solution of spherical triangles 6 Multiple arcs -. 7 Trigonometrical lines 8 Differentials of trigonometrical lines 8 Arcs in parts of radius 9 Signs of trigonometrical lines .. 9 Weights and measures of the United States 10 Miscellaneous measures 12 Weights and volumes of different substances 13 Component parts of the Army ration 15 Metric system of weights and measures 16 Metrical equivalents 17 Foreign measures of length 18 Foreign itinerary measures 19 Table for converting metres into toises and French and English feet.. 20 Table for converting English feet into toises, metres, and French feet. 21 Analytical expressions for lines, surfaces, and solids 22 Progression , 26 Force of gravity , 26 Land-surveying with compass and chain 27 Traverse table 29 Table for converting chains into feet, and vice versa 44 To trace railroad curves , 46 Gauging of rivers c^ Discharge of water through pipes 61 Four-place logarithms of numbers 62 Four-place logarithms of sines, tangents, &c 64 Table of squares and square roots from I to looo 68 Table to facilitate interpolation by differences 79 PART II. GEODESY. Reduction to center of station 81 Correction for phase 81 Spherical excess g 2 Reduction of bases 83 Correction for temperature in metallic rods 85 Vlll CONTENTS. Page. Measurement of distances by sound 86 Problem of the three points 87 Formulae for computing the principal geodetic quantities depending on the spheroidal figure of the earth 88 Bessel s magnitude and figure of the earth 1 89 Relative length of the yard and the metre - 90 Numerical values of Bessel s terrestrial elements in English yards 92 Constant logarithms useful in geodetic computations 92 Formulae for computing geodetic latitudes, longitudes, and azimuths. . 93 Measurement of distances, from astronomical determinations 95 Clarke s magnitude and figure of the earth 97 Forms for record and computation 98 Logarithmic values, in yards, of the normal in different latitudes 100 Logarithmic values, in yards, of the radius of curvature of the me ridian in different latitudes 103 Formulae for the polyconic projection of maps 106 Co-ordinates, in yards, for the projection of maps 108 Values of arcs of parallel, in yards 127 Values of meridional arcs, in yards 127 Lengths, in miles, of degrees of latitude and longitude 137 Co-ordinates, in statute miles, for the projection of maps 138 Trigonometrical measurement of heights . .... 14 Apparent and true level 144 Table of corrections for curvature and refraction 144 , Table for reducing inclined measures to horizontal 145 / Table for the ratio of slopes 145 Barometrical measurement of heights 146 Thermometrical measurement of heights 181 Table for converting Fahrenheit s scale of the thermometer to Reaumur s and the centesimal 182 PART III. ASTRONOMY. Of sidereal and solar time 185 -/ To find the time by an altitude of the sun or of a star 187 Horizontal sun-dial 188 Table for converting sidereal into mean solar time 19 Table for converting mean solar into sidereal time .., 191 Table for converting space into time 192 Table for converting time into space 196 Table for converting AR. in arc into mean time 1 98 Table for converting mean time into AR. in arc 201 Form for record and computation of the determination of the time by altitudes of stars 204 Computation of an example of this method 206 To find the time by equal altitudes of the sun 207 Table for determining the equation of equal altitudes 208 Computation of an example of this method 217 CONTENTS. IX Page. Form for record and computation of this method 218 Table of the sun s parallax in altitude 220 Table of decimals of an hour 220 The transit-instrument, correction to observed transits 221 Form for record and computation of observed transits 224 Example of the method of computing transit corrections 225 Table to facilitate the reduction of transit observations 226 Reduction of transits by least squares 228 Refraction 230 Table of mean refraction 23 1 Bessel s refraction table 235 To determine the latitude by meridional altitudes 237 To determine the latitude by circum-meridian altitudes 238 Tables for reduction to the meridian 240 Form for record and computation of the determination of the latitude by circum-meridian altitudes 248 Computation of some of the quantities in the preceding method 250 To determine the latitude by altitudes of circumpolar stars 251 Form for record and computation of the method of determining the latitude by altitudes of Polaris 252 Computation of an example of this method 254 To determine the latitude with the transit-instrument by transits of stars over the prime vertical 255 To determine the latitude- with the zenith and equal-altitude telescope. 258 Form for record and computation of this method 264 To find the azimuth of the sun or of a star 265 To find the amplitude of the sun or of a star 266 To determine the true meridian by equal altitudes of the sun 266 To find the azimuth of Polaris at its greatest elongation 267 Corrections to observed azimuths 268 Correction for run in reading microscopes 270 To determine the longitude by lunar culminations 271 To determine the longitude by the electric telegraph 280 Formulae for probable error and precision 288 Peirce s criterion for the rejection of doubtful observations 289 Geographical Positions 292 APPENDIX. Field magnetic observations , 295 ERRATA To the 3d edition of Tables and Formula, Professional Papers No. 12. Page 10. For See. XXXIV, read Sec. XXX VII. Page 1.4. Limestone, for 197.5, read 179.5, and for 11.355, read 12.462. Page 47. In Chord-Deflection, for is "therefore double", read "may be assumed as double" the tangent deflection. Page 85. Hassler s expansion of iron bar, for 0.000240687260, read 0.000250687260. Page 96. In value u" Cos. Z, strike out Cos 2 !/ from the numerator. Page 99. For log. rf M =2.9060529, read log. rf M=2.9060531. rfZ=2.7619618, log. JZ=2.761 9620. Page 134. In 4th column, 5th line of table, for 449.4, read 499.4. Page 135. In 7th column, 4th line of table, for 331.0, read 231.0. Page 142. For 0.000000667, read 0.0000000667. Page 159. Omit minus sign at foot of last column. Page I OS. Top line of last column for +0.0958, read +0.0956. Page 168. 10th line of table, for +0.9967, read +0.0967. Page 170. At head of table, read 1). ^60384.3 log. X &c. Page 229. In last line and last figure, for .06, read .09. Page 250. In lines 13, 15 and 18, increase the values by one tenth of a second. Page 268. In value of r in example, for 2m. 30.3s, read 2m. 20.3*. Page 288. For E, read e. Page 288. Forjp, p / etc., read p , p" etc. Page 289. For /ce>y, read <u. Page 290. For 1 and 10 in first example, read 1 and 11. ERRATA. Page 10. For "See XXXIV" read "See XXXVII." Page 85. Hassler s expansion of iron bar, for "0.000240687260" read 0.000250687260." Page 288. For E read e. Page 288. For/, /, &c., read/, /", &c. TABLES AND FORMULAE PART I. MISCELLANEOUS. TRIGONOMETRY. I. Equivalent Expressions. sin 2 x -f cos 2 x = i sin x = cos x tan x __cos# ~~ cotx = V I COS 2 X V i 4- cot 2 x = 2 sin J # cos J _ tan # V i + tan 2 # i cosec x ~ tan x = sin # cot x =:: i sin 2 x = i 2 sin 2 J # = cos 2 x sin 2 x tan # = sec x sin ic COS iC i cot x sin a V i sin 2 ,7 i cos 2 # sin 2 x sin 2 # i cos 2 a? TRIGONOMETRY. I . Equivalent Expressions C ontin u ed . cot x = tan x sec x = - cos x cosec x = sin .r versin x = i cos x = 2 sin 2 J x co-versin x = i sin x chord x = 2 sin J x sin (A i B) = sin A cos B i sin B cos A cos (A =L B) = cos A cos B ^ sin A sin B sin 2 A = 2 sin A cos A cos 2 A = 2 cos 2 A i = 12 sin 2 A = cos 2 A sin 2 A 2 cos 2 J A = i + cos A 2 sin 2 J A = i cos A tan A ^ tan B tan(Ad.B)= tan A tan B 1 1 cos A -j- cos A _ i cos A sin A sin A sin B = 2 sin J ( A i B) cos J (A =p B) cos A + cos B = 2 cos J (A + B) cos J (A B) cos A cos B = 2 sin (A -f B) sin J (B A) sin 2 A sin 2 B = sin (A -f B) sin (A B) cos 2 A sin 2 B = cos (A -f B) cos (A B) TRIGONOMETRY. I. Equivalent Expressions Continued. -n sin (A 4: B) tan A dL tan B = 1-, cos A cos B sin A sin B sin A + sin B _ tan \ (A + B) sin A sin B ~~ tan f(A B) "~ A =tan 2 (4 5 dLJA) sin A sin A cos A JA) 1 1 . Solution of Plane Triangles. In the following formulas, A, B, C, represent the angles, and a, ft, c, the sides opposite, respectively. i. Any plane triangle: sin A _ sin B _ sin C a b c tan_JJA_+B) __ cot ^ C tan I (A -^B)" "~ tan J (A - B) cos A A ( s (s a] } = J -- i I ( ^<r ) . 2 TRIGONOMETRY. II. Solution of Plane Wangles Continued. 2. Right-angled triangles : Making A = 90 in the preceding, they become a 2 = b~ + ^ b a sin B = # cos C, c = (i sin C = a cos B tan B=- c tanC = - HI Solution of Spherical Triangles. .a, b, c, represent the arcs, and A, B, C, the angles opposite, i. Oblique spherical triangles: sin A _sinJB _ sin_C ~sm"rtr"~ smT ~~ sin c cos b sin (c + y) ^ cos a = - ^ - ^ cot 9" = tan b cos A cos B sin (C c) - cot (f = tan B cos a sin (C + ?) cot a tan b = sin cot c? = cot A v cos b Napier s Analogies. cosJ(A-B) tan J (<z + /;) = tan J- ^ AB sinJ(A-B) tan 4 ( - ^) = tan J ^ sJjTf (A + B) cos J( ^) tan J (A + B) = cot J C ^J(^fT) sin J( -^) tan i (A - B) = cot J C ^y ( ^- + ^ ) TRIGONOMETRY. III. Solution of Spherical Triangles Continued. sin S sin (A S) sin2 J" = sin B sin~CT- cos a = sin B sin sin (B-S)sin (C-S) sin B sin C~ 2 sin S sin (A S) tan * a ~ "sin (B-S) sin (C - S) s- - sin A= sin b sin c sin s sin (s a] cos * A = = 7^ sin b sin c 2 sin (s b) sin (.r c} tan J A = sin s s i n ( j 0) In which S and s represent the half-sum of the three an gles diminished by 90 and the half-sum of the three sides, respectively. 2. Right-angled spherical triangles, a being the hypothenuse: cos a = cos b cos c cot B = cot b sin c cos a = cot B cot C cot C = cot c sin b cos B = sin C cos b tan b = tan B sin c cos C = sin B cos c tan c tan C sin b tan b = tan a cos C sin b = sin sin B tan c = tan cos B sin c = sin <? sin C \\.-Multiple Arcs. sin 2 a? = 2 sin # cos ,r sin 3 x = 2 sin x . cos 2 x -{- sin # cos 2 x = 2 cos .r . cos # i cos 3 x = 2 cos .r . cos 2 .r cos j; 2 tan x tan 2 .r = i tan 2 ,r i tan x . tan 2 .r TRIGONOMETRY. V. Trigonometrical Series. A3 A 5 A 7 sin A = A-- A +^ A __ _+etc. 2-3 2,3.4.5 2.3 7 A 2 A 4 A 6 cos A = i - A+_l_ i + etc. 2 2.3.4 2 6 tan A = A + ^+-^+ V+ etc. 3 3-5 3 2 -5-7 A 4- s * n3 A -3 sm5 ^ 3*5 - n ~- 4- etc 2.3 "^2.4.5 " " 2.4.6.7 = tan A i tan 3 A + J tan 5 A l tan 7 A. + etc. -.9. A. log sin A = lc M = logarithmic modulus = 0.43429 45 log M = 9,63778 43 IJ 3 Differentials of Trigonometrical Lines. d sin # = + d x cos x d cos ^ = d x sin # cos^ x d cot x = . ~ sm- 5 x d sin 2 j? = 4- 2 d # sin # cos = 2 d j? sin ^ cos 2 d # tan x cos 2 # d tan 2 x = + 2 d x cot ,o d COt 2 Xi= sin 2 x TRIGONOMETRY. VI. Ratio of the Circumference of a Circle to its Diameter. ?r = 3. 14159 26535 898 log 71 = 0.49714 98726 941 The radius being unity, the number of degrees in an arc equal to radius =r=^- = < D = 57- 2 957 8 = 57 i7 / 44 // -8. TT arc i The number of minutes = r 1 =- - = r , or sin i ,,_6 4 8ooo"_ The number of seconds r" =" H 21_ = .-._* - sin i" log r = 1.75812 26324 09172 comp log r =8.24187 73675 90828 logr / = 3 . 53627 3 882 7 9 2Sl6 comp log / = 6.46372 61172 07 1 84 = log sin i log;-" = 5.31442 51331 76459 comp log r" 4.68557 48668 23541 = log sin i" Let a be the length of an arc of a circle whose radius is i , and a" the number of seconds in that arc, as r 1 -J: and R : r" : : a : a" or a" = r" a; a = a" sin i" sin i" In an equation, therefore, any arc a of a circle whose radius is i is expressed in seconds by changing a into a" sin \" . Signs of Trigonometrical Lines. Quadrants. Sin. Cos. Tan. Cot. Sec. Cosec. 1, Si 9. C + + + + + + j 2, 6, 10, N + - + ( 3 7 " ) - - + 4- - C 4, 8, 12, &c. C - + - - + - > 10 WEIGHTS AND MEASURES. VII. Weights and Measures of the United States. The standards of length and weight of this country and Great Britain are theoretically identical. The United States gallon and bushel represent old English measures. The standard of linear dimensions, adopted by the Treasury Department in the construction of standards for distribution to the custom-houses and States, is a brass scale of 82 inches in length, made in London by Troughton, which formed part of the instruments collected in 1815 by Mr. Hassler for the Survey of the Coast, and was supposed identical with the Schuckburg scale, one of the old English standards. The standard temperature is 62 Fahrenheit, and the yard-measure is between the 271)1 and 63d inches of its scale. This length has not been legalized by act of Congress. (See XXXIV.) Linear Measure. The unit of linear measure is the yard. The yard is divided into 3 feet, and the foot subdivided into 1 2 inches. The multiples of the yard are \hzpole or perch, \hefurlong, and the mile; but the pole and furlong are now scarcely ever used, itinerary distances being reckoned in miles and yards. The following are the relations : Inches. Feet. Yards. Poles. P\irlongs. Miles. , 0.083 0.028 o. 00505 o. 00012626 0.0000157828 12 i. -333 o. 06060 0.00151515 o. 00018939 36 3- i. o. 1818 o. 004545 0.00056818 I 9 8 16.5 5-5 I. o. 025 0.003125 7920 660. 220. 40. i. o. 125 63360 5280. 1760. 320. 8. I. log 5280 = 3.7226339 log 1760 = 3.2455127 WEIGHTS AND MEASURES. I I VII. Weights and Measures of the United States Continued. Square Measure. In square measure the yard is subdivided, as in general measure, into feet and inches ; 144 square inches being equal to a square foot. For land-measure the multiples of the yard are the pole, the rood, and the acre. Very large surfaces, as of whole countries, are expressed in square miles. The following are the relations of square measure : Sq. feet. Sq. yards. Poles. Roods. i. O. IIII 0.00367309 0.000091827 9- 0.0330579 0.000826448 272.25 30.25 i- 0.025 10890. I2IO. 4 o. * ,43500. 4840. TOO. 4- 27878400. 3097600. 102400. 2560. Measure of Capacity. The units of capacity measure are the gallon for liquid and the bushel for dry measure. The gallon is a vessel containing 58372.2 grains (8.3389 pounds avoirdupois) of the standard pound of dis tilled water, at the temperature of maximum density of water, *the vessel being weighed in air in which the barometer is 30 inches at 62 Fahrenheit. The bushel is a measure containing 543391.89 standard grains (77.6274 pounds avoirdupois) of dis tilled water, at the temperature of maximum density of water, and barometer 30 inches at 62 Fahrenheit. The gallon is thus the wine-gallon, (of 231 cubic inches,) nearly; and the bushel, the Winchester bushel, nearly. The temperature of maximum density of water was determined by Mr. Hassler to be 39.83 Fahrenheit. Acres. Sq. miles. 0.000022957 ! O.OOO2O66l2 ; 0.00625 0.25 I. 640. i . DRY MEASURES. LIQUIDS. ! Pint = F4- bushel. Gill = ^gall. Quart = 2 pints = T, 1 ^ bushel. Pint = 4 gills = i gall. Peck = 8 quarts = J bushel. Quart = 2 pints - igall. Bushel = 4 pecks = i bushel. Gallon = 4 quarts = i gall. Barrel = 3ii gallons = 3 ii galls. Hhd. = 2 barrels = 63 galls. The only legalized unit of weight or measure is a troy-pound, , I2 WEIGHTS AND MEASURES. Mil. Weights and Measures of the United States Continued. (act of May 19, 1828,) copied by Captain Kater, in 1827, from the imperial troy-pound of England, for the use of the Mint of the United States, and there deposited. This pound is a standard at 30 inches of the barometer and 62 of the Fahrenheit ther mometer. The standard avoirdupois-pound, as determined by Mr. Hassler, is the weight of 27.7015 cubic inches of distilled water. It is greater than the troy-pound in the proportion of 7000 to 5760; that is, the avoirdupois-pound is equivalent in weight to 7000 grains troy. Weights. AVOIRDUPOIS. TROY. Dram = ^ lb - i Grain = 7Tinr lb - Ounce =i6drs. = ^ lb. | Pennyweight = 20 grs. ,, ^ lb. Pound = 1 6 ozs. = I lb. Quarter = 25 Ibs. = 28 Ibs. Hundred-wt. = 4 qrs. == 1 12 Ibs. Ton 20 cwt. = 2240 Ibs. Short ton = 2000 Ibs. Ounce 24 dwt. = T V Pound = 12 ozs. == i Ib. VIII . Miscellaneous. /. Gunter s chain = 66 feet = 4 poles = 100 links of 7.92 inches. i fathom = 6 feet; i cable-length = 120 fathoms. i hand = 4 inches; i palm = 3 inches; i span = 9 inches. Solid. i cubic foot = 1728 cubic inches. i cubic yard = 27 cubic feet = 46656 cubic inches. i reduced foot (board-measure) = i square foot X i inch thick = 144 cubic inches. i perch of masonry = i perch (i6J feet) long X i foot high x ij foot thick = 24.75 cubic feet ^ 2 5 cubic feet has generally been adopted for convenience. i cord fire-wood = 8 feet long X 4 feet high X 4 feet deep = 128 cubic feet. i chaldron coal = 36 bushels = 57.25 cubic feet. Paper. 24 sheets = i quire. 20 quires = i ream = 480 sheets. WEIGHTS AND VOLUMES. IX. Weights and Volumes of various Substances. METALS. Substances. Cubic foot. Cubic inch. f Copper.. ..67 i Pounds. 488. 75 Pounds. .2829 f Zinc ."U \ 54?. 7? . 3147 547. 25 .3179 543. 625 .3167 450.437 .2607 466. 5 . 27 wrought bars .* - . ........... 486. 75 .2816 709. 5 .4106 rolled 711. 75 .4119 848. 7487 . 491174 Steel plates - .................. 487. 75 .2823 soft 489. 562 .2833 Tin 455. 687 . 2637 Zinc cast .. ........... 428.812 .2482 rolled . 449.437 .2601 WOODS. Substances". Cubic foot. Cubic feet in a ton. Ash . Pounds. 52.812 42. 414 Cedar K. O62 63. 886 Chestnut . 38. 125 58. 754 49. 5 45 2 5 2 shell-bark 41. I2S 51. 942 Lignum- vitcC 8l. ^12 26. 886 $ 35- 64- Oak Canadian . ...... \ 66.437 C4. c 33- 7H 41. 101 English ...... ...... ...... .... .... 58.25 38.455 live seasoned 66.75 11. ">S8 white dry .............. ... ci. 71; 41. 674 upland ....... ...... ...... ...... 42.937 52. 169 WEIGHTS AND VOLUMES. IX. Wrights and Volumes of "carious Substances Continued. WOODS Continued. Substances. Cubic foot. Cubic feet in a ton. Pine, yellow Spruce Walnut, black, dry Willow, dry Pounds. 33-812 3I-25 3 r -25 30- 375 66. 248 71.68 71.68 73-744 MISCELLANEOUS. Substances. Cubic foot. Cubic feet in a ton. Pounds. Air ; . 07529 r ! Brick, fire ; 137.562 16.284 mean i 102. 21.961 Coal, anthracite .. \ ^ " 24 f 8 i ) 102. 5 21.854 bituminous, mean 1 80. 28. cannel 94. 875 23. 609 Cumberland : 84.687 26.451 Coke | 62.5 35.84 Cotton, bale, mean ^ 14. 5 1 54. 48 !^ 20. 114. pressed < _ , ( 25. 89.6 Earth, clay j 120.625 18.569 common soil | 137.125 16. 335 gravel j 109.312 20.49 dry sand . ; 120. 18.667 lo <> s e 93-75 23.893 Granite, Quincy ! 165.75 ^-SH Susquehanna , 169. 13. 254 Limestone j 197.25 H 355 Marble, mean 167.875 13. 343 Mortar, dry, mean 97. 98 22. 862 Water, fresh 62.5 35-84 salt ! 64.125 34. 931 Steam .036747! ARMY-RATION. X. The Army -Ration. TABLE SHOWING THE WEIGHT AND BULK OF IOOO RATIONS. One thousand rations of Net weight. Gross weight. Bulk. 100 rations consist of Pork Pounds, 750. 750. IMS- 750- IOOO. 155. 100. 100. 80. 150. 92.5 15. 40. " 33.75 Pounds. 1218.75 903. 19 1234.06 921.69 1228.91 177.32 114.50 122. 108. 161. 107. 50 17.5 46.89 38.63 Barrels. 3-75 4.90 5-74 9.03 12.05 0.71 0.46 0.65 0.83 0.6 0-33 0.09 o. 19 o. 16 75 Ibs. or i 75 Ibs. } 112.5 Ibs. or 75 Ibs. or I oo Ibs. in the field 8 quarts, or | 10 Ibs. ) 10 Ibs. 8 Ibs. 15 Ibs. 4 quarts. i| Ibs. 4 Ibs. 2 quarts. 1 .1 Bacon Flour Pilot-bread Do Beans Rice Coffee, green .. roasted. Sugar ..... Vinegar Candles.. . . Soap .. Salt Forage. 1 4 Ibs. hay or fodder) , (hay, when pressed, 1 1 Ibs. to cub. ft. / per horse \ *\ c 12 quarts oats, or > ^32 Ibs. to bushel, 25.71 to cub. ft. 8 quarts corn ) pei ay (56 Ibs. to bushel, 45.02 to cub. ft. Three beeves or 15 sheep consume the forage of 2 horses. Wheat , 60 Ibs. | Corn and rye 56 Ibs. J Weights of Grain per Bushel. Oats ....... Barley 32 Ibs. 48 Ibs. A box 1 6 x 1 6. 8 x 8. inches contains i bushel \ 12 x 1 1. 2 X 8. " bushel > dry measure. 8 x 8.4 x 8. " i peck ) 6 X 6 x 6.4 " i gallon 4 X 4 X 3.6 " i quart i6 METRIC SYSTEM. XI. Metric System. By an act of Congress, approved July 28, 1866, the metric system of weights and measures is made optional in the United States; and the act provides that the tables in a schedule an nexed shall be recognized "as establishing, in terms of the weights and measures now in use in- the United States, the equivalents of the weights and measures expressed therein in terms of the metric system; and said tables may be lawfully used for computing, de termining, and expressing, in customary weights and measures, the weights and measures of the metric system." Schedule annexed to act of July 28, 1866. MEASURES OF LENGTH. Metric denominations. Values in metres. Equivalents in denominations in use. Myriametre . ... 6.2137 niiles*. Kilometre IOOO. 0.62137 mile, or 3280 feet and 10 inches. Hectometre ,00. 328 feet and i inch Decametre Metre :: 393.7 inches. 39.37 inches. Decimetre O.I 3.937 inches. Centimetre Millimetre MEASURES OF SURFACE. Metric denominations. Values in square metres. Equivalents in denominations in use. Hectare IOOOO 2.471 acres. Are 119 6 square yards. Centare MEASURES OF CAPACITY. Metric denominations and values. Equivalents in denominations in use. Names. No. of litres. Cubic measure. Dry measure. Liquid or wine measure. Kilolitre or stere. Hectolitre Decalitre IOOO. IOO. IO. O. I 0.01 O.OOI i cubic metre o.i cubic metre 10 cubic decimetres i cubic decimetre . . o.i cubic decimetre 10 cubic centimetres i cubic centimetre . 1.308 cubic yards . 2 bus. and 3.3spks. 9 08 quarts 264.17 gallons. 26.417 gallons. 2.6417 gallons. 1.0567 quarts. 0.845 gill. 0.338 fluid-ounce. 0.27 fluid-drachm. Litre 0.908 quart 6.1022 cubic inches 0.6102 cubic inch. . 0.061 cubic inch. .. Decilitre Centilitre Millilitre. . METRIC SYSTEM. XI . Metric System Continued. WEIGHTS. Metric denominations and values. Equivalents in denom inations in use. Names. Number of grammes^ Weight of what quantity of water at maximum density. Avoirdupois weight. Millieror tonneau. . Quintal Myriagramme Kilogramme, or kilo Hectogramme ...... 1000000. IOOOOO. IOOOO. IOOO. i cubic metre 2204.6 pounds. 220 46 pounds. 22.046 pounds. 2.2046 pounds. 3.5274 ounces. 0.3527 ounce. 15.432 grains. 1.5432 grains. 0.1543 grain. 0.0154 grain. i hectolitre 10 litres i litre i decilitre Decagramme Gramme Decigramme Centigramme Milligramme 10. I . O. I 0.01 O^OOI 10 cubic centimetres i cubic centimetre . o. i cubic centimetre 10 cubic millimetres i cubic millimetre. . . . ADDITIONAL METRICAL KQUIVALEXTS. I surveyor s chain in metres.. =: 20.11662 log = 1.3035550 i metre in surveyor s chain .. = 0.04971 log 8.6964450 I square foot in square metres = 0.09290 log 8.9680221 I acre in hectares = 0.40467 log = 9. 6071100 I square mile in hectares = 258.994 log = 2.4132900 I square metre in square feet. I hectare in acres I hectare in square miles I0 - 7 6 4io log = 1.0319779 2. 47i 9 log o. 3928900 0.00386 log = 7.5867100 I cubic foot in sfceres = o. 0283 1 log = 8. 4520332 I cord in steres = 3.62445 log = o. 5592432 i stere in cubic feet = 35.31561 log = i. 5479668 I stere in cords = 0.27590. log = 9. 4407568 I gram in grammes = 0.064798 log = 8. 8115680 18 FOREIGN MEASURES. & t^ ^ ^- i? C? < ON ICg; CN o" VO ro O " , CO CN M CO -* ro^ !_/-) M OO N CO^ r O^ z - ^^"cc" ^ O ^ CNI O LO W r rj > 1 o ^ 1 O OO Q ci C O *: Co S 2"^ o" o ~ 3" 5 O O hH O M f,s co o hH 6 hH 6 hH O HH O HH HH hH . rt ^ HH 10 O O ON ON oo 1 ^" M !g CN ON 00 i vO 10 Of?^> X-0 ?J CN to CO ON ro ^ ll HH O di 5 CO hn O 0^ O ON C ON O CN ON O ON 1| CO t^ CO 10 N ON coco CN <J 00 ^ Jj CO "~> O o O O OO oo If it CN ON O o^ . CO ro O CO co r C? OO II K, CO 6 HH 6 HH 6 M O O ON ON hH O O CN rt > ^ r- N c^ OH CN IN t~ O CN CN HH Oj; N ON II 11 CN ro vo ro $1 .0 LO IH CN ^ CN ro hH ON &| p rj^-u-^tn HH O O O - 6 O HH CN CN cS CO 6 HH 6 HH 6 HH O HH HI hH ON O ^o O -* O t-^ HH O "^ CN f> CN^ ^0_ CO ON *~ VC? 3 CO ^ ^ ^ ^ CO o~ "^ oo HH O vo" 1 ^* M^ to^ OO vo rt ^ } S w o" do ?& ^^00 ON o^ C o ^ S ^ pq CO O HH O HH O HH ON M hH 5 ^ II I1 11 || 82 O <N % o" 1 rt * CN^ON CN ON cJ^ O o O o CN ON 1" *x, co O MO O ^ ON ON O ON hH O CN 5 . ci J| || - 029722 OI27I99 CN ON CN vo 984270 993 1 MI 0*3 || CN vo~ rt ^ S CO HH 6 i i O ON ON O CN hH O ON H *f m CO -* . < i- CNOO oo c? C4 t-. % CO 10 CO w O co CO gs o o :~ CO oo 00 r? Ss^ t^OO HH CO t^ <,. , ^ ON ON Scf CN ON oil 00 ON SI 00 g p^ g CO O O* ON ON ON O CN O CN O ON ON cu fO^o" CN g- M ct M o"oo" CO ir > CO ^" U 1! CN 00* CN *" 00 fo ci * ro CO -? O 10 CO CO 10 CO O ON ON C ON O ON ON C CN CN FOREIGN MEASURES. 19 <LV s X I- ^ 2 c rt rt || 1 a i if n if I . W t-*O QO> H- ( O H^O QC> OO> 1-nO S ri S ^ OaN^OvCOM OvoCOo Thm Cro co o co t^- ro ir "^r t> ^ *O vN O> "^ o M o C II r "> . 8 F cJ J O 3 "M cj O* O O MO M" O* CN| o M" o MO i ii t^co MosOn VOvo C^^t-rJ-o ON CN. o MO t^. o !> o O ON O o l-O o rt 1 1 CO o) ^" ro O^ -o co ^- cOco ^^ ^- O vo < MO Oooo Oooooooo y. s ? ^ PI i ~" t 4 1 1! - 1 !! I! 1 1 To it II - II I! It 1! II IS H "rt || C/5 VO O r^~ T}~ O O O O Tj" O CO O 8 rt Myriametre i oooo M. to i-O CO M M CO CO M O COro^-Coo CO cOo tj-i M M ^j. CO ^_ ^co^ O^ t^toCOoo Mro ^4: HH 04 1^.00 i>.CO M O ^CJ"O t^.CO \O ^ do do do do do do do Eng. stat. miles, jj Eng. stat. miles. Modern Roman mile = 0.925 j Portugal league = 3.841 Tuscan mile = 1.027 < Flanders league = 3. 900 Old Scottish mile Irish mile . ., = 1.127 = 1-273 Spanish common league . = 4. 214 Hungarian mile. French posting league .. --= 2.422 | S\yedish mile - 5.178 = 6.648 20 COMPARATIVE MEASURES. Table for converting Metres into Toises and French and English Feet and Inches. Metres. Toises. French. English. Feet. Inches. Lines. Feet. inches. I 0.51307 3 o 11.296 3 3.3708 2 1.02615 6 i 10.592 6 6. 7416 3 1.53922 9 2 9.888 9 10. 1124 4 2. 05230 12 3 9. 184 13 1.4832 2. 5 6 537 15 4 8.480 10 4- 8539 6 3. 07844 18 5 7.776 19 8. 2247 7 3-59I52 21 6 7.072 22 11-5955 8 4- 10459 24 7 6.368 . 26 2. 9653 9 4.61767 2 7 8 5.664 2 9 6-337* 10 5- T 374 30 9 4.960 3 2 1 9- 7079 20 10, 26148 61 6 9.920 65 ! 7-4158 30 15.39222 92 4 2.880 98 5- I2 37 40 20. 52296 123 i 7.840 131 2.8316 25.65370 153 ii o. 800 164 o. 5395 60 3 0. 78444 184 8 5- /^o 196 10.2474 70 80 35-9I5J9 4i 4593 2I 5 246 5 3 10. 720 3.680 229 262 7-9553 5.6632 9 46. 17667 277 8. 640 295 3-3/11 100 -0741 37 10 i 600 328 1.0790 200 102.61481 615 8 3. 200 656 2. 1580 300 153.92222 923 6 4. 800 984 3-2370 400 205. 22963 1231 4 6. 400 I 3 I2 4.3160 600 256. 53704 307. 84444 1539 1847 o 8.000 9. 600 1640 5.3950 1968 6.4740 700 800 359.15^85 410.45926 2154 2462 10 9 ! 1 1. 200 o. 800 2296 7. 5530 2624 8. 6320 900 46 1. .76667 2770 7 2. 400 2952 9.7110 IOCO 513.07407 3078 5 4. oco 3280 10. 7900 20CO 1026. 14815 6156 10 , 8. O;)0 6561 9. 5800 3000 1539.22222 9235 4 o. ooo 9842 8. 3700 4000 2052. 29630 12313 9 4. ooo i3 I2 3 7. 1600 5000 25 6 5.37037 15392 2 8.000 16404 5- 95o 6000 3078. 44444 18470 8 o. ooo 19685 4. 7400 7000 3591 51852 21549 i 4. ooo 22966 3- 53 8foo 4104. 5925.9 24627 6 8.000 26247 2. 32OO 9000 4617. 66667 27706 0. 000 29528 i. i 100 IOOOO 5130. 74074 30784 5 4.000 - | 32808 i 11.9000 COMPARATIVE MEASURES. 21 Table for converting English Feet into French Toises, Metres, and Feet. English feet. Toises. Metres. Ffench . Feet. . Inches. Lines. 2 O o. 15638 0.31276 0.46915 o. 30479 o. 60959 0.91438 o 2 II 3.114 10 ; 6. 228 9 9- 343 4 6 0.62553 o. 78191 o. 93829 I. 21918 1.52397 1. 82877 3 4 5 9 0.457 8 3-571 7 6. 685 8 9 I. 09468 I. 25106 1.40744 2. 13356 2.43836 2. 743 i 5 6 69. 799 7 ! 6 0.913 8 5 - 4.028 10 20 30 1.56382 3- 12764 4. 69146 3- 4794 6. 09589 9- H3 8 3 9 : 4 j 7. 142 18 : 9 ; 2.284 28 ; i 9.425 40 6. 25529 7. 81911 9- 3 82 93 12. 19178 15-23972 18.28767 37 6 ! 4.567 46 10 ii. 709 5^J 3 I 6.851 90 10. 94675 12. 5105? 14- 7439 2i.3356i 24- 38536 27-4315 65 i 8 L993 75 o i 9. 134 84 ; 5 4.276 IOO 200 3OO 15.63822 31- 2 7 6 43 46.91465 3. 47945 60. 95850 9L43835 93 9 ! 11.418 187 ; 7 10. 836 281 5 ; 10.254 40O c;oo 600 62. 55286 78. I9I08 93. 82929 121. 91780 152.39725 182.87670 375 3 9-672 469 i , 9.090 562 ii 8.508 700 800 900 109.46751 125.10572 140. 74394 2I3.356I5 243- 83559 274.31504 656 750 844 9 7. 926 7 7-344 5 6. 762 IOOO 20OO 3000 156.38215 312.76431 469. 14646 304. 79449 609. 58899 914.38348 938 1876 2814 3 6. i So 7 o. 360 10 6.539 4000 5000 6000 625.52861 781. 91076 938. 29292 1219.17797 1523. 97246 1828. 76696 3753 4691 5629 2 o. 719 5 6- 899 9 I 79 7000 8000 9000 1094. 67507 1251. 05722 1407.43937 2i33.56i45 2438. 35594 2743- !5 44 6568 756 8444 o 7. 259 4 1.438 7 7.618 IOOOO 1563-82153 3047. 94493 9382 ii 1.798 2 MENSURATION. j , XIV. Analytical Expressions for different Lines, Surfaces, and Solids. i. LINES. Ratio of diagonal to side of square = \/ 2 - = I-4H = Y-, nearly. Log i /2 = 0. 1 55 I 4997 8 Side of inscribed square : R : : \? 2 : i Side of inscribed equilateral triangle : R : : \/3 : i Side of inscribed regular hexagon = R Side of inscribed regular decagon = J R (- - 1 + Vs) = 0.618 R Circle. Ratio of circumference to diameter = 3.1415926 = f^ g, nearly. Length of an arc : = llo : r be!ng the radius of the circle and a the number of degrees in the arc ; or nearly = _ 8 c 1 c c being the chord of the arc, and c 1 (the chord of half the arc) = V i <? + ver sin 2 Ellipse. ly; a and b being the Circumference = ~ OS-" V4 ( a + I}-} near axes. Lengths of Circular Arcs, taking the Base of Segments as Unity. "x "33 ti p5 t^ c ^ "~Ti i I* t o - 1 1 oj ^5 J I-H > 3 3 1 J . OI I. OOO . II .032 .21 1.114 .; 51 1.239 .41 1.401 . O2 I. 000 . 12 . 038 i . 22 1.124 : 52 1.254 .42 1.418 .03 I. OOO .13 .044 i .23 1-135 : 53 1-269 43 1.437 .04 I. OOO . 14 .051 .24 1.147 ; 54 1-284 44 1.455 .05 I. 000 .15 59 -25 1-159 : 55 i-3 .45 1.474 .06 1. 006 .16 . 067 i . 26 1.171 .; }6 1.316 .46 1-493 .07 1.014 17 .075 ! .27 1.184 .; 57 L33 2 47 1.512 .08 1.018 .18 . 084 ;! . 28 1-197 ; 58 1.349 .48 I-53I .09 1.020 .19 .093 .29 i. 212 .; 59 i-3 66 49 J .55i . 10 I. 026 . 2O .103 . 30 1.225 .< 10 1.383 .50 I-57I MENSURATION. 23 XIV. Analytical Expressions, &c. Continued. 2. SURFACES. i. Triangle in terms of b A its base and its altitude = - 2 a b sin C two sides, and the included angle = its three sides = [s (s a) (s b} (s c} } * where A = the altitude; (7, b, c = the three sides; C = the angle included between a and b ; and a + b + c 2. Parallelogram in terms of its base and its altitude .............. = b A two sides and the included angle ........ = a b sin C two sides and their corresponding diagonal = 2 [s(s-a)(s-t)(s-c)]* where C = the angle included between two adjacent sides a, b ; <r=the diagonal opposite; and 2 3. Trapezium in terms of its two parallel bases and its altitude ....... = -A its two parallel bases, one of its oblique sides, and the angle between one of these bases and this side . . . = / sin C 2 where A the distance between the two parallel bases B, b; 7= the length of one of the oblique sides; and C = the angle between one of these bases and this side. 4. Any quadrilateral = half the product of its two diagonals multiplied by the sine of the included angle. MENSURATION. XIV. Analytical Expressions, &c. Continued. 5. Regular polygon where tan i8o c n = the number of sides; and a = the length of one of them. 6. Circle = - R 2 7. Ellipse = - a I) a and b being the semi-axes. 8. Right cylinder, exclusive of its bases . . . . = 2 - R A 9. Sphere = 4 - R 2 10. Zone = 4 - R 2 sin -J (L L) cos A (L 7 + I,) 1 1 . Right cone = - R L 12. Frustum of cone with parallel bases . . . = - / (R -f- ") where R and r the radii of the bases of these solids; and L and / = the lengths of their generating elements. 13. Spherical quadrilateral, formed by two parallels of latitude and two meridians = - Q o (M 7 - M) R 2 sin A (L 7 - L) cos -J (L 7 + L) where R = the radius of the sphere; L, L 7 = the latitudes of the bases of the zone, + when north, when south; and INI 7 , M = the longitudes of the extreme meridians of the quadrilateral, (M 7 M) being expressed in degrees and decimals. In the place of R, the normal N, of the mean latitude , can be used. MENSURATION. 2 5 XIV. Analytical Expressions, &c. Continued. 3. SOLIDS. 14. Prism ........... ........ = B A where B = the area of the base; and A = the altitude. 15. Rectangular parallelopepidon ...... = p x q X 1 Cube ................ =/ 3 where/, </, r, = the lengths of the three contiguous edges. "R A 1 6. Pyramid ................ = ------ The area, B, being found from Xo. 5. 17. Right cylinder ............. = - R 2 A 18. Right cone .............. = - R 2 A 19. Sphere ................ = A - R 3 20. Prismoid, or solid figure, similar to that which is formed in excavations or embankments of roads, terminated by parallel cross -sections. Solid content = area of each end, added to four times the middle area, and the sum multiplied by the length divided by 6, or where b = the breadth at the bottom of the cutting ; h = the perpendicular depth of cutting at higher end; h 1 =. the perpendicular depth of cutting at lower end; /= the length of the solid; and r = the ratio of the perpendicular height ot the slope to its horizontal base. 26 MISCELLANEOUS. X V . Progression . 1. Arithmetical: a = z ( 11 i ) d z = a + (n i)d z a z (? , a -4- z d = - n = , + i s = n i d 2 2, Geometrical : z = at" 1 n = log. .. H. 1 OI! . a = s a = z r (r i ) s r 1 i .$ z where # = the least term; ;- = the common ratio; z = the greatest term; n = the number of terms; d = the common cliff. ; s = the sum of the terms. XVI. Force of Gravity. The velocity acquired at the end of one second by a body falling in vacuo, at the level of the sea, in the latitude of Lon don = 32.1915 feet. The force of gravity at the latitude of 45 = 32.17 feet per second being represented by g, for any other latitude, /, g = g (j _ 0.002588 cos 2 /) If ^represents the force of gravity at the height //, and r the radius of the earth, the force of gravity at the level of the sea Length, in inches, of a pendulum vibrating seconds at the level of the sea : Equator .... =39.0152 I 1 London, lat. 51 31 - - 39.1393 New York, lat. 40 43 =39.1017 [| Spitzbergen, lat. 75 50 =39.2147 LAND-SURVEYING. XVII. Land- Survey ing with Compass and Chain. 2o calculate the Area or Content of Land. If the sum of each adjacent pair of distances perpendicular to a meridian (departures] assumed without the survey be multi plied by the northing or southing between them in succession round the figure in the same order, the difference between the sum of the north products and the sum of the south products will be double the area of the tract. The meridian distance of a course is the distance of the middle point of that course from an assumed meridian. Hence, the double meridian distance of the first course is equal to its departure. And the double meridian distance of any course is equal to the double meridian distance of the preceding course, plus its departure, plus the departure of the course itself, having regard to the algebraic sign of each. Then, to find the area 1. Multiply the double meridian distance of each course by its northing or southing. 2. Place all the //^products in one column, and all the minus product s in another. 3. Add up each column separately, and take their difference. This difference will be double the area of the land. In balancing the work, the error for each particular course is found by the proportion As the sum of the courses is to the error of latitude, (or de parture, ) so is each particular course to its correction. When a bearing is due east or west, the error of latitude is nothing, and the course must be subtracted from the sum of the courses before balancing the columns of latitude. And so with the departures. EXAMPLE. It is required to find the content of a piece of land, of which the following are the field-notes : Sta. Course. Dist. Sta. Course. Dist. i North 46^- west. 20. chains. 4 South 56 east . . 27.60 chains. 2 North 5if east. 13.80 chains. ! 5 South 33^ west. 18.80 chains. 3 East 21.25 chains. I 6 North 74! west. 30.95 chains. 28 LAND-SURVEYING. "5 ^ "^* XVII. Land- 1 1 Surveying, &c\ ( : : ?o OO LT) . . t^ o Continued. o oo oo Tf N ON N vO T}- CO >J-> N OO !>. O LO C\ N rA ro ii t~>. M : : ? ^? 00 ^ Q" i + -S 00 i- ON NT)- N ON t^ <N O O O N ro 1^- N ^f Q <; T X 10 OO CO OO CO C ti Tr ?* rj- O N v5 O CN o d s -O 00 N OO fO O ? 2 N 8 2 1 + + + 1 1 ri rt ^J 3 8 ^ : ^ ^ 3. ro OO i vo uo 00 + + i 1 1 + ^ i ^ : Toe? O ^ ^. S o t~>. u^ N rf rf O* " Tf CN f .+ Tf u-i 00 CO N 00 i- N * * T)- N . TT t^ >O 00 CO < W LO LO I ^ ^ r ; d o o 00 LO i 6 < O t/3 N )H 1-1 o id 5 . . t~>. i-O i t i N ro 00* -co c Jj 8O to O O ^O 00 N O OO O O ro ^ 00 O N 1-1 N N >-i fO c u w X o" 1 ! W *"* 5 S | : %, ^ 3 suoi^S ~ "-, ^ o IOODOO square links of Gunter s chain I acre. I square chain = 65 feet square = i L y - acre. TRAVERSE TABLE, XVIII. Table showing Differences of Latitude and Departures. 45 1 2 QJ C w <D Lat. Dep. Lat. Dep. Lat. Dep. j.2 M C I . OOOOO 0.00000 0.90984 0.01745 0.99939 o. 03490 h I 2 . OOOOO o . ooooo i . 99969 0.03490 1.99878 0.06980 2 3 . ooooo o . ooooo 2-99954 0.05235 2.99817 O.I0470 1 3 4 . ooooo 0.00000 3-99939 0.06980 3.99756 o . i 3960 4 5.00000 o. ooooo 4.99923 0.08726 4.99695 0.17450 5 60 6 . ooooo o . ooooo 5.99908 o. 10471 i 5-99634 0.20940 6 7 . ooooo o . ooooo 6.99893 0.12216 6-99573 0.24430 7 8 . ooooo o . ooooo 7.99878 0.13961 7.99512 0.27920 8 g . ooooo o . ooooo 8.99862 0.15707 8.99451 0.31410 9 0-99999 0.00436 0.99976 0.02181 0.99922 0.03925 i i . 99998 0.00872 1.99952 0.04363 1.99845 0.07851 jl 2 2.99997 0.01308 2.99928 0.06544 2.99768 0.11777 3 ; 3.99996 0.01745 3.99904 0.08725 3.99691 0.15703! 4 4.99995 i o. 0218 1 4.99881 o. 10907 4.99614 0.19629 5 45 5.99994 0.02617 5.99857 0.13089 5-99537 0.23555 6 6.99993 10.03054 6.99833 0.15270 6.99460 0.27481 | 7 7.99992 0.03490 7.99809 o. 17452 7.99383 0.31407 8 8.99991 0.03926 8.99785 0.19633 8.99306 0.35333! 9 0.99996 ! 0.00872 0.99965 0.02617 0.99904 0.04361 i 1.99992 0.01745 1.99931 0.05235 1.99809 0.08723 2 2.99988 0.02617 2.99897 0.07853 2.99714 0.13085 3 3-99984 0.03490 3.99862 o. 10470 3.99619 0.17447; 4 4.99981 j 0.04363 4.99828 0.13088 4.99524 0.21809 5* 30 5.99977 0.05235 5-99794 0.15706 5.99428 0.26171 6 6.99973 0.06108 6.99760 0.18323 6-99333 0-30533; 7 7.99969 0.06981 7.99725 0.20941 7.99238 0.34895 8 8.99965 10.07853 8.99691 0.23559 8.99143 0.39257 9 0.99991 0.01308 0-99953 0.03053 0.99884 0.04797 i 1.99982 0.02617 i . 99906 0.06107 1.99769 0.09595 2 2.99974 0.03926 2.99860 0.09161 2.99654 0.14393 3 i 3. 9996^ 0.0^235 3.99813 o. 12215 3-99539 0.19191 4 4.99957 0.06544 4.99766 o. 15269 4.99424 0.239891! 5 15 i 5.99948 0.07853 5.99720 0.18323 5.99309 0.28786 | 6 16.99940 0.09162 6.99673 0.21376 6.99193 0.33584 7 7-99931 o. 10471 7 . 99626 0.24430 7.99078 0.38382 8 8.99922 0.11780 8.99580 0.27484 8.98963 0.43180 9 Dep. Lat. Dep. Lat. Dep. Lat. O g P c 89 88 87 D n a o en 30 TRAVERSE TABLE. Differences of Latitude and Departures Continued. . o 3 4 5 CJ if. 3 G C c to p Lat. Dep. Lat. Dep. Lat. Dep. uo p g I 0.99863 0.05233 0.99756 0.06975 0.99619 0.08715 I 2 | 1.99726 0.10467 1.99512 O.I395I 1.99238 0.17431 2 3 . 2.99589 0.15700 2.99269 0.20926 2.98858 0.26146 3 4 | 3.99452 0.20934 3.99025 0.27902 3.98477 0.34862 4 o 4.99315 0.26168 4.98782 0.34878 4.98097 0-43577 5 60 6 5.99178 0.31401 5.98538 0.41853 5.97716 0.52293 6 7 6.99041 0.36635 6.98294 0.48829 6.97336 0.61008 7 8 i 7.98904 0.41868 7-98051 0.55805 7.96955 0.69724 8 9 8.98767 0.47102 8.97807 0.62780 8.96575 0.78440 9 i 0.99839 0.05669 0.99725 0.07410 0.09580 0.09150 T 2 1.99678 0.11338 1.99450 o. 14821 1.99160 0.18300 2 3 2.99517 o . 1 7.007 2.99175 0.22232 2.98741 0.27450 3 4 0.22677 3-98900 0.29643 3-98321 0.36600 4 15 4.99 J 95 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 b 7 6.98874 0.39684 6.98075 0.51875 j 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 i 0.99813 0.06104 0.99691 0.07845 : 0.99539 0.09584 i 2 1.99626 0.12209 1.99383 0.15691 } 1.99079 o. 19169 2 3 2 99440 0.18314 2.99075 0.23537 2.98618 0.28753 3 4 3-99253 0.24419 3-98766 0.31383 3-98158 0.38338 4 30 4.99067 0.30524 4.98458 0.39229 14.97698 0.47922 5 30 6 i 5.98880 0.36629 5.98150 0.470751! 5.97237 0.57507 b 8 6.98694 7.98507 0.42733 0.48838 6.97842 7-97533 0.54921 ; 6.96777 0.62/67 i 7.96316 0.67092 0.76676 8 9 8.98321 0.54943 8.97225 0.70613 8.95856 K 0.86261 9 i 9-99785 0.06540 0.99656 0.08280 0.99496 o. 10018 i 2 I-9957I o. 13080 L993I3 0.16561 | 1.98993 0.20037 2 3 2-99357 o. 19620 2.98969 0.24842 ; 2.98490 0.30056 3 4 3.99I43 0.26161 5.98626 0.33123 3-97987 0.40075 4 45 4.98929 0.32701 4.98282 0.41404 4.97484 0.50094 5 15 6 0.39241 5-97939 0.49684 5.96981 0.60112 b 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 3 O on" Dep. Lat. Dep. Lat. Dep. Lat. a En g C &3 P c . o O 86 . S 5 J 84 P c 7) TRAVERSE TABLE. Differences of Latitude and Departures Continued. 0> 6 7 8 1 6 o e c o rt rt 3 G i en s Lat. Dep. Lat. Dep. Lat. Dep. en Q i I o 99452 0.10452 0.99254 o. 12186 0.99026 0.13917 2 1.98904 0.20905 1.98509 0.24373 1.98053 0.27834 2 3 2.98356 0.31358 2.97763 0.36560 2.97080 0.41751 3 4 3.97808 0.41811 3.97018 0.48747 3-96107 o. 55669 4 o 5 4.97261 0.52264 4.96273 0.60934 1 4- 95134 0.69586 5 60 6 0.62717 5.95519 0.73121 i 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 0.11338 8 9 8.95069 0.94075 18.93291 0.09682 8.91241 0.25255 9 j 0.99405 0.10886 0.99200 0.12619 0.98965 0.14349 i 2 1.98811 0.21773 i . 98400 0.25239 1.97930 0.28698 2 ^ 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 4.97028 0-54433 4 . 96002 0.63099 4.94825 0.71746 5 45 6 5.96433 0.65320 5.95202 0.75719 5.93790 0.86095 6 7 6.95839 0.76206 6.94403 0.88339 6.92755 1.00444 7 8 7.95245 0.87093 7.93603 1.00959 7.91721 I.I4794 8 9 8.94650 0.97980 8.92804 I.I3579 8.90686 1.29143 9 i 0-99357 o. 11320 0.99144 0.13052 0.98901 o. 14780 i 2 1.98714 0.22640 1.98288 0.26105 1 1.97803 0.29561 2 3 2.98071 0.33960 2-97433 0.39157 2.90704 0.44342 3 4 3.97428 0.45281 3.96577 0.52210 3.95606 0.59123 4 30 : 4. 96786 o. 56601 4.95722 0.65263 14.94508 0.73904 5 30 6 5.96143 0.67921 5.94866 0.78315 : 5.93409 0.88685 6 7 6.95500 0.79242 6.94011 0.91368 6.92311 1.03466 7 8 7.94857 0.90562 7.93155 1.04420 7.91212 1.18247 8 9 8.94214 1.01882 8 . 92300 I-I7473 i 3.90114 1.33028 9 i 0.99306 0.11753 0.99086 0.13485 0.98836 0.15212 ! i 2 1.98613 0.23507 1.98173 0.26970 1.97672 0.30424 2 2 2.97920 0.35261 2.97259 0.40455 2.06508 0.45637 ! 3 4 3.97227 0.47014 3-96346 0.53940 3.95344 0.60849 4 45 5 6 4-96534 ; 5.95841 0.58768 0.70522 5.94519 0.67425 0.80910 4.94180 5.93016 0.76061 0.91274 6 ID 7 6.95147 0.82276 6. 9" 3606 0-94395 6.91853 1 1.06486 7 8 7.94454 0.94029 7.92692 1.07880 ; 7.90689 1.21698 8 9 3.93761 1.05783 8.91779 1.21365 ! 8. 89525 1 1.36911 9 5 O En Dep. Lat. Dep. Lat. Dep. Lat. g . | P P . c o Cfl 3 fl : 8: 1 S: 1 8 t 3 P TRAVERSE TABLE. Differences of Latitude and Departures Continued. en o 9 1C ) i t , o in ~? c C ^ 3 rt re 3 C i to 5 Lat. Dep. Lat. Dep. Lat. Dep. to |5 C i 0.98768 0.15643 0.98480 0.17364 0.98162 0.19081 j 2 1-97537 0.31286 1.96961 0.34729 1.96325 0.38162 1 .2 3 2 . 96306 0.46930 2.95442 0.52094 2.94488 0.57243 3 4 3-95075 0.62573 3.93923 0.69459 3.92650 0.76324 4 5 4.93844 0.78217 4.92^03 0.86824 4-QQ8I3 0.95405 i 5 60 6 5.92612 0.93860 5.90884 1.04188 5.88976 1.14486 ! 6 7 6.91381 1.09504 6.89365 I-2I553 6.87130 1.33566 i 7 8 7.90150 1-25147 7-87846 1.38918 7-85301 1.52648 8 9 8.88919 1.40791 8.86327 1.56283 8.83464 1.71729 9 i 0.98699 o. 16074 0.98404 0.17794 0.98078 0.19509 i i. 2 1-97399 0.32148 1.96808 0.35588 1.96157 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 o. 78036 1 4 15 5 4.93498 0.80371 4.92020 0.88971 4.90392 0-97545 i 5 45 6 5.92197 0.96445 ! 5-90424 1.06766 5.88471 1.17054 i 6 7 6.90897 1.12519 ; 6. 88828 1.24560 6.86549 1.36563 i 7 8 7.89597 1.28594 7.87232 1.42354 7.84628 1.56072 ; s 9 8.88296 i .44668 8.85636 1.60149 8.82706 i.7558i 9 j i 0.98628 0.16504 0.98325 0.18223 0.97992 0.19936 ! i 2 1.97257 0.33009 : 1.96650 0.36447 1.95984 0.39873 2 3 2.95885 0.49514 1 2. 94976 0.54670 2.93977 0.59810 3 4 3.94514 0.66019 3- 93301 0.72894 3.91969 0-79747 4 30 5 4.93142 0.82523 4.91627 0.91117 1 4.89962 0.99683 5 30 6 5.9I77I 0.99028 ; 5- 89952 1.09341 5.87954 i . 19620 6 7 6.90399 I-I5533 6.88278 1.27564; 6.85947 1-39557 1 7 8 7.89028 1.32038 7.86603 1.45788 7.83939 1-59494 i 8 9 8.87657 1.48542 8.84929 i .64011 8.81932 1-79431 i Q i 0.98555 0.16935 0.98245 0.18652 0.97904 0.20364 i 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 1 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 .01820 ! 5 15 6 5.9*333 i. 01610 5.89470 1.11914 5.87427 .22185 i 6 7 6.89889 1.18545 6.87715 1.30566 6.8^331 .42549 ; 7 8 7.88444 1.35480 7.85960 1.49219 7.83236 .62913 ! 8 9 8.87000 1.52415 8.84205 1.67871 8.81140 .83277 9 g Dep. Lat. Dep. Lat. Dep. Lat. ^ 3 3 y ! P C c en 3 o Sc > 7< | 7* 3 O (/> TRAVERSE TABLE. 33 Differences of Latitude and Departures Continued. 45 12 14 Lat. Dep. Lat. Dep. Lat. Dep. jO. 97814 0.20791 0.97437 0.22495 0.97029 0.24192 1.95629 0.41582 1.94874 0.44990 1.94059 0.48384 2-93444 0.62373 1 2.923II 0.67485 2.91088 0.72576 3.91259 0.83164 1 3- 89748 0.89980 3.88118 0.96768 14.89073 03955 . 4-87185 1.12475 4.85147 1.20961 5.86888 .24747 : 5.84622 1.34970 5.82177 I.45I53 6.84703 .45538 ; 6.82059 1.57465 6.79206 1-69345 : 7.82518 .66329 7.79496 i . 70960 7-76236 1-93537 8.80332 .87120 8.76933 2-02455 8.73266 2.17729 0.97723 0.21217 1 0.97337 0.22920 0.96923 ! 0.24615 1.95446 0.42435 1 1.94675 0.45840 1.93846 0.49230, 2.93169 0.63653 2.92013 0.68760 2.9076910.73845 13.90892 0.84871 3-89351 0.91680 3.87692 0.98461 4.88615 1.06088 4.86689 i . 14600 .4.84615 1.23076 5.86338 1.27306 5-84027 1-37520 5.81538 1.47691 6.84061 1.48524 6.81365 i . 60440 6.78461 | 1.72307 7.81784 1.69742 7.78703 1.83360 7.75384 1.96922 ,8.79507 1.90959- 8. 76041 2.06280 8.72307 2.21537 0.97629 0.21644 ; 0.97237 0.23344 0.96814 ; 0.25038 1.95259 0.43288 I 1-94474 0.46689 1.93629 0.50076 2.92888 0.64932 12.91711 0.70033 2.90444 0.75114 3.90518 0.86576 ; 3.88948 0.93378 3.87259 1.00152 4.88148 .08220 ,4.86185 1.16722 4.84073 1.25190 5.85777 .29864 15.83422 i . 40067 5.80888 1.50228 6.83407 .51508 , 6.80659 1.63411 6.77703 1.75266 7.81036 .73152 17.77896 1.86756 7.74518 2 . 00304 8.78666 .94796 ! 8. 75133 2. IOIOO S.7I332 2.25342 0-97534 0.22069 1.95068 0.44139 0.97134 , o 1 . 23768 1.94268 0.47537 0.96704 0.25460 1.93409 0.50920 2.92602 0.66209 I 2.91402 0.71305 2.90113 0.76380 3.90136 0.88278 3.88536 0.95074 3.86818 I.OI84O t 4.87671 . 10348 14.85671 1.18843 4.83523 I.2730I 5-85205 .32418 i 5.82805 I.426II i 5.80227 1.52761 6.82739 .54488 i 6. 79939 1.66380 6.76932 I.7822I 7-80273 .76557 7.77073 1.90148 i 7.73636 2.03681 8.77808 .98627 8.74207 2.13917 8.70341 2.29141 Dep. Lat. Dep. Lat. 1 Dep. Lat. 77 76 75 60 45 34 TRAVERSE TABLE. Differences of Latitude and Departures Continued. en o 15 16 17 , o c 03 - | C s en Q Lat. Dep. : Lat. Dep. Lat. Dep. co Q c i 0.96592 0.25881 0.96126 0.27563 0.95630 0.29237 i 2 1.93185 0.51763 ; 1.92252 0.55127 1.91260 0.58474 2 . ^ 2.89777 [0.77645 ;; 2.88378 0.82691 2.86891 0.87711 3 4 3.86370 1.03527113.84504 1.10254 3.82521 i . 16948 4 O 5 4.82962 1.29409 4.80630 1.37818 i 4-78152 1.46185 | 5 60 6 5.79^55 1.55291 ;! 5.76757 1.65382 5.73782 1.75423 6 7 6.76148 I.8II73! 6.72883 1.92946 6.69413 2.04660 7 a 7.72740 2.07055 i 7.69009 2.20509 7-65043 2.33897 8 9 8.69333 2.32937 8.65135 2.48073 8.60674 2.63134 9 1 1 i 0.96478 0.26303 0.96005 0.27982 0.95502 0.29654 i 2 1.92957 O. 52606 ; 1 .92010 0.55965 , 1.91004 0.59308 2 ^ 2.89436 0.78909! 2.88015 0.83948 2.86506 0.88962 3 15 4 5 3.85914 4.82393 I.052I2 I.3I5I5 3.84020 4.80025 I.H93I 1.39914 3.82008 4-77510 i. 18616 1.48270 4 5 45 6 5.78872 I.578I8 5 76030 1.67897 5-73012 1.77924 6 7 6.75351 I.S4I2I 6.72035 1.95880 6.68514 2.07579 7 8 ;7.7i82q 2.10424 7.68040 2.23863 i 7.64016 2.37233 8 9 8.68308 2.36728 8.64045 2.51846 8.59518 2.66887 9 i 0.96363 0.26723 0.95882 0.28401 0.95371 0.30070 i 2 1.92726 0.53447 L9I764 o. 56803 1.90743 0.60141 2 2.89089 0.80I7I 2.87646 0.85204 2.86115 o . 902.1 1 3 4 3.85452 1.06895 : 3.8352S 1.13606 ! 3.81486 1.20282 4 30 5 4.81815 1.33619 4.79410 1.42007 14.76858 1.50352 5 30 6 5.78178 1.60343! , 5.75292 i . 70409 1 5.72230 1.80423 6 7 6.74541 1.87066 i 6.71174 1.98810 6.67601 2.10494 7 8 7 . 70Q04 2.13790 ; 7.67056 2.27212 7.62973 2.40564 8 9 18.67267 2.40514 | 8.62938 2.55613 8.58345 2.70635 9 || 1 i 0.96245 0.27144 0-95757 0.28819 0.95239 0.30486 i 2 1.92491 0.54288 1.91514 0.57639 1.90479 0.60972 o o 2.88736 0.81432 2.87271 0.86458 2.85718 0.91459 3 4 3.84982 1.08576 3.83028 1.15278 1 3.80958 1.21945 4 45 5 i 4.81227 1.35720 14.78785 i . 44098 i! 4.76197 1.52432 | 5 15 6 5-77473 1.62864 15.74542 1.72917 j! 5- 71437 1.82918 b 7 6.73718 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 I 8.61814 2.59376 \ 8.57156 2.74377 9 :K srll De P- Lat. ; Dep. Lat. Dep. Lat. g En" 5 3 H P 3 1 3 (5 O 73 . 72 C/l o o 74 CD TRAVERSE TABLE. 35 Differences of Latitude and Departures Continued. c/2 0) 18 19 20 1 c ! ^ .8 rt VI 3 |5 Lat. Dep. Lat. Dep. Lat. Dep. 5 c j 0.95105 0.30901 0.94551 0.32556 0.93969 ! 0.34202 i 2 i. 9021 i 0.61803 1.89103 0.65113 1.87938 10.68404 2 3 2.85316 0.92705 2.83655 0.97670 2.81907 1.02606 3 i 4 3.80422 1.23606 i; 3.78207 1.30227 3.75877 1-36808 4 i 5 4.75528 1.54508 4.72759 1.62784 ! 4.69846 j 1.71010 60 b 5 70633 1.85410 5-67311 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 7.51754 2.73616 8 9 8.55950 i I 2.7SII5 8.50966 2.93011 8.45723 3.07818 9 i 0.94969 0.31316 0.94408 0.32969 0.93819 0.34611 i 2 1.89939 0.62632 : : I.S83I7 0.65938 1.87638:0.69223 2 .3 2.84909 0.93949 2.83226 0.98907 2.81457 1.03835 3 4 3.79879 L25265 || 3.77635 1.31876 3.75276 1.38446 4 15 5 4.74849 1.56581 4.72044 1.64845 4.69095 1.73058 45 6 5.69819 1.87898 5-66453 1.97814 5.62914 2.07670 6 7 6.64789 2.19214 6.60862 2.30783 6.56733J2.4428I 7 8 7-59759 2.50531 7.55271 2.63752 7.50553 2.76893 8 9 8.54729 2.81847 | 8.49680 2.96721 8.44372 3-11505 9 | i 0.94832 0. 31730 : ! 0.94264 0.33380 O.93667 ! O.35O2O i 2 1.89664 0.63460:: 1.88528 0.66761 1.87334 0.70041 2 3 2.84497 0.95191 j 2.82792 1.00142 2.8IOOI 1.05062 3 4 3.79329 I.2692I ! 3.7/056 1.33522 3.74668 1.40082 4 30 i 4.74161 5-68994 1.58652 4.71320 1.903821 5.65584 1.66903 2.00284 4.68336 5.62003 1.75103 2.10124 5 6 30 7 6.63826 2.22113 ! 6.59849 2.33664 6.55670 2.45145 7 8 7-58658 2.53843! 7.54H3 2.67045 7.49337 2.80165 8 9 8-5349 1 2.85574 8.48377 3.00426 8 . 43004 3.15186 9 i 0.94693 0.32143 0.94117 0.33791 0.93513 0.35429 i 2 1.89386 0.64287, 1.88235 0.67583 1.87027 0.70858 2 3 2.84079 0.96431 2.82352 LOI375 2.80540 1.06287 2 4 3.78772 1-28575 3.76470 1.35166 3.74054 1.41716 4 45 5 4.73465 1.60719 4-70588 1.68958 4-67567 I.77I4* 5 15 b 5.68158 1.92863 ; 5.64705 2.02750 5.6loSl j 2. 12574 6 7 6.62851 2.25007 6.58823 2.36541 6.54594 2.48003 7 y 7-57544 2.57151 1 7.52940 2.70333 7.48108 2.83432 8 9 8.52237 2.89295 j 8.4/058 3.04125 8.41621 3.18861 9 i 3 s Dep. Lat. Dep. Lat. Dep. Lat. c 53 P c . w 71 70 69 (0 o M 36 TRAVERSE TABLE. Differences of Latitude and Departures Continued. . . 21 22 23 . C/J O O ! o c G i rt CO 5 Lat. Dep. j Lat. Dep. Lat. Dep. rt tfl 5 3 C s I r 0.93358 0.35836 |j 0.92718 0.37460 ! 0.92050 0.39073 , 2 .86716 0.71673 ! 1.85436 O.7492I ; I.84IOO 0.78146 2 3 . 80074 1.07510 i 2.78155 I.I238I 2.76151 1.17219 3 4 .73432 1-43347 3-70873 1.49842 3.6S20f 1.56292 4 O 5 4.66790 1.79183 4.6359 1 I.S7303 4.60252 1.95365 5 60 6 .60148 2. I502O 5.56310 2.24763 i 5.52302 2.34438 b 7 6.53506 2.50857 6.49028 2.62224 6.44353 2.735H 7 8 7.46864 2.86694; 7.41747 2.99685 7.36403 3.12584 8 9 8.40222 3.22531 8.34465 3.37145 8.28454 3-51657 9 T 0.93200 0.36243 0.92554 0.3/864 0.91879 0.39474 i 2 1.86401 0.72487 1.85108 0.75729 j 1.83758 0.78948 2 2 . 79602 I.OS73I 2.77662 I.I3594 : 2.75637 i. -i 8423 3 4 3.72803 1-44975 3.70216 I.51459 3.675 16 1.57897 4 15 =1 4 . 66004 1.81219 i 4.62770 1.89324 14-59395 1.97372 5 45 6 5.59204 2.17462 ! 5 . 55324 2.27189 5.51274 2.36846 b 7 6.52405 2.53706 6.47878 2.65054 6.43153 2.76320 7 8 7.45606 2.89950 7-40432 3.02918 7.35032 3-15795 8 9 8.38807 3.26194 8.32986 3.40783 8.26912 3-55269 9 i 0.93041 0.36650 0.92388 0.38268 0.91706 0.39874 i 2 1.86083 0.73300 1.84776 0.76536 1.83412 0.79749 2 2.79125 1.09950 2.77164 1.14805 2.75118 1.19624 O 4 3.72167 1.46600 I3.69552 1.53073 3.66824 1-59499 i 4 30 5 6 4.65208 5.58250 1.83250 4-61940 I.9I34I 2.29610 5-50236 1-99374 2.39249 6 30 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 i 0.92881 0.3/055 0.92220 0.38671 0.91531 0.40274 i 2 1.85762 0.74111 1.84440 0.77342 | 1.83062 0.80549 2 12.78643. 1.11167 2 . 76660 I.I60I3 : 2.74593 1.20824 , 3 4 3.7I524 1.48222 3.68880 1.54684 3.66124 1.61098 4 45 5 6 4.64405 5.57286 1.85278 2.22334 4.6IIOO 5-53320 1-93355 2.32O26 4-57^55 5.49186 2.01373 2.41648 5 6 15 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 ;s. 35929 3; 33501 8.29980 3.48039 8.23780 3-62472 , 9 5 5 (3 O on p* Dep. Lat. Dep. Lat. Dep. Lat. g 5 rt t/i n ft) 68 i 67 66 n o o 73 TRAVERSE TABLE. 37 Differences of Latitude and Departures Continued. 45 24 25 26 w B rt "2 Lat. Dep. Lat. Dep. Lat. Dep. to 5 C 0.91354 0.40673 0.90630 0.42261 0.89879 0.43837 i 1.82709 0.81347 1.81261 0.84523 1.79758 0.87674 i 2 2.74063 I.22O2O 2.71892 1.26785 2.69638 1.31511 3 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 : 5 60 5-48127 2.44041 5.43784 2.53570 5.39276 2.63022 \ 6 6.39481 2.84715 6.344T5 2.95832 6.29155 3.06859 7 7.30836 3.25389 7.25046 3-38094 7.19035 3.50696 ! 8 8.22190 3 . 66062 8.15677 3.80356 8.08914 3-94533 ! Q 0.91176 0.4^071 0.90445 0.42656 0.89687 0.44228 j 1.82352 0.82143 1.80891 0.85313 1-79374 0.88457 : 2 2.73528 1.23215 2.71336 1.27970 2.69061 1.32686 3 3-64704 1.64287 3.61782 1.70627 3.58749 1.76915 4 4.55881 2.05359 4.52227 2.13284 4.48436 2.21144 5 45 5.47057 2.46431 !! 5.42673 2.55941 5.38123 2.65373 6 6.38233 2.87503 1 6.33118 2.98598 6.27810 3.09602 : 7 7.29409 3.28575 7-23564 3-41254 7.17498 3-53830: 8 8.20585 3.69647 18.14009 3. 83911 8.07185 3.98059 9 i I. 0.90996 0.414691:0.90258 0.43051 0.89493 0.44619 i 1.81992 0.82938 : i .80517 0.86102 1.78986 0.89239 2 2.72988 1.24407 2.70775 1.29153 2.68480 I.33859 i 3 3.63984 1.65877 3.61034 1.72204 3.57973 1.78479 1 4 4.54980 2.07346 4.51292 2.15255 4.47467 2.23098 , 5 30 5.45976 2.48815 5.4155112.58306 5.36960 2.67718 j 6 6.36972 2.90285 6.31809 3-01357 6.26454 3-12338 7 7.27969 3-31754 7.22068 3.44408 7.15947 3.56958 ; 8 8.18965 3.73223 8.12326 3.87459 8.05440 4.01578 9 0.90814 0.41866 0.90069 0.43444 0.89297 0.45009 i 1.81628 0.83732 1.80139 0.86889 1.78595 O.gOOig 2 ; 2. 72442 1.25598 2.70209 1.30333 2.67893 1.35029 3 3.63257 1.67464 j 3.60279 1.73778 3.57191 1.80039 4 4.54071 2.09330 4-50349 2.17222 4.46489 2.25049! 5 15 S5-44885 2.51196 5.40418 2.60667 5.35787 .2.70059 6 6.35700 2.93062 6.30488 3-04111 6.25085 3. 15068 1 7 7.26514 3-34928 7.20558 3.47556 7.14383 3.60078 8 8.17328 3.76794 8.10628 3.91000 8.03681 4.05088 9 i j Dep. Lat. Dep. Lat. Dcp. Lat. on 5" I P 65 6 4 63 P en 3 TRAVERSE TABLE. Differences of Latitude and Departures Continued. i/i 22 o o c 27 1 28 29 o 3 in o 3 ri rJ 3 c i Q Lat. Dep. Lat. Dep. Lat. Dep. "t/2 Q 3 i 0.89100 0.45399 0.88294 0.46947 ! 0.87462 0.48481 i 2 1,78201 0.90798 1.76589 0.93894 1.74924 0.96962 2 3 2.67301 1.36197 2.648841 1.40841 2.62386 1-45443 3 4 3-56402 1.81596 3.53179 1.87788: 3-49848 1.93924 4 o 5 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 1:6.99696 3-87848 8 9 8.01905 4-08591 7.94652 4.22524 7-87156 4-36329 9 i 1 r i 0.88901 0.45787 1 0.88089:0.47332 0.87249 0.48862 i 2 1.7780310.915741 1.76178 (0.94664 1.74499 0.97724 2 3 2.66705 1.37362 2.64267 1.41996 2.61748 1.46566 3 4 3.556o6 1.83149 3.52356 1.89328 3.48998 r.95448 4 15 5 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 6.22311 3.20511 6. 16623 3.31324 6.10747 3.42034 7 8 7.11213 3.66299 7.04712 3.78656 6.97996 3.90896 8 9 8.00115 4- 12086 7.92801 4.25988, 7.85246 4-39759 9 i : 0.88701 0.46174 0.87881 0.47715 0.87035 0.49242 i 2 1.77402 0.92349 1.75763 0.95431 |: 1.74071 0.98484 2 3 2.66103 ! 1.38524 2.63645 I.43I47 j, 2. 6IIO6 1.47727 3 4 3.54804 1.84699 3.5I526 1.90863 : 3. 48142 i . 96969 4 . 30 5 4.43505 2.30874 4o94o8 2.38579! 4.351/7 2.46211 =; 30 6 5.32206 2.77049 5.27290 2.86295 - : 5.22213 2-95454 6 7 6.20907 3.23224 6.15171 3.34011 6.09248 3.44696 7 8 9 7.09608 7-98309 3-69398; 4.15573 7-03053 7.90935 3.81727 : 6.96284 4.29442 7.83320 3.93938 4.43i8i 8 9 i ; 0.88498 0.46561 0.87672 0.48098 O.S68I9 0.49621 i 2 1.76997 0.93122 1-75345 0.96197! 1.73639 0.99243 2 3 2.65496 i . 39684 2.63018 1.44296 i 2.60459 I 1.48864 3 4 3-53995 1.86245 3.50690 1.92395 , 3.47279 1.98486 4 45 5 4.42493 2.32807 408363 2.40494! 4.34099 | 2.48108 5 15 6 5-30992 2.79368 5 . 26036 2.885931 5-20919 2.97729 6 7 ! 6. 19491 3.25930 6.13708 3.36692,!6.07739 3-47351 7 8 i 7-07990 3.72491 7.01381 3.84791; 6.94559 3.96973 8 4- 19053 7-89054 4.32889 7.81378 | 4.46594 9 3 O en Dep. Lat. Dep. Lat. Dep. Lat. 2 en g 3 p* p 3 * 3 3 r? ;/) n> 62 61 60 P 00 TRAVERSE TABLE. 39 Differences of Latitude and Departures Continued. irt U <u u 53 30 31 32 o o in <L> 3 rt i rt 3 C s tfl Q Lat. Dep. Lat. Dep. Lat. Dep. o> 5 I 0.86602 0.50000 0.85716 0.51503 0.84804 0.52991 i 2 1.73205 I . 00000 I.7I433 03007 T . 69609 1.05983 2 3 2.59807 1.50000 2.57150 545II 2.54414 L5S975 3 4 3.46410 2.OOOOO 3.42866 .06015 3.39219 2. 11967 4 5 4-33012 2.50000 4-28583 .57519 4.24024 2.64959 5 60 6 5-19615 3 . ooooo 5.14300 .09022 5.08828 3.I795I 6 7 6.06217 3 . 50000 6.00017 .60526 5.93633 3.70943 7 8 6.92820 4.00000 6.85733 4.12030 6.78438 4.23935 8 9 7.79422 4.50000! 7.71450 4.63534 7.63243 4.76927 9 i 0.86383 ; 0.50377 0.85491 0.51877 0.84572 0.53361 i 2 1 1.72767 i 1.00754!! 1.70982 1.03754 1.69145 1.06722 2 3 12.59150 1.51132 2.56473 1.55631 2.537IS I .60084 3 4 3-45534 2.01509 3.41964 2.07509 3.38291 2.13445 4 T 5 5 2.51887 4.27456 2.59386 4.22863 2.66807 5 45 6 5.18301 3.02264 5-12947 3.11263 5.07436 3.2OI68 6 7 6.04684 3.52641 5.98438 3-63141 5 - 92009 3.73530; 7 8 6.91068 4.03019 1 6.83929 4. 15018 6.76582 4.26891 8 9 7-77451 4.53396 : 7.69420 4.66895 7-6ii55 4.80253 9 j ; 0.86162 0.50753 1 0.85264 0.52249 0.84339 0-53730 i 2 1.72325 1.01507 1.70528 1.04499 1.68678 I .07460 2 3 2.58488 1.52261 j 2.55792 1.56749 2.53017 I. 6ligO 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 4-21695 2.68650 5 30 i 6 5-16977 3.04523 5.11584 3.13499 5.06034 3.22380 6 7 6.03140 3-55276 ! 5-96948 3.65749 5.90373 3.76110 7 8 6.89303 4.06030 | 6.82112 4.17998 6.74713 4.29840 8 9 7.75466 4.56784 7-67376 4.70248 7-59 52 4-83570 9 i 0.85940 0.51129 0.85035 0.52621 0.84103 0.54097 i 2 I.7I88I 1.02258 1.70070 1.05242 ! 1.68207 1.08194 2 3 2.57821 1.53387 2.55105 1.57864 2.52311 I .62292 3 1 4 I3.43762 2.04517 3.40140 2.10485 3-36415 2.16389 4 45 ! 5 4.29703 2.55646 4.25176 2.63107 4.20519 2.70487 5 15 6 5.15643 3.06775 5.10211 3.15728 5.04623 3.24584 6 7 6.01584 3.57905 5.95246 3-68349 5.88827 3.78682 7 S 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 >" 3 2 1 Dep. (/) Lat. Dep. Lat. Dep. Lat. 2 e as P c e? vi 1 59 58 57 \ 8 Cfl TRAVERSE TABLE. Differences of Latitude and Departures Continued. 15 30 45 <u 33 34 35 oJ u o c e<j rt to 5 Lat. Dep, Lat. Dep. Lat. Dep. tf3 Q T 0.83867 0.54463 0.82903 0.559-19 0.81915 0-57357 I 2 1.67734 1.08927 1.65807 1.11838 1.63830 I.I47I5 2 3 2.51601 1-63391 2.48711 1.67757 2-45745 1.72072 3 4 3.35468 2.17055 3.31615 2.23677 3.27660 2.29430 4 5 4.19335 2.72319 4.I45I8 2.79596 4.09576 2.86788 5 6 5.03202 3-26783 4-97422 3.35515 4.91491 3.44I45 6 7 5.8706913.81247 5.80326 3.91435 5.73406 4.01503 7 8 6.70936 4-357II 6.63230 4.47354 6.55321 4.58861 8 9 7.54803 4.90175 7.46133 5-03273 7.37236 5.16218 9 i 0.83628 0.54829 0.82659 0.56280 0.81664 0.57714 i 2 1.67257 1.09658 1.65318 1.12560 1.63328 1.15429 2 3 2.50885 1.64487 2.47977 1.68841 2.44992 I.73I43 3 4 3.34514 2.19317 3.30636 2.25121 3.26656 2.30858 4 5 4.18143 2.74146 4.13295 2.81402 4.08320 2.88572 5 6 5-01771 3-28975 4.95954 3.37682 4.89984 3-46287 6 7 5.85400 3-83805 5-78613 3-93963 5.71649 4.04001 7 8 6.69028 4.38634 6.61272 4-50243 6.53313 4.61716 8 9 7-52657 4.93463 7-43931 5.06524 7-34977 5 . 19430 9 i 0.83388 0.55193 0.82412 0.56640 0.81411 0.58070 i 2 1.66777 1.10387 1.64825 1.13281 1.62823 i. 16140 2 3 2.50165 1.65581 :2. 47237 1.69921 2.44234 1.74210 3 4 3-33554 2.20774 3.29650 2.26562 3.25646 2.32281 4 5 4.16942 2.75968 4. I2O63 2.83203 4-07057 2.90351 5 6 5.00331 3.31162 14-94475 3.39843 4.88469 3.48421 6 7 5.83720 3.86355 5.76888 3.96484 5.69880 4.06492 7 8 6.67108 4.41549 6.59300 4.53124 6.51292 4.6-f562 8 9 7.50497 4.96743 7-4I7I3 5.09765 7-32703 5.22632 9 | i 0.83147 0.55557 0.82164 0.56099 0.81157 ,0.58425 i 2 1.66294 1. 1 1114 1.64329 1.13999 1.62314 1.16850 2 3 2.49441 1.66671 2.46494 i . 70999 2.43472 1.75275 3 4 3.32588 2.22228 3.28658 2.27998 3.246291 2.33700 4 5 4-15735 2.77785 4.10823 2.84998 4.05787 2.92125 6 4.98882 3.33342 14.92988 3.41998 4.86944 3-50550 6 7 5.82029 3.88899 5.75152 3.98997 5.68101 4.08975 7 8 6.65176 4-44456 6.57317 4-55997 6.49260 4.67400 8 9 7.48323 5.00013 7.39482 5.12997 7.30416 5.25825 9 | Dep. Lat. Dep. Lat. Dep. Lat. D E 5? ? 3 3 3 56 55 54 o 60 45 30 15 o i g TRAVERSE TABLE. Differences of Latitude and Departures Continued.. M OJ a) o 36 37 38 t 3 -2 2 3 a 3 tfl ! 5 Lat. Dep. Lat. Dep. Lat. Dep. to Q c i j 0.80901 0.58778 0.79863 o. 60181 0.78801 0.61566 i 2 1.61803 I.I7557 1.59727 1.20363 1.57602 1.23132 2 3 2.42705 I.76335 2.39590 1.80544 2 . 36403 1.84698 3 4 3.23606 2.35H4 3-19454 2.40726 3.15204 2.46264 4 5 4.04508 2.93892 3.99317 3-00907 3.94005 3.07830 5 60 6 4.85410 3.52671 4.79181 3.61089 4.72806 3-69396 6 7 5-66311 4.11449 5.59044 4.21270 j 5.51607 4.30963 7 8 6.47213 4.70228 6.38908 4.81452 6.30408 14.92529 8 9 7.28115 5 . 29006 7.18771 5.41633 7.09209 5.54095 9 i 0.80644 0.59130 0.79600 0.60529 1.78531 0.61909 i 2 1.61288 1.18261 1.59200 1.21058 | 1.57063 1.23818 2 3 2.41933 1.77392 2.38800 1.81588 2.35595 1.85728 3 4 3-22577 2.36523 3.18400 2.42117 3.14126 2.47637 4 15 5 4.03222 2.95654 3.98001 3-02647 3.92658 3.09547 5 45 6 4.83866 3. 54785 4.77601 3.63176 4.7H90 3.7T456 6 7 5.645H 4.13916 5-57201 4-23705! 5-49721 4.33365 7 8 6.45155 4.73047 6.36801 4.84235 6.28253 4.95275 8 9 7.25800 5-32178 1 7.16401 5.44764 7.C6785 5.57184 9 I i 0.80385 : 0.59482 0-79335 0.60876 0.78260 0.62251 i 2 I.6077I 1.18964 1.58670 1.21752 I.5652I 1.24502 2 3 2.4II57 1.78446 2.38005 1.82628 2.34782 1.86754 3 4 3.21542 2.37929 3. 17341 2.43504 I3.I3043 2.49005 4 30 5 4.01928 2.974II 13.96676 3.043801 3- 9*304 3.H257 5 30 6 4.82314 3.56893 4.76011 3.65256 4.69564 3.73508 6 7 5.62699 4.16375 iS-55347 4.26132 ; 5.47825 4.3576o 7 8 6.43085 4.75858 6.34682 4.87OO9 6.26O86 4.98011 O 9 7.23471 5.35340 7.14017 5.47S85 7-04347 5.60263 9 i O.8OI25 0.59832 0.79068 0.61221 O.-7798S 0.62592 i 2 I.6O25O 1.19664 1-58137 1.22443 L55946 1.25184 2 3 2.40376 1.79497 (2.37206 1.83665 2.33965 1.87777 3 4 3.20501 12.39329 13.16275 12.44886 3-IT953 2.50369 4 45 5 4.00626 2.99162 3-95344 I 3.06108 3 89942 3.12961 5 15 6 4-80752 3.58994 4.74413 3.67330 4.67930 3-75554 , 6 7 5-60877 4.18827 15.53482 4.28552 5.45919 4.38146 i 7 8 6.41003 4.78659 6.32551 4.89773 6.23907 5.00738 8 9 7.21128 5.38492 7. 11620 ; 5.50995 7.01896 5.63331 ; 9 g VI Dep. Lat. Dep. Lat. Dep. Lat. g g c i e of CO p CD 53 52 p 51 I 8 o tn 42 TRAVERSE TABLE. Differences of Latitude and Departures Continued. t/3 4) 39 40 4i d t/5 o ~J o 3 g c rt 3 a W C S Q Lat. Dcp. Lat. Dep. Lat. Dep. Q s I 0.77714 0.62932 0.76604 0.64278 0.75470 0.65605 i 2 1-55^29 1.25864 i. 53208 1.28557 1.50941 1.31211 2 3 2.33143 1.88796 2.29813 1.92836 2.26412 1.96817 3 4 3.10858 2.51728 3.06417 2.57H5 3.01883 2.62423 4 o 5 3.38573 3.14660 3-83022 3-21393 i3-77354 3.28029 5 60 6 4.66287 3.77592 4.59626 3.85672 4.52825 3:93635 6 X" 7 5.44002 4.40524 5-36231 4-49951 5.28296 4.59241 ; 7 8 6.21716 5-03456 6.12835 5.14230 6.03767 5.24847 8 9 6.99431 5.66388 6.89439 5.78508 6.79238 5.90453 9 i 0-77439 0.63270 0.76323 0.64612 0.75184 0.65934 i 2 1.54878 1.26541 1.52646 1.29224 i. 50368 1.31869 ; 2 3 2.32317 1.89811 2.28969 I.93S37 2.25552 1.97803 3 4 3-09757 2.53082 3-05293 2.58449 3.00736 2.63738 4 15 5 3.87196 3-16352 3.81616 3.23062 3.75920 3.29672 5 45 6 4.64635 3.79623 4-57939 3.87674 .4.51104 3-95607 6 7 5.42074 4.42893 5.34262 4.52286 5.26288 4-61542 7 8 6.19514 5.06164 6.10586 5. 16899 || 6.01472 5.27476 8 Q 6.9 6 953 5.69434 6.86909 5.8I5II 6.76656 5-934II 9 r 0.77162 0.63607 0.76040 0.64944 0.74895 0.66262 i 3 1.54324 2.31487 1.27215 1.90823 1,52081 2.28121 1.29889 I.4979 1 1.94834 2.24686 1.32524 2 1.98786 3 4 3-08649 2.54431 3.04162 2-59779 2 .99582 2.65048 4 30 5 3.85812 3.18039 3.80203 3.24724 3-74477 3-31310 5 30 6 4.62974 3.81646 4-56243 3.89668 4-49373 3-97572 6 7 5.40137 4.45254 5.32284 4.54613 5.24268 4-63834 ! 7 8 6.17299 5.08862 6.08324 5.19558115-99164 5 - 30096 , 8 9 6.94462 5.72470 6.84365 5.84503 6.74060 5-96358 9 i 0.76884 0.63943 0.75756 0.65276 ! 0.74605 0.66588 i 2 1.53768 1.27887 L5I5I3 1.30552 ; I.492II I.33I76 2 3 2.30652 1.91831 2.27269 1.95828 | 2.23817 1.99764 1 3 4 3.07536 2-55775 3.03026 2.6II04 2.98422 2.66352 4 45 5 3 . 84420 3.I97I9 3-78782 2.26380 3.73028 3-32940 5 15 6 4-61305 3.83663 4-54539 3.91656 \ 4. 47634 3.99529 6 7 5.38189 4.47607 5-30295 4.56932 15.22240 4.66117 1 7 8 6.15073 5.H55I 6.06052 5.22208 5.96845 5-32705 8 9 6.91957 5-75495 6.81808 5.87484 6.71451 5.99 2 93 ; 9 3 3 O Cfl Dep. Lat. Dep. Lat. Dep. Lat. GO g 5 B p ! P c (t on n> 50 49 48 ; o a> en TRAVERSE TABLE. 43 Differences of Latitude and Departures Continued. C/3 o 42 43 44 . ; ; o in CJ C a rt 3 a i tfl 5 : Lat. Dep. Lat. Dep. Lat. Dep. OQ Q 1 1 jl i o 74314 0.66913 0.73135 0.68199 0.71933 0.69465; I 2 1.48628 1.33826 1.46270 1.36399! 1.43867 1-38931 2 fj 2.22943 2.00739 2. 19406 2.04599 i 2.15801 2.08397 3 4 2.97257 2.67652 2.92541 2.72799 2.87735 2.77863, 4 5 3.71572 3.34565 3.65676 3-40999! 3.59669 3-47329; 5 60 4.45886 4.01478 4.38812 4.091991 4.31603 4.16795! 6 7 5.20201 4.68391 5.H947 4.77398 5.03537 4.86260 7 8 5.94515 5.35304 5.85082 5.45598 5.75471 5.55726! 8 9 6.68830 6.02217 6.58218 6.13798 6.47405 6.25192 9 i 0.74021 0.67236 0.72837 0.68518 0.71630 0.69779 i 2 1.48043 1-34473 1.45674 1.37036 1.43260 I.3955S 2 3 2.22065 2.01710 2.185II 2.05554; 2.14890 2.09337 3 4 2.96087 2.68946 2.91348 2.74073 ; 2.86520 2.79116 i 4 15 5 3.70109 3-36183 3.64185 3.42591 ! 3.58I5I 3-48895;; 5 45 6 4.44130 4.03420 U.37022 4.11109! 4.29781 4.18674 6 7 5.18152 4.70656 5.09859 4.79628 5.01411 4.88453 7 8 5.92174 5.37893 5.82696 5.48146 5-73041 5-58232 8 9 ,6.66196 6.05130 6-55533 6.16664 6.44671 6.28011 9 i 0.73727 0.67559 0.72537 0.68835 0.71325 o. 70090 !| i 2 r. 47455 I.35H8 : i. 45074 1.37670 1.42650 I.40I8I 2 3 2.21183 2.02677 2.I76I2 2.06506 ! 2.13975 2. IO272 3 4 2.94910 2.70236 2.90149 2.75341 i 2.85300 2 . 80363 4 30 5 3.68638 3-37795 3.62687 3.44177 3.56625 3.50454 5 30 6 ; 4- 42366 4-05354 4.35224 4.13012 14.27950 4.20545 6 7 5.16094 4.72913 5.07762 4.81848 4-99275 4.90636 7 8 5.89821 5.40472 5.80299 5 . 50683 5 . 70600 5.60727 8 9 6.63549 6.08031 16.52836 6.19519 6.41925 6.30818 | 9 | i 0.73432 0.67880 0.72236 0.69I5I 0.7IOI8 0.70401 i 2 ; i. 46864 i.3576o 1.44472 1.38302 1.42037 1.40802 2 3 2 . 20296 2.03640 2.I6/O9 2-07453 2.13055 2. II2O4 3 4 2.93729 2.71520 2.88945 2.76605 2.84074 2.81605 4 45 .5 3.67161 3.39400 3.6II82 3.45756 3.55092 3-52007: 5 15 6 4.40593 4.072801:4.33418 4.14907 4.26111 4.22408; 6 7 5.14025 4.75i6o 5.05654 4.84059 4.97129 4.92810 7 8 5.87458 5.43040 5.77891 5.53210 5.68148 5.63211 O 9 6.60890 6. 10920 6.5OI27 6.22361 6.39166 6.33613 9 g 5 S Dep. Lat. Dep. Lat. Dep. Lat. 2 3" c *r ; ^3 c ? 47" 46 45 8 en 44 TRAVERSE TABLE. Differences of Latitude and Departures Continued. . 45 Lat. Dep. I o. 70710 o. 70710 i 2 1.41421 1.41421 2 3 4 2. 12132 2. 82842 2. I2I32 2. 82842 3 4 5 6 3-53553 4. 24264 3-53553 4, 24264 5 6 7 8 9 4- 94974 5.65685 6.36396 4- 94974 5. 65685 6. 36396 7 8 9 Dep. Lat. 45 Chains, Yards, and Feet, WITH THEIR RECIPROCAL EQUIVALENTS. Link 7. 92 inches. Chain = 66 feet = 792 inches. CHAINS INTO FEET. FEET INTO LINKS. c/3 "3 (3 Yards. Feet. Feet. Yards. Links. CJ a o I 0.22 0.66 O. IO 33 0.15 o 2 0.44 1.32 o. 20 .066 o. 30 o 3 o. 66 1.98 0.25 .082 0.38 o 4 0.88 2.64 o. 30 i . oio 0-45 o 5 I. IO 3.30 o. 40 133 o. 60 o 6 1.32 3-96 o. 50 .166 o. 76 o 7 1-54 4.62 o. 60 . 200 0.91 o 8 1.76 5.28 o. 70 . 233 1. 06 o 9 1.98 5-94 0-75 .250 1-13 o IO 2. 2O 6.60 o. 80 . 266 I. 21 MISCELLANEOUS. 45 Chains, Yards, and Feet Continued. CHAINS INTO FEET. FEET INTO LINKS. rt *? Yards. Feet. Feet. Yards. Links. 6 a 20 4-40 13. 20 0.9 -3 1.36 o 30 6.60 19.80 1.0 33 I-5I o 40 8.80 26. 40 2.0 .66 3-o o 50 II. OO 33- . 3- I. OO 4-5 o 60 13.20 39. 60 4. o i-33 6.0 o 70 15.40 46.20 5.0 1.66 7-5 o So 17. 60 52. So ; 6. o 2.00 9.1 o 90 19.80 59-40 ,7- 2-33 10.6 I OO 22. OO 66. oo 8. o 2.66 12. I 2 00 44-00 132 9.0 3.00 I 3 .6 3 66 198 10 3- 33 15. I 4 88 264 15 5.00 22.7 5 IIO 330 20 6.66 30.3 6 132 396 24 8.00 36.3 7 J 54 462 27 9.00 40.9 8 176 528 30 IO. OO 45-4 9 198 594 33 II. OO 50. o 10 220 660 36 12. OO 54-5 20 440 1320 39 13.00 59- i 30 660 1980 40 13-33 60.6 35 770 2310 42 14.00 63-3 4 880 2640 45 15.00 68.2 45 990 2970 48 16. oo 72.7 5 IIOO 33 5 16.66 75-7 55 1210 3 6 3 51 17.00 77-3 60 1320 3960 54 18.00 81.8 65 143 4290 57 19. oo 86.3 70 1540 4620 60 20. oo 90. 9 75 1650 4950 63 21. OO 95-4 80 i 760 5280 66 22. OO IOO 46" RAILROAD CURVES. XIX. To trace Railroad Curves by means of Deflections. GENERAL PROPOSITIONS. 1. The angle formed by a tangent and a chord is equal to half the angle at the center of the circle subtended by the chord. 2. The angle of deflection formed by any two equal chords meeting at the circumference is equal to the angle at the center, subtended by either cord. 3. A line bisecting the angle of deflection formed by any two equal chords is a tangent to the arc at the point where the two chords meet. 4. If an arc of a circle be subdivided into any number of equal parts, and lines be drawn from the several points of subdivision so as to meet at any point in the circumference, these several lines will form equal angles at the point of meeting, and the angles thus formed will be respectively measured by one-half the subdivided arc. METHOD BY DEFLECTION-ANGLES. * The degree of a curve is determined by the angle subtended at its center by a chord of 100 feet. The deflection-angle of a curve is the acute angle formed at any point between a tangent and a chord. It is, therefore, half the degree of the curve. In order to unite two straight lines by a curve, the angle of intersection is measured, and then a radius for the curve may be assumed and the . tangent calculated, or the tangent may be assumed of a certain length and the radius calculated. Let I = angle of intersection of the two lines; R = radius of circle; T = length of tangent, or distance from point of inter section to point where the curvature is to commence; and D = angle of deflection. Then T = R tan J- I R = T cot \ I sin D=i = 5_tanJ_I R I RAILROAD CURVES. 47 XIX. To trace Railroad Currcs, erY. Continued. To lay out a curve, set the instrument at the point at which the curvature is to commence, lay off the given deflection-angle, and the first point in the curve will be at the end of 100 feet measured on this new direction. Then lay off another deflection-angle equal to the first; attach the loo-foot chain to the point last found, and swing it, stretched, until its extremity intersects tile new direction, which will be the second point; and so on. Should it be found necessary to remove the instrument from its first position, either on account of the length of the curve or of some obstruction to the sight, the first deflection at the new position of the instrument will be equal to the total deflection from the preceding position. METHOD BY TANGENT AND CHORD DEFLECTION. Tangent-deflection is the distance between the extremities of a tangent and a chord, each 100 feet long. Chord-deflection is the distance from the extremity of the first chord, produced an additional 100 feet, to the extremity of the next, and is, therefore, double the tangent-deflection. To lay out a curve, stretch the loo-foot chain from the point of beginning in the direction of the tangent, and mark its extrem ity ; swing the chain toward the direction of the curve, keeping trie initial point fixed, until it has diverged a distance equal to the tangent-deflection , which will be \\\e first point of the curve. Produce the first chord an additional 100 feet, and swing the chain (round the extremity of the first chord as a pivot) until it has diverged a distance equal to the chord-deflection, which will be the second point of the curve. Continue to lay off -the chord-deflection from the preceding chord produced until the curve is finished. LENGTH OF CIRCULAR ARCS IX PARTS OF RADIUS. / ;/ I .01745 32925 i .00029 08882 .00000 48481 2 .03490 65850 2 .00058 17764 . ooooo 96962 3 .05235 98775 3 .00087 26646 .00001 45444 4 . 0698 i 31 700 4 .00116 35528 .ooooi 939 2 5 5 .08726 64625 5 .00145 444!0 .00002 42406 6 .10471 97551 6 .00174 53 2 9 2 6 .00002 90888 7 .12217 30476 7 .00203 62174 7 .00003 .39369 8 9 .13962 63401 .15707 96326 8 9 .00232 71056 .00261 79938 8 9 .OOOO3 87850 .00004 36332 48 RAILROAD CURVES. XIX. To trace Railroad Curres,.&c. Continued. ,j ORDINATES. c o c "o Q <u c "o Degree. Radii. To circular arcs on a chord of 100 feet. o 12$ 25 371 5 o t/3 rt r- 1 O Fat . o 5 : 68754.94 .008 . 014 .017 .018 .073 145 io 34377.48 .Ol6 .027 .034 . 036 .145 .291 15 22918.33 .024 .041 .051 055 .218 .436 20 17188.76 .032 055 .068 .073 . 291 .582 25 I375 1 . 02 . 040 .068 .085 . 091 .364 .727 30 : 11459-19 .048 .082 . 102 . 109 -436 .873 35 ! 9822. 18 056 95 . 119 .127 .509 i.oiS 40 8594. 41 . 064 . 109 .136 145 .582 i. 164 45 7639. 49 .072 .123 . T 53 .164 -654 1.309 50 6875. 55 .080 .136 .170 -.182 .727 1-454 55 6250.51 .087 .150 .187 . 200 .800 i. 600 i o 5729-65 095 .164 .205 .218 873 1-745 5 5288.92 103 .177 .222 .236 945 1.891 10 ! 4911. 15 . Ill .191 .239 255 i. 018 2. 036 15 4583. 75 .119 .205 .256 273 i. 091 2.182 20 4297. 28 .127 .218 .273 .291 i. 164 2.327 25 4044.51 .135 .232 .290 .309 1.236 2.472 30 3819.83 .143 .245 .307 327 1.309 2.618 35 3618.80 . 151 .259 .324 345 1.382 2.763 40 3437.87 .159 .273 .341 .364 1.454 2.909 45 3274.17 .167 .286 .358 .382 1.527 3-054 5 3125-36 .175 .300 .375 . 400 i. 600 3. 200 55 2989.48 .183 3H .392 .418 1.673 3-345 2 2864. 93 .191 .327 -409 .436 1-745 3.490 5 2750.35 .199 . 341 . 426 455 i.SiS 3-636 10 2644. 58 .207 355 .443 473 1.891 3- 78i 15 2546. 64 215 .368 .460 .491 1.963 3-927 20 2455- 7o .223 .382 477 509 2. 036 4. 072 25 2371.04 .231 395 494 .527 2. 109 4.218 30 2292.01 .239 .409 5 11 545 2. iSl 4-363 35 2218.09 247 .423 .528 064 2.254 4.508 40 2148.79 .255 .436 545 .582 2.327 4-654 45. 2083. 68 263 45 .562 .600 2. 4OO 4-799 50 2022.41 .270 .464 .580 .618 2.472 4-945 55 1964.64 .2 7 8 477 597 .636 2.545 5.090 RAILROAD CURVES. 49 XIX. To trace Railroad Curves, rv. Continued. ORDINATES. ci o c Q Degree. Radii. To circular arcs on a chord of 100 feet. J ^ "o V p! "f ! O t/3 f 124- 25 37i 5 EH o / 3 1910.08 .286 .491 .614 .655 2.618 5-235 5 1858.47 .294 505 .631 .673 2.690 5.381 10 1809.57 .302 .518 .648 . 691 2- 763 5.526 15 1763.18 .310 532 665 .709 2.836 5.672 20 1719. 12 .318 545 .682 .727 2.908 5-817 25 1677.20 .326 559 .699 -745 2.981 5.962 3 1637.28 .334 573 .716 .764 3- 54 6.108 35 I599-2I .342 .586 733 .782 3.127 6.253 40 1562.88 35 .600 75 .800 3.199 6.398 45 1528. 16 .358 . 614 .767 .818 3.272 6.544 5 1494- 95 .366 .627 -784 .836 3-345 6.689 55 1463. 16 374 .641 .801 855 ! | 3.4I7 6.835 4- o 1432. 69 .382 .655 .818 -873 3.490 6. 980 5 1403. 46 .390 .668 .835 .891 3-563 7.125 10 I 375-4Q .398 .682 .852 .909 3.635 7.271 15 1348.45 .406 .695 . 869 .927 3.708 7.416 20 1322.53 .414 .709 .886 945 3-78I 7-561 25 1297.58 .422 723 .903 . 964 3-853 7./07 3 1273.57 .430 .736 .921 .982 3-926 7.852 35 1250.42 .438 .750 .938 I. 000 3-999 7-997 40 1228. ii .446 .764 955 1.018 4.071 8.143 45 1206. 57 454 777 972 1.036 4.144 8.288 5 1185.78 .462 .791 .989 055 \ 4.217 8-433 55 1165. 70 .469 .805 i. 006 1.073 4.289 8-579 5 1146.28 477 .818 1.023 i. 091 4.362 8.724 5 1127.50 -485 .832 i. 040 1. 109 4.435 8.869 10 1109.33 .493 -846 1.057 1,127 4.507 9.014 15 1091. 73 .501 .859 1.074 i. 146 4.580 9. 1 60 20 1074. 68 59 .873 1.091 1. 164 4.653 9- 35 . 25 1058. 16 5*7 .887 1. 108 I. 182 ; 4.725 9-45 3 1042. 14 525 .900 1.125 1.200 4.798 9.596 35 1026. 60 533 .914 1. 142 1.218 , 4.870 9.741 40 1011.51 .541 .928 I-I59 I - 2 37 4-943 9.886 45 996. 87 .549 .941 i. 176 1.255 ; 5.016 10.031 5<3 RAILROAD CURVES. XIX. To trace Railroad Curves, &c. Continued. ORDINATES. .2 G g _0 tC 3 Degree. Radii. To circular arcs on a chord of 100 feet, j *"O u .* 25 37* SO I <u fcJD C rt H o 6 / 5 50 982. 64 ! -557 .955 i.i93 1.273 5.088 10.177 55 968. 81 -565 .968 I. 2IO 1.291 ;: 5. 161 10.322 6 o 955-37 573 .982 1.228 1.309 5-234 10.467 5 942. 29 .581 .996 1.245 1.327 5-306 10. 612 10 929-57 ! .589 i . 009 . 1.262 1.346 5-379 10. 758 15 917.19 -597 1.023 1.279 1.364 5-451 10. 903 20 905. 13 ; .605 1.037 1.296 1.382 5.524 11.048 25 893.395 -613 1.050 I -3 I 3 1.400 5-597 11.193 3 881.95 || .621 1.064 1.330 1.418 5.669 11.339 35 870. 79 j .629 1.078 1.347 1-437 ! 5-742 11.484 40 859.92 .637 i. 091 1.364 1-455 5-814 II. 629 45 849- 3 2 . 645 1. 105 1.381 1-473 5.887 11.774 5 838. 97 i 653 i. 118 1.398 1.491 i 5.960 11.919 55 828.88 .661 1.132 i.4i5 1.510 6. 032 12.065 7 819.02 .669 i. 146 1.432 1.528 6. 105 12.210 5 809. 40 .677 I.I59 1.449 1.546 6.177 12-355 10 800. oo .685 I.I73 1.466 1.564 6.250 12. 500 15 790.81 693 i. 187 1.483 1.582 6.323 12.645 20 781.84 ! . 701 1.200 1.501 i. 600 6-395 12. 790 2 5 773- 7 709 I.2I4 I.5I7 i. 619 6.468 12.936 3 764. 49 .717 1.228 1-535 1.637 6.540 I3.08I 35 756. 10 . 725 1.242 1-552 1.655 ! 6.613 13.226 40 747- 89 -733 1.255 1.569 1.673 6. 685 I3.37I 45 739- 86 .740 I. 269 1.586 i. 691 6.758 13.516 5 732.01 .748 1.283 i. 603 i. 710 6.831 13.661 55 724.3 1 .756 1.296 i. 620 1.728 6.903 13. 806 8 o 716. 78 .764 I.3IO 1.637 1.746 6.976 13.951 5 79- 40 . 772 1.324 1.654 1.764 7.048 14. 9 6 10 702. 1 8 .780 1-337 1.671 1.782 7. 121 14.241 15 695- 09 .788 I.35I i. 688 1.801 7- J 93 14.387 20 688. 16 -796 I. 365 i. 705 1.819 7.266 14. 532 2 5 681.35 .804 1.378 1.722 1.837 7.338 14.677 3 674. 69 .812 1.392 1.739 1.855 7.411 14. 822 35 668. 15 ; .820 1-757 1.873 7-483 14. 967 RAILROAD CURVES. 51 XIX. To trace Railroad Curves, &c. Continued. ORDIXATES. d o "o ,0 Degree. Radii. To circular arcs on a chord of 100 feet. o 1 12* 25 371 5 5 . o H o 8 40 661. 74 .828 1.419 1-774 1.892 7.556 15. 112 45 655.45 .836 L433 1.791 i. 910 7.628 15.257 50 649. 27 .844 1.447 i. 808 1.928 7.701 15.402 55 643. 22 .852 1.460 1.825 1.946 7-773 15.547 9 o 637.27 .860 1.474 1.842 1.965 7.846 15.692 5 631.44 .868 1.488 1.859 1.983 7.918 15.837 10 625. 71 .876 i. 501 1.876 2. OOI 7.991 15.982 15 620. 09 .884 L5I5 1.893 2. OI9 8.063 16. 127 20 614. 56 .892 1.529 i. 910 2.037 8.136 16.272 25 609. 14 . 900 1.542 1.927 2. 056 8.208 16.417 30 603. 80 .908 1.556 1.944 2.074 8.281 16. 562 35 598.57 . 916 I -57 1.961 2. 092 8-353 16. 707 40 593-42 .924 1-583 1.979 2. no 8.426 16. 852 45 588. 36 .932 1.597 1.996 2. 128 8.498 1 6. 996 5 583-38 .940 i. 6n 2.013 2.147 8.571 17. 141 55 578.49 .948 i. 624 2. 030 2. 165 8.643 1 7. 286 10 573.69 .956 1.638 - 2. 047 2.183 8.716 I7.43I 10 564.3 1 972 1.665 2. 081 2. 219 8.860 17.721 20 555.23 .988 1.693 2.115 2. 256 9.005 18. on 3 546.44 1.004 1.720 2.149 2.292 9.150 1 8. 300 40 537.92 i. 020 1.748 2. 184 2.329 9-295 18. 590 5 529. 67 1,036 1-775 2. 2l8 2.365 9.440 18.880 II 521.67 1.052 1.802 2.252 2.4O2 9.585 19. 169 10 5I3.9I i. 068 1.830 2.286 2.438 9.729 19.459 20 506.38 1.084 1-857 2.320 2-475 9.874 19. 748 30 499. 06 I. IOO 1.884 2.354 2.5II 10. 019 20. 038 40 491.96 i. 116 1.912 2.389 2-547 10. 164 20. 327 50 485. 05 1.132 1.938 2.423 2.584 10. 308 20.6l6 12 O 478. 34 i. 148 1.967 2-457 2. 620 io.453 2O. 906 IO 471.81 i. 164 1.994 2.491 2.657 10. 597 21. 195 2O 465.46 1. 180 2.021 2.525 2.6 9 3 10. 742 21.484 30 459- 28 1. 196 2.049 2.560 2.730 10. 887 21. 773 40 453- 26 I. 212 2.076 2-594 2.766 11.031 22. 063 50 447.40 1.228 2. 104 2.68 2.803 n. 176 22.352 5 2 RAILROAD CURVES. XIX. To trace Railroad Curves, &c. Continued. ORDINATES. . .2 "o o Degree. Radii. To circular arcs on a chord of I oo feet. i .* 25 37* bX) 50 : o 6 1 441.68 1.244 2.131 2.662 2.839 11.320 22. 641 IO 436. 12 1 1.260 2.159 2.697 2.876 i 11.465 22. 930 20 430.69 1.277 2.186 2.731 2.912 : 11.609 23.219 3 425.40 | 1.293 2.213 2.765 2.949 ; 11.754 23. 507 40 420.23 1.309 2. 241 2.799 2.985 j 11.898 23. 796 5 415.19 1.325 2.268 2-833 3.022 ! 12.043 24. 085 14 o 4IO. 28 1.341 2. 296 2.868 3. 058 ij 12. 187 24- 374 10 405- 47 1.357 2.323 2.902 3.095 jj 12.331 24. 663 20 400. 78 1.373 2.351 2.936 3.I3I 12.476 24.951 3 396. 20 1.389 2.378 2.970 3. 1 68 1 12.620 25. 240 40 391. 72 1.405 2.406 3- 00 5 3.204 : 12.764 25-528 50 387- 34 1.421 2.433 3.039 3.241 12.908 25.817 15 o 383. 06 1.437 2.461 3-073 3-277 13.053 26. 105 10 378.88 1-453 2.488 3.107 3-3H 13- 197 26. 394 20 374- 79 i. 469 2.515 3- 142 3-350 13-341 26. 682 3 370. 78 1.486 2-543 3.176 3-387 13.485 26. 970 40 366. 86 1.502 2.570 3.210 3.423 13.629 27.258 5 363. 02 1.518 2.598 3-245 3.460 13. 773 27.547 16 o 359- 26 1-534 2.625 3-279 3.496 13-917 27.835 IO 355-59 1-55 2.653 3.313 3-533 14. 061 28. 123 20 35I-98 1.566 2.680 3-347 3.569 14.205 28.411 3 348. 45 1.582 2.708 3-382 3.606 14-349 28. 699 40 344- 99 1.598 2.736 3.416 3- 6 43 14- 493 28. 986 341.60 1.615 2.763 3.450 3-679 14-637 29-274 17 o 338.27 i . 63 1 2.791 3.485 3.716 ! H. 781 29. 562 10 335- OI 1.647 2.818 3.519 3-752 j 14-925 29. 850 20 331-82 1.663 2.846 3.553 3-789 15. 069 30. 137 3 328. 68 1.679 2.873 3.588 3-825 15.212 3. 425 40 325. 60 1.695 2.901 3.622 3.862 15.356 30. 712 5 322. 59 1:711 2. 928 3.656 3.898 15.500 31. ooo 18 o 319.62 i. 728 2.956 3.691 3-935 15.643 31.287 IO 316.71 1.744 2.983 3.725 3-972 15. 787 31.574 20 313-86 i. 760 ^. 01 1 3-759 4.008 i! I5-93 1 31.861 RAILROAD CURVES. 53 XIX. To trace Railroad Curves, &c. Continued. ORDIXATES. *J3 a i ^ o 1 ! Degree. Radii. j.To circular arcs on a chord of 100 feet. ; ^ \\ ~Z 1 i2| 25 37}- bfl 50 o . 1 || H CJ o 18 30 311.06 1.776 3.039 3.794 4-045 16.074 32. 149 40 308.30 ; 1.792 3.066 3.828 4.081 16.218 32.436 50 3O5.6O \ 1.809 3.094 3.862 4.118 16.3^1 32. 723 19 o 302.Q4 1.825 3.I2I 3.897 4.155 16.505 33.010 IO 300.33 I.84I 3. 149 3.931 4. 191 16.648 33. 296 20 297-77 1.857 3- I 77 3-9 6 5 4.228 16.792 33.583 30 295.25 i 1.873 3.204 4.000 4.265 16.935 33.870 4 292.77 . 1.890 3.232 4.034 4.301 17.078 34.157 5 290.33 <i 1.906 3.259 4.069 4.338 17.222 34- 443 LONG CHORDS. Degree : of curve. 2 stations. 3 stations. 4 stations. 5 stations. 6 stations. 10 2OO. 000 299. 999 399- 998 499- 996 599- 993 20 199. 999 997 992 .983 .970 3 .998 .992 .981 .962 933 40 997 .986 .966 932 .882 50 995 979 947 894 .815 I 199. 992 209. 970 309. 924 499. 848 599- 733 IO .990 959 .896 . -793 .637 20 .986 .946 .865 .729 .526 3 983 932 .829 .657 .401 40 979 915 .789 -577 . 260 50 .974 .898 744 .488 -105 2 199. 970 209. 878 399- 6 95 499- 39i 598. 934 IO .964 857 .643 .285 750 20 959 834 .586 .171 55 30 952 .810 524 .049 .336 4 .946 .783 459 498.918 . 106 50 : 939 756 . 389 . 778 597- 862 54 RAILROAD CURVES. XIX. To trace Railroad Curves, &c. Continued. LONG CHORDS Continued. Degree of curve. 2 stations. 3 stations. 4 stations. 5 stations. 6 stations. / 3 o 199.931 299. 726 399- 3*5 498. 630 597.604 IO .924 695 237 474 331 20 .915 .662 154 .309 043 3 .907 .627 .068 136 596. 740 40 .898 591 398.977 497- 955 423 50 .888 553 .882 765 . 091 4 o 199.878 299.513 398. 782 497. 566 595- 744 IO .868 .471 .679 .360 383 20 .857 .428 571 145 .007 30 .846 .383 459 496.921 594.617 4 .834 337 343 .689 . 212 50 .822 .289 .223 449 593- 792 5 o 199. Sio 299. 239 398. 099 496. 200 593.358 10 797 .187 397. 970 495- 944 592. 909 20 ,783 134 837 . 678 .446 30 .770 .079 .700 .405 591-968 40 .756 .023 559 .123 476 50 .741 298. 964 .413 494. 832 590. 970 6 o 199. 726 298. 904 397- 264 494- 534 590. 449 10 .710 843 . no .227 589. 9 J 3 20 .695 779 396. 95 2 493.912 .364 3 .678 .714 .790 .588 588. 800 40 .662 .648 .623 257 . 221 5o .644 579 453 492.917 587.628 7 o 199. 627 298. 509 396. 278 492. 568 587.021 10 . 609 .438 .099 .212 586. 400 20 591 3 6 4 395- 9 l6 491.847 585. 765 30 572 .289 .729 ! .474 "5 4 553 .212 538 093 584-451 50 533 .34 342 490. 704 583- 773 8 o 199.513 298.054 395- H2 49 0. 306 583. 08 i 1 . GAUGING OF RIVERS. 55 XX. To ascertain the Discharge of Water in any Stream. i. For practically gauging large rivers a locality is selected in a straight portion of the stream where the water flows smoothly and without obstruction. A base-line about 200 feet long is laid out parallel to the current, and the exact cross-section in front of this base is determined by careful sounding. To obtain the discharge, two theodolites are established, and the angular distance from, and the times of transit past, each end of the base, of numerous floats, well distributed between the banks, are noted. The floats should be made double, the surface-float being a minute tin ellipsoid, a piece of cork, or some other small light body, bearing a small flag. The lower float may be a large box or keg without top or bottom, kept upright by lead ballasting; or better, because lighter, two sheets of tin bent at right angles, and soldered together at the bend, so as to make all the angles between the four faces right angles; the essential conditions being that the lower float shall so greatly preponderate in area over the upper, and shall be connected by so fine a wire or cord, that its rate of movement will govern the whole combination. The center of the lower float should be placed at the mid- depth of the stream, in each vertical plane of transit, because the rate of movement will then be unaffected by wind. As it is sometimes troublesome to adjust to mid-depth in the different planes of transit, when there is a tolerably uniform and symmetrical cross-section, the average mid-depth of the river may be adopted for all the floats without sensible error. If floats passing near the surface are used, errors in the com puted discharge may be caused by an ordinary breeze, and as these errors are positive or negative according to the direction of the wind, discrepancies may result in the measurements of differ ent days when there is no real variation in the discharge. All this uncertainty is avoided by using mid-depth floats. The exact level of the water-surface on a permanent gauge- rod should be carefully noted when the observations begin and terminate. 5 6 GAUGING OF RIVERS. XX. To ascertain the Discharge of Water, &c. Continued. Upon a sheet of section-paper the base-line and the two per pendiculars across which the times of transit were noted are then laid down, and, from the recorded angles and a table of natural tangents, the distances from the base-line to the points at which each float passed both lines are plotted. These points, being connected, indicate the paths of the floats. Upon each path the difference between the two recorded times of transit is written in seconds. These seconds of transit are next examined, and the total width of the river is marked off into as many "divisions" as it seems proper to assume are traversed by water moving with sensibly unvarying veolcity, say, for instance, that each division is about Jy of the width of the stream. A mean of the seconds of transit of all the floats in each " division " is next taken, and, when reduced to velocity in feet per second, is adopted as the mid-depth velocity in that "division." A mean of all these mean mid-depth velocities interpolations being made if any are missing closely approximates the mean velocity of the river, provided the " divisions " are equal in width. This method involves two errors, which nearly balance each other, viz: the inequality in area of the divisions, and the differ ence between the mid-depth velocity and the mean velocity in any vertical plane, giving a resulting mean velocity of about 0.95 times its true value. The mean velocity in each plane is obtained from the mid- depth velocities, V J D, by the formula 7 = 1.075 Vi D + 0.004 0.093 (V -J I), b)* and b- T 69 B {D+i. S )* where D = the depth of the stream at any point of the surface. If a is the area of cross-section, and a , a", etc., the partial division areas, the discharge may be found by where denotes the sum of similar quantities. GAUGING OF RIVERS. 57 XX. To ascertain the Discharge of Water , &c. Continued. 2. Determination of the mean velocity in terms of the dimen sions of the cross-section and the slope. Humphreys-Abbot Formula. (Not applicable to water flowing in smooth artificial channels.) = V 0.0081 b+ (225 /- 1 ^)i_ 0.09 !+/ where the symbols have the following signification, all expressed in English feet : . v = mean velocity =- ; a = area of cross-section ; W = width ; r = mean radius, or - 7/ = value of first term in expression for T\ p = wetted perimeter ; Q = discharge in cubic feet per second; a = 7Tw ; 5- = sine of slope of water surface .corrected for bends; b = function of the depth, for small streams = ^ (r+ 1.5)* For rivers whose mean radius exceeds 12 or 15 feet, b may be assumed to be 0.1856, which will make the numerical value of the term involving b so small that it may be generally neglected, reducing the above equation to :=([ 225^]* -0.0388) The following formulae give the value of each variable in terms of the others and known quantities : z = 0.93 v -f- 0.167 (P l )* > and when -p is not known by measurement it may, for ordinary natural channels, be assumed to be 1.015 ^. + W)*\ 195 a ) a = (P + W) z* 95 W* p + W = T 95 (^ 5 8 GAUGING OF RIVERS. XX. To ascertain the Discharge of Water, err. Continued. APPLICATION. The variables which enter these formulas require a knowledge of the mean cross-section of the stream, and a map of the course of the channel between two selected points of the water-surface, whose difference of level should be exactly known. Whenever practicable, the two points should be located on a straight and regular portion of the river to eliminate the effects of bends. As this is not always possible, the general case is con sidered in the above fomiulce, and bends are assumed to exist between the points selected. The field-operations consist in a survey of the channel, with numerous soundings between permanent bench-marks placed near the water, and in running a line of levels between those marks, so as to give their relative level with the most extreme accuracy. These points should be located with care, as far apart as prac ticable, distant from any eddy, and placed where the current on the banks flows with equal velocity. This latter condition is neces sary, because, as water in motion exerts less pressure than when at rest, if it moved rapidly past one bench-mark and was nearly stationary at the other, a difference of level, which has nothing to do with the motive power of the stream, would vitiate the observation. In determining the mean dimensions of cross-sections, care must be taken to extend the soundings throughout the entire distance between the bench-marks, and it must be borne in mind that measured fall in water-surface between two stations corre sponds to the mean channel between them. When the soundings are made, the water-level should be referred to the bench-marks in order to determine the area corresponding to any subsequent stand of the river. These soundings completed, frequent gauging of the river can be made by referring at any time, by accurate levels, the water- surface at the two points to their respective bench-marks, thus determining, the fall and corresponding cross-section of the river from which the discharge is computed. The observations must be simultaneous in order to avoid the effect of any oscillation in the river, and calm days should be selected because waves render it difficult to determine the exact level of the water-surface; and, also, changes of level result from the general piling up or lowering of the water under the influence of winds. GAUGING OF RIVERS. 59 XX. To ascertain the Discharge of Water , &c. Continued. Correction for Bends. A line following the mid-channel is drawn on the map, composed of straight lines with angular changes, wherever necessary, of 30. A mean velocity is assumed, to be corrected subsequently if required, and the value of h is computed in the following formulas, in which N represents the number of deflections : N sin 2 30 134 The deduced value of h is next subtracted from the total fall in the water-surface between the two stations; the remainder divided by the distance in feet between these stations, measured on the middle line of the river, is the true value of s in the formula for mean velocity. If any material error has been made in assuming r, the com putation should be repeated until the requisite approximation has been made. By expressing the formula in the form M M V The following table will facilitate its application: r M I/ At P M Log M i o. 0087 o. 0930 5 o. 400 9. 602060 2 73 855 6 343 9- 535 2 94 -> 65 803 7 .300 9.477121 4 58 764 8 .267 9.426511 c 54 733 9 . 240 9. 380211 6 ^o 707 10 .218 9- 338456 7 i 47 685 12 .185 9.267172 8 44 666 14 . 160 9. 204120 9 42 649 16 .141 9. 149219 10 4 634 18 .126 9. 100371 12 37 610 20 .114 9- 056905 14 35 590 22 . 104 9.017033 . 16 33 573 24 . 096 8.982271 18 558 26 .089 8. 949390 20 29 544 28 .083 8.919078 3 24 494 3 .078 8. 892095 50 Q. 0019 0.0437 50 0.047 8. 672098 For streams larger than 50 or 100 feet in cross-section the term involving M may be dropped, and for larger rivers, exceeding 1 2 or 20 feet in mean radius, M, but not \l M, may be neglected. 6o GAUGING OF RIVERS. 3 G Q J~ QJ ^ Q I p 3 ,nduio D | i & *o u-> * CO G cS o p3Aiosqo o ; S 1 "> ro ro co VO <? bjo 1 o co" s- . | ^ CO oo : N o ^ X 5, ^" jD fcJC cn $ oo Cfl - H (M _ 8 V) . 6 6 o 6 rt CL, 1 ? ^ U O $ - u ^ J u 1 8 </: 1 ^ 1? 1 J? ? ^ o S 5 ci o ^ ro ^*- 85 ON CO CO OO o ^ . Q C MM i > | H a S jj u -; "3 d o : u J "o 3 S O ?> Q U 525 ; o o s o 3 iquemin ke&Oh feeder. 8 1 E 3 O - 1 3 O S K MEASUREMENT OF FLOWING WATER. 6 1 XX. To ascertain the Discharge of Water, &c. Continued. 3. Formulae for the mean velocity, from other authorities : f Downing s and others co-efficient v = 100.0 (rs)* Chezy . . . <( Eytelwein s co-efficient v = 93.4 (rs) i. Young s co-efficient v= 84.3 (rs) !For canals v = (0.0556 + 10593 rs)* - 0.235 7 For canals and pipes ....... v = (0.0237 + 9966 rs)* o. 1542 Eytelwein s co-efficient . .... v = (0.0119 + 8963 rs)* 0.1089 Weisbach s co-efficient v = (0.00024 + 8675 rs)^ 0.0154 / IOOO S Darcy-Bazin Z/ = ^0^8534^+^0^5 XXI. Motion of Water in Conduit- Pipes. Discharge through pipes of uniform dimensions, and having no sudden changes of direction : For ordinary cases : Q = 38.436 - - 0.070862 D In great velocities : Q = 36.7*9 If the velocity be required, divide the discharge by the area of the section (0.7854 D 2 ). To find the diameter of a conduit-pipe for a given discharge under a given head : Where Q = discharge in cubic feet per second; H, the head, and D and L the diameter and length of the pipe in feet. The resistance of curves is proportional to the square of the velocity of the fluid, to the number of angles of reflextiqn, and to the square of their sine, Q2 or, in function of Q, =0.006079 4 j2 s 2 being the sum of the squares of all the sines of the angles of reflexion. 1 62 LOGARITHMS. XXII. Logarithms of Numbers. 8 Proportional parts. ^ 1 2 3 4 5 6 y 8 9 fc I * 3 1 I 6 1 8 9 10 .0000 .0043 .0086 .0128 .0170 .0212 0253 .0294 0334 0374 4 8 12 7 21 25,29 33 37 II .0414 453 .0492 053 1 .0569 .0607 .0645 .0682 .0719 0755 4 8 I 1 i g 19 23 26^30 34 12 .0792 .0828 .0864 .0899 934 .0969 . 1004 .1038 . 1072 .1106 3 7 IO 1 17)212428 3 1 T 3 "39 "73 .I2o6[ .1239 .1271 1303 J 335 .1367 J 399 .1430 3 6 10 , ; 1619232629 14 .1461 .1492 1523 1553 .1584 .1614 .1644 .1673 .1703 1732 3 6 9 12 15 1821 24 27 15 .1761 .1790 .1818 .1847 1875 .1903 1931 1959 .1987 .2014 q 6 8 i i 14 17 20 22 25 16 .2041 .2068 .2095 .2122 .2148 2175 .2201 .2227 2253 .2279 3 5 8 i i 13 i6 i8 21 24 17 .2304 2330 2355 .2380 .2405 .2430 2455 .2480 .2504 .2529 2 5 ;; ii > I2 l5 17 2022 18 2553 2577 .2601 .2625 .2648 .2672 .2695 .2718 .2742 .2765 2 5 7 9 12^4 16 19 21 19 .2788 .2810 2833 .2856 .2878 .2900 .2923 2945 .2967 .2989 2 4 - 9 n i3 16 IS 2O 20 . 3 oio| .3032 3054 3075 3096 3"8 3139 .3160 .3181 3201 2 4 < 8 II 13 is J 7 19 21 .3222 3243 3263 3284 .3304 3324 3345 33 6 5 3385 344 2 4 ! IOJI2JI4 (6 18 22 3424 3444 3464 0483 3502 3522 354 1 3560 3579 3598 2 4 < 10 12 14 15 17 23 3 6l 7 3636 3 6 55 -3 6 74 .3692 37" 3729 3747 .3766 3784 2 4 7 9" J 3 15 17 2 4 .3802 .3820 3838 3856 3874 .3892 3909 3927 3945 3962 2. 4 1 9 " \ - i 16 2 5 3979 -3997 .4014 .4031 .4048 .4065 .4082 .4099 .4116 4i33 2 3 7 9 10 12 H 15 26 .4150 .4166 .4183 .4200 .4216 .4232 .4249 .4265 .4281 .4298 2 3 - 8 10 ii 1 .: 15 2 7 43 J 4 -433 4346 .4362 4378 4393 .4409; .4425 .4440 .4456 2 3 5 6 8 9" 13 14 28 4472 .4487 .4502 .4518 4533 4548 4564 4579 4594 .4609 2 3 g < 8 9" 12 4 29 .4624 4 6 39 .4654 .4669 .4683 .4698 .4713 .4728 4742 4757 I 3 4 6 - 910 1 . 13 30 477 1 .4786 .4800 .4814 .4829 4843 4857 .4871 .4886 .4900 I 3 6 7 I II ,3 31 .4914 .4928 .4942 4955 .4969 4983 4997 .5011 .5024 .5038 I 3 i 7 S IQ II 12 S 2 505 1 5065 579 .5092 5 I0 5 5"9 5132 5 I 45 5i59 5 I 7 2 I 1 7 8 9 II 12 33 0185 .5198 .5211 .5224 5237 5250 .5263 .5276 .5289 .5302 I 3 i 6| 8 9 10 12 34 5315 .5328 5340 5353 .5366 5378 539 1 543 .5416 .5428 I 3 6 8 9 IO II 35 5441 5453 54 6 5 5478 549 5502 5514 5527 5539 5531 I 2 6 7 9 IO II 36 5563 -5575 5587 5599 .5611 5623 5 6 35 5647 5658 .5670 I 2 g 6 7 ;- IO II 37 .5682 5694 575 57 I 7 5729 574 5752 57 6 3 5775 .5786 I 2 2 g 6 7 I 9 10 38 .5798 .5809 .5821 5832 .5843 5855 .5866 5877 .5888 5899 I 2 ; 6 7 8 9 10 39 59" .5922 5933 5944 5955 .5966 5977 .5988 5999 .6010 I 9 ! - 7 8 . , IO 40 .6021 .6031 .6042 .6053 .6064 .6075 .6085 .6096 .6107 .6117 I 2 , 5 6 8 " IO 4 1 .6128 .6138 .6149 .6160 .6170 .6180 .6191 .6201 .6212 .6222 I a -i i ( 7 9 42 .6232 .6243 6253 .6263 .6274 .6284 .6294 .6304 .6314 6325 I a ., 6 7 8 9 43 .6335 6 345 6355 6365 .6375 6385 6395 .6405 .6415 .6425 I 2 i - 6 7 8 9 44 6435 .6444 6 454 .6464 .6474 .6484 6493 .6503 6513 .6522 I a 3 ! -> 6 7 8 9 45 6532 6542 6551 .6561 .6571 .6580 .6590 6 599 .6609 .6618 I 2 , I -, 6 7 8 9 46 .6628 .6637 .6646 .6656 .6665 .6675 .6684 .6693 .6702 .6712 1 2 , s 6 7 7 8 47 .6721, .6730 6739 .6749 .6758 .6767 .6776 .6785 .6794 .6803 1 a 3 ., s S 6 7 8 48 49 .6812 .6821 .6902! .6011 .6830 .6920 .6839 .6928 .6848 .6857 6046 .6866 .6875 .6884 .6893 .6981 I 2 3 4 1 5 6 7 8 50 .6990 .6998 .7007 .7016 .7024 u y4 u 7033 ^955 .7042 .7050 .OQ72 759 .7067 z 9 i s ... 7 8 51 .7076 .7084 793 . 7101 .7110 .7118 .7126 7*35 7*43 7152 I 2 3 ; r . 6 7 8 52 .7160 .7168 7177 7185 7*93 .7202 .7210 .7218 .7226 7235 I 2 . . 1 S 6 7 7 53 7243 7251 7259 .7267 7275 .7284 .7292 .7300 .7308 .7316 I 2 3 ; 5 , 54 7324 7332 7340 7348 7356 73 6 4 7372 738o 7388 7396 I 2 a 1 S 6 67 _ _ LOGARITHMS. 63 XXII. Logarithms of Numbers Continued. Proportional parts. ^ Q -| Q 3 4 5 6 7 8 9 cd 1 2 { J 3 <; 7 8 9 ^ 1 55 .7404 .7412 .7419 .7427 7435 7443 7451 7459 .7466 7474 ! 3 3 | 4 5 5 6 7 56 .7482 7490 .7497 7505 7513 7520 .7528 753 6 7543 7551 I 3 -i 5 5 6 7 57 7559 .7566 7574 .7582 .7589 7597 .7604 .7612 .7619 .7627 I . 2 | 4 : 5 6 7 58 7 6 34 .7642 .7649 7657 .7664 .7672 .7679 .7686 .7694 .7701 1 1 3 | 4 4 5 6 7 59 .7709 .7716 7723 773i 7738 7745 7752 .7760 .7767 7774 I 1 2 I 4 4 5 6 7 60 .7782 .7789 .7796 .7803 .7810 .7818 .7825 7832 7839 .7846 I i 2 -I 4 5 6 6 61 7853 .7860 .7868 7875 .7882 .7889 .7896 7903 .7910 .7917 I I 2 , 4 4 5 6 6 62 .7924 7931 7938 7945 7952 7959 .7966 7973 .7980 7987 I i . 3 i 5 6 6 63 7993 .8000 .8007 .8014 .8021 .8028 .8035 .8041 .8048 8055 1 I 3 4 5 5 6 i . 64 .8062 .8069 8075 .8082 .8089 .8096 .8102 .8109 .8116 .8122 I I 4 5 5 6 65 .8129 .8136 .8142 .8149 .8156 .8162 .8169 .8176 .8182 .8189 I 1 : 4 5 5 6 66 8195 .8202 .8209 .8215 .8222 .8228 .8235 .8241 .8248 .8254 I I 2 ! 4 5 5 6 67 .8261 .8267 .8274 .8280 .8287 .8293 .8299 .8306 .8312 .8319 I I . 3 4 5 5 6 68 .8325 8331 8338 .8344 8351 .8357 -8363 .8370 .8376 .8382 ; J i I 2 3 4 4 5 6 69 .8388 .8395 .8401 .8407 .8414 .8420 .8426 .8432 .8439 .8445 i 2 i 4 -1 56 70 .8451 8457 .8463 .8470 .8476 .8482 .8488 .8494 .8500 .8506 i I 2 3 4 4 56 71 8513 .8519 8525 8531 853 8543 .8549 .8555 .856 .8567 i i 1 2 4 4 5| 5 72 8573 8579 8585 .8591 859 .8603 .8609 .8615 .862 .8627 i I 2 2 4 4 5| 5 73 8633 .8639 8645 8651 .865 .8663 .8669 8675 .868 .8686 i I 2 4 4 5 5 74 .8692 .8698 .8704 .8710 .871 .8722 .872 8733 -8739 .8745 i I 2 4 4 5 5 75 8751 .8756 .8762 .8768 .877 .8779 .878 .8791 .879 .8802 i i 2 3 4 5 5 76 .8808 .8814 .8820 .8825 .883 .8837 .884 .8848 .885 8859 i i 2 3 4 5 5 77 .8865 .8871 .887 .8882 .888 8893 .8899 .8904 .891 .8915 i I 2 4 4 5 78 .892 .8927 .893 8938 894 .8949 895 .8960! .896 .8971 i i 2 3 4 4 5 79 .8976 .8982 .898 8993 .899 .9004 .900 .9015: .902 .9025 1 2 3 4 4 5 80 93 .9036 .904 .904 95 .9058 .906 .9069 .907 .9079 i I 2 3 4 4 5 81 .908 .9090 .909 .910 .910 .9112 .911 .9122 .912 9 X 33 i I 2 3 ; 4 5 82 913 9H3 .914 9154 915 .9165 .917 9^75 .918 .9186 i 1 2 3 4 4 5 8 .919 .9196 .920 .9206 .921 .921 .922 .9227 923 .9238 i I 2 3 .( 4 5 8 .924 .9248 925 925 .926 .926^, .927 9279 .928 .9289 I 2 3 4 4 5 8 .929 9299 93 93 931 932C 932 933 933 9340 i I 3 3 i 4 5 8 934 935c 935 93 6 .936 937C 937 .9380 .938 9390 i 1 Z 3 i 4 5 8 939 .9400 .940 .941 .941 .9420 .942 943 943 .9440 I ] 3 4 4 8 944 945C 945 .946 .946 .946 947 9479 .948 .9489 I 3 1 4 4 8 949 .9499 950 95 95 1 951 952 9528 953 9538! o i I ; 3 1 4 4 9 954 9547 955 955 .956 956 957 957 6 958 .9586 o ! I . - 4 4 9 959 959: .960 .960 .960 .961 .961 .9624 .962 9 6 33 o I i , * 3 4 4 9 963 .964; .964 9 6 5 .965 .966 .966 .9671 .967 .9680 o I I 3 4 4 9 .968 .9689 .969 .969 .970 .970 .971 .9717 .972 .9727 ( 1 2 1 3 4 4 9 973 973^ 974 974 975 975 975 9763 .976 9773 < i 3 4 4 1 9 977 978? .978 979 979 .980 .980 .9809 .981 .9818 ( ] ! 1 4 4 9 .982 .982; 983 983 .984 .984 .985 .9854 985 9863 [ 2 3 4 4 9 .986 .987= .987 .988 .988 .989 .989 .9895 .990 .9908 < I . . 4 4 9 .991 .991} .992 .992 993 993 993 9943 994 9952 1 3 4 4 9 995 .9961 996 .996 997 997 .998 .9987 .999 .9996 o; i i - - 3 . 4 I SINES AND TANGENTS. XXIII. Logarithms of Sines and Tangents. o i" Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. o I 2 3 4 5 6.4637 .7648 6 . 9408 7-0658 .1627 o.oooo .0000 .0000 .0000 .0000 .0000 6.4637 .7648 6.9408 7.0658 . 1627 3-5363 2352 3-0592 2.9342 8373 8.2419 .2490 .2561 .2630 .2699 .2766 9-9999 9999 9999 9999 9999 -9999 8.2419 .2491 .2562 .2631 .2700 2767 1-7581 759 7438 7369 .7300 7233 60 59 58 57 56 55 6 I 9 10 .2419 .3088 .3668 .4x80 4637 .0000 .0000 .0000 .0000 .0000 .2419 .3088 .3668 .4180 4637 7581 .6912 6332 .5820 5363 .2832 .2898 .2962 3 02 5 .3088 9999 9999 9999 9999 9999 2833 .2899 2963 .3026 3089 .7167 .7101 .7037 6974 .6911 54 53 52 50 ii 5051 .0000 5051 4949 .3150 9999 -3150 .6850 49 12 5429 .0000 5429 457 1 .3210 9999 .3211 .6789 48 3 5777 .6099 6398 .0000 .0000 .0000 5777 .6099 6398 .4223 .3901 .3602 .3270 3329 .3388 9999 9999 9999 3271 3330 3389 .6729 .6670 ,66n 47 46 45 16 J 7 .6678 .6942 .0000 .0000 .6678 .6942 -3322 3058 3445 3502 9999 9999 3446 353 6 554 .6497 44 43 18 .7190 .0000 .7190 .2810 3558 9999 3559 ~y< .0441 4 2 7425 .7648 .0000 .0000 7425 .7648 2575 2352 3613 .3668 9999 9999 .3614 3669 .6386 6331 4 1 40 21 22 23 24 7859 .8061 8255 8439 .0000 .0000 .0000 .0000 .7860 .8062 8255 .8439 .2140 .1938 1745 1561 3722 3775 .3828 .3880 9999 9999 9999 9999 3723 3776 .3829 .3881 .6277 .6224 .6171 .6119 39 38 2 5 .8617 .0000 .8617 1383 393 1 9999 3932 .6068 35 26 27 .8787 .8951 .0000 .0000 .8787 .8951 .1213 .1049 3982 .4032 9999 9999 -3983 4033 .6017 5967 34 33 28 .9109 .9261 .0000 .0000 .9109 .9261 .0891 739 .4082 4 I 3 1 9999 9999 .4083 .4132 59*7 .5868 S 2 3 1 30 .9408 .0000 .9409 .0591 .4179 9999 .4181 5819 30 CO CO Co CO 955 1 .9689 .9822 7-9952 .0000 .0000 .0000 .0000 9551 .9689 9823 7-9952 .0449 .0311 .0177 2.0048 .4227 4275 4322 .4368 9998 .9998 .9998 .9998 .4229 .4276 4323 437 5771 5724 5677 .5630 28 27 26 35 8.0078 .0000 8.0078 1.9922 .4414 9998 .4416 -5584 2 5 36 .0200 .0000 .0200 .9800 4459 9998 .4461 5539 24 37 .0319 .0000 .0319 .9681 4504 .9998 .4506 5494 2 3 39 40 435 .0548 .0658 .0000 .0000 .0000 435 .0548 .0658 9565 9452 934 2 4549 4593 4 6 37 .9998 9998 9998 455 r .4638 -5449 5405 5362 22 21 2O 4 1 .0765 .0000 .0765 9235 .4680 .9998 .4682 .5318 ig 42 .0870 .0000 .0870 .9130 4723 .9998 4725 5275 1 8 43 .0972 .0000 .0972 .9028 4765 9998 4767 5 2 33 17 44 .1072 .0000 .1072 .8928 .4807 9998 .4809 16 45 .1169 .0000 .1170 .8830 .4848 .9998 .4851 5149 15 46 47 48 1265 1358 .1450 .0000 .0000 . .1265 *359 .1450 8735 .8641 8550 .4890 4930 .4971 9998 .9998 .9998 .4892 -4933 4973 .5108 5067 5027 13 12 49 50 .1627 .0000 .0000 .1540 .1627 .8460 8373 .5011 5050 9998 9998 5013 5053 4987 4947 II 10 : I 7 I 3 .0000 I 7 I 3 .8287 .5090 9998 .5092 .4908 .1797 o.oooo .1798 .8202 .5129 .9998 5 1 3 I .4869 8 53 54 . 1880 .1961 9-9999 9999 .1880 .1962 .8120 .8038 5167 .5206 9998 .9998 5 1 ? .5208 .4830 4792 7 6 55 .2041 9999 .2041 7959 5243 9998 .5246 4754 5 56 .2119 9999 .2120 .7880 .5281 .9998 .5283 . -47 T 7 4 57 .2196 9999 .2196 .7804 5318 9997 5321 -4679 3 58 59 .2271 2346 9999 9999 .2272 2346 7728 7654 5355 5392 9997 9997 5358 5394 .4642 .4606 2 I 60 8.2419 9-9999 8.2419 1.7581 8.5428 9-9997 8-5431 1.4569 O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 89 88 SINES AND TANGENTS. XXIII. Logarithms of Sines and Tangents Continued. 2 3 4 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. o I 8.5428 54 6 4 9.9997 9997 8.5431 5467 1.4569 4533 3. 7 i88 .7212 9-9994 9994 8.7194 .7218 i . 2806 .2782 8.8436 8454 9.9989 9989 8.8446 8465 I.IJ54 !535 60 59 2 550 9997 5503 4497 .7236 9994 .7242 2758 .8472 .9989 8483 J5I7 58 3 5535 9997 5538 .4462 .7260 9994 .7266 2734 .8490 9989 8501 .1499 57 4 557 1 9997 5573 4427 .7283 9994 .7290 .2710 .8508 9989 .8518 .1482 56 5 5605 9997 .5608 4392 737 9994 73i3 .2687 8525 9989 .8536 .1464 55 6 .5640 9997 5 6 43 4357 733 9994 7337 .2663 8543 9989 .8554 .1446 54 7 5674 9997 5677 4323 7354 9994 .7360 .2640 .8560 .9989 .8572 .1428 53 8 .5708 9997 57 11 .4289 7377 9994 .7383 .2617 .8578 .9989 .8589! .1411 5 2 9 5742 9997 5745 4255 .7400 9993 .7406 2594 8595 .9989 .8607 1393 51 10 577 6 9997 5779 .4221 7423 9993 7429 257 1 .8613 .9989 .8624 1376 50 ii .5809 9997 .5812 .4188 7445 9993 7452 2548 .8630 .9988 .8642 1358 49 it .5842 9997 .5845 4*55 .7468 9993 7475 2525 .8647 .9988 .8659! .1341 48 13 .5875 9997 .5878 .4122 .7491 9993 7497 .2503 .8665 .9988 .8676 .1324 47 *4 597 9997 59" .4089 7513 9993 .7520 .2480 .8682 .9988 .8694] .1306 46 15 5939 9997 5943 457 7535 9993 7542 .2458 .8699 .9988 .8711 .1289 45 16 5972 9997 5*975 .4025 7557 9993 7565 2435 .8716 .9988 .8728 .1272 44 *7 .6003 9997 .6007 3993 .7580 9993 .7587 .2413 8733 .9988 8745 1255 43 1 8 6035 .9996 .6038 3962 .7602 9993 .7609 .2391 .8749 .9988 .8762 I2 38 42 J 9 .6066 .9996 .6070 393 .7623 9993 7631 .2369 .8766 .9988 .8778 .1222 4 1 20 .6097 .9996 .6101 3899 7645 9993 .7652 .2348 .8783 .9988 8795 .1205 40 21 .6128 .9996 .6132 .3868 .7667 9993 .7674 .2326 .8799 9987 .8812 .1188 3Q 22 .6159 .9996 .6163 3837 .7688 .9992 .7696 2304 .8816 . 99 8 7 j .8829 .1171 38 2 3 .6189 .9996 .6193 3807 .7710 .9992 .7717 .2283 .8833 9987 .8845 "55 37 2 4 .6220 .9996 .6223 3777 7731 .9992 .7739 .2261 .8849 .9987 .8862 .1138 36 2 5 .6250 .9996 .6254 .3746 7752 .9992 .7760 .2240 .8865 9987 .8878 .1122 35 26 .6279 .9996 .6283 3717 7773 .9992 .7781 .2219 .8882 .9987! .8895 .1105 34 27 .6309 .9996 6313 .3687 7794 .9992 .78021 .2198 .8898 9987 .8911 .1089 33 28 29 6 339 .6368 .9996 .9996 6 343 -3657 .6372 .3628 7815 7836 9992 9992 .7823! .2177 .7844! .2156 .8914 .8930 .9987 .9987 .8927 .1073 32 .8944 .1056 31 30 6397 .9996 .6401 3599 7857 .9992 .7865 .2135 .8946 .0987 .8960 .1040 3 3 1 .6426 .9996 .6430 357 .7877 .9992 .7886 .2114 .8962 .9986 .8976 .1024 29 32 .6454 .9996 6459 354 1 .7898 9992 7906J .2094 .8978 .9986 .8992 .1008 28 33 .6483 .9996 .6487 35 I 3 .7918 .9992 7927 .2073 8994 .9986 .9008 .0992 27 34 .6511 .9996 6515 3485 7939 9992 7947 2053 .9010 .9986 .9024 .0976 26 35 6539 .9996 6544 3456 7959 9992 .7967 2033 .9026 .9986 .9040 .0960 25 36 .6567 .9996 657 1 3429 7979 .9991 .7988 .2012 9342 .9986 .9056 .0944 24 37 6595 9995 6599 .3401 7999 .9991 .8008 .1992 957 .9986 .9071 .0929 3 38 .6622 9995 .6627 3373 .8019 .9991 .8028 I 97 2 973 .9986 .9087 .0913 2 39 .6650 9995 6654 .3346 .8039 .9991 .8048 .1952 .9089 .9986 .9103 .0897 I 40 .6677 9995 .6682 33i8 .8059 .9991 .8067 ^933 .9104 .9986 .9118 .0882 O 4 1 .6704 9995 .6709 .3291 .8078 .9991 .8087 1913 .9119 9985 9 I 34 .0866 g 42 6 73 r 9995 .6736 3264 .8098 .9991 .8107 .1893 9*35 9985 .9150 .0850 8 43 .6758 9995 .6762 3238 .8117 .9991 .8126 .1874 .9150 9985 .9165 0835 7 44 .6784 9995 .6789 .3211 .8137 .9991 .8146 .1854 .9166 9985 .9180 .0820 6 45 .6810 9995 .6815 3185 .8156 .9991 .8165 1835 .9181 9985 .9196 .0804 5 46 .6837 9995 .6842 3158 8i75 .9991 .8185 .1815 .9196 9985 .9211 .0789 4 47 .6863 9995 .6868 3132 .8194 .9991 .8204 .1796 .9211 9985 .9226 .0774 3 48 .6889 9995 .6894 .3106 .8213 .0990 .8223 .1777 .9226 9985 .9241 759 2 49 .6914 9995 .6920 .3080 .8232 .9990 .8242 .1758 .924! 9985 .9256 .0744 i 5 .6940 9995 6945 3055 8251 999 .8261 I 739 .9256 9985 .9272 .0728 o 51 .6965 9995 .6971 .3029 .8270 .9990 .8280 .1720 .9271 .9984 .9287 OT^S 9 52 .6991 9995 .6996 .3004 .8289 .9990 .8299 . 1701 .9286 . 99 84 .9302 .0698 8 53 .7016 9994 .7021 .2979 .8307 999 8317 .1683 .9301 .9984 .9316 .0684 7 54 .7041 9994 .7046 2954 .8326 .9990 8336 .1664 9315 9984 933 1 .0669 6 55 .7066 9994 .7071 .2929 8345 999 .8355 .1645 933 9984 .9346 .0654 5 56 .7090 9994 .7096 .2904 8363 .9990 8373 .1627 9345 9984 .9361 .0639 4 57 7S 9994 .7121 .2879 8381 .9990 .8392 .1608 9359 .9984 9376 .0624 3 58 .7140 9994 7*45 2855 .8400 999 .8410 .1590 9374 .9984 Q3QO .0610 2 59 60 .7164 8.7188 9994 9-9994 .7170 8.7194 2830 1.2806 .8418 8.8436 .9989 9.9989 .8428 8.8446 1572 I - I 554 9388 8.9403 9984 ! .9405 9.99838.9420 0595 1.0580 I Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 87 86 85 66 SINES AND TANGENTS. XXIII. Logarithms of Sines and Tangents Continued. Arc Sin. Df. Cos. Df Tan. Df. j Cot. Arc Arc Sin. Df. Cos. Df. Tan. Df. Cot. Arc / / / / 5 o 8.9403 1429.9983 8.9420 143 1.0580 85 o *5 o 9.4130 47 9.9849 3 9.4281 50 0.5719 75 o IO 9545 137 .9982 9563 138 .0437 So IO 4*77 46 .9846 3 433i 5 5669 5 20 .9682 J 34 .9981 .9701 135 .0299 4 20 .4223 46 9843 4 .4381 49 .5619 40 3 .9816 29 .9980 .9836 130 .0164 3 30 .4269 45 9839 3 4430 49 557 30 408.9945 2 = 9979 8.9966 I27JI.0034 20 40 4314 45 .9836 4 4479 48 5521 20 50 9.0070 22 9977 9.0093 1230.9907 IO 50 4359 44 .9832 4 4527 48 5473 IO 6 o .0192 19 997f> .0216 1 20 9784 840 16 o 443 44 .9828 3 4575 47 5425 74 o IO .0311 15 9975 0336 117 .9664 50 IO 4447 44 .9825 4 .4622 47 5378 5 20 .0426 13 9973 453 114 .9547 40 20 .4491 42 .9821 4 .4669 47 533 1 40 30 40 539 .0648 09 07 .9972 .9971 .0567 .0678 I" -9433 io8j .9322 3 20 3 40 4533 457 6 43 42 .9817 .9814 3 4 .4716 .4762 46 46 .5284 .5238 3 20 5 0755 04 .9969 .0786 105 .9214 IO 50 .4618 41 .9810 4 .4808 45 .5192 10 7 o .0859 102 .9968 .0891 104 .9109 83 o 17 o .4659 41 .9806 4 4853 45 5H7 73 o IO .0961 99 .9966 0995 101 .9005 5 IO .4700 4 1 .9802 4 4898 45 .5102 5o 20 .1060 97 .9964 . 1096 98 .8904 40 20 .4741 40 .9798 4 4943 44 5057 40 3 "57 95 99 6 3 194 97 .8806 30 3 478i 40 9794 4 .4987 44 5013 3 4 .1252 93 .9961 . 291 94 .8709 20 40 .4821 40 .9790 4 5031 44 .4969 20 50 1345 9i 9959 385 93 8615 10 50 .4861 39 .9786 4 5075 43 4925 10 8 o .1436 89 9958 478 91 .8522 82 o 18 o .4900 39 .9782 4 .5118 43 .4882 72 o 10 1525 8 7 .Q956 569 89 .8431 50 IO 4939 3 8 .9778 4 .5161 42 .4839 50 20 . 1612 85 9954 658 87 8342 40 20 4977 S 8 9774 4 5203 42 4797 40 3 .1697 84 9952 I 745 86 8255 3 3 5 OI 5 37 .9770 3 5245 4 2 4755 3 40 .1781 82 9950 .1831 84 .8169 20 40 5052 38 97 6 5 4 .5287 42 47 I 3 20 50 .1863 80 .9948 I 9 I 5 82 .8085 IO 5 .5090 36 .9761 4 5329 4 1 .4671 10 9 o 1943 79 .9946 .1997 81 .8003 81 o 19 o 5 I 26 37 9757 5 537 4 1 4630 71 o IO .2022 78 9944 .2078 80 .7922 So IO 5163 36 9752 4 54" 40 4589 50 20 .2IOO 76 .9942 .2158 78 .7842 40 20 5199 36 9748 5 5451 40 4549 40 30 .2176 75 .9940 .2236 77 .7764 3 3 5235 35 -.9743 4 5491 40 4509 30 4 .2251 73 9938 2313 76 .7687 20 4 5270, 36 9739 s 553 1 40 4469 20 5 .2324 73 9936 .2389 74 .7611 IO 5 53o6 35 9734 4 5571 40 4429 IO 10 2397 7 1 9934 .2463 73 7537 80 o 20 5341 34 9730 5 .5611 39 4389 70 o IO .2468 70 993 1 .2536 73 .7464 50 IO 5375 34 9725 4 .5650 39 4350 50 20 .2538 68 .9929 .2609 7i 7391 40 20 5409 34 .9721 5 5689 38 43" 40 3 .2606 68 9927 3 .2680 7 .7320 3 30 5443 34 97 l6 5 5727 39 4273 30 40 .2674 66 9924 3 .2750 69 .7250 20 40 5477 33 97" 5 .5766 38 4234 20 50 .2740 66 .9922 3 .2819 68 .718! IO 60 5510 33 .9706 4* 5804 38 .4196 IO II O .2806 64 .9919 2 .2887 66 7"3 79 o 31 O 5543 33 .9702 5 .5842 37 .4158 690 IO .2870 64 .9917 3 2953 67 747 50 IO 5576 33 .9697 5 5879 38 .4121 5 20 2934 63 .9914 2 .3020 65 .6980 40 2O 5609 32 .9692 r > 59i7 37 .4083 40 3 .2997 61 .9912 3 3085 64 .6915 30 30 .5641 32 .9687 r ^ 5954 37 .4046 30 40 .3058 61 .9909 2 3*49 63! .6851 20 40 5673 3i .9682 5 599 1 37 .4009 20 50 3"9 60 .9907 3 .3212 63 .6788 IO 5 .5704 32 .9677 5 .6028 36 3972 IO 12 O 3*79 59 994 3 3 2 75 61 .6725 78 o 22 O 5736 3i .9672 5 .6064 36 3936 68 o 10 3238 58 .9901 2 333 6 6 1 .6664 50 10 5767 3i .9667 6 .6100 ^ .3900 5 2O .3296 57 .9899 3 3397 61 .6603 40 2O .5798 30 .9661 5 .6136 36 .3864 40 3 3353 57 .9896 3 3458 59 -6542 30 3 5828 31 .9656 s .6172 36 .3828 30 40 50 .3410 .3466 56 55 9893 .9890 3 3 3517 357 6 59 .6483 .6424 20 10 40 50 5859 30 .5889! 30 .9651 .9646 I .6208 .6243 35 36 3792 3757 20 IO 13 o 3521 54 .9887 3 3 6 34 57 .6366 77 o 23 o .5919! 29 .9640 5 .6279 35 3721 6 7 IO 3575 54 .9884 ) .3691 57| - 6 39 50 IO 5948| 30 9 6 35 6 .6314 34 .3686 5 20 .3629 53 .9881 3 3748 56 .6252 40 20 5978 29 .9629 5 .6348 35 .3652 40 3 .3682 5 2 .9878 1 .3804 55 .6196 30 30 .6007] 29 .9624 6 .6383 34 3617 30 40 3734 52 9875 3 3859 55 -6141 20 40 . 6036 29 9618 5 .6417 35 3583 20 50 .3786 5i 9872 1 39 Z 4 54 .6086 IO 50 .6065 28 .9613 6 .6452 34 .3548 IO 14 o IO 20 .3837 .3887 3937 SO 5 49 .9869 .9866 .9863 3 1 4 .3968 .4021 .4074 53 1 -6032 53 1 -5979 53j -5926 76 o 4 50 40 24 o IO 20 .6093 .6121 .6149 28 28 28 .9607 .9602 .9596 S 6 6 .6486 34 .6520 33 .6553 34 35 I 4 .3480 3447 66 o 5o 40 3 .3986 49 9859 .4127 52 .5873 3 30 .6177 28 959 6 6587! 33 .3413 3 40 .4035 48 .9856 .; .4178 51 .5822 20 40 .6205 2 7 .9584 5 .6620; 34 338o 20 50 .4083 47 9853 4 .4230 5i -5770 10 50 .6232 27 9579 i .6654 33 3346 10 15 09.4130 47 9.9849 3 9.4281 500.5719 75 o 25 o 0.6259 27 9-9573 7 9-6687 33 o-33 I 3 650 Arc Cos. Df. Sin. Df. Cot. Df. Tan. Arc Arc Cos. Df. Sin. D Cot. Df. Tan. Arc SINES AND TANGENTS. 67 XXIII. Logarithms of Sines and Tangents Continued. Arc Sin. Df. Cos. Df. Tan. Df. Cot. Arc Arc Sin. Df. Cos. I)i Tan. Df. Cot. Arc / / / / 25 09.6259 27 9-9573 6 9.6687 33 0-3313 65 o 35 o 9.7586 18 19.9134 9 9.8452 2 7 0.1548 55 o 10 .6286 27 9567 6 .6720 3 2 .3280 5 IO .7604 18 9 I2 5 9 .8479 27 .1521 50 20 6313 27 .9561 6 .6752 33 .3248 40 20 .7622 18 .9116 9 .8506 27 .1494 4 3 .6340 26 9555 6 .6785 32 3215 30 3 .7640 *7 .9107 9 8533 26 .1467 3 40 .6366 26 9549 6 .6817 33 .3183 20 40 .7657 18 .9098 9 8559 2 7 .1441 20 5 .6392 26 9543 6 .6850 32 .315 IO 50 7675 *7 .9089 9 .8586 27 .1414 IO 26 o .6418 26 9537 7 .6882 3 2 .3118 64 o 36 o .7692 18 .9080 IO .8613 26 1387 54 10 .6444 26 953 6 .6914 3 2 .3086 50 IO .7710 *7 .9070 9 .8639 2 7 .1361 So 20 .6470 25 9524 6 .6946 3i .3054 4 20 7727 17 .9061 9 .8666 26 1334 4 3 6 495 26 .9518 6 .6977 3 2 3023 30 30 7744 17 .9052 10 .8692 26 .1308 3 40 .6521 25 .9512 7 .7009 3i .2991 20 40 .7761 *7 .9042 9 .8718 27 .1282 20 50 .6546 24 9505 6 .7040 3 2 .2960 IO 50 7778 17 933 zo 8745 26 1255 IO 27 o .6570 25 9499 7 .7072 3i .2928 63 o 37 o 7795 16 .9023 9 .8771 26 .122953 o 10 .6595 25 9492 6 7103 3 1 .2897 50 10 .7811 17 .9014 IO 8797 27 .1203 50 20 .6620 24 .9486 7 -7 I 34 3i .2866 4 20 .7828 16 .9004 9 .8824 26 .1176 40 3 .6644 24 9479 6 7 l6 5 3 1 .2835 30 3 .7844 i7 .8995 10 .8850 26 .1150 3 40 .6668 24 9473 7 .7196 3 .2804 40 .7861 16 .8985 IO .8876 26 .1124 20 So .6692 24 .9466 7 .7226 3 1 2774 10 50 .7877 16 .8975 IO .8902 26 .1098 IO 28 o .6716 24 9459 6 .7257 30 2743 62 o 38 o .7893 I? .8965 IO .8928 26 i .1072 52 o IO 20 .6740 .6763 23 24 9453 9446 7 7 .7287 7317 3 3i 2713 .2683 50 40 IO 20 .7910 .7926 16 15 .8955 .8945 IO 10 8954 .8980 26 26 .1046 .1020 50 40 3 .6787 23 9439 7 .7348 30 .2652 3 3 .7941 16 .8935 10 .9006 26 .0994 30 40 .6810 23 9432 7 7378 3 .2622 20 40 7957 16 .8925 10 .9032 26 .0968 20 50 6833 23 94 2 5 7 .7408 30 .2592 IO 5 7973 16 .8915 to .9058 26 .0942 10 29 o .6856 22 .9418 7 7438! 29 .2562 61 o 39 .7989 15 .8905 IO .9084 26 .0916 51 o IO .6878 2 3 .9411 7 74 6 7I 3 2533 50 IO .8004 16 .8895 ii .9110 2 5 .0890 50 20 .6901 22 .9404 7 .7497 29 2503 40 20 .8020 15 ^.8884 10 9*35 26 .0865 40 3 .6923 23 9397 7 7526) 30 2474 3 30 8035 i5 .8874 10 .9161 26 .0839 3 40 .6946 22 9390 7 .7556 29 -.2444 20 40 .8050 16 .8864 II .9187 2 5 .0813 20 5 .6968 22 9383 8 7585 29 .2415 IO 50 .8066 15 .8853 I0 .9212 26 .0788 10 30 o .6990 22 9375 7 -7614 30 .2386 00 40 o .8081 = .8843 II .9238 26 .0762 50 o IO .7012 21 .9368 7 .7644 29 .2356 50 IO .8096 15 .8832 1 ! .9264 2 5 .0736 50 20 .7033 22 .9361 8 .7673 28 .2327 40 20 .8m H .8821 1 1 .9289 26 .0711 40 3 755 21 9353 7 .7701 29 .2299 3 30 .8125 15 .8810 I ) 9315 26 .0685 30 40 .7076 21 9346 8 .7730 29 .2270 20 40 .8140 15 .8800 II 934 1 25 .0659 20 5 .7097 21 9338 7 7759 29 .2241 IO 5 8i55 14 .8789 I I .9366 26 .0634 IO 31 o .7118 21 9331 8 .7788 28 .2212 59 *> 41 o .8169 15 .8778 1 1 9392 2 5 .060849 10 .7139 21 9323 8 .7816 29 .2184 5 10 .8184 M .8767 i 1 .9417 26 .0583 50 20 .7160 21 93 I 5 7 .7845 28 2155 40 20 .8198 15 .8756 : I 9443 25 .0557 40 3 .7181 2O .9308 8 .7873 29 - .2127 3 3 .8213 14 8745 ! 2 .9468 26 0532 3 4 5 .7201 S 7 222 21 20 .9300 .9292 8 8 .7902 793 28 28 .2098 .2070 20 IO 40 So .8227 .8241 14 14 8733!" .8722 II 9494 95 J 9 25 25 .0506 .0481 20 IO 32 o .7242 2O .9284 8 7958 28 .2042 58 o 42 o 8255 14 .8711 i a 9544 26 .0456 48 IO .7262 2O .9276 8 .7986 28 .2OI4 50 10 .8269 M . 8699 1 1 957 25 .0430 5 20 .7282 20 .9268 :-, .8014 28 .1986 40 20 .8283 *4 .8688, 12 9595 26 .0405 4 o 3 .7302 20 .9260 8 .8042 28 .1958 30 30 .8297 14 .8676 ii .9621 2 5 0379 3 40 .7322 20 .9252 8 .8070 27 .1930 20 40 .8311 *3 .8665 i : .9646 25 354 20 So 734 2 19 .9244 8 .8097 28 .1903 IO 50 8324 14 .8653 a .9671 26 .0329 IO 33 o .7361 19 .9236 8 .8125 28 1875 57 o 43 o 8338 *3 .8641 i 3 .9697 2 5 .0303:47 o 10 .7380 2O .9228 9 8153 27 .1847 5 10 8351 14 .8629 i i .9722 25 .0278 50 20 . 7 400 19 .9219 8 .8180 28 .1820 40 20 .8365 13 .8618 i 3 9747 25 -0253 40 3 .7419 J 9 .9211 8 .8208 27 1792 30 30 .8378 13 .8606 i < .9772 26 .0228 3 40 .7438 19 .9203 9 .8235 28 1765 20 4 .8391 *4 8594 1 9798 2 5 .0202 20 50 7457 19 .9194 8 .8263 27 J 737 IO 50 .8405 13 .8582 13 .9823 25 .0177 IO 34 o .7476 18 .9186 9 .8390 27 .1710 56 o 44 o .8418 13 .8569! 12 .9848 26 .0152 46 o IO 7494 19 .9177 8 8317 27 .1683 50 IO .8431 *3 8557 I -j 9874 25 .OI26 5 20 -75*3 18 .9169 9 .8344 2 7 .1656 4 20 .8444 *3 8545 13 .9899 2 5 .OIOI 40 3 7531 19 .9160 9 8371 27 .1629 3 3 .8457 12 8532 ia .9924 25 .0076 30 40 7550 18 9*5* 9 .8398 27 .1602 20 4 .8469 *3 .8520 i ^ 9949 26 .0051 20 5 .7568 18 .9142 8 .8425 27 1575 IO 50 .8482 13 .8507 u: 9-9975 25 .0025 10 35 o 9.7586 18 9-9I34 9 9.8452 27 0.1548 55 o 45 o 9.8495 9.8495 0.0000 o.oooO|45 o Arc Cos. Df. Sin. Df. Cot. Df. Tan. Arc Arc Cos. 1 Df. Sin. Df. Cot. Df. Tan. Arc 68 SQUARES AND SQUARE ROOTS. XXIV. Squares and Square Roots. No. Square. Square root. No. Square. Square root. i i 1. 000 5i 2601 7.141 2 4 1.414 52 2704 7. 211 3 9 1.732 53 2809 7.280 4 16 2.000 54 2916 7.348 5 25 2.236 55 3025 7.416 6 36 2.449 56 3136 7-483 7 49 2.646 57 3249 7.550 8 64 2.828 58 3364 7.616 9 Si 3.000 59 348i 7.681 10 100 3.162 60 3600 7.746 ii 121 3.3T7 61 3721 7.810 12 144 3.464 62 3844 7.874 13 169 3.606 63 3969 7-937 14 196 3-742 64 4096 8.000 15 225 3.873 65 4225 8.062 16 256 4.000 66 4356 8.124 17 289 4.123 67 4489 8.185 18 324 4-243 68 4624 8.246 19 361 4.359 69 4761 8.307 20 4OO 4.472 70 4900 8.367 21 441 4o83 71 5041 8.426 22 484 4.690 72 5184 8.485 23 529 4.796 73 5329 8.544 24 576 4.899 74 5476 8.602 25 625 5.OOO 75 5625 8.660 26 676 5.099 76 5776 8.718 27 729 5.196 77 5929 8.775 28 784 5.292 78 6084 8.832 29 8 4 I 5.385 79 6241 8.888 30 900 5.477 80 6400 8.944 31 961 5.568 Si 6561 9.000 3 2 1024 5.657 82 6724 9-055 33 1089 5-745 83 6889 9.110 34 1156 5.831 84 7056 9.165 35 1225 5.916 85 7225 9.220 36 1296 6.000 86 7396 9.274 37 1369 6.083 87 7569 9.327 33 1444 6.164 88 7744 9.38r 39 1521 6.245 89 7921 9.434 40 I6OO 6.325 90 Sioo 9.487 41 1681 6.403 9 1 8281 9.539 42 1764 6.481 92 8464 9-592 43 1849 6-557 93. 8649 9.644 44 1936 6.633 94 8836 9.695 45 2025 6.708 95 9025 9-747 46 2116 6.782 96 9216 9. 798 47 2209 6.856 97 9409 9-849 48 2304 6.928 98 9604 9-899 J9 2401 7.000 99 9801 9-950 50 2500 7.071 IOO IOOOO IO.OOO SQUARES AND SQUARE ROOTS. 69 XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 101 I020I 10.050 151 22801 12.288 102 10404 IO. IOO 152 23104 12.329 103 10609 10.149 153 23409 12.369 104 10816 10.198 154 23716 12.410 105 11025 10.247 155 24025 12.450 I O6 11236 10.296 156 24336 12.490 107 11449. 10.344 157 24649 12.530 108 11664 10.392 158 24964 12.570 109 11881 10.440 159 25281 12.610 - IIO I2IOO 10.488 160 25600 12.650 III I232I 10.536 161 25921 12.689 112 12544 10.583 162 26244 12.728 113 12769 10.630 163 26569 12.767 114 12996 10.677 164 26896 12.806 115 13225 10.724 165 27225 12.845 116 13456 10.771 166 27556 12.884 117 13689 10.817 167 27889 12.923 nS 13924 10.863 168 28224 12.961 119 14161 10.909 169 28561 13.000 120 14400 .10.954 170 | 28900 13.038 121 14641 I I . OOO 171 29241 13.077 122 14884 11.045 172 29584 I3.U5 123 15129 1 1 . 09 1 173 29929 I3.T53 124 15376 II. 136 174 30276 13.191 125 15625 II. I 80 175 30625 13-229 126 15876 11.225 176 30976 13.266 127 16129 11.269 177 31329 13.304 128 16384 11.314 i/8 31684 13.342 I2 9 16641 11.358 179 32041 13-379 130 16900 11.402 1 80 32400 13-416 131 I7l6l 11.446 181 32761 13.454 132 17424 11.489 182 33124 I3.49I 133 17689 11-533 183 33489 13.528 134 17956 11.576 184 33856 13-565 135 18225 11.619 185 34225 13.601 136 18496 11.662 186 34596 13-638 137 18769 11.705 187 34969 13-675 138 19044 n.747 188 35344 I3.7H 139 19321 11.790 189 35721 13.748 140 19600 11.832 190 36100 13.784 141 igSSi 11.874 191 36481 13.820 142 20164 11.916 192 36864 13-856 143 20449 11.958 193 37249 13-892 144 20736 I 2 . OOO 194 37636 13-928 14-5 21025 I2.O42 195 38025 13.964 146 2I3IO 12.083 196 38416 14.000 147 2l6og 12. 124 197 38809 14.036 I 4 8 21904 I2.I66 198 39204 14.071 149 222OI 12.207 199 39601 14.107 150 22500 12.247 200 40000 14.142 7 SQUARES AND SQUARE ROOTS. XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 2OI 40401 14.177 251 63001 15.843 2O2 40804 14.213 252 63504 15.875 203 41209 14.248 253 64009 15.906 204 41616 14.283 254 64516 15-937 2O5 42025 14-318 255 65025 15.969 206 42436 14-353 256 65536 16.000 2O7 42849 14-387 257 66049 16.031 208 43264 14.422 258 66564 16.062 2O9 43681 14.457 259 67081 16.093 2IO 44IOO 14.491 260 67600 16. 125 211 44521 14.526 261 68121 16.155 212 44944 14-560 262 68644 16.186 213 45369 14.595 263 69169 16.217 214 45796 14.629 264 69696 16.248 215 46225 14.663 265 70225 16.279 . 216 46656 14.697 266 70756 16.310 217 47089 14.731 267 71289 16.340 218 47524 14-765 268 71824 16.371 219 47961 14.799 269 72361 16.401 220 48400 14-832 270 72900 16.432 221 48841 14.866 271 73441 16.462 222 49284 14.900 272 73984 16.492 223 49729 14-933 273 74529 16.523 224 50176 14.967 274 . 75076 16.553 225 50625 i 5 . ooo 275 75625 16.583 226 51076 15.033 276 76176 16.613 227 51529 15.067 277 76729 16.643 228 51984 15.100 278 77284 16.673 229 52441 I5-T33 279 77841 16.703 23O 52900 15. 166 280 78400 16.733 231 5336i 15.199 281 78961 16.763 232 53824 15-232 282 79524 16.793 233 54289 15.264 283 80089 16.823 234 54756 15-297 284 80656 16.852 235 55225 15.330 285 81225 16.882 236 55696 15-362 286 81796 16.912 237 56169 15.395 287 82369 16.941 238 56644 15.427 288 82944 16.971 239 57121 i 5 . 460 289 83521 i 7 . ooo 240 57600 15-492 290 84100 17.029 241 58081 15.524 291 84681 17.059 242 58564 15.556 292 85264 17.088 243 59049 15.588 293 85849 17-117 244 59536 15.620 294 86436 17.146 245 60025 15.652 295 87025 17.176 246 60516 15-684 296 87616 17-205 247 61009 15.716 297 88209 17.234 248 61504 15.748 298 88804 17-263 249 62001 15.780 299 i 89401 17.292 250 62500 15.811 300 j 90000 17.321 SQUARES AND SQUARE ROOTS. 71 XXIV. Squares and Square Roots Continued. No. Square. Square root. No . Square. Square root. 301 90601 17-349 35i 123201 18.735 302 91204 17.378 352 123904 18.762 303 91809 17.407 353 124609 18,788 304 92416 17.436 354 125316 18.815 305 93025 17.464 355 126025 18.841 306 93636 17.493 356 126736 18.868 307 94249 17.521 357 127449 18.894 308 94864 17.550 358 128164 18.921 309 95481 17.578 359 128881 18.947 310 96100 17.607 360 129600 18.974 311 96721 17-635 361 130321 19.000 312 Q7344 17.664 362 131044 19.026 313 97969 17.692 363 131769 I9.053 3U 9 8 596 17.720 364 132496 19.079 315 99225 17.748 365 133225 19.105 316 99856 17.776 366 133956 19.131 317 100489 17.804 367 134689 ig^s? 318 101124 17.833 368 135424 19.183 319 101761 17.861 369 136161 19.209 320 102400 17.889 370 136900 19-235 321 103041 17.916 37i 137641 19.261 322 103684 17.944 372 138384 19.287 323 104329 17.972 373 139129 19-313 324 . 104976 18.000 374 139876 19-339 325 105625 18.028 375 140625 19-365 326 106276 18.055 376 141376 19.391 327 106929 18.083 377 142129 19.416 328 107584 18.111 378 142884 19.442 329 108241 18.138 379 143641 19.468 330 108900 18.166 380 144400 19.494 331 109561 18.193 38i 145161 19-519 332 110224 18.221 382 145924 19-545 333 110889 18.248 383 146689 19.570 334 111556 18.276 384 147456 19.596 335 112225 18.303 385 148225 19.621 336 112896 18.330 386 148996 19.647 337 113569 18.358 387 149769 19.672 338 114244 18.385 388 150544 19.698 339 114921 18.412 389 151321 19.723 340 115600 18.439 390 152100 19.748 34i 116281 18.466 39i 152881 19-774 342 116964 18.493 39 2 153664 19.799 343 117649 18.520 393 154449 19.824 344 118336 18.547 394 155236 19.849 345 119025 18.574 395 - 156025 19-875 346 119716 18.601 396 156816 19.900 347 120409 18.628 397 157609 19.925 348 121104 18.655 398 158404 19.950 349 121801 18.682 399 159201 19.975 350 % 122500 18.708 400 160000 ; 20.000 SQUARES AND SQUARE ROOTS. XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 401 160801 20.025 45i 203401 21.237 402 161604 20.050 452 204304 21.260 403 162409 20.075 453 205209 21.284 404 163216 20.100 454 206116 21.307 405 164025 20. 125 455 207025 21.331 406 164836 20. 149 456 207936 21.354 407 165649 20.174 457 208849 21.378 408 166464 20.199 458 209764 21.401 409 167281 20.224 459 210681 21.424 410 168100 20.248 460 211600 21.448 411 168921 20.273 461 212521 21.471 412 169744 20.298 462 213-144 21.494 413 170569 20.322 463 214369 21.517 414 171396 20.347 464 215296 21.541 415 172225 20.372 465 216225 21.564 416 173056 20.396 466 217156 21.587 417 173889 20.421 467 218089 21.610 418 174724 20.445 468 219024 21.633 419 I7556I 20.469 469 219961 21.656 420 176400 20.494 47 220900 21.679 421 177241 20.518 471 221841 21.703 422 178084 20.543 472 222784 21 .726 423 178929 20.567 473 223729 21.749 424 179776 20.591 474 224676 21.772 425 180625 20.616 475 225625 21.794 426 181476 20.640 476 226576 21.817 427 182329 20.664 477 227529 21.840 428 183184 20.688 4/8 228484 21.863 429 184041 20.712 479 229441 21.886 430 184900 20.736 480 230400 21.909 43r 185761 20.761 481 231361 21.932 432 186624 20.785 482 232324 21-954 433 187489 20.809 483 233289 21.977 434 188356 20.833 484 234256 22.000 435 189225 20.857 485 235225 22.023 436 190096 20.881 486 236106 22.045 437 190969 20.905 487 237169 22.068 438 191844 20.928 488 238144 22.091 439 192721 20.952 489 239121 22.113 440 193600 20.976 490 240100 22.130 441 194481 2 1 . OOO 491 241081 22.159 442 195364 21.024 492 242064 22.I8I 443 196249 21.048 493 243049 22.2O4 444 197136 21.071 494 244036 22.226 445 198025 21.095 495 245025 22.249 446 198916 21.119 496 246016 22.271 447 199809 21.142 497 247009 22.293 448 200704 21.166 498 248004 22.316 449 201601 21.190 499 249001 22.338 450 202500 21.213 500 1 250000 22.361 SQUARES AND SQUARE ROOTS. 73 XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. SGI 251001 22.383 55i 303601 23.473 502 252004 22.405 552 304704 23.495 503 253009 22.428 553 305809 23-516 504 254016 22.450 554 306916 23-537 505 255025 22.472 555 308025 23.558 506 256036 22.494 556 309136 23.580 507 257049 22.517 557 310249 23.601 508 258064 22.539 558 311364 23.622 509 259081 22.561 559 312481 23.643 5io 260100 22.583 560 313600 23.664 BII 261121 22.605 56i 314721 23-685 512 262144 22.627 562 315844 23.707 5i3 263169 22.650 563 316969 23.728 5i4 264196 22.672 564 318096 23-749 515 265225 22.694 . 565 319225 23.770 5i6 266256 22.716 566 320356 23.791 5i7 267289 22.738 567 321489 23.812 5i8 268324 22.760 568 322624 23-833 519 269361 22.782 569 323761 23-854 520 270400 22.804 570 324900 23.875 521 271441 22.825 57i 326041 23. 896 522 272484 22.847 572 327184 23.917 523 273529 22.869 573 328329 23-937 524 274576 22.891 574 329476 23.958 525 275625 22.013 575 330625 23.979 526 276676 22.935 576 331776 24 . ooo 527 277729 22.956 577 332929 24.021 528 278784 22.978 578 334084 24.042 529 279841 23.000 579 335241 24.062 530 280900 23.022 580 336400 24.083 53i 281961 23-043 5Si 33756i 24.104 532 283024 23.065 582 338724 24.125 533 284089 23.087 583 339889 24.145 534 285156 23. 108 584 341056 24.166 535 286225 23.130 585 342225 24.187 536 287296 23.152 586 343396 24.207 537 288369 23.173 587 344569 24.228 538 289444 23.195 588 345744 24.249 539 290521 23.216 589 346921 24.269 540 291600 23.238 590 348100 , 24.290 54i 292681 23-259 59i 349281 24.310 542 293764 . 23.281 592 350464 24.331 543 294849 23.302 593 351649 24.352 544 295936 23.324 594 352836 24.372 545 297025 23-345 595 354025 24.393 546 298116 23.367 59 6 355216 24.413 547 299209 23.388 597 356409 24.434 548 300304 23.409 598 357604 24.454 549 301401 23.431 599 358801 24.474 550 302500 23.452 600 360000 24.495 74 SQUARES AND SQUARE ROOTS. XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 601 361201 24.515 651 423801 25.5T5 602 362404 24-536 652 425104 25.534 603 363609 24.556 653 426409 25.554 604 364816 24-576 654 427716 25.573 605 366025 24-597 655 429025 25.593 606 367236 24.617 656 430336 25.612 607 368449 24-637 657 431649 25.632 608 369664 24-658 658 432964 25.652 609 370881 24.678 659 434281 25-671 610 372100 24.698 660 4356oo 25.690 611 373321 24.718 661 436921 25.710 612 374^44 24.739 662 438244 25.720 613 375769 24.759 663 439569 25.749 614 376996 24.779 664 440896 25.768 615 378225 24.799 665 442225 25.788 616 379456 24.819 666 443556 25.807 617 380689 24-839 667 444889 25.826 618 381924 24.860 668 446224 25.846 619 383161 24.880 669 447561 25.865 620 384400 24 . 900 670 448900 25.884 621 385641 24.920 671 450241 25.904 622 386884 24 . 940 672 451584 25.923 623 388129 24.960 673 452929 25.942 624 389376 24.980 674 454276 25.962 625 390625 25.000 675 455625 25.981 626 391876 25.020 676 456976 26.000 627 393129 25.040 677 458329 26.019 628 394384 25.060 678 459684 26.038 629 395641 25.080 679 461041 26.058 630 396900 25.100 680 462400 26.077 631 398161 25.120 681 463761 26.096 632 399424 25.140 682 465124 26.115 633 400689 25.160 683 466489 26.134 634 401956 25.180 684 467856 26.153 635 403225 25.200 685 469225 26.173 636 404496 25.220 686 470596 26.192 637 405769 25.239 687 471969 26.211 638 407044 25-259 688 473344 26.230 639 408321 25.278 689 474721 26.249 640 409600 25.298 690 476100 26.268 641 410881 25.318 691 477481 26.287 642 412164 25.338 692 478864 26.306 643 413449 25-357 693 480249 26.325 644 414736 25.377 694 481636 26.344 645 416025 25.397 695 483025 26.363 646 417316 25-417 696 484416 26.382 647 418609 25.436 697 485809 26.401 648 419904 25.456 698 487204 26.420 649 421201 25-475 699 488601 26.439 650 422500 25.495 700 490000 26.458 SQUARES AND SQUARE ROOTS. 75 XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 701 491401 26.476 751 564001 27.404 702 492804 26.495 752 565504 27-423 703 494209 26.514 753 567009 27.441 704 495616 26.532 754 568516 27-459 705 497025 26.552 755 5/0025 27-477 706 498436 26.571 756 571536 27.495 707 499849 26.589 757 573049 27.514 708 501264 26.608 758 574564 27.532 709 502681 26.627 759 ! . 576081 27.550 710 504100 26.646 760 | 577600 27.568 711 505521 26.665 761 579121 27.586 712 506944 26.683 762 580644. 27.604 713 508369 26.702 763 582169 27.622 7 J 4 509796 26.721 764 583696 27.641 715 511225 26.739 765 585225 27.659 716 512656 26.758 766 586756 27.677 7 J 7 514089 26.777 767 588289 27.695 718 515524 26.796 768 289824 27.713 719 516961 26.814 769 591361 27.731 720 518400 26.833 770 592900 27.749 721 519841 26.851 77i 594441 27.767 722 521284 26.870 772 595984 27-785 723 522729 26.889 773 597529 27.803 724 524176 26.907 774 599076 27.821 725 525625 26.926 775 600625 27-839 726 527076 26.944 776 602176 27.857 727 528529 26.963 777 603729 27.875 728 529984 26.981 778 605284 27.893 729 53I44I 27.000 779 606841 27.911 730 532900 27.019 780 608400 27.928 731 534361 27.037 781 609961 27.946 732 535824 27-055 782 611524 27.964 733 537289 27-074 783 613089 27.982 734 538756 27.092 784 614656 28.000 735 540225 27. in 785 616225 28.018 736 541696 27.129 786 617796 28.036 737 543169 27.148 787 619369 28.054 738 544644 27.166 788 620944 28.071 739 546121 27.185 789 622521 28.089 740 547600 27.203 790 624100 28.107 741 549081 27.221 791 625681 28.125 742 550564 27.240 792 627264 28.142 743 552049 27.258 793 628849 28.160 744 553536 27.276 794 630436 28.178 "745 555025 27.295 795 632025 28.196 746 556516 27.313 796 633616 28.213 747 558009 27.331 797 635209 28.231 748 559504 27.350 798 636804 28.249 749 561001 27.368 799 638401 28.267 750 562500 27.386 800 640000 28.284 7 6 SQUARES AND SQUARE ROOTS. XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. Soi 641601 28.302 851 724201 29.172 802 643204 28.320 852 725904 29. 189 803 644809 28.337 853 727609 29.206 804 646416 28.555 854 729316 29.223 805 648025 28.373 855 731025 29.240 806 649636 28.390 856 732736 29.257 807 651249 28.408 857 734449 29.275 808 652864 28.425 858 736164 29.292 809 654481 28.443 859 737881 29.309 8 10 656100 28.460 860 739600 29.326 Sn 657721 28.478 861 741321 29-343 812 659344 28.496 862 743044 29.360 813 660969 28.513 863 744769 29-377 814 662596 | 28.531 864 746496 29.394 815 664225 28.548 865 748225 29.411 Si6 665856 28.566 866 749956 29.428 817 667489 28.583 867 751689 29.445 818 669124 28.601 868 753424 29.462 8ig 670761 28.618 869 755i6i 29.479 820 672400 28.636 870 756900 29.496 821 674041 28.653 871 758641 29-513 822 675684 28.671 872 760384 29-530 823 677329 28.688 873 762129 29.547 824 678976 28.705 874 763876 29-563 825 680625 . 28.723 875 765625 29.580 826 682276 28.740 876 767376 29.597 827 683929 28.758 877 769129 29.614 828 685584 28.775 878 770884 29.631 829 687241 28.792 879 772641 29.648 830 688900 28.810 SSo 774400 29.665 S3i 690561 28.827 881 776161 29.682 832 692224 28.844 882 777924 29.698 833 693889 28.862 883 779689 29-7I5 834 695556 28.879 884 781456 29.732 835 697225 28.896 885 783225 29.749 836 698896 28.914 886 784996 29.766 837 700569 28.931 887 786769 29.783 838 702244 28.948 888 788544 29.799 839 703921 28.965 889 790321 29.816 840 705600 28.983 890 792100 29-833 841 707281 29.000 891 793881 29.850 842 708964 29.017 892 795664 29.866 843 710649 29.034 893 797449 29.883 844 712336 29.052 894 799236 29.900 845 714025 29.069 895 801025 29.917 846 715716 29.086 896 802816 29.933 847 717409 29.103 897 804609 29.950 848 719104 29.120 898 806404 29.967 849 720801 29.138 899 808201 29.983 850 722500 29.155 goo 810000 30.000 SQUARES AND SQUARE ROOTS. 77 XXIV. Squares and Square Roots Continued. No. Square. Square root. No. Square. Square root. 901 811801 30.017 95i 904401 30.838 902 813604 30-033 952 906304 30.854 903 815409 30.050 953 908209 30.871 904 817216 30.067 954 910116 30.887 905 819025 30.083 955 912025 30.903 906 820836 30. TOO 956 9^936 30.919 907 822649 30.116 957 915849 30.935 908 824464 30.133 958 917764 30.952 909 826281 30.150 959 919681 30.968 910 828100 30.166 960 921600 30.984 911 829921 30.183 961 923521 3 i . ooo 912 331744 30.199 962 925444 31.016 913 833569 30.216 963 927369 31.032 914 835396 30.232 964 929296 31.048 9 J 5 837225 30.249 965 931225 31.064 916 839056 30.265 966 933156 31.081 917 840889 30.282 967 935089 31-097 918 842724 30.299 968 937024 3i.ii3 919 844561 30.315 969 938961 31.129 920 846400 30.332 970 940900 3I.I45 921 848241 30.348 971 942841 31.161 922 850084 30.364 972 944784 3LI77 9 2 3 851929 30.381 973 946729 3LI93 924 853776 30.397 974 948676 3 i 209 925 855625 30.414 975 950625 31-225 tL 926 857476 30.430 976 952576 31.241 927 859329 30.447 977 954529 3L257 928 861184 30.463 978 956484 3L273 929 863041 30.480 979 958441 31.289 930 864900 30.496 980 960400 31.305 931 866761 30.512 981 962361 31-321 932 868624 30.529 982 964324 31-337 933 870489 30.545 983 966289 31-353 934 872356 30.561 984 968256 31-369 935 -j *j +j 874225 30.578 985 970225 3L385 936 876096 30.594 986 972196 31.401 937 877969 30.610 987 974169 3I.4I7 93S 879844 30.627 988 976144 3L432 939 881721 30.643 989 978121 31.448 940 883600 30.659 990 980100 31.464 941 885481 30.676 991 982081 31.480 942 887364 30.692 992 984064 31.496 943 889249 30.708 993 986049 31.512 944 891136 30.725 994 988036 31-528 945 893025 30.741 995 990025 3L544 946 894916 30.757 996 992016 31-560 947 896809 30.773 997 994009 31-575 948 898704 30.790 998 996004 3I.59I 949 900601 30.806 999 998001 31.607 950 902500 30.822 IOOO I 000000 31.623 78 INTERPOLATION BY DIFFERENCES. BessePs Co-efficients. f.i 2d diff. 3d diff. 4th diff. !i 2d diff. 3d diff. 4th diff. o ." o ^ f ~~ T t t ~ l *~ l t *~ 2 t/) / - 1 t J-i t-\ f _ -* ii 2 2 3 4 11 2 2 3 4 O. OI .00495 .00081 . 00083 0.51 - 12495 . 00042 02343 . 02 . 00980 .00157 . 00165 52 . 12480 . 00083 . 02340 .03 01455 . 00228 . 00246 53 12455 .00125 02334 .04 . OI92O . 00294 . 00326 54 . I242O . 00166 .02327 .05 02375 . 00356 . 00405 55 12375 . 00206 .02318 .06 . O282O . 00414 . 00483 56 . 12320 . 00246 . 02306 .07 03255 . 00467 . 00560 57 .12255 . 00286 . 02293 .08 .03680 .00515 . 00636 .58 . I2l8o . 00325 . 02278 .09 04095 . 00560 . 00711 59 . 12095 . 00363 . 02260 . 10 . O45OO . 00600 . 00784 . 60 . I20OO . 00400 . 02240 . II .04895 . 00636 . 00856 .61 .11895 . 00436 . 02218 . 12 . 05280 . 00669 . 00927 .62 .11780 . 00471 . 02194 13 05055 . 00697 . 00996 63 .11655 . 00505 .02169 . 14 . O6O2O . 00723 . 01064 .64 . U520 . 00538 . 02141 J 5 06375 00744 .01130 .65 ."375 . 00569 . O2III . 16 . 06720 . 00762 .01195 .66 . II22O . 00598 . O2o8o .17 07055 . 00776 .01259 .67 .11055 . 00626 . 02046 .18 .07 3 80 . 00787 .01321 .68 . 10880 . 00653 . O2OIO .19 . 07695 . 00795 .01381 .69 . 10695 .00677 .01973 .20 . 08000 . 00800 . 01440 .70 . 10500 . 00700 .01934 .21 . 08295 . 00802 . 01497 71 . 10295 .00721 .01893 . 22 . 08580 .00801 .01553 .72 . IOO8O . 00739 .01850 2 3 .08855 . 00797 .01606 73 .09855 . 00756 . 01805 .24 . O9I2O . 00790 .01658 74 . 09620 . 00770 .01758 25 09375 . 00781 .01709 75 09375 . 00781 .01709 .26 . 09620 . 00770 .01758 76 .09120 . 00790 .01658 27 .28 09855 . 10080 . 00756 00739 .01805 . 01850 77 78 .08855 . 08580 . 00797 . 00801 . Ol6o6 OI 553 .29 . 10295 .00721 .01893 79 . 08295 . 00802 .01497 30 . 10500 . 00700 .01934 .80 . O8OOO . 00800 . 01440 31 . 10695 . 00677 01973 .81 . 07695 . 00795 . 01381 3 2 . 10880 . 00653 . O2OIO .82 . 07380 . 00787 .01321 33 II055 . 00626 . 02046 83 .07055 . 00776 .01259 34 . II22O . 00598 . 02080 84 . 06720 . 00762 .01195 35 .H375 . 00569 . O2III .85 06375 . 00744 . 01130 36 . H520 . 00538 .02141 .86 . O6O2O . 00723 . 01064 37 ."655 . 00505 .02169 .87 05655 . 00697 . 00996 38 . II78O . 00471 .02194 .88 . 05280 . 00669 . 00927 39 .11895 . 00436 . O22l8 .89 04895 . 00636 . 00856 .40 . 12000 . 00400 . O224O .90 . 04500 . 00600 . 00784 4i 12095 . 00363 . O226o .91 04095 . 00560 .00711 42 . I2I80 . 00325 .02278 92 . 03680 .00515 . 00636 43 12255 . 00286 . 02293 93 03255 . 00467 . 00560 44 . 12320 . 00246 . 02306 .94 . O282O . 00414 . 00483 45 12375 . 00206 .02318 95 02375 . 00356 . 00405 .46 . 12420 . 00166 . 02327 .96 . OI92O . 00294 .00326 47 12455 . 00125 02334 97 .01455 . 00228, . 00246 .48 . 12480 . 00083 . 02340 .98 . 00980 .00157" . 00165 .49 . 12495 . 00042 02343 99 00495 . 00081 . 00083 50 -. I2 5 00 . 00000 .02344 I. 00 .00000 . ooooo . ooooo 1 TABLES AND FORMULAE. PART II. GEODESY. GEODESY. XXV. Reduction to Center of Station. Call P the place of the instrument ; C the center of the station ; O the angle at P, between two objects, A and B ; y the angle at P, between C and the /(//-hand object, B ; r the distance, C P ; C the unknown angle at C ; D*the distance A C ; and G the distance B C, then c = o 4- ; sin IP "i" ~ v ) - r ^_y D sin i" Gsin i" In the use of this formula proper attention should be paid to the signs of sin (O +J ) and sin_y; for the first term will \>Q posi tive when (O +j} ; ) is less than 180, (the reverse with sin y } ) D being the distance of the /7g///-hand object, the graduation of the instrument running from left to right. r being small, the lengths of D and G are computed with the angle O. XXVI. Reduction to Center of Signal Observed, or Correction for Phase in Tin Cones Used as Signals. , r cos 2 i Z Correctio = i - . ^ D sin i" where r = radius of the signal ; Z = angle at the point of observation between the sun and the signal ; and D = the distance. 82 GEODESY. XX VI I. Spherical Excess. ^ _ S a b sin C ~ pTirn" ~ : T7- 2 sin i" S being the area of the triangle ; r, the radius of the earth. a b sin c = V s (s-a) (7^7(7^7) tf _|_ _j_ f j bemer = ! - 2 Between latitudes 45 and 25 the spherical excess amounts to about i" for an area of 75.5 square miles. Hence, if the area in square miles be known, a close approxi mation to the spherical excess will be had by dividing the area by 75-5- log mean radius of the ear.h in yards = 6.8427917 If the three angles of a triangle are assumed to have been equally well determined, the previous determination of the spher ical excess is not necessary for. the. calculation of the sides, though it will be required for estimating the relative accuracy of the ob servations; for the sides of a spherical triangle may be com puted as if they were rectilineal when one-third the excess of the sum of .the three angles above 180 is deducted from each of the three observed angles. Then side b side a sin (B J E) 4- sin (A J E) For large triangles : a b sin C (i + e 2 cos 2 L) 2 A 2 sin i" A being the equatorial radius, and L the mean latitude of the three stations. GEODESY. 83 XXVIII. To Reduce the Length of an Inclined Base to Horizontal Measure. Let B be the length of the base on the inclined plane ; b that reduced to the horizontal plane ; and the inclination, then b = B cos O But as is generally a small angle, and need not be known with extreme precision, it is better to compute the excess of B above b ; and, supposing to be given in minutes, B b = B ( 1 -cos 0) = 2 B sin 2 - = B O 2 sin 2 I/ = ^l 1 tf B 2 2 or, B /> = 0.00000004231 O 2 B or, by logarithms, log (B b) = const log 2.626422 -j- 2 log -f log B XXIX. To Reduce a Broken Base to a Straight Line. Let- a and b be the given sides ; and C the contained angle, very nearly 180. Make C = 180 0; being small, and cos r O 2 , then = c7 -j- ^ - O.OOOOOOO42^I X -- being expressed in minutes. log 0.00000004231 = 2.626422 84 GEODESY. XXX. To Find the Length, B D = x, of a Portion of a Straight Line, A H, Knowing the Two Other Portions, A B = a, D H = b, and also the Angles a, ft, y,from any Exterior Station, C, between B and A, D and A, and H and A. The problem being intended to supply by observation any por tion of a base which cannot be directly measured tan 2 y = - 4 -^A x sin P sin (r ) (a i>f sin a sin (Y ft) a -j- b , a b 2 2 COS tp XXXI. To Reduce a Measured Base to the Level of the Sea. Let r represent the radius of the earth (or better, the normal, N,) corresponding to the base b at the level of the sea ; and r 4- a the radius referred to the level of the measured base B, then r+a : r\\ B : ^ == B x - r+a and But the radius of the earth being very great in comparison to the difference of level, a, we have the correction 8 sufficiently ac curate by retaining only the first term ; hence GEODESY. 85 XXXII. Correction for Temperature in Metallic Rods. Let e the linear expansion for i of Fahrenheit; / = the length of the rod before expansion ; / = the length of the rod after expansion ; / = the number of degrees Fahrenheit, then- total expansion = e t and The following expansions were adopted by Mr. Hassler in his comparisons of weights and measures, (report of 1832 :) Expansion for i F. = c For I in a yard s length. Platinum . .0.0000051344. .. .0.0001848384 English inches. Brass bar. .0.00001050903. . .0.00037832508 Iron bar. . .0.000006963535 . .0.000240687260 Other authorities : Expansion for 1 Y.c /or I in a yard s length. Brass bar. .0.000010480 ..... 0.0003772800 Eng. in. Bailey. Brass rod. .0.0000105155 ... .0.0003785580 " Roy. Brass rod. .0.0000106666. .. .0.0003839976 " Troughton. Brass wire . . 0.0000107407 .... 0.0003866652 " Smeaton. Iron bar. . .0.0000069907 ... .0.0002516652 " Smeaton. Steel rod. . .0.0000063596. .. .0.0002289456 " Roy. Glass barom eter-tubes .0.0000043119 ... .0.0001552284 " Roy. White Nor way pine . .0.0000022685 ... .0.0000816660 " Kater. 86 GEODESY. XXXIII. Measurement of Distances by Sound. The velocity of sound, in one second of time, at 32 Fahrenheit, is about 1090 English feet. For any higher temperature, 7>=io89 ft .42 V i + (/ 32) x 0.00208" / being the temperature in degrees Fahrenheit. The velocity of sound through the air is independent of the barometric pressure, and experiments show it to be sensibly un affected by its hygrometrical state of moisture and dryness ; by the nature of the sound itself, whether produced by a blow, gun shot, the voice, or a musical instrument ; by the original direction of the sound, whether, for instance, the muzzle of a gun is turned one way or the other ; or. by the nature and position of the ground over which the sound is conveyed. It is affected by the wind ; but, in ordinary cases, likely to be selected for experiment, its influence would be almost inappre ciable. Velocity and Force of the Wind. Velocity in Pressure on I square foot. Common designations of the force of the winds. I hour. i second. , Miles. Feet. Pounds. i 1.47 0.005 Hardly perceptible. 2 3 2-93 4.40 O.O2O 0.044 Just perceptible. 4 5 5-87 7-33 0.079 0.123 Gentle, pleasant wind. 10 15 14.67 22.0O 0.492 1.107 Pleasant, brisk breeze. 20 25 29-34 36.67 1.968 3-075 Very brisk. 30 44.01 4.429 Pli^h wind. 35 51.34 6.027 40 45 58.68 66.01 7-873 9-9 6 3 Very high. 5o 73-35 12.300 A storm or tempest. 60 88.02 I7-7I5 A great storm. 80 117.36 3I-490 A hurricane. IOO 146.70 49.200 A hurricane that tears up trees, carries buildings before it, &c. GEODESY. 87 XXXIV. For Reconnaissances. "THREE-POINT PROBLEM." At a point, P, from whence are to be seen three points, A, C, B, forming a triangle, the elements (/. <?., the angles and sides) of which are known, measure the angles A P C and C P B; then, required to determine the direction and distance of the point P from each object. Make A C=<z; B C = J; B C A = C ; A P C = P ; and C P B = P ; also, make R = 360 - P - P - C ; x=C A P;, }> = P B C. Then will . sm P cos R R _ x J The use of these formulae need not be embarrassing if care is taken in properly applying the signs of cos R and cot R. When R is less than 90 both cos R and cot R are plus; between 90 and i8oboth are minus; between i8oand 270 cos R is minus and cot R plus; between 270 and 360 cos R is plus and cot R minus. This problem is indeterminate when P falls upon the circumfer ence of the circle passing* through A B C. A case of this nature is of rare occurrence, however, in practice. For the more general form of this problem, where the angles are measured from the point sought to any number of given points, to fix its position, see Coast Survey Report of 1864. 38 GEODESY. XXXV. For Computing the Principal Geodetic Quantities De pending on the Spheroidal Figure of the Earth at any Green. Latitude. Eccentricity of the Earth = e = - Ellipticity = E =- = i - or, very nearly : Normal ending at minor axis (or radius of curvature of a section perpendicular to the meridian) = N a = (T^~sir?T)"i Normal ending at major axis ..... = N = N (i e 2 } Tangent ending at minor axis. . . = / = N cot L Tangent ending at major axis. . . . = T = N tan L (i e 2 } Radius of the parallel .......... = /> = N cos L N 3 Radius of curvature of the meridian = R = (i e 2 ) = a (i--* 2 } (i <f 2 sin 2 L)3 Radius of curvature of a section making an angle 7. with the meridian ~ N Radius of the earth .......... = ;- Equatorial radius Polar radius The given latitude GEODESY. XXXVI. Numerical Values of Some of the Preceding Quanti ties, from a Discussion by BESSEL in the " Astronomische Nach- richtcn" No. 438. a = equatorial radius =3272077.14 toises log = 6.5148235337 b = polar radius = 3261 139.33 toises log = 6. 5133693539 Ratio of the toise to the metre, law of France, December 10, 1799: T = i 1 ". 9490363; log = 0.2898 1 99300 whence in metres rt=6 3 77397 m .i5; log = 6.8046434637 b = 6356078" . 96 ; log = 6.8031892839 Ratio of the axes : a : b : : 299.1528 : 298.1528 Mean uncertainty = i 4.667 units. Length of the earth s quadrant = 5131179*. 81 = 10000855 ". 76 Mean uncertainty = =t 498 ". 23 - 7 2 X. 4 c = eccentricity = ( i ----- J =0.0816967 log = 8.9122052271 E = ellipticity = i e 2 log = 7.5233789824 Length, in toises, of a meridional degree whose middle lati tude is (?: D, H = 57013^109 286*. 337 cos 2 <p -f- o ^n cos 4 <p -f- o f .ooi cos 6 <p Length of a degree of the parallel, in toises : D p = 57 156*. 285 cos <p 47 t .825 cos 3 <p -f- o t .o6o cos 5 <p or, making sin / = e sin <f log D ;) = 4.7567009.0 -f- log cos <p log cos (," 90 GEODESY. XXXVII. Relative Lengths of the Yard and the Metre. i. -From Clarke s comparisons referred to the present parlia mentary standard, (Comparison of Standards of Length made at the Ordnance Survey Office by Captain A. R. Clarke, R. E.,F. R. S., published by authority, 1866:) Values Adopted in the Measurement, now in Progress, of an Arc of Parallel Extending from Ireland to the Rtier Ural in Russia, as "the Exact Relative Lengths of Standards" used as the Units of Measure in the Triangulations of England, France, Belgium, Prussia, and Russia : i Expressed in Ex din | Expressed in Standards. terms of the i . , i lines of the standard yard. inches. toise. The yard i . oooooooo 36. oooooo 405. 34622 The toise 2.13151116 76.734402 i 864.00000 The metre 1.09362311 39.370432] 443.29600 Expressed in millimetres. 914.39180 1949. 03632 IOOO. OOOOO ] g 39-3743 2 = i-595 1 7 lSl6 2. From Rater s comparisons with the Shuckburg scale, (/%//. Trans, for 1818:) i metre at 32 F. = 39.370790 inches of the old imperial standard at 62 F. 3. From Hassler s comparisons of the Troughton 8 2 -inch scale, with the iron standard committee metre of the American Philosophical Society, (Report of the Secretary of the Treasury on the Comparison of Weights and Measures, Twenty -Second Congress, First Session, June 20, 1832 :) Value adopted by the United States Coast Survey. i metre at 32 F. = 39.36850535 inches of the Troughton 82- inch scale at 62 F. a value materially smaller than the preceding. There is a doubt whether this discordance is to be attributed to inaccuracy in the length of the Troughton scale or in errors in the co-efficients of expansion used by Mr. Hassler. GEODESY. 91 XXXVII. Relative Lengths of the Yard and the Metre Con d. Logarithms to Reduce Metres to Yards. Clarke. Kater. Coast Survey. 0.0388676809 0.0388716286 0.0388464579 log 3 = 0.4771212547 log 12 = 1.0791812460 log 5280 = 3.7226339225 Kater s length of the metre in English inches was adopted in the preparation of the first edition of this Collection as being at the time most generally in use. It has been retained through out the present volume, although the results of Clarke s compari sons should now be universally adopted. The committee metre of the American Philosophical Society is the unit of length to which all linear measures of the Coast Sur vey are referred. It was compared August 24, 1867, at Paris, directly with the standard platinum metre of the Conservatoire des arts et metiers, and was found (at the temperature of melting ice) = i m .ooooo336 of the platinum metre of the archives. The French standard metre has its normal length at zero centi grade, or the freezing-point. It was intended to be a natural standard, and to -represent the ten-millionth part of the terrestrial arc between the equator and the pole, which was assunred to be 5130740 toises, and the length of the metre 443.29596 lines of the toise du Perou; which quantity was declared by law in 1799 to be the length of the legal metre, and is the length of the stand ard platinum metre of the archives. The toise du Perou, made in 1735, is a bar of iron, and has its standard length at 13 Reaumur, (6i.25 Fahrenheit .) It was used by La Condamine in the measurement of an arc of the meridian in Peru in 1744. As the above determination of the length of the quarter of the meridian is now known to be errone ous, the legal metre becomes, in fact, but a legalized part of the toise du Perou, and this last remains the primitive standard. It is the unit of length in which the greater part of the European geodetic measurements are expressed. The standard Klafter of Vienna has its normal length at 13 Reaumur, and is = 840.76134 lines of the toise du Perou. The standard Prussian foot is a standard also at 13 Reaumur, and was declared by law to be = 139.13 lines of the toise du Perou. 92 GEODESY. XXXVIII. Numerical Values of BesseVs Terrestrial Elements in English Yards, adopting Rater s Value-of the Metre, viz : 39.37079 English Inches ; log 1.5951 741 293. Log. to reduce toises to yards = 0.3286915586 Log. to reduce metres to yards = 0.0388716286 a = equatorial radius = 6974532^ . 339 log = 6. 8435150923 l> = polar radius =695121 8^.059 log = 6.8420609125 Length, in yards, of a meridional degree whose middle lati tude is <p : D,, ( = 1215255". 1 83 610^.336 cos 2 cr 4- 1^.302 cos 4 <r ) -f 0^.002 cos 6 cr j Length, in yards, of a degree of the parallel : D^= 121830^.366 cos (f 101^.941 cos 3 cr-fo> .i2S cos 5 <p or, making sin y = e sin ^ log D, = 5- 8 539 2 5 + lo g cos V lo S cos / or, using the logarithms of the numerical co-efficients D,,, = 121525^.183 (2.7855691) cos 2 ^ -f (0.1147) cos 4 ^ > 4- (7.3287) cos 6 <p J D y =(5- 8 5755 6 ) cos V ( 2 - 8 35) cos 3 <F +(9- Io6 9) cos 5 V or D ^- cos V COS / Constant Logarithms. r = 0.00667435 ........................ log = 7.8244104542 ^ = E = ellipticity = --- = 7-5 2 337 8 9 82 4 299.66 sin i" ............................. = 4-6855748668 J Sin i" ..... : ........................ =4.3845448711 3 ^_sin i 7/ ............................. =2.6860751039 2 (I _ (*) = 0.99332565 ............. = 9.9970916404 (l ^) ............................ = 6.8406067325 rt sin i /x ............................ - - = i-5 2 9 o8 9959 r a sin \" , (arithmetical complement) ..... . . = 8.4709100409 GEODESY. 93 XXXIX. For Computing the Geodetic Latitudes, Longitudes, and Azimuths of Points of a Triangulation. i. For distances not exceeding one hundred miles : - a L = K B cos Z + K 2 C sin 2 Z + (8 L) 2 D - K 2 h E sin 2 Z where T B = Rarci" tan L 2 N R arc i /7 y, f e* sin L cos L arc \" (i <? 2 sin 2 L)* _ i + 3 Um JjL ~6~N*~ h = K B cos Z, or first term ; d L an approximate value for d L, computed from first and second term ; -V K sin Z ~cosL ; A / _ referred to second point : N arc i" ~~^ Z= ^cosV^L log F for latitude 25 = 7.8324; for latitude 45 = 7.8404. 2. For distances not exceeding twenty miles : In terms of the sides of the triangles : N sin i" a sin i". L = L (i + e 2 cos 2 L) u" cos Z > - (i + e 1 cos 2 L) (H" sin Z) 2 tan L X J sin i" f cos Z/=:i8o +Z.- /// -- I J- / Z sin J (L+ L ) cos L 7 94 GEODESY. XXXIX. For Computing Gtodttic Latitudes. ,3-V. Continued. In terms of the co-ordinates of rectangular axes referred to one of the points of the triangulation, the latitude and longitude of which are known : _r being the ordinate in the direction of the meridian, and x the ordinate erendicular to it : L = L = ^-^-"(N^ y *K L = R^T = (N- :.:) -: K = distance in yards between two station, the latitude and :._---" - . : - - - : " - - distance convened to seconds of arc : L = latitude of ist station: : := .:.-. . :">-- - : Z = azimuth of 2d station at ist, counted from the south round by the west, from o to 360: the algebraic signs of the sine, and cosine of this angle must be carefully L 7 . M , Z/ the same things at 2d station, or quantities required : section r*en>er.d:cuiar to the sn (M -M^sin J; by which the azimuth at one end of at the other, is called the cmrtrxfiK XL. To Compute the Length and Direction of a Line Joinmg Two Points, the Latitudes and Longitudes of which are KiKncn, or Measurement of a Base by Astronomical Observations. -.? _ r (L L 7 ) cos 2 ^ (L + I/) ~2~ 2 V= . /=L = . i sin i .v - tan / -? x tan Z = = .r X sin sin Z cos Z _-. = ; . X sir. : K = u" X sin i in which L. L . M. M 7 . represent the latitudes and longitudes of the two points : u". the distance between these points in seconds of arc ; K. the distance between these points in linear units : x", the number of seconds in the arc passing through the point of which L 7 is the latitude, and perpendicular to the meridian of the point of which L is the latitude: \". the seconds in the portion of this meridian between L and the foot of this perpendicular ; ,\. v, the same quantities in linear units : Z. the azimuth of the second point, L . from the first. L ; and N, the normal at the middle latitude. 96 GEODESY. XL. To Compute the Length and Direction of a Line, erV. Con d. Particular attention must be paid to the sign (L L ), for upon this depends the sign of- , and also to that of (/ I } in the value of y , so as to know whether the small quantity ( J sin i"#" 2 tan /) is to be added to or substracted from (/ / ). The azimuth Z is counted from the south, round by the west, from o to 360. The azimuth Z (if required) is to be computed from Z, as in XXXIX, (2.) This can be presented in a different form, thus : as, (M 7 M) cos L 7 = u" sin Z and, r/j y rf/w I " o T \ // rj \ -- -"-* j y ** out t.j \^\j^ \.j liin JL* sin i \^\ | t ~ cos*" i^j Substituting, in this last, the value of u" sin Z, and dividing one by the other : (M - M) cos L 7 (i + e l cos 2 L) ~ (L-L ) - j" (M / -M) 2 cos 2 L 7 tan L sin i" ( i + "^ cos 8 L) Then, knowing Z (M 7 - M) cos L 7 ?/ = ^ sin Z and K =: n" N sin i 7/ N being the normal for the mean latitude. XLI. To Compute the Distance between Two Points, knowing theit Latitudes and the Azimuth of one from the other. t 3_e 2 (L - L 7 ) cos 2 j (L + L 7 ) e 2 sin 2 J (L + L 7 )]i sin(^ //") = -= j sin ; See the note to the preceding formulae. The algebraic sign of the azimuth, Z, will determine the sign of <p, and consequently whether the quantity u" is to be added to or subtracted from p. GEODESY. 97 XLII. To Compute the Distance between Two Points, knowing the Latitude of one, the Azimuth from this to the other, and the Difference of their Longitudes. tan ( = sin L tan Z tan L" = tan L sin (y ~ sin <p - L") cos 2 J (L + L") L/ _ L// _ .3 / _ L __ if // j / i ^ u i-i ._ ; ^ cos - / K = " N sin i /x sin Z ;;/ = the difference of longitude. The azimuth, Z, is, as before, counted from the south round by the west; its algebraic sign will determine the sign of y, and consequently whether it is to be increased or diminished by m. Elements of the Figure of the Earth, Deduced by Captain A. R. Clarke, Royal Engineers, in Computing the Figures of the Meri dians and of the Equator for Several Measured Arcs of Meri dian, (Comparison of Standards of Length, &c., London, 1866. ) Semi-axes. Major semi-axis = , 34 east) of equator, (longitude 15 Feet. Toiscs. Metres. Minor semi-axis = <5, 34 east) of equator, (longitude 105 Polar semi-axis c.. 6356068 I Length. a c i c = 285^97 bc a b 313-38 3 26 9 -5 The length of the meridian quadrant passing through Paris is. and the minimum quadrant, in longitude 105 34 , is. ,10001472.5 metres. .10000024.5 metres. For a Spheroid of Revolution more nearly Corresponding to the Same Geodetic Measurements. Semi-axes. Length. Equatorial semi-axis a Feet. 20926062 20855121 Toises. 3272492.3 3261398.4 Metres . 6378206.4 6356583-8 Polar semi-axis b . . b 293.98 a b i llipticity. a ~ 294. 98 a ~294. 9 8~ 9 8 GEODESY. FORMS FOR RECORD Survey of ^6 3 Position. Names of stations. 6 Observed angles. 14J ^ .b ~S _C rC "3 . cLrt in Spherical excess. Final plane angles. / // ,, u n o / // Sought . Cedar Point.. 18 66 34 04.80 O.36 04.44 1.58 66 34 02.86 XIII Right . . Buck Hill . . . 18 64 08 37.78 -0.36 37-43 i. 5 8 64 08 35.84 R || Left . . . Fort Flats . . . 18 49 17 23.24 -0.36 22.88 1.58 j 47 17 21.30 180 o o. oo Survey of Names of stations. Latitudes. L = L- u".(i + sin i x/ sin 2 Z //2 ( os 2 L) cos Z ^- cos 2 L) tan L Fort Flats . . Latitude L ....... = 45 39 i3"- ! log K (yards). .= 4.7295212 .=18.4701676 log u" = 3.1996888 log (i + ^3 cos 2 L) =0.0014140 log cos Z ( ) = 9.971 1240 -4.38454 2 log sin Z = 9.09522 2 log u" = 6.39936 . = 0.00141 log tan L = 0.00991 Cedar Point . log 1st term .......... = 3.1722268 log 2d term ..... = 9.89034 o ;/ .77 ist term 2d term ( + ) = + i486".7i ( ) = o".77 2d term L ................ = 45 39 i3"-89 L+ L f ....= 9i 43 "3 -72 Latitude L ..... = 46 03 59" -83 ~ GEODESY. 99 AND COMPUTATION. Calculation of Triangles. bJ3 P Logarithms of their sines. Calculation of the sides. Sides in yards. Designation. s R L 9.9626198 9.9541886 9.8796760 log R L 4 737CK24. = 54695.61 = 53644.00 = 4518649 $ Buck Hill Fort \ Flats. ( Fort Flats Cedar \ Point. f Buck Hill Cedar \ Point. comp log sin S . . log sin R o. 0373802 = 9.9541886 loo- L S = 4.7295212 log R L + > comp log sin S ^ log sin L = 4- 77533 26 = 9.8796760 loe R S = 4.6550086 Geodetic Determination of Positions. (Secondary/ Longitudes. Azimuths. Remarks. L + 1/ ; Longitude M =^84 42 22". 19 Azimuth Z = 159 20 i3". log sin Z ( + )=9. log// = 3. 1 80 2.7473005 log cos L = 9.8412474 log 6 M 6 M M .. . = 2.9060529 180 -j- Z = 339 20 13". 62 20 39 46". 3 8 log sin L + L/ = 9-8559089 ..( + ) = 2. 9060529 log (5 Z .=12.7619618 .= o : 13 25". 48 .= 84 42 22". 19 Longitude M =84 55 47". 67 r5Z 180 + Z = 339 20 13". 62 Azimuth Z .. .. = 339 10 35". 5 7 100 GEOD ESY. Normal or Radius of Curvature of the Perpendicular to the Meridian. Ellipticity = ^o ; equatorial radius = 6974532 yards; log = 6.8435151. Log. to reduce yards to metres = 9.9611283714. N.= a Latitude. (I ^ sin 2 L)* log i+^cos L) Difference for 10 . .logN Com. difif. for 10 . lop- * 5 N sin i" 20 15 3 45 6. 8436847 6888 6929 6971 27-3 27.6 27.8 28.1 8. 4707404 7363 7322 7280 o. 0025521 5439 5356 5274 55 . 56 55 5 6 21 O 7013 7056 28.4 28 7 7238" 7196 5106 57 3 0- 7099 zo - / 7153 5021 j?8 45 7142 29-3 7109 4934 58 22 15 3 45 7186 7230 7274 7319 29-5 29-7 30.0 30.2 7066 7021 6977 6932 4847 4760 4671 4582 58 59 59 60 23 o 15 30 45 7365 7410 7457 7503 30.5 3-7 3 I.O 31.2 6887 6841 6795 6748 4492 4401 439 4217 61 62 62 62 24 o 15 30 45 755 7597 7645 7693 3*. 5 3 1 -? 32.0 32.2 6701 6654 6607 6559 4124 43 3935 3840 63 63 s 25 o 15 3 45 774i 7790 7839 ,,000 7000 32.5 32.7 32-9 33-2 6510 6462 6413 6363 3744 3648 355 3452 64 65 3 26 o 15 30 45 7938 7988 8038 33-4 33.6 33-8 34-0 6313 6263 6213 6162 3353 3 2 54 3154 353 66 67 67 68 27 o 15 3 45 8140 8192 8243 8295 34-3 34.5 34 7 34-9 6111 6060 6008 5956 295 1 2849 2746 2643 68 69 69 69 28 o 8348 5904 2539 7 15 8400 5851 2434 70 3 8453 l\ \ : 5798 2329 45 6. 8438507 glj 8.4705745 o. 0022223 " GEODESY. 101 Normal, &c. Continued. Latitude. N- a log (i+t y2 cos 2 L) Difference for 10 . (i t> 2 sin 2 L)* logN Com. diff. for 10 . log L_ b N sin i" / 29 o 15 6. 8438560 8614 35.9 8. 4705691 5637 o. 0022117 2OIO 71 3 45 8668 8723 36. i 36.3 36.5 5583 5529 1902 1794 7 2 72 72 3 8777 oA *T 5474 1686 15 3 45 8832 8888 8943 3. 7 3 6 .9 37.i 37-2 5419 5364 53 8 1576 1466 1356 73 73 73 74 31 o 15 8999 9055 37 6 5252 1245 H34 74 3 45 9111 9168 37.7 37-9 5084 IO22 0910 75 75 32 o 15 9225 9282 38.1 ^8 -7 5027 4970 0797 0684 75 3 9339 Jj;~ 4912 0570 n*r 45 4855 0455 / / 77 33 o 15 3 45 9454 9512 957i 9629 38.7 38.9 39-0 39- i 4797 4737 4681 4622 0340 0225 . 002OI09 .0019993 77 77 77 77 34 o 15 3 45 9688 9747 9806 9865 39-2 39-4 39-5 39-7 45 6 4 455 4446 4387 9877 9760 9043 9525 78 78 79 79 35 o 15 30 45 9924 . 8439984 . 8440044 0104 39-8 40. o 40. o 40. i 4327 4267 4208 4148 9407 9288 9169 9050 80 80 80 80 36 o 15 3 0164 0224 0285 40.3 40.3 4087 4027 3966 8931 8811 8690 80 81 45 0346 40! 6 3906 8570 81 37 o 15 3 45 0406 0467 0529 0590 40.7 40.7 40.9 41. o 3845 3784 3g3 3661 8449 8328 8206 8084 81 8r 81 81 38 o 15 0651 0713 41.1 AI I 3600 3538 7840 82 82 3 45 0775 0837 41 . i 41.2 41-4 3477 3415 7717 7594 82 82 1 39 15 30 45 8 f 4I 4 3353 0961 3291 1023 3229 6.8441085 41-5 8.4703166 747 1 7348 7224 o. 0017101 82 11 83 102 GEODESY. Normal, &c. Continued. N = a Latitude. ( I 6 2 sin 2 L)* - log Difference logN Com. diff. for 10 . **Nab<- (i+t -cos 2 L) for 10 . o / 40 o 6.8441147 A T *7 8. 4703104 o. 0016977 15 3 45 I2IO 1273 1335 41. / 41.8 41.8 41.9 34i 2979 2916 6853 6728 6604 84 83 84 41 o 15 3 45 1398 1461 1524 1587 41.9 41.9 42.0 42.1 2853 2791 2728 2665 6479 6354 6229 6104 84 84 84 84 42 o 15 30 45 1650 1713 1776 1839 42. I 42.1 42.2 42.1 2602 2539 2475 2412 5979 5853 5728 5602 84 84 84 84 43 15 3 45 1903 1967 2029 2093 42.2 42.2 42.3 42.3 2349 2286 2222 2159 5477 535i 5225 599 84 84 84 84 44 o 15 30 45 2156 2219 2283 2346 42.3 42.3 42.3 42.3 2095 2032 1969 4973 4847 4721 4595 84 84 84 84 - 45 15 30 45 2410 2473 2537 2600 42.3 42.3 42.3 42.3 1842 T 77 8 T 5 1651 4469 4343 4217 4091 84 84 84 84 46 o 15 3 45 2663 2727 2790 2854 42.3 42.3 42.3 42.2 1588 1525 1461 1398 3965 3839 37J3 3587 84 84 84 84 47 o 15 30 45 2917 2980 3043 42.2 42. I 42.1 42.1 1334 1271 1208 1145 3336 3210 3084 84 84 84 84 48 o 15 3 45 3170 3233 3296 3359 42. I 42. o 42.0 41.9 1082 1018 0955 0892 2959 2833 2708 2583 84 84 84 84 49 o 15 3 45 3422 3485 3547 3610 41.9 41.9 41.8 41.8 0830 0767 0704 0641 2458 2333 2209 2084 84 83 84 83 50 o 6. 8443673 8.4700579 o. 0011960 GEODESY. I0 3 Radius of Curvature of the Meridian. Elliptieity = ^,-y ; equatorial radius = 6974532 yards. log R Com. diflf. for 10 . log 33 Rsin i" o / 20 15 6.8411155 1278 81.9 82 7 ^2973 3 45 1402 1527 * / 83.5 84.3 2849 2724 21 O 1654 85 i 2598 15 1781 86 . o 2470 30 1910 86^8 2341 45 2040 2211 22 2172 88.4 2080 15 2304 3 2438 45 2573 89.1 90.0 91. o 1947 1679 23 o 15 3 45 2709 2846 2984 3124 91-5 92.3 93- 93-6 1543 H05 1267 1128 24 o 15 3264 3406 94.6 0987 0845 3 3549 ofi* 07O2 45 3693 96.7 559 25 o 3838 15 3984 97.4 0414 0268 30 4I3 1 yo. i . 4730120 45 , 4279 99.4 . 4729972 26 o 15 3 4428 4578 4730 IOO. I 100. 9 101,5 9823 9673 9522 45 4882 IO2. I 937 27 o 5035 102.8 9216 15 30 AC 5 l8 9 5344 103.4 104. I PC? 43 3D 104.7 75 28 o 15 3 45 5 6 57 5974 6.8416134 i5-3 106. o 106.5 107. i 8594 8436 8277 8.4728117 io 4 GEODESY. Radius of Curvature of the Meridian Continued. Latitude. R ^ ( 7f^L)t log R C f om - di / f - for 10 . lo Rsin i " 29 o 15 6. 8416295 6456 107. i 8.4727956 7795 30 45 6619 6782 1 08. 6 109. 4 7632 7469 30 o 6946 I IO O 7305 15 30 7111 7277 no. 5 T T T T 7140 6974 45 7444 111.1 in. 6 6808 31 o 7611 T T >> 1 6640 15 3 7779 ;;;; 7948 6472 6303 45 8118 JJ 3<I 3 2 o 15 30 45 8288 8460 88? 114. I 114. 6 115.1 115.6 5963 5792 5620 5447 33 15 3 i !ai 5274 5100 4925 45 CKO2 1 1 7. 3 475 34 o 15 3 9678 . 8419854 .8420031 117.7 118. i TTX C 4574 4397 4220 45 O2OQ J . II9.I 4042 35 0387 3864 15 3 45 0566 0746 0926 II9.7 120.0 120.4 3685 3325 36 o 15 3 1107 1288 1469 120. 7 121. I 3H5 2964 2782 45 1651 121. 7 2600 37 o 1834 2417 15 201 7 ! IS , 2234 3 22OO 2051 45 2384 122.7 122. 9 1867 38 o 15 30 45 2569 2753 2939 3124 I2 3 .I 123.5 123.7 124. o 1683 1498 1313 1127 39 o 3310 0941 15 3496 ^ 0755 3 3Oo7 : O^Oo 45 6. 8423870 I2 4 <-, 8. 4720^82 1 2A O 1 GEODESY. I0 5 Radius of Curvature of the Meridian Continued. Latitude. R = a(i-e*) a. (i ^ 2 sin 2 L) 2 log R Com. diff. for 10 . *nh 40 o 15 30 45 6.8424057 4244 4432 4620 125.0 125.1 125.3 125.5 8.4720194 . 4720007 .4719819 9631 41 o 4808 12$. 7 9443 3 4997 5186 :lo ! ?* 45 5375 126^2 8877 42 o 15 5564 5753 126.2 ! f 7 126 - 8498 3 5943 8709 126.4 Q 45 6132 126.6 43 6322 126.6 7929 15 6512 3 6702 , , 7549 45 6892 ^26 8 7359 44 15 3 7082 7273 7463 5! il 45 7653 lll .g 6 59 8 45 o 15 3 45 7844 8034 8224 8415 127. o 126. 9 126.9 126.9 6408 6217 6027 5837 46 o 15 30 8605 8795 8985 126.8 126.8 126 7 5 6 47 5456 5266 45 9175 ;: 126. 7 5076 47 o 15 3 45 9365 9555 9745 . 8429934 126.6 126.6 126. 4 126.3 4886 4696 4506 43^7 48 o 15 .8430124 0313 126. 2 126. I 4127 3938 3 45 0502 0691 126.0 125.8 3749 49 15 3 45 0880 1068 1257 1445 125.7 125.5 125.3 125.0 3371 3183 2995 2807 50 o 6. 8431632 8.4712619 io6 GEODESY. XLIII. Projection of Maps. POLYCONIC PROJECTION. In this development of the earth s surface each parallel of lati tude is supposed to be represented on a plane by the develop ment of a cone having the parallel for its base and its vertex in the point where a tangent to the parallel intersects the earth s axis. The map thus becomes the development of the surfaces of several successive cones, and the degrees of the parallel preserve their true length. Normal (i <? 2 sin 2 L)- v (i e 2 } Radius of the meridian , . . R = N 3 -- a 2 Radius of the parallel R y , = N cos L Degree of the meridian D , = - * R /rt IOO = 3600 R ni sin i" Degree of the parallel D /( = - ," - R p = 3600 R yJ sin i" Radius of the developed parallel or side of tangent cone r = N cot L Designating by n any arc of the parallel, or difference of longi tude, to be developed, and by the corresponding angle sub tended by the developed parallel at the vertex of the cone; then the length of the given arc will be : n R^ = // N cos L and also Or N cot L whence Angle of the developed parallel, = n sin L and as the developed parallels are circular arcs, the co-ordinates of curvature are : J M) difference of meridians, = x = r sin d p , difference of parallels, = y = r ver sin = x tan J For surfaces of small extent the arc of the parallel may be con sidered coincident with its chord ; and as the angle between a PROJECTION OF MAPS. 107 XLIII. Projection of Maps Continued. tangent and a chord is half the angle at the center subtended by the chord, d ltl , difference of meridians, = x = D p cos J d p , difference . of parallels, = y = T> p sin J The values of d m and d p and of D w and D^ will be found in the following tables. Example of their Use. Let it be required to make a projection containing 40 of longitude between the parallels of 41 30 and 42 io 7 , to be subdivided to 5 . Assume the center of the sheet to be the intersection of the middle parallel with the middle meridian of the proposed map, which point call A; in this case a point in the parallel of 41 50 . Through A draw the central meridian and a line at right angles to it. Beginning at A, lay of above and below, on the central merid-. ian, the values of D, tt from 41 50 to 41 55 ; 41 55 to 42; 42 to 42 5 , etc.; and from 41 50 to 41 45 ; 41 45 to 41 40 , etc. ; these values to be taken from the table of Meridional Arcs Values of D m in Yards, by interpolation from the values there given for the middle latitudes of 41 and 42. Through each of the points . . . , A", A 1 , A, Aj, AH, . . . , thus found, lay off perpendiculars to the central meridian. Now turn to the table of Co-ordinates, O M and d p , in Yards, and lay off, from each of the points . . . , A", A 1 , A, Aj, A ii} . . . , to the right and left of the central meridian, the values of d m for succes sively 5 , 10 , 15 , and 20 , corresponding (by interpolation from the columns of 41 30 and 42) to each parallel of latitude re quired; and, from the points thus found, the corresponding values of d p at right angles to the lines already drawn. Lines passing through the extremities of d p will be the required meridians and parallels. The projection being made, any point whose latitude and longi tude are known will be projected on the map from elements taken from the tables of values of D Ml and D^, which are measured from the meridians dj\& parallels, and not from the axes of co-ordinates used in making the projection. IOS GEODESY. Poly conic Projection Co-ordinates, d in , d jt , in Yards. o Latitude 22 o . Latitude 22 30 . Latitude 23 o . 3 Sb c 4. *. 4, ,.,. 4. 4, I 1882.0 O. I 1875.3 O.I 1868.5 o.i 2 3763-9 0.4 3750.6 0.4 3737.0 0.4 3 5645-9 0.9 5625.9 I.O 5605.4 I.O 4 7527.8 1.6 7501.2 1.7 7473-9 1.7 5 9409.8 2.6 9376.4 2.6 9342.4 2-7 6 11291.8 3.7 H25I.7 3.7 11210.9 3-8 7 13173.7 5.0 I3I27.0 5-1 13079.4 5-2 8 15055.7 6.6 I5OO2.3 6.7 14947.8 6.8 9 16937.6 8-3 16877.6 8-5 16816.3 8.6 10 18819.6 10.3 18752.9 10.5 18684.8 10.6 ii 20701.6 12.4 20628.2 12.6 20553-3 12.8 12 22583.5 14.8 22503.5 15-0 22421.8 15.3 13 24465.5 17-3 24378.8 17.6 24290.2 17.9 14 26347.4 20.1 26254. I 20.5 26158.7 20.8 15 28229.4 23.1 28129.3 23.5 28027.2 23.9 16 3OIII.4 26.2 3OOO4.6 26.7 29895.7 27.2 17 31993.3 29.6 31879.9 30.2 31764.2 30.7 18 33875.3 33-2 33755-2 33.3 33632.6 34.4 19 35757.2 37.o 35630.5 37-7 35501.1 38.3 20 37639.2 41.0 37505.8 41.7 37369-6 42.5 25 47049.0 64.1 46675.8 65.2 46712.0 66.4 30 56458.7 92.3 56258.7 93-9 56054.3 95-6 40 75278.2 164.1 750H.5 167.0 74739-0 169.9 50 94097.7 256.3 93764.2 260.9 93423.7 265.5 I 00 II29I7.O 369.1 II25I6.9 375-8 112108.2 382.3 I 20 150555.4 656.2 I5OO2I.9 668.0 149476.9 679.6 I 30 169374.4 830.5 168774.2 845.4 168161. i 860.1 I 40 188193.3 1025.4 187526.3 1043.8 186845.1 I 061.8 2 OO 225830.5 1476.5 225030.0 1503-0 224212.5 1529.1 2 30 282284.7 2307.1 281284.0 2348.5 280261.9 2389-2 3 oo 338736.6 3322.2 337535-6 33SI.8 336309.0 3440.4 3 30 395186.0 4521.9 393784.5 4603.0 39 2 353.i 4682.8 4 oo 451632.0 5906.2 450029.9 6012.1 448393.7 6116.2 PROJECTION OF MAPS. 109 Poly conic Projection Coordinates, d m , 8 p , in Yards. Longitude. Latitude 23 30 . Latitude 24 o . Latitude 24 30 . d m *, <*m 8* dm &* i 1861.5 O. I 1854.4 o.i 1847.2 O.I 2 3723.1 0.4 3708.9 0.4 3694.4 0.4 3 5584.6 I.O 5563-3 i.o 554L6 I.O 4" 7446.1 1.8 7417.7 1.8 7388.8 1.8 5 9307.6 2.7 9272.2 2.7 9236.0 2.8 6 11169.2 3-9 11126.6 ! 3.9 11083.2 4.0 7 13030.7 5-3 12981.0 5-4 12930.4 5.4 S 14892.2 6.9 14835-5 7.o 14777-6 7-1 9 16753.7 8.7 16689.9 8.9 16624.8 9.0 10 18615.3 10.8 18544-3 II. 18472.0 11. i ii 20476.8 13-1 20398.8 13.3 20319.2 13.5 12 22338.3 15.6 22253.2 15-8 22166.4 16.0 13 24199.9 18.2 24107.6 18.5 24013.6 18.8 14 26061.4 21 .2 25962.1 21-5 25860.8 21.8 15 27922.9 24-3 27816.5 24.7 27708.0 25-1 16 29784.4 27.7 29670.9 28.1 29555.2 28.5 17 31646.0 3L2 31525.4 31.7 31402.4 32.2 18 33507.5 35-0 33379-8 35.5 33249.6 36.0 19 35369.0 39-0 35234.2 39.6 35096.8 40.2 20 37230.5 43-2 37088.7 43.9 36944.0 44-6 25 46538.1 67.5 46360.8 68.6 46179.9 69.6 30 55845.8 97.2 55632.9 98.7 55415.9 100.3 40 74460.9 172.7 74I77.I 175.5 73887.7 178.3 50 93076.0 269.9 92721.3 274-3 92359.5 278.5 I 00 111691.0 388.7 111265.3 394-9 110831.2 401 .1 I 20 148920.6 690.9 148353-0 702. i I47774-I 713-0 I 30 167535.2 874.5 166896.6 888.6 1*66245.4 902.4 I 40 186149.7 1079.6 185440.1 1097.0 184716.5 1114. i 2 00 223377.9 1554.6 222526.4 1579.7 221658.0 1604.3 2 30 279218.6 2429.1 278154-1 2468.3 277068.4 2506.8 3 oo 335056.8 3497-9 333779-1 3554-4 332476.1 3609.8 3 30 390892.0 4761.1 389401.1 4837.9 387880.6 49!3-3 4 oo 446723.4 6218.5 445019.2 6318.9 44328i.i 6417.4 no GEODESY. Poly conic Projection Co-ordinates, d m , d f , in Yards. Longitude. Latitude 25 o . Latitude 25 30 . Latitude 26 o . tJ ** 6m 6p 6 m SP / I 1839.8 O. I IS32.3 O. I 1824.7 O.I 2 3679.6 0.5 3664.6 0.5 3649-3 0.5 3 55I9-5 I.O 5496.9 i .0 5474-0 I.O 4 7359-3 1.8 7329.2 1.8 7298.6 1.9 5 9199.1 2.8 9l6l.5 2.9 9 I2 3-3 2.9 6 11038.9 4.1 10993.8 4.1 10947.9 4-2 7 12878.8 5.5 I2826.I 5.6 12772.6 5.7 8 14718.6 7.2 14658.5 7-3 14597-2 7-4 9 16558.4 9-2 16490.8 9-3 16421.9 9-4 10 18398.2 ii-3 I8323.I ii. 5 18246.5 ii. 6 ii 20238.0 13-7 20155.4 13.9 20071.2 14.1 12 22077.9 16.3 21987.7 16.5 21895.8 16.8 13 23917.7 19.1 23820.0 19.4 23720.5 19.7 14 25757-5 22.2 25652.3 22.5 25545-1 22. S 15 27597-3 25-4 27484.6 25.8 27369.8 20.2 16 29437.1 29.0 29316.9 29.4 29194.4 2 9 .S 17 31277.0 32.7 3II49.2 33-2 31019.1 33-C 18 33116.8 36.6 32981.5 37-2 32843-7 37- > 19 34956.6 40.8 34813.8 41-4 34668.4 42. c 20 36796.4 45-2 36646.1 45-9 36493.0 46.; 25 45995-5 70.7 45807.6 7L7 45616.2 72-: 30 55194.6 101.8 54969.1 103.2 54739-5 104.- 40 73592.7 180.9 73292.0 183.6 72985-8 186.1 50 91990.7 282.7 91614.9 286.8 91232.1 290.5 I OO 110388.6 407.1 109937.6 413.0 109478.3 418.$ I 20 147184-0 723.8 146582.7 734-3 145970.3 744- < I 30 165581.5 916.0 164905.0 929-3 164216.0 942.: I 40 183978.8 1130.9 IS3227.I H47.3 182461.5 1163.. 2 OO 220772.7 1628.5 219870.6 1652.1 218951.9 1675.1 2 30 275961.6 2544-5 274833.9 2581.4 273685.3 2617. ( 3 o 33H47.8 3664.1 329794.3 3717.3 328415.8 3769-: 3 30 386330.6 4987.2 38475L3 5059. 6 383142.7 | 5130. 4 oo 441509.4 6513-9 439704.0 6608.5 437865.3 6701. ( PROJECTION OF MAPS. Ill Poly conic Projection Co-ordinates ^ d m , d p , in Yards. Latitude 26 30 . Latitude 27 o . Latitude 27 30 . . *. <r ro * / I 1816.9 O.I 1808.9* o.i 1800.9 O. I 2 3633.7 0.5 3617.9 i 0.5 3601.7 0.5 3 5450.6 i.i 5426.8 i.i 5402.6 1. 1 4 7267.4 1.9 7235.7 1.9 7203.4 1.9 5 9084.3 2.9 9044.6 3.o 9004.3 3-0 6 10901.2 4-2 10853.6 4-3 10805.1 4-4 7 12718.0 5-8 12662.5 5-9 12606.0 5-9 8 14534.9 7-5 14471.4 7-6 14406.9 7-7 9 16351-7 9-5 16280.3 9-7 16207.7 9.8 10 18168. 6 ii. 8 18089.3 11.9 18008 6 12. r ii 19985.5 14.3 19898.2 14.5 19809.4 14.6 12 21802.3 17-0 21707.1 17.2 21610.3 17.4 13 23619.2 19.9 23516.0 20.2 23411.1 20.4 14 25436.0 23.1 25325.0 23.4 25212.0 23.7 15 27252.9 26.5 2/133.9 j 26.9 27012.8 27.2 16 29069.8 30.2 28942.8 : 30.6 28813.7 30.9 17 30886.6 ; 34.1 3075L7 34-5 30614.6 34-9 18 32703.5 38.2 32560.7 38.7 32415-4 39-2 19 34520.3 42.6 34369.6 43-1 34216.3 43-6 20 36337.2 47-2 36178.5 47-8 36017.1 48.4 25 45421.4 73-7 45223.1 74-7 45021.4 75-6 30 54505.6 106.1 54267.7 107.5 54025.6 108.8 40 72674.1 188.6 72356.8 191. i 72034.0 193-5 50 90842.4 294.8 90445 . 8 298.6 90042 . 4 302.4 I OO 109010.7 424-5 108534.8 430.0 108050.6 435-4 I 20 145346.7 754-6 144712.1 764.4 144066.5 774-0 I 30 163514.5 955.1 162800.5 967.5 162074.2 979-6 I 40 181682.0 j 1179. i 180888.7 1194.4 180081.7 1209.4 2 OO 218016.4 1697.9 217064.4 1720.0 216095.9 1741-6 2 3 272515-9 2652.9 271325-7 2687.5 270114.9 2721.2 3 00 327012.2 3820.2 325583-8 3870.0 324130.7 3918.6 3 30 381505.0 5199.8 379838.2 5267.5 378142.5 5333-6 4 oo 435993-2 679 1 - 5 434088.0 6880.0 432149.7 6966.3 112 GEODESY. Polyconic Projection Co-ordinates, d m , d jt , in Yards. Longitude. Latitude 28 o . Latitude 28 30 . Latitude 29. <*, " * 6 m ** dm 6 P i 1792.7 O. I 1784-3 O. I 1775-8 O. I 2 3585.3 0.5 3568.6 0.5 355L7 0.5 3 5378.0 i i.i 5352.9 i . i 5327.5 i.i 4 7I7O.6 I 2.O 7I37-2 2 .O 7I03-3 2 .O 5 8963.3 1 3-i 8921.5 3-1 8879.1 3-i 6 10755.9 4.4 10705.8 4-5 10655.0 4.5 7 12548.6 ! 6.0. 12490.2 6,1 12430.8 6.1 8 14341.2 7-8 14274-5 7.9 14206.6 S.o 9 16133.9 9-9 16058.8 10. 15982.5 IO. I 10 17926.5 12.2 17843.1 12.4 17758.3 12.5 ii I97I9.2 14.8 19627.4 15-0 10534- i 15.2 12 2I5II.8 17.6 21411.7 17.8 21309.9 iS.o 13 23304.5 20.7 23196.0 20.9 23085.8 21.2 14 25097.1 24.0 24980.3 24.3 24861.6 24-5 15 26889.8 27.5 26764.6 27.9 26637.4 28.2 16 28682.4 31.3 28548.9 31-7 28413.2 32.1 17 30475.1 35-4 30333 2 35-8 30189.1 36.2 18 32267.7 39.7 32II7.5 40.1 31964.9 40.6 19 34060.3 44.2 33901.8 44-7 33740.7 45.2 20 35853.0 49-0 35686.1 49-5 355i6.6 50.1 25 44816.2 | 76.5. 44607.6 78.4 44395-7 78.3 30 53779-4 no. i 53529.1 in. 4 53274.8 112. 7 40 71705.8 195.8 71372.1 198.1 71032.9 200.3 50 89632.0 306.0 89214.9 309.6 88790.9 313.0 I OO 107558.2 440.7 107057.6 445-8 106548.8 450.8 I 20 143410.0 783-4 142742.5 792.5 142064.1 801.4 I 30 I6J335.6 1 99 r -5 160584.6 1003.0 159821.5 1014.3 I 40 179260.9 1224.1 178426.5 1238.3 177578.6 1252 . 2 2 00 215110.9 1762.6 214109.6 1783.2 213092.0 IS03.I 2 30 268883.6 275-4-1 267631.8 2786.2 266359.6 2817.4 3 oo 322652.8 3965-9 321150.5 4012.1 319623.7 4057.1 3 30 376418.1 5398.1 374665.0 5460.9 372883.4 5522.1 4 oo 430178.5 : 7050.6 428174.6 7132 7 426138.2 7212.6 PROJECTION OF MAPS. Poly co nic Projection Co-ordinates^ d m , o , in Yards. Longitude. Latitude 29 30 . SP Latitude 30 o . Latitude 30 30 . s m $m 6* 6 m &* / I 1767.2 O.I 1758.5 O.I 1749.6 O. I 2 3534-4 0.5 3516.9 0-5 3499.2 0.5 3 5301.6 I.I 5275.4 1.2 5248.8 I .2 4 7068.9 2 ,O 7033.9 2.O 6998.3 2.1 5 8836.1 3-2 8792.3 3-2 8747.9 3-2 6 10603.3 4.6 10550.8 4.6 10497-5 4.6 7 12370.5 6.2 12309.3 6-3 12247. i 6-3 8 I4I37.7 S.i 14067.8 8.2 13996.7 8.2 9 15904.9 10.3 15826.2 10.4 15746.3 10.5 10 17672.2 12.7 17584.7 12.8 17495-9 12.9 TI 19439-4 15-3 I9343-I ; 15-5 19245-4 15.6 12 21206.6 lS.2 21101.6 18.4 20995.0 18.6 13 22973.8 21.4 22860.1 | 21.6 22744.6 21.8 14 24741.0 2 4 .8 24618.5 25.1 23494-2 25-3 15 26508.2 28.5 26377.0 i 28.8 25243-8 29.1 16 28275.4 32.4 28135.5 j 32.7 26993-3 33-1 17 30042.6 36.6 29893-9 37-0 28742.9 37-4 iS 31809.9 41.0 31652.4 41.4 30492.5 41.8 19 33577-1 45-7 33410.9 46.2 32242.1 46.6 20 35344-3 5 1- -6 35169.3 51-2 3399!-7 51-7 2 5 44180.3 79.1 43961.6 ,79-9 43239-6 80.7 30 53016.4 II3-9 52753.9 115.1 52487-4 116.3 40 70688.4 202.5 70338.4 204.6 69983. i 206.6 50 88360.2 316.4 87922.8 3I9-7 87478.7 322.9 I 00 106032.0 455-6 105507.1 460.4 104974.1 464-9 I 20 I4I375.0 810.0 140675-1 818.4 139964.4 826.6 I 30 159046.2 1025.2 158258.7 1035-8 I 57459- 2 1046.1 I 40 176717.1 1265.7 175842.2 1278.8 174953.8 1291.5 2 00 212058.1 1822.6 211008. i 1841.5 209942. i 1859.8 2 30 265067. i 2847-8 263754.5 2877.3 262421.8 2905 . 9 3 oo 318072.5 4100.8 316497.1 4143.3 314897.6 4184.5 3 30 371073.4 5581.7 369235-2 5639-5 367368.9 5695-6 4 oo 424069.3 7290.3 421968.0 7365.9 419834.7 7439-1 GEODESY. Polyconic Projection Co-ordinates, 3 m , d p , in Yards. a> "O d Latitude 31 o . Latitude 31 30 . Latitude 32 o . So 6 m 6 P c 6 m 6 P 6 m ** i 1740.6 O. I I73I-4 O. I 1722. I O. I 2 348I.I 0.5 3462.8 0.5 3444-3 0.5 3 5221.7 I .2 5194.3 1.2 5166.4 1.2 4 6962.3 2.1 6925.7 2.1 6888.6 2. I 5 8702.9 3 3 8657.1 3 3 8610.7 3 3 6 10443.4 4-7 10388.5 4-7 10332.8 4.8 7 12184.0 6.4 12119.9 6.4 12055 .0 6.5 8 13924.6 8-3 13851-4 8.4 I3777-I 8-5 9 15665.1 10.6 15582.8 10.7 15499-3 10.8 10 17405.7 13.0 17314.2 13-2 17221.4 13-3 ii 19146.3 15-8 19045.6 15-9 18943-6 16.1 12 20886.8 18.8 20777.1 18.9 20665.7 19.1 13 22627.4 22. O 22508.5 22.2 22387.8 22.4 14 24368.0 25-6 24239.9 25.8 24110.0 26.0 15 26108.5 29-3 2597L3 29.6 25832.1 29.9 . 16 27849.1 33-4 27702.7 33-7 27554-3 34-0 1 17 29589.7 37-7 29434.2 38.0 29276.4 38.4 18 31330.3 42.2 31165.6 42.6 30998.5 43-0 19 33070.8 47-i 32897.0 47-5 32720.7 47-9 20 34811.4 52.2 34628.4 52.6 34442.8 53-1 25 43514.2 8i.f 43286.0 82.3 43053.5 83.0 30 52217.0 II7-3 51942.5 118.4 51664.1 II9-5 40 69622.5 208.6 69256.6 210. 5 68885.4 212.4 50 87027.9 326.0 86570.5 328.9 86106.5 331-8 I OO 104433.2 469-4 103884.3 473-7 103327.4 477-8 I 2O 139243.1 834-5 138511.2 842.1 137768.8 849-5 I 30 156647.8 1056.1 155824.4 1065.8 154989.1 1075.1 I 40 174052.2 1303-8 173137.2 1315-8 172209.1 I327-3 2 00 208860.0 I877-5 207762.0 1894.7 206648.2 1911.3 2 30 261069. i i 2933.7 259696.5 2960.5 258304.1 2986.5 3 oo 313274.2 I 4224.5 311626.9 4263.1 309955.S 4300.5 3 30 365474.6 5750.0 363552.4 5802.6 361602 ..5 5853.5 4 oo 417669.4 i 7510.2 415472.3 7578.9 4I3243-4 PROJFXTION OF MAPS. Poly conic Projection Co-ordinates, d m , d jt , in Yards. 0> T3 3 1 hJ Latitude 32 30 . Latitude 33 o . Latitude 33 30 . <J* 3* dm *, 6 m <** I 1712.7 O. I 1703.2 O.I 1693-5 O.I 2 34250 0.5 3406.4 0.5 3387-0 0.5 3 5138.2 I .2 5 i 09 . 6 1.2 5080.6 1.2 4 6850.9 2.1 6812.8 2.2 6774.1 2.2 5 8563-7 3-3 8515-9 3-4 8467.6 3-4 6 7 10276.4 11989. i 4.8 6.6 10219. i 11922.3 4-9 6.6 10161. i 18154.6 4-9 6.7 8 13701.8 8.6 13625.5 8.6 I3548.I 8-7 9 15414.6 10. S 15328.7 10.9 15241-7 II. 10 17127.3 13.4 17031.9 13-5 16935.2 13-6 ii 18840.0 16.2 18735-1 16.3 18628.7 16.4 12 20552.8 , 19.3 20438.3 19.4 20322.2 19.6 13 22265.5 22.6 22141.4 22. S 22015.7 23.0 14 15 23978.2 26.2 25690.9 30.1 23844.6 25547.8 26.4 30-4 23709-2 25402.7 26.6 30.6 16 27403.7 34-3 27251.0 34-5 27096.3 34-9 17 29116.4 38.7 28954.2 39-0 28789.8 39-3 18 30829.1 43.4 30657,4 43-7 30483-3 44-0 19 32541.9 i 48.5 32360.6 49.0 32176.8 49.1 20 34254.6 : 53-5 34063.8 54.0 33870.3 54-4 25 42818.2 83.6 42579.6 4-3 42337-9 85.0 30 51381.8 120.5 51095.5 121.4 50805.4 122.3 40 68508.9 ; 214.1 68127.2 215.9 67740.4 217-5 50 85635.9 334-6 85158.8 337-3 84675-2 339-9 I OO 102762.7 481.8 102190.2 485.7 101609.9 489.4 I 20 137015.8 856.6 136252.4 863.5 135478.6 870.1 I 30 154142.0 1084.1 153283.1 1092.8 152412.6 IIOI.2 I 40 171267.9 1338.4 170313.6 1349-2 169346.3 1359-5 2 OO 205518.7 1927.4 20*373.5 1942.8 203212.7 1957-7 2 30 256892.0 3011.5 255460.4 3035-6 254009.2 3058.8 3 oo 308261.1 ; 4336.6 306542.9 437L3 304801.4 4404 7 3 30 359625.1 5902. 6 357620.3 5949-9 355588.2 5995-3 4 oo 410983.2 7709-5 408691.6 7771.2 406368.8 7830.6 116 GEODESY. Poly conic Projection Co-ordinates, d in , d p , in Yards. <J T3 a Latitude 34 o . Latitude 34 30 . Latitude 35^ o . So j 6m > S m *, 6 m ** 1 I 1683.7 O. I 1673.8 O. I 1663.7 , o.i 2 3367.4 0.5 3347-6 0.6 3327.5 0.6 3 505LI 1.2 5021.4 I . 2 4991.2 1.2 4 6734.9 2.2 6695.1 2.2 6654.9 2.2 5 8418.6 3-4 8368.9 3-4 8318.7 3-5 6 IOI02.3 4-9 10042.7 5-0 9982.4 5-o 7 II786.0 6-7 11716.5 6.8 11646. i 6.8 8 13469.7 8.8 13390.3 S.8 13309.8 8.9 9 I5I53.4 II. I 15064.1 II. 2 14973-6 II .2 . 10 16837.2 13.7 16737.9 - 13-8 16637.3 13-9 ii 18520.9 16.6 18411.6 I6. 7 18301 .0 16.8 12 20204.6 19.7 20085.4 19.9 19964.8 20. o 13 21888.3 23.1 21759-2 23-3 21628.5 23-5 14 23572.0 26.8 23433-0 27.0 23292.2 27.2 15 25255-7 30.8 25106.8 31.0 24955-9 31.2 16 26939-4 35-1 26780.6 35-3 26619.7 35.5 17 28623.1 39-6 28454-4 39-8 28283.4 40.1 18 30306.9 44.4 30128.1 44-7 29947.1 45.o IQ 31990-6 49-4 31801.9 ! 49-8 31610.9 50.1 20 33674.3 54-3 33475-7 55-2 33274.6 55-5 25 42092.8 85.6 41844-6 86.2 4I593.2 86.7 30 50511.4 123.2 50213.5 125.1 49911-8 124.9 40 673^8.3 219.1 66951.1 220. 6 66548.9 222.1 50 84185.1 342.3 83688.7 344-7 83185.8 . 347-0 I 00 IOIO2I .8 493-0 100426.0 496.4 99822 . 6 499-7 I 20 134694.5 876.4 I33900.I ! 882.5 133095.5 888.3 I 30 151530.5 1 1109.2 150636.8 1116.9 I4973L5 1124.2 I 40 168366.1 ! 1369.4 I67373.I 1378.9 166367.3 1387-9 2 OO 202036.4 I97L9 200844.7 1985.6 199637.7 1998.6 2 30 252538.7 3081.1 251049.0 3102.5 249540. i 3122.8 3 oo 303036.6 4436.8 301248.7 4467.5 299437.8 4496.9 3 30 353529.0 6039.0 351442.8 6080.8 349329.8 6120.8 4 oo 404015.1 7887.7 401630.5 i 7942.3 399215.4 7994-5 PROJECTION OF MAPS. 117 Poly conic Projection Co-ordinates, d m , o 2> , in Yards, 3 4 5 6 7 8 9 10 ! I 12 13 4 15 16 17 18 J 9 2; s 25 5 40 50 oo 20 30 40 2 OO 2 30 3 oo 3 30 4 oo Latitude 35 30 . Latitude 36 o . Latitude 36 30 . 4. 4, 1 "<: 4, 1653-5 i o.i 1643.2 i o.i 1632.8 O.I 3307.1 0.6 3286.5 0.6 3265.6 0.6 4960.6 1.3 4929-7 i-3 4898.4 i-3 6614.2 ! 2.2 6572.9 2.2 6531-2 2-3 8267.7 3.5 8216.2 3.5 8164.0 3-5 9921.3 5.0 9859-4 5-i 9796.8 5-r 11574.8 6.8 II502.7 6.9 11429.6 6.9 13228.4 8.9 I3I45.9 9.0 13062.4 9.0 14881.9 11.3 14789.1 : ii. 4 14695.2 ii. 4 16535.5 14.0 16432.4 14.0 16328.0 14.1 18189.0 16.9 18075.6 i 17.0 17960.8 17.0 19842.5 20. i 19718.8 20.2 19593-6 20.3 21496.1 23-6 2I362.O 23.7 21226.4 23.9 23149.6 27.4 23005.3 1 27.5 22859.2 27.7 24803.2 31-4 24648.5 31,6 24492.0 31. S 26456.7 35-7 26291.8 36.0 26124.8 36.2 28110.3 40.4 27935-0 40.6 27757.6 40.8 29763.8 45-2 29578.2 45.5 29390.4 45.8 3I4I7.3 50.4 3T22I.5 50.7 31023.2 51.0 33070.9 55-9 32864.7 56.2 32656.0 i 56.5 41338.6 87.2 4IOSO.S 87.8 40819.9 88.3 i 49606 . 2 125.7 49296.9 126.4 48983.9 127.1 66141.5 223.4 65729.1 224.8 653H.7 226.0 82676.6 349-1 82l6l.I 351-2 81639.3 353-1 992II.5 502.7 98592.9 505.7 97966.7 508.5 132280.7 893-8 I3M55.9 899.1 130621.0 904.1 148814.9 1131.2 147886.9 II37-9 146947.7 1144.2 165348.8 1396.6 164317.7 1404.8 163274.0 1412.6 198415.4 2011. I 197178.0 2022.9 195925.6 2034.1 2480I2.I 3M2.3 246465.3 3160.8 244899.6 3178.3 297604.0 4524-9 295747.6 4551-6 293868.6 4576.8 347190.2 | 6158.9 345024.1 6195.2 342831.7 6229.5 396769.7 8044.3 394293.8 8091.6 391787.8 8136.5 Il8 GEODESY. Foly conic Projection Co-ordinates, <J M , d jt , in Yards. c. U Latitude 37 o . Latitude 37 30 . Latitude 38 o . c .. *, . * . * / I 1622.2 O. I 1611.6 O. I 1600.7 | o.i 2 3244.5 0.6 3223.1 0.6 3201.5 0.6 3 4866.7 i-3 4834.7 i-3 4802.2 i-3 4 6489.0 2-3 6446.2 2-3 6403.0 2-3 5 8111.2 3-5 8057.8 3.6 8003.7 3.6 6 9733-4 5.i 9669.3 5-i 9604.5 5-2 7 H355.7 7.0 11280.9 7-o 11205.2 7-0 8 12977-9 9.1 . 12892.4 9.1 12806.0 9-2 9 14600.2 ii. 5 14504-0 ii. 6 14406.7 ii. 6 10 16222.4 14.2 16115.6 I4o 16007.5 14-3 ii 17844.6 17.2 I7727.I 17.3 17608.2 17-3 12 19466.9 20.4 19338.6 20.5 19209.0 20.6 13 21089.1 24.0 20950.2 24-1 20809.7 24.2 14 22711.4 27.8 22561.8 28.0 22410.5 28.1 15 24333-6 31-9 24173.3 32.1 24011.2 32.3 16 25955-8 36.4 25784-9 36.5 25612.0 36.7 17 27578.1 41.0 27396.4 41.2 27212.7 41.4 18 29200.3 46.0 29008 . o 46.2 28813.4 46.4 19 30822.6 51-3 30619.5 51.5 30414.2 51.7 20 32444.8 56.8 32231. i 57-1 32015.0 57.3 25 40555.9 88.7 40288.8 89.1 40018.6 89.6 30 48667.1 127.8 48346.5 128.4 48022.3 129.0 40 64889.2 227.2 64461.9 228.3 64029.6 229.3 50 81111.3 355-0 80577.1 356.7 80036.7 358.3 I OO 97333-1 SIT- 2 96692.0 513.7 96043 - 6 516.0 I 20 129776.1 908.8 128921.3 913.2. 128056.7 917.4 I 30 145997-2 1150.2 145035.5 II55-8 144062.8 1161.0 I 40 162217.9 1419.9 161149.4 1426.9 160068.5 1433-4 2 OO 194658.2 2044.7 193375-9 2054-7 192078.9 2064. i 2 30 243315.2 3194.9 241712.2 3210.5 240090.8 3225.1 3 oo 291967. i 4600.7 290043.4 4623.1 288097.5 4644.1 3 30 340613.1 6262.0 338368.4 6292.6 336098.0 6321.2 4 oo 389251-9 8178.9 386086.3 8218.8 384091.2 8256.3 PROJECTION OF MAPS. 119 Polyconic Projection Co-ordinates, o,, ( , o,., in Yards. 4 Latitude 38 30 . Latitude 39 o . Latitude 39 30 , 3 c Oro S P rfm 6p <Jm dp / I 1589.8 i O.I 1578.8 O. I 1567.6 O.I 2 3179.6 0.6 3157.5 0.6 3135.2 0.6 3 4769-5 r -3 4736.3 i.3 4702.8 1-3 4 6359.3 2,3 63I5.I 2-3 6270.4 2-3 5 7949.1 3-6 7893.8 3.6 7838.0 3-6 6 9538.9 : 5.2 9472.6 5-2 9405.6 5.2 7 11128.7 7.1 H05I.4 7-i 10973.2 7-i 8 12718.6 9.2 12630. I . 9.2 12540.8 9-3 9. 14308.4 11.7 14208.9 ii. 7 14108.4 ii. 7 10 15898.2 14.4 I57S7.7 14-5 15676.0 14-5 ii 17488.0 17.4 17366.4 17-5 17243.6 17-5 12 19077.8 20.7 18945.2 20.8 iSSn.i 20.9 13 20667.6 24.3 20524.0 24-4 20378.7 24-5 14 22257.4 28.2 22IO2.7 28.3 21946.3 28.4 15 23847-3 32.4 23681.5 32.5 235TJ- 9 32.6 16 25437.1 36.8 25260.3 37.0 25081.5 37-1 17 27026.9 41.6 26839.0 41.8 26649.1 41.9 18 28616.7 46.6 28417.8 46.8 28216.7 47-0 19 30206.5 52.0 29996.6 52.2 29784.3 52.3 20 31796.3 57-6 31575.3 57-8 3I35L9 58.0 25 39745-4 90.0 39469.1 90.3 39189.8 90.6 30 47694.4 129.5 47362.9 130.1 47027.7 130.5 40 63592.4 230.3 63150.3 231.2 62703.4 232.0 50 79490.2 359-9 78937.6 361.3 78379.0 362.6 I OO 95387.8 518.2 94724.7 520.2 94054.4 522.1 I 20 127182.3 921.2 126298. I 924.8 125404-3 928.2 I 30 143079.0 1165.9 142084.4 1170.5 141078.8 1174.7 I 40 158975.5 1439-4 157870.3 I445-I 156753.0 1450.2 2 OO 190767.1 2072.8 189440.8 2080.9 188100.1 2088.3 2 30 238451.0 3238.7 236793.0 325L4 235116.9 3263.0 3 oo 286129.6 4663.8 284139.8 4682.0 282128.3 4698.8 3 30 333801.8 6347.9 331480.2 6372.7 329133-2 6395.6 4 oo 381466.7 1 8291.2 3788I3.I 8323.6 376130.5 8353.4 120 GEODESY. Polyconic Projection Co-ordinates, 3 IH) <5 JJ} in Yards. o T3 Latitude 40 o . Latitude 40 30 . Latitude 41 o . So 3 3,* ** d m *, 6m &P I 1556.3 ; O. I 1544-9 O.I 1533-4 O.I 2 3II2.6 0.6 3089.8 0.6 3066.7 0.6 3 4668 . 9 1-3 4634.7 i-3 4600. I 1-3 4 6225.2 2-3 6179.6 2 > 3 6133.5 2-3 5 7731.5 3-6 7724.5 3-6 7666.8 3-7 c 9337-8 5-2 9269.4 5-3 9200.2 5-3 7 10894.1 7-1 10814.3 7-2 10733.6 7-2 8 12450.4 9-3 12359-2 9-3 12266.9 9-4 9 14006.7 u. 8 13904.0 1 1. 8 13800.3 11. g 10 15563-0 14.6 15448.9 14.6 15333.7 14.6 ii 17119.3 17.6 16993.8 17-7 16867.0 17-7 12 18675.6 21 .O 18538.7 21 .O 18400.4 a 1. 1 13 20231.9 2 4 .6 20083.6 24-7 19933-7 24-7 14 21788.2- 28.5 21628.5 28.6 21467.1 28.7 15 23344-5 32.7 23173.4 32.8 23000.4 32.9 16 24900.8 37-2 24718.3 37-4 24533.8 37-5 17 26457.1 42.0 26263.2 42.2 26067.2 42-3 18 28013.4 47.1 27808.1 47-3 27600.5 47-4 19 29569.8 52.5 29352.9 52.7 29133-9 52.8 20 31126.1 58.2 30897.8 58.4 30667.3 58.5 25 38907-5 90.9 38622.2 91.2 38334.0 91.4 30 46689.0 130.9 46346.7 I3I.3 46000 . 8 I3L7 40 62251.8 232.8 6I795.3 233.5 61334.2 234.1 50 77814.4 363-7 77243.9 : 364-8 76667.4 365.8 I OO 933/6-9 523-8 92692.2 525-3 92000.4 526.7 i 20 124500.9 931.2 123588.0 ( 933-9 122665.7 936.4 i 30 140062.5 1178.5 I39035.5 1182.0 137997.8 1185.1 i 40 155623.7 1455-0 154482.6 1459.3 I53329-6 1463.1 2 OO 186744.9 2095.2 185375.4 2101.4 183991-8 2106.9 2 30 233422.9 3273.7 231710.9 3283.4 229998.3 3292.0 3 oo 280095.3 4714.1 278040.9 | 4728.1 275965-1 4740.6 3 30 326761.1 6416.5 324364.1 6435.4 321942.2 6452.4 4 oo 3734I9-3 8380.7 370679.4 8405.5 367911.3 8427-7 PROJECTION OF MAPS. 121 Poly conic Projection Co-ordinates, d m , S tt1 in Yards. 0) "O Latitude 41 30 . Latitude 42 o . Latitude 42 30 . Si) o J d* $v 6 m 6 P dm * I 1521.7 o.i I5IO.O O. I 1498.1 O. I 2 3043.4 0.6 3019.9 0.6 2996.2 0.6 3 4565-2 1-3 4529.9 i-3 4494.2 i-3 4 6086.9 2 -3 6039.8 2.4 5992.3 2.4 5 7608.6 3.7 7549-8 3-7 7490.4 3-7 6 9*30.3 5-3 9059.7 5-3 8988.5 5-3 7 10652.0 7.2 10569.7 7-2 10486.5 7-2 8 12173.8 9.4 12079.6 9-4 11984.6 9-4 9 13695.5 11.9 13589.6 n. 9 13482.7 n. 9 10 15217.2 14.7 15099.6 14.7 14980.8 14.7 ii 16738.9 17.7 i 6609 . 5 17.8 16478.8 17.8 I 2 IS260.6 21. I 18119.5 21 .2 17976.9 21.2 13 19782.3 24.8 19629.4 2 4 .8 19475-0 24-9 14 21304.0 28.7 21139.4 28.8 20973.1 28.8 15 22825.8 33.0 22649.3 33-1 22471.1 33-1 16 24347-5 37-5 24159-3 37-6 23969-2 37-6 17 25869.2 42.4 25669.2 42.5 25467-3 42.5 18 27390.9 47.5 27179.2 47-6 26965.4 47-7 19 28912.6 52.9 28689.1 53-0 28463.4 53-i 20 30434-3 58.6 30199.1 58.8 29961.5 58.9 25 38042.9 91.6 3774S.8 91.8 37451-3 92.0 30 45651.4 132.0 45 2 93.5 132.3 44942.2 132.5 40 60868.3 234.6 60397.8 235-1 59922.7 235-5 50 76085.1 366.6 75496-9 367-4 74903.0 368.0 I 00 91301.6 527.9 90595-9 529.0 89883.2 529-9 I 20 121733.9 933.6 120792.9 940.5 119842.6 942.1 I 30 136949.6 * 1187.9 135890.9 1190.3 134821.8 1192.3 i 40 152164.9 1466.5 150988.5 1469.5 149800.6 1472.0 2 00 182594.1 2III.8 181182.5 2116.1 179756.9 2119.7 2 30 228234.1 ; 3299.7 226469.4 3306.4 224687.4 3312.0 3 oo 273868.3 4751-6 271750.5 4761.2 269612.0 4769-3 3 30 319495.6 6467.5 317024.7 6480.5 3I4529.5 6491 .6 4 oo 365115.0 ! 8447.3 362290.8 8464.4 359438.9 8478.8 122 GEODESY. Polyconic Projection Co-ordinates, d m , d jt , in Yards. 3 So c 2 Latitude 43 o . Latitude 43 30 . Latitude 44 o . *, 6 m * *, j I 1486.1 O. I 1474.0 O. I 1461.8 O.I 2 2972.2 0.6 2948.0 0.6 2923-5 0.6 3 4458.3 1-3 4421.9 i.3 4385.3 1.3 4 5944.3 2.4 5895.9 2.4 5847.0 2.4 5 7430.4 3-7 7369-9 3-7 7308.8 3-7 6 8916.5 5-3 8843.9 5-3 8770.5 5-3 7 10402.6 7-2 10317.8 7-2 10232.3 7-2 8 11888.7 9-4 11791.8 9-5 11694.1 9-5 9 13374-8 11.9 13265.8 12.0 I3I55.8 12.0 10 14860.8 14.7 14739-8 I 4 .8 14617.6 I 4 .8 ii 16346.9 I7.S 16213.7 I 7 .S 16079.3 17.9 12 17833.0 2T.2 17687.7 21.2 I754LI 21.3 13 19319.1 24.9 19161 .7 24-9 19002.8 25.O 14 20805.2 28.9 20635.7 28.9 20464.6 28.9 15 22291.2 33-2 22109.6 33-2 21926.3 33-2 16 23777-3 37.7 23583.6 37.8 23388.1 37-8. 17 25263.4 42.6 25057-6 42.6 24849.9 42.7 18 26749-5 47-8 26531.6 47-8 26311.6 47-9 19 28235.6 53-2 28005.5 53-3 27773-4 53-3 20 29721.7 59-o 29479.5 59-o 29235.1 59-1 25 37102.0 92.1 36849-3 92.2 36543.8 92.3 30 44582.4 132.7 44219.2 132.8 43852.6 132.9 40 50 59443-0 74303.4 235-8 368.5 58958.7 73698.0 236. i 368.9 58469.9 73087.0 236.3 369-2 I OO 89163.6 530.7 88437.1 531-2 87703.9 53L7 I 20 I 30 118883.1 133742.4 943-4 1194.0 H79M-5 132652.7 944-4 H95.3 116936.9 I3I552.9 945-2 1196.3 I 40 148601.2 1474.1 I47390.5 1475-7 146168.4 1476.9 2 OO 178317.6 2122.7 176864.7 2125.0 175398.2 2126.7 2 30 222888.2 3316.7 221071.9 3320.3 219238.7 3323-0 3 oo 267452.8 4776.0 265273.1 4781.3 263073.1 4785.1 3 30 312010.3 6500.7 309467.1 6507.9 306900.3 6513-0 4 oo 356559.5 8490.7 353652.8 8500.1 350719- 8506-8 PROJECTION OF MAPS. I2 3 Poly conic Projection Co ordinates, d m , d lt , in Yards. o> TD 3 1 O J Latitude 44 30 . Latitude 45 o . Latitude 45 30 . 6 m *, d m *, <U 6* o / I 1449.4 O. I t437.0 O.I 1424.4 O. I 2 2898.9 0.6 2874.0 0.6 2848.9 0.6 3 4348.3 i-3 4311.0 , 1.3 4273.3 1-3 4 5797-7 2.4 5747-9 2 . 4 5697.7 2.4 5 6 7247.1 8696,6 3-7 5-3 7184.9 8621 .9 3-7 5-3 7122.2 8546.6 3-7 5-3 7 10146.0 7-2 10058.9 7-2 9971.0 7-2 8 11595.4 9.5 11495.9 9-5 II395.4 9-5 9 13044.8 12. O 12932.9 12.0 12819.9 12.0 10 14494-3 I 4 .8 14369.8 14-8 14244.3 I 4 .8 ii 15943-7 17.9 15806.8 17-9 15668.7 17-9 12 I7393-I 21.3 17243.8 21.3 17093.1 21-3 13 18842.5 25.O 18680.8 25.0 18517.6 25.0 14 20292.0 29.0 20117.7 29.0 19942.0 20. O 15 21741.4 33-2 21554-7 33-3 21366.4 33-2 16 23190.8 37-8 22991.7 37-8 22790.8 37-8 17 24640 . 2 42.7 24428.7 42-7 24215.3 42.7 18 26089.7 47-9 25865-7 47-9 25639.7 47-9 19 27539-1 53-3 27302.7 53-4 27064.1 53-3 20 28988.5 59.1 28739.6 59-1 28488.6 59-1 25 36235.6 92.3 35924.5 92.4 35610.6 92-3 30 43482.6 133-0 43109.3 i33-o 42732.7 i33.o 40 57976.6 236.4 57478.9 236.5 56976.8 236.4 50 I 00 I 20 I 30 72470.4 369-4 86964.0 531.9 115950.3 ! 945.7 130442.9 [ 1196.8 71848.3 86217.4 H4954-9 129323.0 369-5 532-0 945-8 1197.1 71220.6 85464-2 II3950.6 128193.2 369-4 532.0 945-7 1196.9 I 40 144935.2 1477.6 143690.8 1477.9 142435.5 1477-7 2 OO i739 l8 -3 2127.7 172425.0 2128.1 I709I8.5 2127.8 2 30 217388.7 3324.6 215522.0 3325.2 213638.8 3324.8 3 oo 260853.0 4787.4 258612.0 4788.3 256352.9 4787.7 3 30 304310.0 6516.2 301696.3 6517-3 299059.6 6516.5 4 oo 347758.4 8510.9 344771-2 8512.5 34r757.6 8511.4 124 GEODESY. Poly conic Projection Co-ordinates, d m , d p , in Yards. 6 Latitude 46 o . Latitude 46 30 . Latitude 47 o . fc/i c ,3 4. *, 4, | * 4. v I I4II.8 O. I 1399.0 O.I I386.I O. I 2 2823.5 0.6 2798.0 0.6 2772.2 0.6 3 4235.3 i-3 4197.0 1-3 4153.4 1.3 4 5647.1 2-4 5596.0 2.4 5544-5 2.4 5 7058.8 3-7 6995.0 3-7 6930.6 3-7 6 8470.6 5-3 8394.0 5-3 8316.7 5-3 7 9882.4 7-2 9793-0 7-2 9702.8 7- 2 8 II294. I 9-5 11192.0 9-4 IloSg.O 9-4 9 12705.9 12. O 12591.0 12. O I2475-I II. 9 10 I4H7.7 I 4 .8 13990.0 , I 4 .8 I386I.2 14.7 ii 15529.4 17.9 15389-0 17.9 15247.3 17.8 12 16941.2 21.3 16788.0 21.3 16633.4 21.2 13 18353.0 25.0 18186.9 24.9 18019.5 24.9 14 19764.7 28.9 19585-9 28.9 19405.7 28.9 15 2II76.5 33-2 20984.9 33-2 2O79I.8 33-2 16 22588.3 37.8 22383.9 37-8 22177.9 37-7 17 24000.0 42.7 23782.9. 42.7 23564.0 42.6 18 254II.8 47-9 25181.9 47-8 24950.1 47.8 J 9 26823.6 53-3 26580.9 53-3 26336.2 53-2 20 28235.3 59-1 27979.9 59-0 27722.4 59.0 25 35294.1 92-3 24974-8 92.2 34652.9 92.2 30 42352.9 132.9 41969.8 132.8 41583.4 132.7 40 56470.3 236.3 55959-5 236.2 55444-3 235.9 50 70587.5 369-3 69949-0 369.0 69305.1 368.6 I OO 84704.5 531.7 83938.2 531-3 83165.6 530.8 I 20 II2937-6 945-3 111915.9 944-6 110885.7 943-6 I 30 127053.6 1196.4 125904.2 II95-5 124747.2 II94-3 I 40 I4II69.2 1477-0 139892.1 1476.0 138604.3 1474.4 2 OO 169399.0 2126.9 167866.4 2125.4 166321.0 2123.2 2 30 2H739-3 3323.3 209823.5 3320.9 207891.7 3317.5 3 oo 254073.4 4785.6 251774.4 4782.1 249456.0 4777-2 3 30 296400.0 6513-8 293717.6 6509.0 291012.8 6502.7 4 oo 338717.8 8507.8 335652.1 8501.5 332560.6 8492.7 PROJECTION OF MAPS. I2 5 Poly conic Projection Co-ordinates, d m , 8 , in Yards. 03 -a _a Latitude 47 30 . Latitude 48 o . Latitude 48 30 . | 2 * &P 8m <* *, / I I373-I O.I 1360.0 o.i 1346.9 O. I 2 2746.3 0.6 2720.1 0.6 2693.7 0.6 3 4119-4 L3 4080.1 1.3 4040.6 1-3 4 5492.5 2.4 5440.2 2.4 5387.4 2.4 5 6865.7 3-7 6800.2 3.7 6734.3 3-7 6 8238.8 5-3 8160.3 5-3 8081. i 5-3 7 9612.0 7.2 9520.3 7.2 9428.0 7-2 8 10985.1 9.4 10880.4 9.4 10774.8 9-4 9 12353 2 ii. 9 12240.4 ii. g 12121.7 ii. 9 TO I373L4 .4-7 13600.5 ! 14.7 13468.5 14.7 II 15104.5 I 7 .8 14960.5 17.8 14815.4 17.8 12 16477.6 21.2 16320.5 I 21.2 16162.2 21. I 13 17850.8 24.9 17680.6 | 24.8 17509.1 24.8 14 19223.9 28.9 19040.6 28.S 18855.9 28.8 15 20597.0 33-1 20400.7 33.1 20202.8 33-0 16 2I970.I 37-7 21760.7 37-6 21549.6 37-6 17 23343-3 42.6 23120.7 42.5 22896.5 42.4 18 24716.4 47-7 24480.8 47-6 24243.3 47-5 T 9 26089.5 53-1 25840.8 53-1 25590.2 53-0 20 27462.7 58.9 27200.9 58.8 26937.0 58.7 25 34328.3 92.0 34001.0 91.9 33671.2 91.7 30 41193.9 132.5 40801.2 132.3 40405.4 132.0 40 54925.0 235-6 54401.4 235-2 53873.6 234.7 50 68655.8 368.1 68COI.4 | 367.5 6734L7 366.8 I 00 82386.5 530.1 8l6oi. I 529.2 80809. 5 528.2 I 20 109846.9 942.4 108799.7 940.8 107744.2 939- I 30 123576.6 1192.7 I223 9 8.5 1190.7 I2I2II.O 1188.4 I 40 137305.8 1472.4 135996.8 1470.0 134677.3 1467.1 2 OO 164762.8 2120.3 163191 .9 2116. 8 161608. 6 2112.7 2 30 205943-9 33I3.0 205980.3 3307.5 2O2OOI.O 330I.I 3 oo 247118.6 4770.7 244762.2 4762.9 242387.0 4753-5 3 30 288285.6 6493.5 285536.3 i 6482.8 282765.2 6470. i 4 oo 329443.7 | 8481.3 326301.5 8467.3 323134.3 8450-7 126 GEODESY. Poly conic Projection Co-ordinates, d m , 8 p ; in Yards. Td 3 Latitude 49 o . Latitude 49 30 . Latitude .50 o . C *. * d m dp 6m * I 1333.6 O. I 1320.2 O.I 1306.7 o.i 2 2667.1 0.6 2640.3 0.6 2213.3 .6 3 4000.7 i-3 2960.5 i-3 3920.0 1.3 4 5334-2 2-3 5280.6 2-3 5226.6 2-3 5 6667.8 3-7 6600.8 3-7 6533.3 3.6 6 8001.3 5-3 7920.9 5-3 7839-9 5-2 7 9334-9 7.2 9241.1 7-2 9146.6 7-1 8 10668.4 9-4 10561.2 9-3 10453.2 9.3 9 I2OO2.O II.Q 11881.4 ii. 8 11759.9 n. 8 10 13335-5 . T 4-6 13201.6 14.6 13066.5 14.6 ii 14669.1 17.7 I452I.7 17.7 14373.2 17-6 12 16002.7 21. I 15841.8 21. 15679-8 21.0 13 17336.2 24.7 17162.0 24.7 16986.5 24.6 14 18669.7 \ 28.7 18482.1 28.6 18293.1 28.5 T5 20003.3 32.9 19802.3 32-8 19599.8 32.8 16 21336.8 37-5 21122.4 37-4 20906 . 4 37- 3 17 22670.4 42.3 22442.6 42.2 22213.1 42.1 18 24004.0 47.4 23762.8 47-3 235I9.7 47.2 19 25337-5 52.8 25082.9 52.7 24826.4 52.6 20 26671.1 58.6 26403. i 58.4 26133.0 58.2 25 33338.8 91-5 33003.8 91.2 32666.2 91. c 30 40006.5 - 131.7 39604.5 I3L4 39199.4 131.0 40 53341-7 234.2 52805.7 233.6 52265.7 232. (] 50 66676.8 365.9 66006 . 8 365-0 6533 T -7 364-c I 00 80011.6 527.0 79207.6 525-6 78397.5 524-1 I 20 106680.4 936.8 105608.3 934-4 104528.2 931.7 I 30 120074.2 1185.7 118808.2 1182.6 II7593-0 H79-- I 40 133347.5 , 1463-8 132007.5 1460.0 130657.4 I455-? 2 CO 160012.8 2107.9 158404.8 2102.5 156784.7 2096.4 2 3O 200006.3 3293-6 197996.2 3285.1 195970.8 3275-f 3 oo 239993-2 4742.8 237581.0 4730.5 235150.6 4716. c 3 30 279972.3 6455.4 277158.0 6438.8 274322.4 6420.2 4 oo 319942.3 8431.6 316725.8 8409.9 313485.0 8 3 3 5 - I PROJECTION OF MAPS. 127 Arcs of Para He! Values of D,, in Yards. L. 20 30 . L. 21 . L. 21 30 . L. 22 O . ; L. 22 30 . L. 23 o 7 . / // 7 221.8 221. I 220. 3 219.6 218. 8 218.0 8 253.5 252.6 251.8 250. 9 250. o 249. i 9 285.2 284. 2 283.3 282.3 281.3 280.3 10 316.9 315.8 S 4-7 3I3.7 312.5 311. 4 2O 633.7 631.6 629.5 627.3 625.1 622.8 3 40 950.6 1267.4 947-4 1263. 2 944-2 1259.0 941.0 937.6 1254.6 1250.2 934.2 1245. 7 5 1584.3 I 579- 1 1573-7 1568.3 1562.7 60 1901. 1 1894.9 1888. 5 1881.9 1875.3 1868.5 7 oo 13307- 8 13264. i 13219.4 I3I73-7 13127.0 T 3079-4 8 oo 9 oo 10 OO 20 00 30 oo 40 oo 15208. 9 17110.0 19011. i 38022. i 57033.2 76044. 3 15159.0 1 7053- 8 18948. 7 37897. 4 56846. i 75794.8 15107.9 16996. 4 18884. 9 37769. 7 56654.6. 75539- 5 i555- 7 15002.3 16937.6 16877.6 18819. 6 ; 18752. 9 37639.2 ; 37505.8 56458. 8 36258. 8 75278.4 75011.7 M947- 9 16816.3 18684. 8 37369.6 56054. 4 74739- 3 50 oo 60 oo 95055.4 114066.4 94743- 4 113692. i 94424- 3 113309.2 94098. o 93764. 6 112917. 7 \ 112517.6 93424. i 112108. 9 Meridional Arcs Values of I) w /// Yards. i L. 21 . L. 22 O ; . L. 23 o . 8 235-4 269.0 235.4 269. I 235-5 269. i 9 302. 7 302. 7 302.8 10 336.3 336.4 336.4 20 672.6 672.7 672.8 30 1008. 9 1009. i 1009. 2 40 1345. 2 I345.4 r 345-6 lo - 1 681.6 2017.9 1 681.8 2018. i 1682.0 2018.4 14125.0 14126. 7 14128.5 8 oo 9 oo 16142. 9 18160.8 16144. 8 18162. 9 16146.8 18165.2 10 oo 20 oo 30 oo 40 oo 20178.6 40357.2 60535. 9 80714.5 20181. o 40362. i 60543. i 80724. i 20183. 5 40367. i 60550. 6 80734. i 50 oo 60 oo 100893. i 121071. 7 100905. 2 I2I086. 2 100917. 6 I2IIOI. 2 Intermediate minutes and seconds will be found by moving the decimal point. 128 GEODESY. Arcs of Parallel Values of D^, in Yards. L. 23 30 . L. 24 o . L. 24 30 . L. 25 o . L. 25 30 . L. 26 o . 217.2 216.4 215.4 214. 6 213.8 212. 9 g 248.2 247.3 246.3 245-3 244-3 243-3 9 279.2 278.2 277. i 276. o 274.8 273.7 10 309. i 307.9 306.6 305.4 304.1 20 620.5 618. i 613-3 610.8 608. 2 . 30 40 930.8 1241. o 927.6 1236. 3 1545-4 1854.4 923.6 1231-5 1539-3 1847. 2 919.9 1226.5 1533-2 1839.8 916.2 1221. 5 1526.9 1832.3 9 I2.3 I2I6.4 1520.5 1824. 7 7 oo 8 oo 9 oo 10 oo 20 00 30 oo 40 oo 50 oo 60 oo 13030.7 14892. 2 16753. 8 18615.3 37230. 6 55845.8 74461. i 93076. 4 111691. 7 12981. o 14835- 5 16689.9 18544-3 37088. 7 55633. o 74177.4 92721. 7 111266.0 12930.4 14777.6 16624. 8 18472.0 36944- o 55416.0 73887. 9 92359. 9 110831. 9 12878.8 14718.6 16558.4 18398.2 36796. 5 55194.7 73592. 9 91991.1 110389. 4 12826. I 14658. 5 16490. 8 18323. i 36646. i 54969. 2 73292. 3 91615-3 109938.4 12772.6 H597- 2 16421 9 18246. 5 36493. o 54739- 6 72986. i 91232. 6 109479. i Meridional Arcs Values of D m /// Yards. L. 24 o . I, 25 o . L. 26 o . 235-5 " 2 ! 5 5 235-6 8 269. i 269.2 269. 2 O 302.8 302.8 302.9 10 336.4 336.5 336.5 20 672.9 673.0 673- 1 3 1009.3 1009.4 1009. 6 40 50 1345. 7 1682. 2 1345-9 1682.4 1346. i 1 682. (5 60 2018.6 2018. 9 " 2019. 2 7 oo 8 oo 9 oo 14130.3 16148. 9 18167.5 14132. i 16151.0 18169.9 I4I34. I I6I53.2 18172.4 10 oo 20186. i 20188.8 20I9I.5 20 00 30 oo 40 oo 50 oo 60 oo 40372. 2 60558. 3 80744. 4 100930. 5 121116. 6 40377- 5 60566. 3 80755- i 100943. 9 121132.6 40383. o 60574. 6 80766. i 100957.6 121149. i Intermediate minutes and seconds will be found by moving the decimal-point. . PROJECTION OF MAPS. I2 9 Arcs of Parallel Values of T) p in Yards. L. 26 30 , L. 27 o . L. 27 30 . L. 28 o . L. 28 30 . L. 29 o , 7 212.0 211. O 2IO. I 205. i 208.2 207.2 8 242.2 241 .2 240. I 239-0 237.9 236.8 9 272.5 271.3 270.1 268.9 267.6 266.4 10 302.8 301.5 30O. I 298.8 297.4 296.0 20 605.6 603.0 6OO.3 597-6 594-8 591.9 30 908.4 904.5 900.4 896.3 892.2 887.9 40 I2II.2 1206.0 I2OO.6 1195.1 1189.5 1183.9 50 I5I4.0 1507.4 1500.7 1493-9 1486.9 T 479-9 Co 1816.9 1808.9 1800.9 1792.7 1784-3 1775-8 7 oo I27I8.0 12662.5 12606.0 12548.6 12490. i 12430.8 8 oo 14534-9 M47I.4 14406.9 14341.2 M274.5 14200.6 9 oo I635L7 16280.3 16207.7 16133-9 16058.8 15982.5 10 00 18168. 6 18089.3 18008.6 17926.5 17843-1 17758.3 20 00 36337.2 36178.5 36017.1 35S53-0 35686.2 35516.6 30 oo 54505.3 54267.8 54025.7 53779-5 5352Q.3 53274.9 40 oo 72674.4 72357.1 72034.3 71706. i 71372.4 71033.2 50 oo 90843.0 90446 . 3 90042 . 9 89632.6 89215.4 88791.5 60 oo 109011. 5 108535.6 108051.4 107559.1 107058.5 106549.8 Meridional Arcs Values of D.* in Yards. L. 27 o . L. 28 o . L. 29 o . 7 235.6 235-6 2 35-7 8 269.3 269.3 269.3 9 302.9 303-0 303 . o 10 336.6 336.6 336.7 20 673-1 673.2 673-3 30 ] 009 . 7 1009.9 IOIO.O 4 T346.3 T346.5 1346.7 50 1682.9 1683.1 1683.3 60 2019.4 2019.7 2O2O.O 7 oo 14136.0 14138.1 I4I40.I 8 oo 16155-5 16157.8 I6I60.2 9 oo 18174.9 18177-5 l8l80.2 10 oo 20194.3 20197.2 2O2OO. 2 20 oo 40388.7 40394.5 40400.4 30 oo 60583.0 60^91. 7 60600 . 6 40 oo 80777.4 80788.9 80800.8 50 oo 100971.7 100986.2 IOIOOI .O 60 oo 121166.0 121183.4 I2I2OI .2 Intermediate minutes and seconds will be found by moving the decimal-point. 9 130 GEODESY. Arcs of Parallel Values of D I} in Yards. L. 29 30 . L. 30 o . L. 30 30 . L. 31 o . L. 31 30 . L. 32 o . 7 206.2 205.2 204.1 203.1 202. O 200.9 8 235-6 234.5 233-3 232.1 230.9 229.6 9 265.1 263.8 262.4 201 . I 259.7 258.3 10 294.5 293.1 291.6 290. I 288.6 287.0 20 589.1 586.2 ^83.2 580.2 577-1 574-0 30 883.6 879.2 874.8 870.3 865.7 861.1 40 1178.1 1172.3 1166.4 TI60.4 II54-3 1148. i 50 1472.7 1465.4 1458.0 1450.5 1442.9 I435-I 60 1767.2 , 1750.5 1749.6 I74O.6 I73L4 1722.1 7 oo 12370.5 12309.3 12247.1 I2IS4.0 I2I20.0 12055.0 8 oo I4I37.7 14067.7 13996.7 13924.6 I385L4 I3777-I Q oo 15904.9 15826.2 15746.3 I5665.I 15582.8 15499-3 10 00 17672.2 1/584.7 17495.9 17405.7 I/3I4.2 17221.4 20 oo 35344-3 35169.4 3499^-7 348II.4 34628.4 34442.9 30 oo 53016.5 52754.0 52487-6 522I7.I 51942.7 i 51664.3 40 oo 70688.7 70338.7 69983.4 69622.8 69256.9 68885.7 50 oo 88360.8 87923.4 S7479-3 87028.5 86571.1 86107. i 60 oo 106033.0 10550*. i 104975.2 104434.2 103885.3 103328.6 Meridional Arcs Values of D m in Yards. L. 30 o . L. 31 o . L. 32 o . 7 235-7 235-7 235-8 8 269.4 269.4 209.5 9 303 . o 303-1 303-1 10 336.7 336.8 "36.8 20 673-4 673.5 673.6 30 IOI0.2 1010.3 1010. 5 40 1346.9 I347-I 1347-3 50 1683.6 1683.9 1684.1 60 2020.3 2O2O.6 2020.9 7 oo I4U2.3 14144.4 14146.6 8 oo I6I62.6 16165. ! 16167.6 9 oo 18182.9 18185.7 18188.5 10 oo 2O2O3.2 2O2O6.3 20209.5 20 00 40406 . 5 40412.6 40418.9 30 oo 60609.7 60619.0 60628.4 40 oo 80812.9 80825.3 80837.9 50 00 101016.1 IOIO3I.6 101047.4 60 oo 121219.4 121237.9 I I2I256.8 Intermediate minutes and seconds will be found by moving the decimal-point. PROJECTION OF MAPS. 131 Arcs of Parallel Values of D 2> in Yards. L. 3 2. 30 . L. 33 o . L. 33 30 . L. 34 o . L. 34 30 . L. 35 o . 7 199.8 198.7 197.6 196.4 195-3 194.1 8 228.4 227.1 225.8 224.5 223.2 221.8 9 256.9 254-5 254.0 252.6 251.1 249.6 10 285.5 283.9 282.3 280.6 279.0 277-3 20 570-9 567.7 564.5 561 .2 557-9 554-k 30 856.4 851.6 846.8 841.9 836.9 831.9 40 II4I.8 IT35-5 1129.0 II22.5 "15.9 1109.2 50 1427.3 I4I9.3 I4H.3 I403.I 1394.8 1386,4 60 1712.7 1703.2 1693-5 1683.7 1673-8 1663.7 7 oo 11989.1 11922.3 11854.6 II7S6.0 11716.5 11646.1 8 oo 13701.9 13625. .5 I3548.I 13469.7 13390.3 13309.8 9 oo I54I4.6 15328.7 15241.7 I5I53.5 15064.1 14973.6 10 oo 17127.3 17031.9 16935-2 16837.2 16737.9 16637.3 20 00 34254.6 34063.8 33870.4 33674.3 33475-8 33274.6 30 oo 51381.9 51095-7 50805.5 505H.5 50213.6 49911.9 40 oo 68509.3 68127.6 67740.7 67348.7 66951.5 66549.2 50 oo 85636.6 85I59-5 84675.9 84185.8 83689.4 83186.5 60 oo 102763.9 102191.4 101611. i IOIO23.O 100427.3 99823-8 Meridional Arcs Values of D m /// Yards. L. 33 o . L. 34 o . L. 35 o . 7 235-8 235 - 9 235-9 8 269.5 269.5 269.6 9 303-2 303-2 303 3 10 336.9 336.9 337-0 20 673-8 673-9 6/4-0 30 IOIO.6 1010.8 ion .0 40 1347-5 1347.7 1347-9 50 1684.4 1684.7 1684.9 60 2021.3 2021.6 202 i . 9 7 oo 14148.9 I4I5I.2 I4I53.5 8 oo 16170. i 16172.7 16175. a g oo 18191.4 18194.3 18197-3 10 00 20212.7 20215.9 20219.2 20 oo 40425.4 40431.9 40438.5 30 oo 60638.0 60647.8 60657.7 40 oo 80850.7 80863.7 80877.0 50 oo 60 oo 101063.4 121276.1 101079.7 121295.6 101096.2 121315.4 Intermediate minutes and seconds will be found by moving the decimal-point. 132 GEODESY. Arcs of Parallel Values of D , in Yards. L. 35 30 . L. 36 o . L. 36 30 . L. 37 o . L. 37 30. L. 38 o . / ;i | 7 192.9 191.7 190.5 189.3 188.0 186.8 8 220.5 219.1 217.7 216.3 214.9 213.4 9 248.0 246.5 244.9 243.3 241.7 240.1 10 275.6 273-9 272.1 270.4 268.6 266.8 20 551.2 547-7 544.3 540.7 537-2 533-6 30 826.8 821.6 816.4 i sii.i 805.8 800.4 40 1102.4 1095.5 1088.5 1081.5 1074.4 1067.2 50 1378.0 1369.4 1360.7 1351-9 1343-0 1334.0 60 1653.5 1643.2 1632.8 i 1622.2 1611.6 1600.7 7 oo 11574.8 11502.7 11429.6 ; II355-7 11280.9 11205.2 8 oo 13228.4 I3I45.9 13062.4 12977.9 12892.5 12806.0 9 oo 14881.9 14789.1 14695.2 14600.2 14504.0 14406.7 10 oo 16535.5 16432.4 16328.0 16222.4 16115.6 16007.5 20 oo 33070.9 32864.7 32656.0 : 32444.8 32231.1 32015.0 30 oo 49606.4 49297.1 48984.0 48667.2 48346.7 48022.5 40 oo 66141.9 65729-5 65312.1 64889.6 64462.3 64030.0 50 oo 82677.3 82161.8 81640.1 , 81112.0 80577.8 80037.5 60 oo 99212.8 98594.2 97968.1 97334-4 96693.4 96045.0 1 Meridional Arcs Values of D Jrt in Yards. L. 36 o . L. 37 o . L. 38 o . 7 235-9 236.0 236.0 8 269.6 269.7 269.7 9 303.3 303-4 303-4 10 337-0 337-1 337-2 20 674.1 674.2 674-3 30 IOII. I ion .3 1011.5 40 1348.2 1348.4 1348.6 50 1685.2 1685.5 1685.8 60 7 oo 8 oo 2022^1 14155^^ 16178.1 2022.6 I4I5S.2 16180. 8 2022.9 14160.6 16183.5 9 oo 18200.3 18203.4 18206.5 10 00 2O222.6 20226.0 20229.4 20 oo 40445.2 40451.9 40458.8 30 oo 60667.5 60677.9 60688.2 40 oo 80890.3 80903.9 80917.6 50 oo IOIII2.9 101129.9 101147.0 60 oo I2I335-5 I2I355.8 121376.4 Intermediate minutes and seconds will be found by moving the decimal-point. PROJECTION OF MAPS. 133 Arcs of Parallel Values of D 7 , in Yards. L. 38 30 . L.39 o . L.3Q C 30 . L. 40" o . L.40 30 . L.4io . i a 7 185-5 184.2 182.9 181.6 180.2 | 178.9 8 212. O 210.5 i 209.0 207.5 206.0 i 204.4 9 238.5 236.8 235.1 233-4 231.7 230.0 10 265.0 263.1 261.3 259-4 257.5 255.6 20 | 529.9 526.3 522.5 518.8 515-0 511-1 30 ; 794.9 789.4 783-8 77S.2 772.4 766.7 40 1059.9 1052.5 1045.1 1037.5 1029.9 IO22. 2 50 i 1324.9 1315-6 1306.3 1296.9 1287.4 1277.8 60 1589.8 1578.8 1567.6 1556.3 1544.9 1533-4 7 oo 11128.7 11051.4 10973.2 ! 10894.1 10814.3 10733.6 S oo ; 12718.6 12630.2 12540.8 j 12450.4 12359.2 12266.9 9 oo 14308.4 14208.9 14108.4 14006.8 13904.1 13800.3 10 oo 15898.2 15787-7- 15676.0 15563-1 15448.9 15333-4 20 oo 31796.4 31575-4 ! 31351.9 i 31126.1 30897.9 30667.3 30 oo 47694.6 47363.1 47027.9 46689.2 46346.8 46OOI.O 40 oo j 63592.8 63150.8 62703.9 ! 62252.2 61795-8 61334.6 50 oo 79491.0 78938.4 783/9-9 : < 77815-3 77244.7 76668.3 60 oo 95389.2 94726.1 94055.8 93378.3 92693.7 92OO1.9 Meridional Arcs Values of D m in Yards. L. 39 o . L. 40 o . L. 41 o . 1 236.0 236:1 236.1 8 269.8 269.8 269.9 9 303-5 303-5 303.6 10 337-2 337-3 337-3 20 674.4 674.5 674.7 30 ion .6 ion .8 IOI2.0 40 1348-9 I349-I 1349-3 50 1686.1 1686.4 1686.7 60 2023.3 2023.6 2024.0 7 oo 14163.0 14165.4 14167.9 8 oo 16186.3 16189.1 16191.9 9 oo 18209.6 18212.7 I82I5.8 10 00 20232.8 20236.3 20239.8 20 oo 40465 . 7 40472 . 7 40479.7 30 oo 60698.5 60709 . o 60719.5 40 oo 80931.4 80945-3 80959.3 50 oo 101164.2 101181.6 IOI199.2 60 oo 121397-1 121418.0 I2I439.0 Intermediate minutes and seconds will be found by moving the decimal-point. J 34 GEODESY. Arcs of Parallel Values of D p in Yards. L.4I 30 . L. 42 o . L.42 30 . L. 43= o . L-43 30 . L. 44 o . 7 177-5 176.2 174-8, 173-4 172.0 170.5 8 9 202.9 228.3 201.3 226.5 199.7 224.7 198.1 196.5 222.9 221. I 194.9 219.3 10 253-6 251-7 249.7 247-7 j 245.7 243-6 20 507.2 503.3 449-4 495-4 49L3 487-3 30 760.9 755-0 749.0 743.0 i 737.0 730.9 40 50 1014.5 I268.I 1006. 6 1258.3 998.7 1248.4 990.7 982.7 1238.4 1228.3 974-5 1218.1 60 1521.7 1510.0 1498.1 1486.1 1474.0 1461.8 7 oo 10652.0 10569.7 10486.6 10402.6 10317.9 10232.3 8 oo 9 oo I2I73-3 13695.5 12079.7 13589-6 11984.6 13482.7- 11888.7 13374-8 11791.8 13265.8 11694. i I3I55.S 10 00 I52I7.2 15099.6 14980.8 14860.9 14739-8 14617.6 20 oo 30 oo 40 oo 30434.4 45651.6 60868.8 30199.1 45298.7 60398.3 29961.6 44942.4 59923.2 29721.7 44582.6 59443-4 29479.6 44219.4 58959-2 29235.2 43852.8 58470.4 50 oo 60 oo 76086.0 91303.2 75497-9 90597.4 74903.9 89884.7 74304.3 89165.1 73698.9 88438.7 73088.0 87705.6 Meridional Arcs Values of D w in Yards. L. 42 o . L. 43 o . L. 44 o . 7 236.2 236.2 236.3 8 269.9 270.0 270.0 9 303.7 303.7 303.8 10 337-4 337.4 337-5 20 674.8 674.9 675.0 30 IOI2.2 1012.3 1012.5 40 1349.6 1349.8 1350.0 50 1686.9 1687.2 1687.5 60 2024.3 2024.7 2025.0 . 7 oo I4I70.3 14172.8 I4I75.3 8 oo 16194.7 16197.5 16200.3 9 oo 18219.0 18222.2 18225.4 10 oo 20243.4 20246.9 20250.4 20 00 40486.7 40493 8 40500.9 30 oo 60730.1 60740.7 60751.3 40 oo 80973.4 80987.5 81001.7 50 oo IOI2I6.8 101234.4 101252.2 60 oo I2I460.I 1 121481.3 121502.6 Intermediate minutes and seconds will be found by moving the decimal-point. PROJECTION OF MAPS. Arcs of Parallel Values of D p in Yards. L.44 30 . L.45o . L. 4 530 . L. 46 o . L. 4 6 30 . L. 47 o . 7 i6g.i 167.6 166.2 164.7 163.2 161 .7 8 IQ3-3 191.6 189.9 188.2 186.5 184.8 9 217.4 2i5o 213.7 211. 8 i 209.8 207.9 10 241.6 239-5 237-4 235.3 233.2 331-0 20 483.1 479.0 474-8 470.6 j 466.3 . 462.0 30 724.7 7i3.5 712.2 705.9 ! 6 99-5 693-1 40 966.3 958.0 949-6 941.2 ! 932.7 1 924.1 50 1207.9 H97.5 1187.0 1176.5 I 1165.8 "55-1 Go 1449.4 1437-0 1424-4 1411.8 1399- 1386.1 7 oo 10146.0 10058.9 9971-0 gS32.4 | 9793-0 9702.8 8 oo 1 1 595 - 4 11495.9 II395-5 1129.1.2 11192.0 11089. c g oo 13044.8 12932.9 12819.9 12705.9 12591.0 12475.1 10 00 14494.3 14369.8 14244.3 14117.7 13990-0 13861 .2 20 oo 28988.6 28739.7 28488.6 28235.4 27980.0 27/22.4 30 oo 40 oo 50 oo 60 oo 43482.8 57977-1 72471.4 86965.7 43109.5 57479-4 71849.2 86219.1 42733.0 56977.3 71221.6 85465.9 42353.I 56470.8 70588.5 84706.2 4ig7O.o 55g6o.o 69950.0 83940.0 41583. C 55444-5 69306.0 83167.: Meridional Arcs Values of D m in Yards. \ L. 45 o . L. 46" o . L. 47 o . " 236.3 236.3 236.4 8 270. 1 270. i 270.1 9 303.8 303.9 303-9 10 337-6 337-6 337-7 20 675.T 675o 675.4 30 1012.7 1012.9 1013.1 40 1 50 1350.3 1687.8 1350.5 1688.1 1350.7 1688.4 60 2025.4 2025.8 2026.1 7 oo 14177.8 14180.3 14182.8 8 oo 16203.2 , 16206.0 16208. g g oo 18228.6 18231.8 18235.0 10 00 20254.0 20257.5 20261 . i | 20 oo 40508.0 40515.1 40522.2 30 oo 60761.9 60772.6 60783.2 40 oo 81015.9 81030.1 81044.3 50 oo 101269.9 101287.7 101305.4 60 oo 121523.9 121545.2 121566.5 Intermediate minutes and seconds will be found by moving the decimal point. I3 6 GEODESY. Arcs of Parallel Values of D p in Yards. L.47 30 . L. 48 o . L.48 30 . : L. 49 o . 1 L. 49 30 . L. 50 o . | 7 160.2 158.7 157-1 , 155-6 154-0 152.4 8 183.1 181.3 179-6 i 177.8 176.0 174.2 9 206.0 204.0 202. o 20O.O igS.O 196.0 10 228.9 226.7 224.5 222.3 220.0 217.8 20 457-7 453-3 449-0 444-5 ! 440.1 435-6 30 686.6 6So.o 673.4 666.8 660. i 653.3 40 9 J 5-4 906.7 897.9 889.0 880.1 871.1 50 H44.3 "33-4 1122.4 mi. 3 i noo.i 1088.9 60 I373-I 1360.0 1346.9 1333-6 1320.2 1306.7 1 7 GO 8 oo 9612.0 10985.1 9520.3 9428.0 10880.4 i 10774.8 9334-9 10668.5 9241.1 10561.2 9146.6 10453.2 9 oo 12358.2 12240.4 12121.7 I2OO2.O 11881.4 H759-9 IO OO I373I.4 13600.5 13468.5 13335.6 13201.6 13066.5 20 oo 27462.7 27200.9 26937.1 26671. I 26403.1 26133. r 30 oo 41194.1 40801 .4 40405.6 40006 . 7 39604.7 39199.6 40 oo 54925.5 54401.9 53874.1 53342.3 52806.2 52266.2 50 oo 60 oo 68656.8 82388.2 68002 . 4 81602.8 67342.7 808 1 1 . 2 66677.8 80013.4 66007.8 79209.4 65332.7 78399.3 Meridional Arcs Values of D CT in Yards. L. 48 o . L. 49 o . L. 50 o . 7 236.4 236-5 236.5 8 2-0.2 270.2 270.3 g 304.0 304.0 304.1 10 337-7 337-8 337-9 20 675.5 675.6 675.7 30 IOI3.2 1013.4 1013.6 40 I35I.O I35L2 I35L4 50 1688.7 1689.0 1689.3 60 2026.5 2026.8 2027.2 7 oo I4I85.2 14187.7 14190.2 8 oo I62II.7 16214.5 16217.3 9 oo 18238.2 18241.3 18244-5 10 00 20264.6 20268.1 20271.7 20 oo 40529.2 40536-3 40543.3 30 oo 60793.9 60804.4 60815.0 40 oo 81058.5 81072.6 81086.6 50 oo IOI323.I 101340.7 101358.9 60 oo 121587.7 121608.9 121629 9 Intermediate minutes and seconds will be found by moving the decimal-point. PROJECTION OF MAPS. 137 Lengths in Nautical Miles and Statute Miles of Degrees of Latitude and Longitude in Different Latitudes. DEGREE OF THE PARALLEL. DEGREE OE THE MERIDIAN. Latitude of Nautical Statute i Latitude of Nautical Statute Parallel. miles. miles. middlepoint miles. miles. 20 56.404 65.018 20 59-664 68.777 21 56.039 64-598 22 55.657 64-158 23 55.258 63.698 24 54-843 63-219 25 54-411 62.721 25 59-706 68.825 26 53.962 62.204 27 53-497 61.668 2S 53-016 61.113 29 52.518 60.540 30 52.005 59.948 30 59-749 68.875 31 5L476 59.338 32 50.931 58-709 33 50.370 58.063 34 49-794 57-399 35 49-203 56.718 35 59.796 68.929 36 48.597 56.019 37 47-976 55.304 33 47-341 54-571 39 46.690 53-822 . 40 46.026 53-056 40 59-847 68.987 41 45.348 52.274 42 44.654 5L476 43 43-949 50.662 44 43.230 49-833 | 45 42.497 48.988 45 59.899 69.048 46 41.752 48.128 47 40.993 47-254 48 40.222 46.365 49 39-439 45-462 50 38.643 44-545 50 59-951 69.108 A degree of longitude at the equator = 69.163 statute miles. A second of time at the equator = 1521.6 feet. 138 GEODESY. Co-ordinates of Curvature in Statute Miles for Maps of Large Extent. i Latitude 20. Latitude 22. Latitude 24. Latitude 26. fc/3 e d n dp *, 1 4 d m d p d m d p 1-1 1 2 130.0 0.8 I2S.3 0.8 126.4 0.9 124.4 - 0.9 4 260.0 3-i 256.6 3-3 252.8 3-6 248.8 3-8 . 6 390.0 6.9 384.8 7-5 379-2 8.1 373-1 8.6 8 520.0 12.4 513.0 13-4 505.5 14.4 497-3 15-2 10 649.8 19.4 64I.I 21.0 631-. 7 22.4 621.4 23.8 12 779-7 27.8 769.1 30.2 757-9 32.2 745-4 34-2 14 909.2 38.0 896.9 41.0 883.6 43-9 869.2 46.6 16 1039.2 49-6 1024.5 53-6 1009.9 57-4 992.8 60.8 18 1168.1 62.8 II52.2 67-9 1134-8 72.6 1116. i 77.0 20 1298.0 77.6 1279-5 83.8 I20I.2 89.7 1239.2 95-0 r 10892 9813 8905 8130 Longitude. Latitude 28. Latitude 30. Latitude 32. Latitude 34. d m 4 d m 4 d m dp d m d P 2 122.2 I.O 119.8 i .0 117.4 i.i 114.8 i.i 4 244 . 4 4.0 239-7 4.2 234.8 4-3 229.5 4-5 6 366.5 9.0 359-5 9-4 352.0 9.8 344-2 IO. I 8 488.6 16.0 479.2 16.7 469-3 17-3 458.7 17.9 10 610.4 25.0 598.7 26.1 586.3 27.1 573-1 28.0 12 732.4 36.0 718.0 37-6 703.5 39- 1 687.2 40.3 14 853.7 49.0 837-1 51-2 819.6 53-1 801.1 54-8 16 975-7 64.1 956.0 66.9 936.8 69-5 9M-7 71.6 18 1096.0 80.9 1074.6 84.6 1051.9 87.8 1027.9. 9-5 20 1218. 8 IOO. I 1192.9 104.3 1169.2 108.6 1140.7 in. 7 r 7458 6869 6348 5881 PROJECTION OF MAPS. Co-ordinates of Curvature in Statute Miles for Maps of Large Extent. Longitude. Latitude 36. Latitude 38. Latitude 40. Latitude 42. d m d p d m 4 d m d p d m 4 2 112. O 1.2 109.1 1.2 1 06. 1 1.2 102.9 1.2 4 224.0 4.6 218.2 4-7 212.2 4.8 205.8 4.8 6 335-9 10.3 327.2 10.6 3I8.I 10.7 308.6 10,8 8 447-7 18.4 436.0 18.8 423.9 18.9 4II.2 19.2 10 559-2 28.7 544-7 29-3 529.4 29-7 5I3.6 30.0 12 670.5 41-3 653.0 42.2 634.7 42.8 615.7 43-2 14 781.6 56.2 761. i 57-4 739-7 53.2 717.5 58.8 16 892.3 73-4 868.8 74-9 844.3 76.0 818.8 76.7 18 1002 . 6 92.8 976.2 94-7 948.5 96.1 919.8 97-0 20 1112.5 II4-5 1083.0 niG.S 1052.3 118.5 1020.2 119.7 r 5461 5079 4729 4408 Longitude. Latitude 44. Latitude 46. Latitude 48. Latitude 50. .</ d p d m 4 , j * d 2 99-7 1.2 96.2 1.2 92.7 1.2 89.1 1.2 4 6 198.9 298.7 4.8 10.9 192.4 288.5 4.8 10.9 185.4 ! 4-8 277.9 IO -8 178.1 267.0 4.8 10.7 8 398.0 19-3 384-4 19.3 370.3 19-2 355-7 19.0 . 10 497-1 30.2 480.0 30.2 462.3 ! 30.0 444-1 29-7 12 595-9 43-4 575-4 43-4 554-1 j 43-2 532-3 42.8 14 694-3 59-1 670.3 59-1 645.6 58.8 620.0 58.2 16 18 792.3 889.9 77-1 97-5 764.9 859.0 77-1 97-5 736.5 76.7 827.0 97.0 707.3 794.1 75-9 96.0 20 986.9 1 2O. 2 952.5 I2O.2 916.9 119.6 880.3 118.4 r 4110 3833 3575 3332 140 GEODESY. X L I V . Trigonometrical Leveling . Reciprocal zenith-distances measured at two stations at the same time give the best results. When reciprocal, but not simul taneous, the observations should be made on different days, in order to obtain as far as possible an average value of the refrac tion as well as the mean value of the difference between the respective angles; the same with zenith-distances measured at one station only. The refraction being greater and more variable at sunrise and sunset, and comparatively stationary between the hours of 10 a. m. and 4 p. m., the best time for observation is between those hours. The condition of the atmosphere and the relative refraction may be so different at stations more than twenty miles apart, that, as a general rule, the difference of level, determined even by reciprocal observations, cannot be relied upon for much accuracy at greater distances unless a very large number of measurements have been made under the most favorable circumstances. The higher the elevations the more reliable the results. i. To obtain the Difference of Level of Two Points from Reciprocal Zenith- Distances, simultaneous or not. Let Z, Z = the measured zenith-distances of the telescopes at the two stations, of which Z is the smaller ; K = distance in yards between the tw6 stations ; R^= radius of curvature of the arc joining the two stations ; C = angle at the earth s center subtended by the arc ; and dh = difference of level of the two stations ; then K ji K sin fr (Z - Z) = K7*n~7" ; d ~ = cos J (Z " - Z + C) TRIGONOMETRICAL MEASUREMENT OF HEIGHTS. 141 XLIV. Trigonometrical Leveling Continued. If the telescope is not observed upon, but some other object near, the measured zenith-distance can be reduced to the tele scope by the formula : Reduction to telescope, in seconds, = i - K sin i" in which r represents the distance of the object (in yards and decimals) above or below the telescope. Logarithmic values of R 2 , in yards, which depend on the inclina tion to the meridian of the arc joining the two stations and their mean latitude, are given in the following table : I i> - Lat. 25. Lat. 30. Lat. 35. Lat. 40". Lat. 45. Lat. 50. ^ 1 ~" logR, "** log R r log R, logR* log R.Z o 6.841384 6.841695 6.842039 6.842406 6.842785 6.843164 10 M56 1760 2098 2458 2829 3200 20 1663 1950 2267 2606 2955 3304 30 1981 2240 2527 2833 3148 3464 40 2370 2596 2845 3HI 3386 3631 50 2785 2975 3184 3408 3639 3870 bo 3176 3331 3504 3687 3877 4066 70 3494 3622 3764 3915 4069 4227 80 3702 3812 3934 4063 4197 4331 90 6.843774 6.843878 6.843993 6.844115 6.844241 6.844368 log sin i" = 4.685575 2. By the Zenith- Distance measured at one Station. Let Z = the measured zenith-distance of the signal or object; K = the distance between the two stations in yards; m = the co-efficient of refraction = 0.071 ; and dh = difference in height between the two stations ; then c = K sn dh K cos ( Z + m C ~ i_ sin (Z + m~C C} I4 2 GEODESY. XLI V. Trigonometrical Leveling Continued. 3- By the Observed Zenith-Distance of the Sea-Horizon. Let Z = the measured zenith-distance ; R ^ = the radius of curvature of the arc ; and m the co-efficient of refraction = 0.078 ; then = - -,, tan 2 (Z - 90) 21 m\ l h 4- By Observed Angles of Elevation or Depression. Let A = the observed angle expressed in seconds ; and K == the distance in yards between the two stations. dJi =. 0.00000485 K A i 0.000000667 K- 2 [log 4.68574] [log 2.8241^] This gives the difference in heights between stations not more than ten or fifteen miles apart, with a probable error less than the uncertainty in the co-efficient of refraction. Co-efficient of Refraction. The co-efficient of refraction, or proportion of the intercepted arc, is determined from the observed zenith-distances of two sta tions, the relative altitudes of which have been determined by the spirit-level ; or from reciprocal zenith-distances, simultaneous or not, under the assumption that the mean of a number of ob servations taken under favorable conditions will eliminate the difference of refraction which is found to exist, even at the same moment at two stations a few miles apart. The longer the distance the greater is the error caused by any uncertainty in the co-efficient, or in the actual refraction; conse quently there is a limit to the distance for which any assumed mean value of the refraction can be depended on for accurate results. TRIGONOMETRICAL MEASUREMENT OF HEIGHTS. 143 X L I V. Trigonometrical Leveling Continued. The average value of the co-efficient from the Coast Survey observations in the New England States Between primary stations = 0.071 Of small elevations. ..... =0.075 Of the sea-horizon ...... =0.078 In the trigonometrical survey of Massachusetts Mr. Borden used 0.0784 as a mean co-efficient for the sea-co ast, and 0.0697 for the interior of the State. 1. To determine the Co-efficient of Refraction from Reciprocal Zenith- Distances. Let- C = angle at earth s center subtended by arc ; F = angle of refraction ; and m = co-efficient of refraction ; then C = =-^-- /7; F= C _(Z + Z- 180); m = * R, sini" 2 2 v C 2. To determine the Co -efficient from the Zenith-Distance observed at one Station, when the Altitude of the Two Stations above Tide, or their D iff ere tic e in Height, have been determined by the Leve ling- Instrument. Compute the true zenith-distances, Z ( / and Z , of the two given points, and the difference between the true and the observed zenith-distances will be the angle of refraction F. J (Z + Z ) = 90 * (7 7 ) - i- 1 / Z Z = F; m= ~c h and h 1 having been determined by the leveling-instrument. 144 GEODESY. Difference between the Apparent and the 7 rue Level. Correction for curvature . . 2R Correction for refraction = I - m Correction for curvature and refraction = (i 2 w) 2 R where D= the distance; R = mean radius of the earth ; and . ;;/ = co-efficient of refraction ; or m being = 0.075, an <* lo g R in feet = 7.31991307, Correction for curvature, in feet, = log D- const log [7.6209430] Correction for refraction, in feet, log D* -f- const log [1.5551483] Corrections for Curvature and Refraction, showing the Differences of the Apparent and True Level, in Feet and Decimals of a Foot, for Distances in Miles. en <u Difference in feet for 1 Difference in feet for S g o e s Curva ture. Refraction. Curvature and refraction.. Distance, Curva ture. Refraction. Curvature and refraction. 1 0.7 O.I 0.6 13 112. 8 16.9 95-9 2 2-7 0.4 2-3 14 130.8 19.6 III. 2 3 4 6.0 10.6 0.9 1.6 5-1 9.0 15 16 150.2 170.8 22.5 25.6 127.7 145-2 5 16.7 2.5 14.2 17 192.9 28.9 164.1 6 24.0 3-6 20.4 18 216.2 32.4 183.8 7 32.7 4.9 27.8 19 240.9 36.1 204.8 8 42.7 6.4 36.3 20 266.9 40.0 226.9 9 10 54-1 66.7 8.1 10. 44-0 56.7 21 22 294.3 323.0 44.1 48.4 25O.2 274.6 ii 80.7 12. I 68.6 23 353-0 52.9 300.1 12 96.1 14.4 81.7 24 384-4 57-7 326.7 GEODESY. j^r Reduction, in Feet and Decimals, upon i oo Feet, for the following Vertical Angles. Angle. Reduc. Angle. Reduc. Angle. Reduc. Angle. Reduc. / / / / 3 o 137 7 30 .856 12 O 2.185 16 30 4.118 3 15 .161 7 45 .913 12 15 2.277 16 45 4-243 3 30 .187 8 o 973 12 30 2.370 17 o 4-370 3 45 .214 8 15 1.035 12 45 2.466 17 15 4.498 4 o .244 8 30 1.098 13 2-553 17 30 4.628 4 15 275 8 45 1.164 13 15 2.662 17 45 4.760 4 30 .308 9 o 1.231 13 30 2.763 18 o 4.894 4 45 343 9 15 1.300 13 45 2.866 18 15 5-030 5 o .381 i 9 30 i.37i 14 o 2.970 1 8 30 5-168 5 15 .420 9 45 444 14 15 2.077 18 45 5-307 5 30 .460 10 .519 14 30 3-185 19 o 5.448 5 45 6 o .503 10 15 .548 ; 10 30 59 6 675 14 45 15 3.295 3-407 19 15 19 30 5.591 5.736 6 15 594 10 45 755 15 15 3.521 19 45 5.882 6 30 6 45 643 693 II ii 15 .837 .921 15 30 15 45 3-637 3.754 20 o 6.031 7 o 745 ii 30 2.008 16 o 3.874 7 15 .800 ii 45 2.095 16 15 3-995 Ratio of Slopes for the following Vertical Angles. To one To one To one To one Angle. perpen Angle. perpen Angle. perpen Angle. perpen dicular. dicular. dicular. dicular. o ; / / 1 o 15 229 3 35 16 8 8 7 18 26 3 o 30 H5 3 49 15 8 45 J 9 59 2| o 45 76 4 6 14 9 27 6 21 48 I O 57 4 24 13 9 52 5t 23 =;8 2-L i 15 46 4 45 12 10 18 51 26 34 2 i 30 39 5 o Ir l 10 47 5i 29 44 i 45 33 5 12 II ii 19 5 33 42 jJL 2 O 28 5 27 iol ii 53 4f 38 40 jl 2 15 25 5 42 IO 12 32 45 o I 2 30 23 6 o 91 13 15 4i 53 8 f 2 45 21 6 21 9 14 2 4 63 28 3 o 19 6 43 84- 14 55 31 i 75 58 1 3 15 3 28 18 17 7 7 7 36 8 71 15 56 17 6 31 si ; 78 41 . i ,- 10 146 GEODESY. XLV. Barometrical Measurement of Heights. TO OBTAIN THE DIFFERENCE IN THE HEIGHT OF TWO PLACES BY MEANS OF THE BAROMETER. The following tables have been condensed from those in the appendix of Lieutenant-Colonel Williamson s Treatise on the Use of the Barometer, etc., Professional Papers, Corps of Engineers, No. 15, and are those of Plantamour (Guyot s tables D, 72-79) re-arranged and adapted to English measures. They are based upon Bessel s formula, which differs from that of La Place principally in containing a factor depending upon the humidity of the air. The modifications introduced by Plan tamour consist in some slight changes in the values of the baro metric constants in accordance with the supposed more accurate results obtained from the experiments of Regnault. La Place s formula reduced to English measures, as given by Guyot, (page D, 35,) is : = iog x 60158.6 English feet ^ (i + 0.00260 cos 2 L) \ CO 9 / _ _ _ _ 20886860 10443430 Williamson adopts the same convenient form in his reduction of Plantamour s formula to English measures ; thus : 982.2647; 7 (1 + 0.0026257 cos 2 L) Z = log fi X 60384.3 English feet < 7+52252 ~ s __ \ 3430^ where h and H are the heights of the barometer reduced to 32 Fahrenheit; /, / , the temperatures of the air at the two sta tions; and ;, a factor depending upon the humidity of the stratum of air between them; L the latitude of the place. BAROMETRICAL MEASUREMENT OF HEIGHTS. XLV. Barometrical Measurement, &c. Continued. APPLICATION. i. Reduce the readings of the barometer to 32 Fahrenheit by Table I. 2. Representing by A the constant 60384.3, Table II gives the value "of A log h or A log H, and consequently their differ ence, A j ^rf- The numbers have been diminished by a con stant quantity, which does not affect their differences. 3. Table III, column B, gives values of the factor ^+^64 982.2647 of the temperature-term, to be used only in connection with the tables that give the corrections for humidity. Column C gives values of the factor _i ~4 and is used 900 where no observations of atmospheric humidity are made. 4. Table IV gives A log ~ x 0.0026257 cos 2 L> the correc- rl tion due to gravity at the sea-level in the mean latitude between the two stations. It is positive from 45 to the equator, and negative from 45 to the poles. 5. Table V shows the correction A log h x^"^ 2 - 5 - to be H 20886860 added to the approximate difference of altitude, on account of the decrease of gravity on a vertical acting on the density of the mercurial column. 6. Table VI furnishes the small correction A loir - x H 10443430 for the decrease of gravity on a vertical acting on the density of the air; s representing the approximate difference of altitude. It is always additive 7. Table VII gives the relative atmospheric humidity in frac tions of unity. This table is different from any given in Guyot s collection ; for, though based upon Regnault s table of maximum force of vapor, and so far the same as Guyot s, it has been cal culated with factors determined by Glaisher. 8. Tables VIII and IX are intended to give the correction A log x m, due to the relative humidity of the stratum of air between the two stations. I4 8 GEODESY. XLV. Barometrical Measurement, &c. Continued. These hypsometrical tables represent the full formula of Planta- mour. If, as is often the case, the observations for the relative humidity are not given with those of the barometer and dry ther mometer, then the tables III, (column B>) VIII, and IX should not be used, but the temperature-correction must be calculated from the formula A log -^ X ^"t^ 6 - 4 with the aid of column H 900 C of Table III. With the temperature-term so calculated, the results will differ from those by Guyot s table, on account of the different value given to the barometric constant of the pressure term. 5 Abnormal and Horary Oscillations of the Weight of the Atmo sphere. 1\& first is a gradual change generally extending over a period of two to seven days, causing the barometer to rise or fall gradually during that time, although sometimes more or less sud den, and occupying perhaps but a few hours; the second, a regu lar horary oscillation occurring at about the same hours every day, and having a magnitude entirely independent of this gradual change. The abnormal change usually extends over large tracts of coun try, and in settled weather the barometer rises and falls so gradu ally that the forces that produce the motion can be separated with more or less accuracy from the horary changes by assuming the portion of this great wave corresponding to 24 hours to be a straight line; generally inclined, however, since the observations at any time of a barometric day differs from that of the next one at the same hour. To eliminate this movement, subtract the barometric reading (reduced to 32) at the beginning of one day from that at. the same hour on the next succeeding one, and divide the difference by 24. The result is the correction for one hour, to be applied with its proper sign to the hour succeeding the initial hour. The correction for two hours is twice the correction for one hour, etc. EXAMPLE. Barometer at 7 a. m., August 7, = 29. 743 Barometer at 7 a. m., August 8, = 29. 487 Difference for 24 hours = +o. 256 For I hour ==+0.0106 and correction at 8 a. m., = + o.oi I ; at 9, = + 0.021 ; at 10, = + 0.032, etc. This correction Williamson names "reduction to level." BAROMETRICAL MEASUREMENT OF HEIGHTS. 149 XLV. Barometrical Measurement, &c. Continued. In the horary oscillation there are two maximum and two min imum points during the 24 hours. Near the sea-level the barome ter attains its maximum about 9 or 10 a. m. In the afternoon there is a minimum about 3, 4, or 5 p. m. It then rises until from 10 to midnight, when it falls again until about 4 a. m., and again rises to attain its morning maximum, the day-fluctuations being the larger. The oscillation is greatest nearer the equator and diminishes toward the poles. Its amount within the limits of the United States varies from 40 to 120 thousandths of an inch of the barometric column. It is not equal at all places of the same latitude. In a series of barometric observations at any place, the mean barometric reading is better obtained from daily horary curves, by plotting the separate readings of each day reduced to 32, and corrected for the abnormal change by reduction to level. These would present an approximate horary curve for every day of the series, from which erroneous or erratic observations could be detected and rejected if necessary. EXAMPLE OF THE USE OF THESE TABLES. Geneva and the Grand St. Bernard. - h = 28.600 in. t = 48.2 F. Relative humidity, a 0.77 H = 22.191 in. t = 28.6 F. Relative humidity, a = 0.80 Lat=46 / + f = 76.8 F. + ^=1.57 Table II, with argument 7i, gives 2 7557-3 Table II, with argument H, gives 20903.7 Difference = first approximate difference of altitude. . . 6653.6 Table III, col. B, with argument 76.8, gives + 0.0130 0.0130 x 6653. 6 = +86.5 Second approximate difference of altitude = 6740.1 Table IV, arguments 46 lat. and 6700 feet, gives 0.6 Table V, argument 6740, gives 19.0 Table VI, argument 6700 feet and 28.6 inches 0.8 Third approximate difference of altitude = 6759.3 J 5 GEODESY. XLV. Barometrical Measurement, &c. Continued. Table VIII, arguments 22.2 in. and 28.6 in., gives 79 Table IX, arguments 79 and *j6.8 ........... 1 1.9 11.9 x 1.57 vapor correction = 18.7 Difference of altitude ............................ 6778.0 The altitude by level is stated to be 6791.5 feet. The same readings of the barometer and the same air-tempera ture being used, but calculating the 1 value of the temperature- term from column C, table III, temperature-term = 6653.6 x 0.0142 = 94.4 The value of the temperature-term, as calculated in the above example, increased by the vapor-correction, is 105.2, a larger result than by the method of La Place, because the sum of the observed relative humidities of the stations was greater than that assumed by him. In a dry climate the reverse would have been the case. Calculation of the same Observations by Guyofs Tables. First table of Guyot gives, for h ................. 27454.4 First table of Guyot gives, for H ........... . ..... 20825.6 First approximate difference of altitude ........... = 6628.8 Second approximate difference of altitude .......... = 6733.1 Table III of Guyot gives ....................... .6 Table IV of Guyot gives ....................... 19.0 Table V of Guyot gives ........................ 0.8 Difference of altitude ........................... 6752.3 This result is 25.6 less than by the following tables, and 39.2 less than by the spirit-level. Aneroid Barometer. The best aneroids are, as nearly as possible, compensated by the maker for differences of temperature, so that the index shall remain at the same reading on the dial when it is heated and cooled, and are intended to be adjusted to read uniformly inches of mercury at a temperature of 32 Fahrenheit at the level of the sea in 45. latitude. But before using any aneroid for BAROMETRICAL MEASUREMENT OF HEIGHTS. 151 XLV. Barometrical Measurement, &c. Continued. accurate observations it should be tested under an air-pump, together with a mercurial column, at a known temperature, and its scale-errors carefully noted. In many of them there is an additional scale of altitudes in feet outside of the scale, corre sponding to the inches of mercury, generally in the best English instruments divided and marked according to a table prepared for the purpose by Professor Airy. There are some, however, very erroneously marked. As the aneroid is not affected in its reading by the variation in the force of gravity, it needs no correction for the latitude, nor for the decrease of gravity in altitude acting on the mercurial column. The correction for the decrease of gravity in altitude acting on the density of the air, and the correction for humidity, remain; but the first being small in amount, it can be omitted, and the second combined with the correction for temperature, as is done in the formula of La Place. The formula for the aneroid would then be : Z = log jx 60384.3 Eng. feet and the tabular quantities may be taken from these tables. Professor Airy s table, made for the purpose of graduating aneroids to a scale of feet, gives the height of the corrected mer curial column in inches for each fifty feet of altitude at 50 Fahrenheit. The formula is : h t + 1 looN - ~ -- Z = log x 62759 Eng. feet As it is sometimes convenient to have an approximate formula that can be used without any tables whatever, the following may be found useful : Z= 553 2 H~+7* Eng< feet at 55 Fahrenheit > with a correction of i - for each degree of mean tempera- l"O v) ture above 55; or, nearly U-/i . i L = CCOOO TT : - ; H -f /* 500 a formula easily remembered, but only useful for altitudes not exceeding 3000 feet. GEODESY. TABLE I. Reduction of the English Baromctet to tJie Freezing- Point. English inches. g C II .7-5 18 18.5 19 19-5 20 20.5 ii o + .045 + .046 + .047 + .049 + .O5O + .051 + -053 o I 043 045 .046 .047 .048 .049 .051 1 I 2 .042 .043 .044 45 .046 .0 4 8 .049 2 3 .040 .04! .042 .044 045 .046 .047 2 4 .039 .040 .041 .042 043 .044 045 4 5 037 .038 039 .040 .041 .042 043 5 6 035 .036 037 .038 039 .040 .041 6 7 .034 .035 .036 37 .038 .039 .040 7 . 8 .032 033 034 035 .036 37 .038 8 9 .031 .032 .032 33 034 035 .036 9 10 .029 .030 .031 .0 3 2 .032 33 034 10 ii .028 .028 .029 .030 .031 .031 .032 ii 12 .026 .027 .027 .028 .029 .030 .030 12 I 3 .024 .025 . 026 .026 .027 .028 .029 13 I 4 .023 .023 .024 .025 .025 .026 .027 !4 15 .021 .022 .022 .023 .024 .024 .025 15 16 .020 .020 .021 .O2I .022 .022 .023 16 I 7 .018 .019 .Oig .020 .020 .021 .O2I 17 18 .017 .017 .017 .Ol8 .018 .019 .019 18 19 .015 .015 .Ol6 .Ol6 .017 .017 .018 20 21 .013 .012 .012 .OI2 .013 .013 .013 .014 21 23 24 25 .009 .007 .006. .009 .007 .006 .009 .007 .006 .009 .008 .006 .OIO .008 .006 .010 .008 .006 .010 .po8 .006 23 24 25 26 .004 .004 .004 .004 .004 .005 .005 26 27 .002 .002 .003 .003 .003 .003 .003 27 28 + .OOI + .OOI + .001 + .OOI + .001 + .OOI + .001 28 2 9 .OOI .OOI .001 .OOI .OOI .OOI .OOI 29 30 .002 .002 .002 .003 .003 .003 .003 3 3 1 32 33 .004 .005 .007 .004 .006 .007 .004 .006 .007 .004 .006 .008 .004 .006 .008 .004 .006 .008 .005 .006 .008 32 33 34 .009 .009 .009 .009 .OIO .OIO .OIO 34 35 .010 .OIO .Oil .on .on .012 .012 35 36 .012 .012 .012 .013 .013 .013 .014 36 37 .013 .014 .014 .014 .015 .015 .Ol6 37 38 .015 .015 .016 .016 .017 .017 .017 38 39 .016 .017 .017 .018 .018 .019 .019 39 40 .018 .019 .Oig .020 .020 .021 .O2I 40 4 1 .020 .020 .O2I .021 .022 .022 .023 4 1 42. .021 .022 .022 .023 .024 .024 .025 4 2 43 44 .023 .024 .023 .025 .024 .026 .025 .026 .025 .027 .026 .028 .027 .028 43 44 45 .026 .027 .02 7 .028 .029 .030 .030 45 46 .027 .028 .029 .030 .031 .031 .032 46 47 .029 .030 .031 .031 .032 33 034 47 48 .031 .031 .032 33 .034 035 .036 48 49 5 .032 -.034 033 035 034 .036 -035 -037 .036 .038 037 .038 .038 -39 49 5 BAROMETRICAL MEASUREMENT OF HEIGHTS. TABLE I. Reduction of the English Barometer to the Freezing- Point Continued. Degrees of Fahrenheit. English inches. Degrees of Fahrenheit. 17-5 18 I8. 5 J 9 !9-5 20 20.5 51 -035 .036 37 -.038 039 . 040 -.041 51 52 037 .038 039 .040 .041 .042 43 52 53 .038 39 .041 .042 043 .044 045 53 54 .040 .041 .042 043 .044 .046 .047 54 55 .041 043 .044 045 .046 .047 .049 55 56 043 .044 045 .047 .048 .049 .050 56 57 045 .046 .047 .048 .050 .051 .052 . 57 58 .046 .047 .049 .050 .051 053 054 58 59 .048 .049 .050 .052 053 055 .056 59 60 .049 .051 .052 054 055 .056 .058 60 61 .051 .052 054 55 57 .058 .060 61 62 .052 054 055 057 .058 .060 .O6l 62 63 054 055 057 059 .060 .062 .063 63 64 .056 .057 059 .060 .062 .063 065 64 65 057 059 .060 .062 .064 .065 .067 65 66 059 .060 .062 .064 .065 .067 .069 66 67 .060 .062 .064 .065 .067 .069 .071 67 68 .062 .064 .065 .067 .069 .071 .072 68 69 .063 .065 .067 .069 .071 .0 7 2 .074 69 70 .065 .067 .069 .070 .072 .074 .076 70 7 1 72 .066 .068 .068 .070 .O/O .072 .072 .074 .074 .070 .076 .078 .078 .080 7 1 72 73 .070 .072 74 075 .077 .079 .O8l 73 74 .071 73 075 .077 .079 .081 .083 74 75 073 075 .077 .079 .081 .083 .085 75 76 .074 .076 .078 .081 .083 .085 .087 76 77 .076 .078 .080 -.082 .084 .087 .089 77 78 .077 .080 .082 .084 .086 .088 .091 78 79 .079 .081 .083 .086 .088 .090 .092 79 80 .080 .083 .085 .087 .090 .092 .094 80 81 .082 .084 .087 .089 .091 .094 .096 81 82 .084 .086 .088 .091 093 095 .098 82 83 .085 .088 .090 .092 095 .097 . OO 83 84 .087 .089 .092 .094 .097 .099 . 01 84 85 .088 .091 093 .096 .098 . IOI . 03 85 86 .090 .092 095 .097 . oo .103 5 86 87 .091 .094 .096 .099 . 02 .104 7 87 88 93 095 .098 . IOI 3 .106 . 09 88 89 .094 .097 . IOO . IO2 05 .108 . ii 89 90 .096 .099 .101 .104 7 . no . 12 9 9i .097 . IOO .103 .I06 . 09 .III 14 9 1 92 .099 .102 .105 .108 . 10 "3 . 16 92 93 . IOI .103 .106 .109 . 12 "5 . 18 93 94 . IO2 .105 .108 .III 14 .117 . 20 94 95 "3 . 16 .118 . 21 95 96 .114 .117 . I2O .123 96 97 .116 .119 . 122 .125 97 98 .118 . 121 .124 . 127 98 99 -.119 .122 -.126 .129 99 154" GEODESY. TABLE I. Reduction of the English Barometer to the Freezing- Point Continued. 0-3 o 43 English inches. || fj 21 2I -5 22 22.5 23 23-5 24 II o + .054 + 055 + -055 4- .058 + .059 -f .060 + .062 I .052 053 053 .056 .057 .058 059 I 2 .050 .051 .051 054 .055 .056 57 2 3 .048 .049 .049 .052 053 54 055 3 4 .046 .047 .047 .050 .051 .052 053 4 5 .044 045 45 .048 .049 .050 .051 5 6 .042 .044 .044 .046 .047 .0 4 8 .049 6 7 .041 .042 .042 .044 .044 045 .046 7 8 039 .040 .040 .041 .042 043 .044 8 9 37 .038 .038 39 .040 .041 .042 9 10 035 .036 036 37 .038 39 .040 10 ii 033 034 034 .035 | .036 37 .038 ii 12 .031 .032 .032 033 -34 035 .036 12 13 .029 .030 .030 .031 .032 33 033 13 T 4 .027 .028 .028 .029 j .030 .031 .031 15 .025 .026 .026 .027 1 .028 .029 .029 15 16 .024 .024 .024 .025 | .026 .026 .027 16 17 .022 .022 .022 .023 .024 .024 .025 iy 18 .020 .020 .020 .021 .022 .022 .023 18 I 9 .018 .018 .018 .019 .020 .020 .020 ig 20 .016 .016 .016 .017 .018 .018 .018 20 21 .014 .014 .014 .015 .Ol6 .Ol6 .016 21 23 .010 .013 .Oil .013 .Oil .013 .Oil .Oil .012 .012 23 2 4 .008 .009 .009 .009 .009 .OIO .010 24 25 .007 .007 .007 .007 .007 .007 -.008 25 26 .005 .005 .005 .005 .005 .005 .005 26 27 .003 .003 .003 .003 .003 .003 .003 27- 28 + .001 + .001 + .001 + .001 + .001 4- .001 + .001 28 2 9 .001 .001 .001 .001 .001 .001 .001 29 3 .003 .003 .003 .003 .003 .003 .003 30 3 1 .005 .005 .005 .005 .005 .005 .005 3 r 3 2 .007 .007 .007 .007 .007 .007 .008 32 33 .008 .009 .009 .009 . .009 .009 .010 33 34 .010 .on .on .on .Oil .012 .012 34 35 .012 .013 .013 .013 .013 .014 .014 35 05 .01 <5 ,Ol6 ,Ol6 tf 3 37 .016 .016 .017 .017 .018 .Ol8 .018 J^ 37 38 .018 .018 .019 .019 .020 .020 .020 38 39 .020 .020 .021 .021 .022 .022 .023 39 40 .022 .022 .023 .023 .024 .024 .025 40 4i .023 .024 .025 .025 .026 .026 .027 4 1 42 .025 .026 .027 .027 .028 .028 .029 42 43 .027 .028 .029 .029 .030 .031 .031 43 44 .029 .030 .031 .031 .032 33 33 44 45 .031 .032 .032 033 034 035 35 45 46 033 034 034 035 .036 037 .038 46 47 035 .036 .036 037 .038 039 .040 47 48 037 .038 .038 039 .040 .041 .042 48 49 39 39 .040 .041 .042 043 .044 49 50 .040 .041 .042 -.043 .044 045 .046 50 BAROMETRICAL MEASUREMENT OF HEIGHTS. TABLE I. Reduction of the English Barometer to the Freezing- Point Continued. -~ .*- 0-5 CBrt sg Q b English inches. S B-a ?S fra p 21 21.5 22 22.5 23 23-5 24 51 -.042 --043 -.044 -045 .046 .047 -.048 sr 52 .044 045 .046 .047 .048 .049 .050 52 53 .046 .047 .048 .049 .050 .051 053 53 54 .048 .049 .050 .051 .052 054 055 54 55 .050 .051 .052 053 055 .056 57 55 56 .052 053 054 055 057 .058 59 56 57 54 55 .056 057 ^059 .060 .061 57 58 055 057 .058 059 .O6l .062 .063 58 59 57 059 .060 .061 .063 .064 .065 59 60 059 .O6l .062 063 .06 5 .066 .068 60 61 .061 .062 .064 .065 .06 7 .068 | .O7O 61 62 .063 .064 .066 .067 .069 .070 .072 62 63 .065 .066 .068 .069 .071 .072 .074 63 64 .067 .068 .070 .071 73 075 .076 64 6 5 .068 .070 .072 73 75 .077 .078 65 66 .070 .072 .074 75 .077 .079 .080 66 67 .072 .074 .076 .077 .079 .O8l .083 67 68 .074 .076 .078 .079 .O8l .083 .085 68 69 .076 .078 .080 .081 .083 085 .067 69 70 .078 .030 .082 .083 .085 .08 7 .089 70 7i .080 .082 .083 .085 .087 .089 .091 7 1 72 .082 .084 .085 .087 .089 .Ogi 93 72 73 083 .085 .087 .089 .091 .093 095 73 74 .085 .087 .089 .091 .093 .095 .097 74 75 .087 .089 .091 93 .095 j .098 .100 75 76 .089 .0 9 I 093 095 .098 1 .100 . IO2 76 77 .091 93 095 .097 .100 .102 .104 77 73 93 095 .097 .099 . 102 . 104 . 106 78 79 095 .097 .099 .101 .104 . 106 . 108 79 80 .096 .099 . IOI .103 .106 .108 .no 80 81 .098 .101 .103 .105 .108 .no . 112 81 82 . IOO .103 .105 .107 .110 .112 "S 82 83 . IO2 .105 .107 .109 . 112 .114 .117 83 84 .104 .106 .109 .in .114 .116 .119 84 85 .106 .108 .III .113 .116 .118 . 121 85 86 .108 . no Ir 3 .115 .118 j .120 .123 86 87 . no . 112 115 .117 .120 i .123 I .125 87 88 . Ill .114 .117 . 119 . 122 .125 .127 88 89 "3 .116 .119 .121 .124 .127 .129 89 90 ii5 .Il8 .121 .123 .126 .129 .132 90 Qi .117 . I2O .123 125 .128 .131 134 9 1 92 .119 . 122 .124 .I2 7 13 -133 .136 92 93 . 121 .124 .126 .129 132 j .135 .138 93 94 .123 .125 .128 131 134 .137 .140 94 95 .124 .127 .130 133 136 | .139 .142 95 96 .126 .129 .132 135 .138 | ,I 4 I .144 96 97 .128 J3 1 .134 137 .140 *43 .146 97 98 .I 3 133 .136 139 .142 .145 .149 98 99 -.132 .I3S -.138 .141 -.144 -.148 -iS 1 99 56 GEODESY. TABLE I. Reduction of the English Barometer to the Freezing- Point Continued. ll English inches. 0-5 ! Oj Jn Qr^ 240 25 25-5 26 26.5 27 27-5 || + .063 + .06 4 + -065 + .067 + .068 + .069 + .071 o I .061 .062 .063 .064 .066 .067 .068 I 2 .058 .060 .061 .062 .063 .064 .066 2 3 .056 57 .058 .060 .061 .062 .063 3 4 054 055 .056 057 .058 .060 .061 4 5 .052 053 054 55 .056 057 .058 5 6 .050 .051 .052 053 054 55 .056 6 7 .047 .048 .049 .050 .051 .052 053 7 8 045 .046 .047 .048 .049 .050 .051 8 9 043 .044 045 .046 .046 .047 .048 9 10 .041 .042 .042 043 .044 45 .046 10 ii 039 039 .040 .041 .042 .042 043 ii 12 .036 37 .038 39 39 .040 .041 12 I 3 034 035 .036 .036 037 .038 .038 I s J 4 .032 33 033 034 035 035 .036 14 15 .030 .030 .031 .032 .032 33 033 15 16 .028 .028 .029 .029 .030 .030 .031 16 18 .025 .026 .026 .027 .027 .028 .028 18 .021 .021 .022 .022 .023 .023 .023 19 20 .019 .019 .Dig .O2O .020 .021 .021 20 21 .017 .017 .017 .018 .018 .018 .019 21 22 .014 .015 .015 .015 .015 .016 .016 22 23 .012 .012 .013 .013 .013 .013 .014 23 2 5 .008 .008 .008 .008 .008 .009 .009 2 4 2 5 26 .006 .006 .006 .006 .006 .006 .006 26 27 .003 .003 .003 .004 .004 .004 .004 27 28 + .001 + .001 + .001 + .OOI 4- .001 + .001 + .001 28 29 .001 .001 .001 .001 .001 -.001 .001 29 30 .003 .003 .003 .003 .004 .004 .004 30 3 1 .005 .006 .006 .006 .006 .006 .006 3 1 32 .008 .008 .co8 .008 .008 .008 .009 32 33 .010 .010 .010 .010 .Oil .Oil .Oil 33 34 .012 .012 .013 .013 .013 - OI 3 .014 34 35 .014 .015 .015 .015 .015 .016 .016 . 35 36 .016 .017 .017 .017 .018 .018 .018 36 37 .Dig .Dig .019 .020 .020 .021 .021 37 38 .O2I .021 .022 .022 .023 .023 .023 38 39 023 .024 .024 .024 .025 .025 .026 39 40 .025 .026 .26 .027 .027 .028 .028 40 4 1 .027 .028 .029 .029 .030 .030 .031 4i 42 .030 .030 .031 .031 .032 033 33 42 43 .032 .032 033 034 034 035 .036 43 44 034 035 035 .036 037 037 .038 44 45 .036 037 .038 .038 039 .040 .041 45 46 .038 039 .040 .041 .042 .042 043 46 47 .041 .041 .042 043 .044 45 .046 47 48 043 .044 .044 045 .046 .047 .048 48 49 045 .046 .047 .048 .049 .050 49 50 -.047 .048 .049 .050 -.051 .052 -053 50 BAROMETRICAL MEASUREMENT OF HEIGHTS. 157 TABLE I. Reduction of the EnglisJi Barometer to the Freezing- Point Continued. 0-55 gc English inches. I U_rt (31 24-5 25 25-S 26 26.5 27 27-5 II 51 .049 .050 -.051 .052 053 054 55 51 52 .052 053 054 55 .056 57 .058 5 2 53 054 055 .056 057 .058 59 .060 53 54 55 .056 .058 057 059 .058 .060 59 .062 .060 .063 .062 .064 .063 .065 54 55 56 .060 .061 .063 .064 .065 .066 .068 56 57 .062 .064 .065 .066 .068 .069 .070 58 .065 .066 .067 .069 .070 .071 73 58 59 60 .067 .069 .068 .070 .070 .072 .071 073 .072 75 .074 .076 75 .077 59 60 61 .071 73 .074 .076 .077 .078 .080 61 62 073 . 075 .076 .078 .079 .081 .082 62 63 64 . 6 5 .076 .078 .080 .077 .079 .082 .079 .081 .083 .080 .082 .085 .082 .084 .086 .083 .086 .088 .085 .087 .090 s 65 66 67 .082 .084 .084 .086 .085 .088 .087 .089 .089 .091 .090 093 .092 95 66 67 68 .086 .088 .090 .092 093 95 .097 68 69 .089 .090 .092 .094 .096 .098 .099 6q 70 .091 93 095 .096 .098 .100 . 102 70 7 1 093 095 .097 .099 . IOI .102 .104 7 I 72 095 .097 .099 . IOI .103 .105 .107 72 73 -.097 .099 . IOI .103 .105 .107 I0 9 i 73 74 . IOO . IO2 .104 .106 .108 .110 .112 i 7J. 75 . IO2 .104 .106 .108 .110 . 112 MI4 75 76 .104 .106 .108 .110 . 112 .114 .117 76 77 .106 .108 . IIO IJ 3 "5 .117 .119 77 78 .I08 . no "3 "5 .117 .119 . 121 78 79 . no "3 US .117 .119 . 122 .124 79 80 113 US .117 .119 .122 .124 .126 So 81 us .117 .119 .122 .124 .126 .129 81 82 .117 .119 . 122 .124 .126 .129 82 83 84 85 .119 .121 .123 . 122 .124 .126 .124 .126 .128 .126 .129 I3 1 .129 134 134 134 .136 83 84 85 86 87 .126 .128 .128 .130 I3 1 !33 133 .136 .136 .138 .138 .141 .141 . 143 86 87 88 .130 133 135 .138 .141 -143 146 88 89 .132 135 .138 .140 143 .146 .148 89 90 I 34 137 .140 143 145 .148 90 9i .136 139 .142 145 .I 4 8 .150 153 9 1 92 !39 .141 .144 .147 .150 153 .156 92 93 .141 .144 .147 .149 .152 155 .158 93 94 143 .146 .149 .152 155 .158 .160 94 95 145 .148 151 154 I 57 .160 .163 95 96 .147 .150 153 .156 159 .162 .165 96 97 .149 !53 .156 159 .162 .165 .168 97 98 .152 155 158 .161 .164 .167 . 170 98 99 I 154 I 57 -.160 -.163 -.166 .170 99 GEODESY. TABLE I. Reduction of the English Barometer to the freezing- Point Continued. o| English inches. <** -J Si &l ol 28 28.5 29 29-5 30 3-5 31 Q o + .072 + -73 + .074 + .076 + .077 + .078 + .080 o I .069 .071 .072 73 .074 075 .077 I 2 .068 .069 .070 .072 073 .074 2 3 4 5 !o62 059 .065 .063 .060 .067 .064 .061 .068 .065 .062 .069 .066 .063 .070 .067 .064 .071 .068 .066 3 4 5 6 7 057 054 .058 055 059 .056 .060 057 .061 .058 .062 .063 .060 6 7 8 .052 53 .053 054 055 .056 057 8 9 .049 .050 .051 .052 .053 053 054 9 10 .047 .047 .048 .049 .050 .051 .052 10 ii .044 045 .046 .046 .047 .048 .049 ii 12 .042 .042 043 .044 .044 045 .046 12 J 3 039 .040 .040 .041 .042 .042 .043 13 15 .036 .034 37 035 .038 035 .038 .036 039 .036 .040 037 .040 .038 15 16 .031 .032 033 33 034 .034 .035 16 \l .029 .026 .029 .027 .030 .027 .030 .028 .031 .028 .032 .029 .032 .029 i? 18 29 .024 .024 .025 .025 .026 .026 .027 19 20 .021 .022 .022 .023 .023 .023 .024 20 21 .019 .019 .020 .020 .020 .021 .021 21 22 .Ol6 .017 .017 .017 .018 .018 .018 22 23 .014 .014 .OI 4 .015 .015 .015 .015 2 3 24 24 25 .009 .009 .OOg .009 .009 .010 .010 25 26 .006 .006 .007 .007 .007 .007 .007 26 27 .004 ,004 .004 .004 .004 .004 .004 27 28 + .OOI + .OOI + .OOI + .001 + .001 + .OOI + .OOI 28 2 9 .001 .OOI .OOI .001 .OOI .OOI .OOI 29 30 .004 .004 .004 .004 .004 .004 .004 30 .006 .006 .006 .007 .007 .007 .007 31 3 2 .009 .009 .009 .009 .009 .010 .010 32 33 34 .Oil .014 fOI 4 .014 .015 .015 .015 .015 34 35 .016 .017 .017 .017 .017 .018 .018 35 36 .019 .019 .019 .020 .020 .020 .021 36 .021 .022 .022 .022 .023 .023 .024 37 38 .024 .024 .025 .025 .026 .026 .026 38 39 .026 .027 .027 .028 .028 .029 .029 39 40 .029 .029 .030 .030 .031 .031 .032 40 4 1 42 43 .031 .034 .036 .032 .034 037 .032 035 .038 033 .036 .038 .034 .036 039 034 .037 .040 .035 037 .040 4 1 42 43 44 39 .040 .040 .041 .042 .042 043 44 45 .041 .042 043 .044 .044 045 .046 45 46 .044 045 045 .046 .047 .048 .049 46 47 .046 .047 .048 .049 .050 .050 .051 47 .c 48 49 5 .049 .051 -054 .050 .052 -.055 .051 53 .056 .051 .054 -.057 .052 055 -.058 .053 .056 059 .054 057 -.060 40 45 5 C BAROMETRICAL MEASUREMENT OF HEIGHTS. TABLE I. Reduction of the English Barometer to the Freezing- Point Continued. if English inches. 11 tjs II ^ 28 28.5 29 29-5 30 30-5 3i 5 1 -.056 -.057 .058 059 -.060 -.061 -.062 5 1 52 059 .060 .061 .062 .063 .064 .065 5 2 53 .061 . 062 . 064 .065 .066 .067 .068 53 54 .064 .065 i .066 .067 .068 .070 .071 54 55 .066 .068 .069 .670 .071 .072 073 55 56 .069 .070 .071 073 .074 075 .076 56 57 .071 .073 .074 075 .076 .078 .079 57 58 .074 .075 .076 078 .079 .080 .082 58 59 .076 .078 .079 "080 .082 083 .085 59 60 .079 .080 .082 .083 .084 .086 .087 60 61 .081 083 .084 .086 .087 089 .090 61 62 .084 .085 .087 .088 .090 .091 93 62 63 .086 .088 .089 .091 .092 .094 .096 63 64 io8g .090 .092 .094 095 .097 .098 64 f>5 .091 .093 .095 .096 .098 .099 , IOI 6s 66 .094 .095 | .097 .099 .101 . IO2 .104 66 67 .096 .098 .100 . IOI .103 .105 .107 67 68 .099 .101 . IO2 .104 .106 .108 . 109 68 69 .101 .103 .105 .107 .109 . no .112 69 7 . 104 ; . 106 . 107 i . 109 . Ill "3 "5 70 7i .106 ; .108 .no .112 .114 .116 .118 7 1 72 .109 .in .113 .115 .117 .118 .120 72 73 74 .III j .!I 3 .114 .Il6 MS .118 .117 . I2O .119 . 122 . 121 .124 .123 .120 73 74 75 .Il6 .Il8 .I2O . 122 .125 .127 .129 75 76 .119 i .121 .123 .125 .127 .129 131 76 77 .121 ! .123 . 126 .128 .130 .132 134 77 78 .124 ; .126 .128 .130 133 I3S 137 78 79 .126 .128 *3* 133 135 137 .140 79 80 .129 ; .131 *33 .136 .138 .140 143 80 81 I3 1 -133 -136 .I 3 8 .141 .143 145 81 82 134 .136 .138 .141 I 43 .146 .148 82 83 .136 139 .141 143 .146 .148 IS 1 83 84 .139 .141 .144 .146 .148 84 85 .141 ; .144 .146 .149 15*4 156 85 86 .144 .146 .149 151 154 .156 -X59 86 87 .146 .149 151 154 .156 . 162 87 88 .149 .151 154 .156 159 .162 .165 88 89 I 5 I .154 .156 159 .162 .164 .167 89 90 153 -156 159 .162 .164 .167 .170 90 91 .156 .159 .162 .164 .167 .170 I 73 91 92 93 94 ,158 .161 163 .\6\ .166 .164 .167 .169 .167 ,I 7 -I 7 2 .170 .172 J 75 I 73 !i 7 8 .175 92 93 94 95 .166; . 169 .172 175 .178 .181 183 95 96 .168 .171 .174 .177 .180 .183 .186 96 97 .171 .174 .177 .ISO .183 .186 .189 97 98 99 173 -.176 .176 -.179 .180 -.182 .I8 3 -.185 .186 -.188 .189 -.191 .191 -.194 98 -99 i6o GEODESY. M ro T)-VO t> O> O vN ro i sqipinssnoqx i? i ^- IOVO C^OO ON I M 10 ON a- t^.od o- H H N ro * irjvo loo ON O\ "^rt-t^OOO O OO vo ro s 10 ON rovo OO O O 00 . K" O w ro lovo oo Ov n ca ro iovo oo o M cvi -<f iovo co : i N >* Vr * rc^-iOLOio 10 10 10 iovo vo NO vo vo NO i w vo O 00 OM m too * O O O I " ] ^ "0 C O I ^O-^ ! 2? Ov w "000 OO OOt^lOM t-^N^OO lOO J-O-* OOrOt^M^ t^oo O M ^j-\o r^ O Tj--^-u-!ioLn mmio io\ ? I i 2 " 5 l LO invo vo * txvo vOOOHVO OvvO XOVO *O 10 ; po > M 10 t O\ Q O <>oo u N co ro t. O ro ovo tvo m O ! ^vow\Owt NVOMVOH !00-<J-OvfO t^MloOfO i r\^ o H w -^t- 10 r>.oo ^ fo ^vo t^oo o M ro -^- 10 r^ ; ^S 2 2 3 2" 2"?!?!?^? l^^i!? ^"2 ^2 "S ^2 ^ ^2 "5J *O Q 4- in io4-fOOl^ rood rovd (> M N ro w -^ O> H CJ rf- IO t^*OO O M 04 ^J- IO C^CQ O M fO ^ iO tN. i fc< )r r) T j.^.- T j. T j. Tf -^- 10 IA in 10^010 xovo vo *O vo vo vO OOOVOrO Ov -^-OO N VO t^OO OOO t^ ^s"? ^-^^8 J-sss-^R *-Tj-Tt- --^Tl-ioiO loioio iO\O vo vo vo vo vo O w ro it- >OvO txoo ON O H N ro * <O\O r-00 ON qstiSug ui BAROMETRICAL MEASUREMENT OF HEIGHTS. l6l o " 3 ^ II ipui U-B jo sqipuBsnoqx ro\o oo H M- r>. o o in M N ro io\o i>. ON o" H H M CO T- lOVO t^CO ON CO COVO W w TOCO t^ON -t CJ-^-OONC^ ONO^fOlO ON t^ ro ON -^~ O Jjj^> O^ ^ M N Cl "^ ON r^ t^ t^oo co co co co co oo VO ON M H 01 Tt- O rooo vot^ HCOONTOO O-*M rnoo t^ ONMD o O ^ ON O w ssxput ui [^ co" O 2"^ ^J- 100 CO ON O N ro ^v hOO VO CO ^ CM NO roco ro t- H rovo ^ co t^ O ^i- t^ H Tf-co M -^t- [^ CO ON M (N ro 10VD r-, ON O 00 I^VO 10 ( oo oo co co cococ^a> 1 1 1 62 GEODESY. ipui in? jo H ro inoo O H N CO rf- irivO t^OO M ro in t^oo O N ** in H o ro ^ in t^.cd o> o saiptn N!N (MMNtMO txoo ro ro r^ 10 "^ VO O ^ t*- ON M ^ L.OOO O N t* *"* S.& S^ NNiNNN O O o> M in 01 O^ VO 0) >l^ M t^ oo O - | <"*"> mvo oo ^ 0^ " M " w" w 1 ^ "" WOJN MOIMMO) MMN VD in m o oo ro m t^ o^ O S"8 8^2 2 ON N O N oo .00 ovo W C> ^t- O t^ o> M N ^- in N O M ro -^t- U-)\O O O> O- O> C> I" . i MVOW wroto WNWC4C4 01 ro -^- n vo oo ON O -< M o*> invo 0000 OOOWW HMMM oioiojoi eioioioioi oioiNd Ovovooco HoooNinin ^ moo O 01 to r^ ON M ro to S ON c? ON ON ON 8 O o" ? :?fJ>? ScS^ . ON o w W ro ^v BAROMETRICAL MEASUREMENT OF HEIGHTS. 163 -d o ipui TIB jo \o t-^co CN 100 f^oo Os M Tt- rovo O ONOO ro ro ON \ O HI CM ro 10*0 r^co ON O HI N oo moo r^ H H t^co H 0) ro -^ i^>*O t^ r^ CO O\ O<- O f^oo 0\ O M N ro rh io\O tx JS IH ro ^ iovo ( O 1 O [C 1 ON O " ro *N M w N W W (NDNMN vo- f i N t-^co ON rOVO CO O " f^ON OMC?ro? lovo ^Rco" ON (N M ro ro ro ro ro ro ro ro ro ro ^ ^o rovo o w qsiiu^ u .1 9 } 3 UI a B [ inco r~- O ON 164 GEODESY. IpUI UT5JO C U X CO cd o vo ooo ro Tt- o M o txoo moo oo m in rooo co in ro t^ H Acc H ro iri^o t^ oooococor^ \o-^*wo N pi IN PI P) <N(NN04<N P) P) P) CN N HlOrt-ONl-l OOHOvOt^ -*00 00 -*UD m <> O OO ro t^. w 4-00 6 ro Avo i>. ooooooooti. ^;>O I O O M S 5-*-? 8<N Tt-vo t^ co oo oo co t^ vonroiH( NNNN N PI (M N N N N N vD C^oo G>O MOJro-^- O ^O t^co c^^ w" ct N" ct C? (? P? c? M 0) 04 !N N M t-* M ro O ro P) t^.co \O O O v H t^co o H t-. H O \O 00 !>. M S31JOUI ovo vorom -oo c^ moo t^ ro -<t- vo \q ro Tj- m^D t^-co O O * BAROMETRICAL MEASUREMENT OF HEIGHTS. 165 O O X i I 3 .e 5 O ON C^OO CO t^ t-~<O M3 M w <M m * ui\d t^-oo n <N ro CM CM CM 1 oco 00 t^oo O* H -3- fOCO H CO O>CO CM CM CM CM CM CM CM CM >ro f^O ro^O CO O IN ro ^o lom^j-roiN SM M o < ON O w OJ ro -^- u^o t^ I CM IN CM CM CM IN *oo W IT) t^ O N Th 1OVO tN. i ro M M M o Oco t^o M CM ro ^- 10 io\o r^oo CSMWCM (M IN CM M (M Nvo o ro^O O> M ro to rOCM (N^OO^ON CO f~- CM CM <N CM CMM^^S^S CM" CM" CM" c-T CM" CM CM CM CM CM ^ 1-^ * O t- CM CO rOCO CM VO O Tt- t^ O tN Tf MD CO ON O ^ O w 1 0? ro ro CM CM H H O ON ONCO t^VO 10 rj- ** k-1 VO vS \0 O O vc?\ vo"S v? S. t^ ?L S. K tt tC^R t^^. CMCMCMININ CMCMCMCMCM CMCMCMCMtM CMCMCMCMIN CMCM CMCMCMWCM CM i66 GEODESY. .3 bO C ,2 a x 1 ipiu UB jo ONCO vo -3- ro 01 M O 1 ON t^vo -"i- ro 04 O O> O H 04 ro -*i- iovo t^co O M 04. ro ^- iovo vo tx ON vc i ^ O OCQ t^.MD Tf- ro W H o co t^vo -^ ro CJ O O^ t^O r^ O "- 1 M ro -i- iovo t-^co cooO - M ro ^ ^ ^\o ^ ^ ^ "S ^ S^ S^ ^^1?^,? oT^cToTcT 1 6 | ^ OCX) t^-O UO ^-M^CO> OOVO^O ^M M OCO O *O ^t^-COCOOOO^ OOOOCOOOCO COOOOOO OOOOO W N N MMMCNM W N 04 W N ,W(NMMW Hundredths of an inch. s- 6 M 01 f H ^t r^oo OOOOO3 OOCOOOCOCO COCOONONON ONONCNONON N W N N 0)040)C4(N CMCMCSOOi WOilNNO) 6 MN.OO ts o 6 lisi nil! iiiii tiii o 6 c 6 1 Hill fffft Illll Illll t^ 0^ t^ rovo 10 04 ovo <* co O ON 1000 ONVO H ro 01 "^ ^ m^ *% o" ^. -;g ^ ^ ON o " ? ^ ^ ^ fovS ^i t^OO OOCOCO OOOOCOOOCO OOCOO>ONO> ONOONOCN CM04CN1CMCM CN1MCMC401 NMMN04 010404040) 8 6 vo ONCO * tx vo ro t^co vo Hro04ONOi roHvococo 0?^ < 01 C <N C 0) C N C N C So)0 OtCSNCSO) 0404^0404 6 VO ONCO * t-N 00 10 ON O CO -*-VO VO 04 VO t~- 1O H rO rO 3l^f lo^lovlK oSlll^ o?^^S ^ir^^^^ C 8 C 8 S C S C S S^^^oT ^oTo^cToT 8 6 VO ONCO IOCO CNVO H 01 H VO ON O^O O H O> VOOO CO I|i||! Hill llli! llllf HSijSu Pi in J3}9UIOJB{[ M -, ro -f ,ro r-co ON O H 01 ro * .00 -co O 0> " " ?o BAROMETRICLL MEASUREMENT OF HEIGHTS. I6 7 TABLE III. Correction for Temperature. D I = A log ~ X B or C. B C B C t + t * + t 1 64 t + 1 - 64 t + f t + t 64 t + t 64 982.2647 900 982.2647 900 3 -0.0346 0.0378 70 + o . 006 1 + 0.0067 3i 0336 .0367 7i .0071 .0078 32 .0326 0356 72 .0081 .0089 33 .0316 0344 73 .0092 .0100 34 0305 0333 74 .0102 .OIII 35 .0295 .0322 75 .0112 .OI22 36 .0285 .0311 76 .OI22 0133 37 .0275 .0300 77 .0132 .0144 38 .0265 .0289 78 .0142 .0156 39 0255 .0278 79 0153 .0167 40 .0244 .0267 80 .0163 .0178 4 1 .0234 .0256 81 .0173 .0189 42 .0224 .0244 82 .0183 .0200 43 .0214 .0233 8 3 .0193 .02 1 1 44 .0204 .0222 84 .O2O4 .0222 45 .0193 .0211 85 .0214 .0233 46 .0183 .0200 86 .0224 .0244 47 0173 .0189 87 .0234 .0256 48 .0163 .0178 88 .0244 .0267 49 OI 53 .0167 . 89 0255 .0278 So .0143 .0156 90 .0265 .0289 5i .0132 .0144 9 1 .0275 .0300 52 .0122 0133 92 .0285 .0311 53 .0112 .0122 93 .0295 .0322 54 .0102 .OIII 94 0305 0333 55 .0092 .0100 95 .0316 0344 56 .0081 .0089 96 .0326 0356 57 .0071 .0078 97 .0336 .0367 58 .0061 .0067 98 .0346 .0378 59 .0051 .0056 99 35 6 .0389 60 .0041 .0044 IOO .0366 .0400 61 .0030 .0033 101 0377 .0411 62 .0020 .0022 1 02 .0387 .0422 63 o.ooio O.OOII 103 0397 0433 64 .0000 .0000 104 .0407 .0444 65 + O.OOIO + O.OOII I0 5 .0417 .0456 66 .0020 .0022 106 .0428 .0467 67 . .0030 .0033 107 .0438 .0478 63 .0041 .0044 108 .0448 .0489 69 + 0.0051 + 0.0056 109 + 0.0458 + O.O5OO i68 GEODESY. TABLE III. Correction for Temperature Continued. D n A log ~ x B or C. 15 C B C t + t t + t 1 - 64 / + t 64 t + t t + t - 64 t + t 64 982.2647 900 982.2647 900 no + 0.0468 + 0.0511 150 + 0.0875 + 0.0958 III .0478 .0522 151 .0886 .0967 112 .0489 .0533 152 .0896 .0978 113 .0499 544 153 .0906 .0989 114 .0509 0556 J 54 .0916 .1000 II 5 .0519 .0567 155 .0926 .ion 116 .0529 .0578 156 937 .1022 117 .0540 .0589 157 .0947 1033 118 0550 .0600 158 957 .1044 119 .0560 .0611 159 .9967 .1056 I2O .0570 .0622 1 60 .0977 . 1067 121 .0580 .0633 161 .0987 .1078 122 .0590 .0644 162 .0998 .1089 I2 3 .0601 .0656 163 .1008 . IIOO 124 .0611 .0667 164 .1018 .1111 125 .0621 .0678 165 .1028 . 1122 126 .0631 .0689 1 66 .1038 33 127 .0641 .0700 167 .1049 .1144 128 .0651 .0711 168 .1059 1156 I2 9 .0662 .0722 169 .1069 .1167 I 3 .0672 0733 170 .1079 .1178 I 3 I .0682 .0744 171 .1089 .1189 132 .0692 .0756 172 .1099 .1200 *33 .0702 .0767 173 . 1109 .1211 134 .0713 .0778 174 . II2O .1222 135 .0723 .0789 175 .1130 1233 136 733 .0800 176 . 1140 .1244 137 743 .0811 177 .1150 .1256 138 c/53 .0822 178 . 1160 . 1267 139 .0763 o333 179 .1171 .1278 140 .0774 .0844 1 80 .1181 .1289 141 .0784 .0856 181 .1191 . I3OO 142 .0794 .0867 182 . I2OI .1311 143 .0804 .0878 183 .1211 .1322 144 .0814 .0889 184 . 1222 1333 J 45 .0825 .0900 185 .1232 .1344 146 0835 .0911 186 .1242 fjS 6 M7 .0845 .0922 187 .1252 1367 148 0855 933 188 .1262 1378 149 + 0.0865 + 0.0944 189 + o. 1272 + 0.1389 BAROMETRICAL MEASUREMENT OF HEIGHTS. 169 S ooooo ooooo ooooo ooooo oooo o o -^ 10 O o H w CM CM ro ro * <!- in in m^D vo t^ t^co co ON ON O O O o >O ro ON >n N oo * H t-x * -Ht^ COO^O i TJ- -4- io\O t^oo i-a O O O ON ON ONONON ONOO OOOOOOCOOO t^t^t^l^t^ t-^^o VO <O > M N ro rn 4- IOMD t^-oo ON O M N ro voo t^oo ON O w N ro I H cJ ^ 4 ^ VO K O O M CJ j ^00 ^ 00 ON O H CO 4 -OVO *o rovo ON roo ON CM 1000 H -^J-oo H TJ- t^ O rovo ON ro ^O Ox N 10 ^ M N ro LO^O t^ ON O M ro ^-. 10 r^oo ^^Jj^ 1 ^^ cM^roro tJvOCNICOlOM r^rOONiOW COTt*ONOCM ONiOHt^rO ONVO CM OO CO 10 ro O 00 10 ro M oo MD ro H oo MO ro H O^O * M ^J M ro 10 f^co 2^i?^T 2"riCNrTcM (NNrororo H O O oo r-^vo -* ro N H o Oco t^vo * ro 1? ^"^ S " N~"N ^ ^)?n?i^^ ?)^- ! ? ! ? N\O h> ON O H c-i * iovo I-^ON OuNTj-iovor^ONO O (N^-t^ON" rovot^ONM TJ->O co O N * >) oo o <N ^-vo oo H *J roo O ro o O M O ON el loco M loco w -^- t^ ^ t-^ O roo { <N * C^ ON w ro^O oogro ^^^^^ 5-0^^.^^ ^ O ^1-oo oj ^o O ^-oo CM O O -^J-co CM ^O O ^CM-i-t^ONCM 4-t--ONCJ4- t^. ON C\i -4- !>. ON M 4-VD ON H TJ-VO ON o.! OOJ 1000 O co ioco O *J OCJOO-^-O VOMOO ^ O VO(NOO-^-O 1 OWOO ^"O VONCO^ .*** C* lO Ix O CO lOOOOroO OOi-irO l OCN WTfOOOJ ^l-^ON " Jl ^HHMW(N04 (Ncororoco^-^Tj-^-Lnioio u-)^0 . o ro inoo H mo co w ^-o o - M^t^o(N ioc^otN ^ M HHWMM (Nrororoco ^-Tf^-Tj-ir) m LOO vo lOt^O 10 ovc 170 GEODESY. TABLE -V. Correction for Decrease of Gravity on a Vertical actin- on the Density of the Mercurial Column. D IV = 60384.3 log 4 4. 60384.3 log - + 52252 20886860 Approx . cliff, of alt. OOO 100 200 300 400 500 600 700 800 900 Feet. ! Feet. Feet. Feet. Feet. feet. Feet. Feet. Feet. Feet. IOOO 2-5 2.8 3-i 3-3 3-6 3-9 4.1 4-4 4-7 4-9 2OOO 5-2 5-5 5-7 6.0 6-3 6.6 6.8 7 * 7-4 7-7 3OOO 7-9 8.2 8-5 8.8 9.1 9-3 9.6 9-9 IO.2 10.5 4OOO 10.8 II. I 11.4 ii. 6 11.9 12.2 12.5 12.8 I3.I J 3-4 5000 13-7 14.0 14.3 14.6 14.9 J 5-2 15-5 15-8 16.1 16.4 6OOO 16.7 17.0 i7-4 17.7 18.0 I8. 3 18.6 18.9 19.2 19-5 JOOO 8000 19.9 20.2 20.5 20.8 21 . I . 21.5 21.8 22. I 22.4 22.8 oA T 9OOO 26.4 26.7 27. i 20. 1 10000 29.8 30.2 30-5 30.8 31-2 31-5 31-9 32.2 29. i 32.6 29.5 33-o I IOOO 33-3 33-7 34-o 34-4 34-7 35-1 35-5 35-8 36.2 36.5 I2OOO 36.9 37-3 37-6 38.0 38.4 38.8 39-i 39-5 39-9 40.2 13000 40.6 41.0 41.4 41.7 42, i 42.5 42.8 43-3 43-6 44.0 14000 44-4 44-8 45-2 45-6 46.0 46.3 46/7 47.1 47-5 47-9 15000 43.3 48.7 49.1 49-5 49.9 50.3 50.7 51-1 5* -5 51-9 16000 52.3 52.7 53-i 53-5 53-9 54-3 54-7 55-i 55-5 56.0 17000 56.4 56.8 57-2 57-6 58.0 58.4 58.9 59-3 59-7 60. i 18000 60.5 61.0 61.4 61.8 62.2 62.7 63.1 630 64.0 64.4 19000 64.8 65.2 65-7 66.1 66.6 67.0 67.4 67.9 68.3 68.7 2OOOO 69.2 69.6 70.1 70-5 71.0 71.4 71.9 72.3 72.7 73-2 2IOOO 73-6 74.1 74.6 75-0 75-5 75-9 76.4 76.8 77-3 77-7 22000 78.2 78.7 79.1 79-6 80. i 80.5 81.0 81.5 81.9 82.4 23OOO 82.9 83-3 83.8 84-3 84.8 85-2 85-7 86.2 86.7 87.1 24OOO 87.6 88.1 88.6 89.1 89.5 90.0 90-5 91.0 91-5 92.0 NOTE. In Table I the scale of the barometer is supposed to be of brass, extending from the cistern to the top of the mercu rial column, and the difference of expansion of brass and of mer cury is taken into account. The standard temperature of the yard being 62 Fahrenheit and not 32, the difference of expan sion of the scale and of the mercurial column carries the point of no correction down to 29 Fahrenheit. BAROMETRICAL MEASUREMENT OF HEIGHTS. 171 ON i H N N CO * lOVO O C^c N _; _; j -T M N CO CO * tOVO CX 1X00 0. ! co 10 ix oo O co 10 . ts-oo 5 oomrowO vo * N. O t-x loroqoO O O H w CO ro ^ lOvO t^ tx CO ^ O O * ja ; N VOCOONO" N CO O O w N CO mvO H c ro 10VO " ,00 Pi ro 10 N -<J-^O OO O <M t- f^O ^ N IOCO H xoiadv j 1 172 GEODESY. TABLE VII Giving the Relative Humidities in Fractions of Unity. !!* Wet bulb. > *> *8-o| h 10 ^ 20 22 24 26 28 0.0 I. 000 I. OOO I. OOO I. OOO i. ,000 I. 000 i I. OOO . 2 925 . 926 929 935 1 .941 949 958 4 .854 ; .856 .864 .874 .887 . 902 .919 .6 .788 ! .791 .804 .817 .837 .858 .883 .8 .727 : 731 749 .764 .790 .818 .850 I.O .671 .675 .699 .717 747 .781 .819 . 2 .619 j .624 653 . 674 I . 707 747 ; .791 4 570 ; 577 .610 635 .670 .716 .765 .6 525 , -534 571 599 .636 .687 741 .8 483 . 494 535 j -566 .605 .660 .719 2.0 445 456 .502 535 577 635 ! .698 .2 . 410 421 .471 56 552 . 612 .679 4 377 ; 389 443 479 .529 591 .662 .6 346 .360 .417 454 . 507 572 .646 .8 3i8 333 393 432 . 487 554 .631 3-0 .292 .308 371 412 .469 .538 .618 .2 .268 .285 351 394 - 453 524 .605 4 245 .264 -332 377 | .439 593 .6 .224 245 3*4 3 6 i .426 499 .582 .8 .205 .227 .298 346 | .413 .488 572 4.0 .187 . 211 283 332 .401 .478 .562 . 2 .170 . 196 .269 . 320 . 390 .469 553 4 154 . 182 257 39 -379 .461 544 .6 . 140 .I6 9 .246 .299 | .370 453 .536 .8 . 128 157 236 . 290 . 362 .446 .528 5-0 H7 H5 .226 281 .355 .440 521 .2 , .217 273 ; -348 434 .514 4 .209 .266 .342 429 57 .6 .201 2 59 -33 6 .424 .501 .8 .194 253 331 .419 495 6.0 .188 .248 327 415 489 . 2 .182 243 324 .411 .482 4 .I 7 6 239 .321 .407 .476 .6 171 236 319, 403 .469 .8 .167 2 33 3i7 399 463 7.0 1 .164 .231 315 i 396 j 457 BAROMETRICAL MEASUREMENT OF HEIGHTS. 173 TABLE VII. Giv&g the Relative Humidities, &c. Continued. 11 1 Wet bulb. II <L & <u o-e s all 3 32 34 36 38 40 42 Q o I. OOO I. 000 I. OOO I. 000 I. OOO I. OOO I. OOO i .856 .885 .903 .909 .914 .917 . 920 2 .756 .799 .821 .831 .837 .843 .847 3 .684 .728 .750 .761 .768 .776 .781 4 .628 .665 .686 .697 .707 715 .721 5 579 .609 .628 .640 .652 .660 .667 6 533 .558 575 .589 . 601 .610 .6:8 7 .490 512 .528 543 554 .565 573 8 .450 .470 .486 . 500 5 12 523 532 9 .414 .432 .448 . 461 473 .485 495 10 381 .398 413 .426 438 .451 .462 ii 351 .367 381 394 .406 .420 432 12 .324 339 352 3^5 378 .391 .404 J 3 .299 313 .326 339 352 .365 378 14 .277 .289 .302 3 r 5 328 .341 354 i5 257 .268 .280 293 .306 .320 332 16 .238 .249 .261 .274 .286 .300 .312 17 . 221 .232 .244 .256 .268 .281 293 18 .206 .216 .228 .240 .252 .263 275 ^9 .192 .202 .213 .225 237 . 246 .258 20 .ISO .I8 9 . 200 .211 .223 .231 .242 21 .168 .178 .189 . 199 .209 .217 .226 22 .158 . 1 68 178 .187 195 . 204 .213 23 .149 .158 .167 175 183 .192 . 200 24 .140 .149 157 .164 .172 .181 .188 25 .132 .141 .148 154 .162 .170 .177 26 . 1 60 .168 27 151 159 28 143 151 29 135 H3 30 .128 135 174 GEODESY. . TABLE VII. Giving the Relative Humidities, &c. Continued. ^ ~ Wet bulb. g-c 44 46= 48 50 5 2 54 56 o I. OOO I. 000 I. OOO I. 000 I. OOO I. 000 I. OOO i .922 923 925 .928 93 931 933 2 .851 853 .858 .862 .866 .869 .872 3 .786 .791 .796 .802 .807 .811 .816 4 .727 734 740 747 753 .758 .764 5 .674 .682 .689 697 .704 .709 .716 6 .626 .635 .643 .651 .658 .664 .671 7 .582 592 . 60 1 . 609 .616 .622 .629 8 542 552 .562 .570 577 .583 590 9 .506 .516 526 . 534 541 547 554 10 473 483 493 .501 507 513 521 ii 443 453 .462 .470 .476 .482 .491 12 415 425 433 .441 .448 454 463 J 3 389 398 .406 .414 .422 .428 437 14 365 373 381 389 397 .404 15 343 35 358 .366 374 382 391 16 322 329 337 345 353 361 370 J 7 .302 .310 318 326 333 341 350 18 .284 . 292 .300 .308 315 .322 331 19 .267 275 283 .291 .298 305 20 251 259 .267 275 .282 .289 . 296 21 .236 244 .252 .260 .267 .274 .280 22 . 222 230 238 .246 253 .260 .265 23 . 209 .217 .225 2 33 . 240 .246 .251 24 .197 .205 .213 . 221 .227 233 238 25 .186 .194 .202 .209 .215 .221 .226 26 .176 .184 .192 . 198 .204 . 209 .214 27 .167 174 .182 .188 193 .198 .203 28 .I 5 8 .165 .172 .178 .183 .188 193 2 9 .ISO 157 .163 .I6 9 .174 .178 .183 30 . 142 .148 154 .160 .165 . 169 .174 31 151 .156 .161 .166 32 143 . 148 -153 .158 33 .136 .141 .146 .150 34 .129 134 139 143 35 .123 .127 I3 2 .136 BAROMETRICAL MEASUREMENT OF HEIGHTS. 175 TABLE VII. Giving the Relative Humidities, &c. Continued. II Wet bull). > ^2 <D L? 13 IS o ^ SI-SP 53 60 62 64 66 67 68 o I.OOO I.OOO I. OOO I. OOO I.OOO I.OOO I.OOO i 935 1 .936 .937 .938 .940 .940 .941 2 .875 .877 .879 .881 .884 .885 .886 3 .819 .822 .825 .828 832 834 835 4 .767 .771 775 779 .783 .786 .788 5 .719 .724 .729 734 739 .741 -743 6 6/5 .681 .686 .692 1 . 697 [ . 699 I . 702 7 .634 .641 647 -653 . 658 . 660 . 663 8 597 ! .603 . 610 . 616 i .621 .624 . 627 9 .562 .568 575 .582 | .587 590 593 10 529 .536 543 549 555 5581 -561 ii .499 .506 513 -519 ; .526 .528 531 12 471 .478 485 .491 | 497 . .5 53 13 445 .452 459 465 i .471 474 .476 H .420 .428 434 -440 .446 449 451 15 397 .405 .411 .417 .422 .426 .428 16 376 .383 389 395 . 400 . 403 . 406 i7 .356 .362 .368 .374 379 382 .385 18 337 .343 348 . 354 .360 .3631 -366 19 3*9 .325 33 33 6 341 . 344 . 447 20 .302 .308 313 -3*9 324 .327 .330 21 .286 .292 .297 .303 .308 .311 3 T 4 22 .271 .2/7 .282 .288 . 293 . 296 .299 23 .257 .263 .268 .273 .279 .282 . 284 24 .244 .249 . 255 260 .265 .268 .2/1 25 .231 .236 .242 , .247 .252 .255 .258 26 . 219 . 22 4 . 2 3 . 235 .240 . 243 . 246 27 .208 .213 .219 .224 .229 .232 ! .235 28 .198 .203 .208 .213 .219 .221 .224 2 9 .188 193 . 198 . 203 . 209 .211 i .214 30 .179 . 184 .189 . 194 .200 . 2O2 . 204 31 .170 175 .180 .185 .190 . 193 . 195 32 .162 .167 ,172 .177 .182 .184 . ;86 33 154 .159 . 164 .169 173 .176 .178 34 ..147 .152 157 .161 .166 .168 .170 35 HI 145 . 150 . 154 159 .161 .163 I! " i 7 6 GEODESY. TABLE VII. Giving the Relative Humidities, &c. Continued. li . Wet bulb. IJai o "S C r^ g ^ " r- 1 69 70 71 73 75 77 79 I. OOO I. OOO I. OOO I. OOO I. 000 I. OOO I. OOO i .941 .942 .942 943 -945 .946 .946 2 .887 .888 .889 .891 .893 .895 .896 3 .836 .838 .839 842 . 845 .847 .849 4 .789 .791 793 796 . 799 .801 .804 5 745 747 749 753 75 6 759 .762 6 74 .706 .708 .712 .716 .719 .723 7 .665 .668 . 670 .674 .678 .682 .686 8 . 629 .632 634 .638 .642 647 .651 9 595 .598 .600 . 605 .609 . 614 .618 10 563 .566 .569 573 .578 583 .587 ii 533 536 539 544 549 553 558 12 .505 .508 .511 .516 .521 .526 53 1 13 479 .482 484 .490 495 . 500 505 14 454 457 459 465 .47 .476 .481 15 431 434 43 6 442 447 453 45 s 16 .409 .412 .414 .420 425 431 436 17 388 39 1 393 399 405 .411 .416 18 .368 371 374 .380 386 392 397 J 9 350 353 .356 361 .367 373 379 20 333 336 339 344 350 356 361 21 .317 .320 323 .328 334 340 345 22 .301 304 .307 313 .319 324 33 23 .287 . 290 293 299 34 .310 315 24 .274 .276 .279 .285 .290 .296 .301 25 .261 263 .266 .272 277 .283 .288 26 .249 .252 254 .260 .265 .270 275 27 238 .240 243 .248 .-253 .258 263 28 . .227 .229 232 237 .242 247 251 2 9 .217 .219 . 222 .227 .231 ..236 . 240 30 . 207 .209 .212 .217 . 221 .225 .229 31 . 198 .200 . 2O2 .207 . 211 .215 . -219 32 . 189 .191 193 .198 . 2O2 .205 .209 33 .181 I8 3 .185 . 189 193 . 196 . 200 34 173 175 .177 .181 . 184 ,187 .191 35 .165 .167 . 169 173 .176 .179 183 BAROMETRICAL MEASUREMENT OF HEIGHTS. 177 & o be 3 O vo -* d M O j t^. 10 ^ ro 01 w y O OVOO \O 10 ^ - O ONCO ^O 10 ^- ro N M w ^^ G -B O ooo t^.io-^-rorotN M o Oco oo Cx O MD LO ^ TJ- ro ro (M cl "in w" b- D H H H ^ 12 i 7 8 GEODESY. TABLE IX. Second Part of Corrcc ^VII J h z^ v V H > V. (t + 1 ) (t + 1 1 ) 180 F. 170 F. 160 F. 150 F. 140 F. 130 F. 120 F. 110 F. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 10 8.5 7-i 6.0 5-0 4.2 3-5 2.9 20 16.9 14-3 12.0 10. 8.4 6.9 5-8 30 . 25.4 21.4 iS.o 15.0 12.5 10.4 8.6 40 33-8 28.5 23-9 20.0 16.7 13-9 ii. 5 50 42.3 35-6 29.9 25-1 20.9 17.4 14.4 60 50-7 42.8 35-9 3 O.I 25-1 20.8 17-3 70 59-2 49-9 41.9 35-i 29-3 24.3 20.1 80 67.6 57-o 47-9 40.1 33-4 27.8 23.0 9 76.1 64.1 53-9 45-i 37-6 31.3 25-9 100 84.5 7i-3 59-9 50.1 41.8 34-7 28.8 no 93- 78.4 65.8 55-i 46.0 38-2 31-6 120 101.5 85.5 71.8 60. i 50.1 41.7 34-5 I 3 109.9 92.6 77.8 65-1 54-3 45-1 37-4 140 118.4 99-8 83.8 70.1 58.5 48.6 40-3 150 126.8 106.9 89.8 75-2 62.7 52.1 43-1 160 135-3 114.0 95-8 80.2 8- 2 66.9 55-6 46.0 4.8 170 1 80 I 43-7 152.2 128.3 107.7 03.2 90.2 71 o . 75-2 59-O 62.5 ^vty 51-8 1 190 160.6 135-4 "3-7 95-2 79-4 66.0 54-6 200 169.1 142.5 119.7 100.2 83.6 69-5 57-5 210 177.6 149.6 125-7 IOj.2 87.8 72.9 60.4 22O 186.0 156.8 i3i-7 110. 2 91.9 76.4 63-3 230. 194-5 163.9 137-7 II5-2 96.1 79-9 66.1 240 202.9 171.0 M3-7 I2O.2 100.3 83-3 69.0 250 211.4 178.1 149.6 125-3 104.5 86.8 71.9 260 219.8 185-3 155-6 I30-3 108.6 90-3 74.8 2 7 228.3 192.4 161.6 135-3 112. 8 93-8 77.6 280 236.7 199-5 167.6 140.3 117.0 97-2 80.5 290 245.2 206.7 173-6 145-3 121. 2 100.7 83-4 300 253-6 213.8 179.6 150.3 125.4 104.2 S6. 3 3 IO 262.1 220.9 185.6 155-3 129.5 107.6 89.1 3 20 270.6 228.0 191-5 160.3 133-7 in. i 92.0 330 279.0 235-2 197-5 165.3 137-9 114.6 94-9 340 287.5 242.3 203.5 I70-3 I42.I . 118.1. 97.8 35 295-9 249.4 209.5 I 75-4 146.3 121. 5 100.6 360 34-4 256.5 215-5 180.4 150.4 125.0 103.5 37 312.8 263-7 221.5 185.4 154-6 128.5 106.4 380 321-3 270.8 227.5 190.4 158.8 132.0 109.3 BAROMETRICAL MEASUREMENT OF HEIGHTS. 179 tlon for Atmospheric Humidity. V. W. 100 F. 90 F. 80 F. 70 F. 60 F. 50 K 40 F. 30 F. 20 F. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 2.4 1.9 1.6 i-3 i . i 0.8 0.7 0.5 0.4 4-7 3-9 3- 2 2.6 2.1 i-7 i-3 I.O 0.8 7-i 5-8 4.8 3-9 3-2 2-5 2.0 1.6 1.3 9-5 7-8 6. 4 5-2 4.2 3-4 2-7 2.1 r-7 11.9 9-7 8.0 6-5 5-3 4.2 3-3 2.6 2. 1 14.2 11.7 9.6 7-8 6-3 5-0 4.0 3- 1 2-5 16.6 13.6 II .2 9.1 7-4 5-9 5-7 3-7 2.9 19.0 15-6 12.8 10.4 8.4 6-7 5-3 4.2 3-3 21.3 I 7-5 14.4 xx. 7 9-5 7-6 6.0 4-7 3-7 23-7 19-5 16.0 13-0 10.5 8.4 6.7 5-2 4,1 26.1 21.4 17-5 i4-3 ii. 6 9.2 7-3 5-8 4-5 28.5 23-4 19.1 15-6 12.6 10. I 8.0 6-3 5-0 30.8 25-3 20.7 16.9 13-7 10.9 8.7 6.8 5-4 33-2 27-3 22.3 18.2 14.7 11. 8 9-3 7-3 5-8 35-6 29.2 23-9 19-5 15-8 12.6 IO.O 7-9 6.2 38.0 31 .2 25-5 20.8 16.8 13-4 10.7 8.4 6.6 40-3 33-1 27.1 22.1 17.9 14-3 "3 8.9 7.0 42.7 35-1 28.7 23-4 18.9 JS- 1 12.0 9-4 7-4 45-i 37-0 30-3 24.7 20. o 16.0 12.7 IO.O 7-8 47-4 39-o 3L9 26.O 21.0 16.8 13-3 IO -5 8-3 49-8 40.9 33-5 27-3 22.1 17.7 14.0 II. 8.7 52.2 42.9 35- 1 23.6 23.1 18.5 14.7 "5 9.1 54-6 44-8 36.7 29.9 24.2 19-3 15-3 12. I 9-5 56.9 46.8 38.3 31.2 2j.2 20.2 16.0 12.6 9.9 59-3 48.7 39-9 32.5 26.3 21.0 16.7 I3-I 10.3 61.7 50.7 4 I S 33-8 27-4 21.9 17-3 13.6 10.7 64.0 52.6 43-1 35-1 28. 4 22.7 18.0 14.2 ii. i 66.4 54-6 44-7 36.4 29o 23-5 18.7 14.7 xi. 6 68.8 56.5 46.3 37-7 30.5 24.4 19-3 15.2 12. 71.2 58.5 47-9 39-0 3 1.6 25.2 20. o 15-7 12.4 73-5 60.4 49-5 40-3 32.6 26. I 20.7 I6. 3 12.8 75-9 62.4 51.0 41.6 33-7 26.9 21.3 16.8 I 3 .2 78.3 64-3 52.6 42.9 34-7 27.7 22.0 17-3 I 3 .6 80.7 66.3 54-2 44-2 35-8 28.6 22. 7 17.8 14.0 83.0 68.2 55-8 45-5 36.8 29.4 23-3 18.4 14.4 85.4 70.2 57-4 46.8 37-9 30-3 2 4 .0 18.9 14.9 87.8 72 . i 59-o 48.1 38.9 3". i 24.7 19.4 r 5-3 90.1 74.1 60.6 49-4 40.0 3i-9 25-3 19.9 15.7 iSo GEODESY. I <> IH 3 li IS H H H M^ 1 88888888888 888888888888 ore 73 88888888888888888 88888 | 8888888888 8888888 8888 C -f d 8888888 88888 88888 888 c. -i- ? 888888888 8888 8888 88 . 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 1 $ 1 8 888 88 88 888 88 888 8 ; o o - d 8 8 3 8 8 8 8 8 8 8 8 88 88 8 : o ich* 2 88888 88 8 88 88 8 88 8 \ .2 $ 1 8 8 8 88 8 88 8 8 88 8 8 1 o c^ -f 1 8 888 8 88 8 8 8 8 8 8 o S 1 888888888888 rt 3! ^ 8 8 8 8 8 8 8 8 8 8 8 * -a 8 8 8 8 8 8 8 8 8 "o _ -* 5 8 8 8 8 88 8 8 8 o 6 a oofe a 8 8 8 8 8 8 8 8 J -f 3 H g 00 H M H H g | oo oo ooq q q , H5 1 googooogooogooogooogo o g g " T3 | 00 g 0000 g 0000 go OgOOOOgOg * 1 000"00000w00000^00000w0g 8 8 o q q "ft d 13 OOOOwOOOOOOOMOOOOOOOOOQ 8 8 q q * d T3 OOOOOOHOOOOOOOOOOOgOOOOQ | 8 q o _}* | 000000000000"00 000000000 8 q o i H N m -* l^\0 t^CO H ro -* >0.0 t-03 ON t^OO & O " H M THERMOMETRICAL MEASUREMENT OF HEIGHTS. 181 XL VI. TJiermometrical Measurement of Heights. BAROMETRIC PRESSURES CORRESPONDING TO TEMPERATURES OF THE BOILING-POINT OF WATER. o *: o in % Tenths of a degree of Fahrenheit. || 2 4 6 8 I8 5 17.048 17. 122 17. 197 17.272 17.348 1 86 423 499 575 .652 .729 187 .806 .883. .961 18.039 18.117 1 88 IS. 195 18.274 iS.353 432 512 189 592 .672 753 .833 .914 190 .996- 19.077 I9.I59 .19- 241 19- 324 191 19.407 .490 r**-. .657 .741 192 .825 .910 995 20. 080 20. 1 66 193 20. 251 20. 338 20. 424 . 5 II .598 194 .685 773 -861 .949 21.038 195 21. 126 21. 2l6 21.305 21.395 .485 196 .576 . 666 . 758 . 849 . 941 197 22. 033 22. 125 22. 2l8 22.311 22.404 198 .498 592 .686 .781 .876 199 .971 23.067 i 23.163 23.259 23.356 200 23. 453 550 .648 .746 | .845 2OI 943 24. 042 24. 142 24. 241 24. 341 2O2 24.442 .542 .644 745 i 847 203 -.949 25.051 25.154 25-257 25.361 204 25. 465 .569 .674 . 779 . 884 . 205 .990 26. 096 26. 202 26.309 26.416 2O6 26.523 .6 3 I . 740 .848 957 207 27.066 27.176 27.286 27.397 27-507 208 .618 .730 i .842 954 28. 067 209 28. 1 80 28.293 28.407 28.521 .636 210 .751 .866 .982 29. 098 29.215 211 29-331 29. 449 29. 566 .684 .803 212 .922 30. 041 30. 161 30. 281 30. 401 182 GEODESY. Table of Comparison of Fahrenheit s Thermometer with Reaumur s and tJie Centesimal. Fah. Reaum. Centes. Fah. Reaum. Centes. Fah. Reaum. Centes. o o 33 + 0.4 + 0.6 67 + 15-6 +19.4 O 14.2 -17.8 34 0.9 i. i 68 16. o 20. I 13.8 17.2 35 1.3 1-7 69 16.4 20. 6 2 13.3 16.7 3 6 1.8 2.2 70 16.9 21. I 3 12.9 1 6. i 37 2. 2 2.8 7i J 7.3 21. 7 4 12.4 15.6 38 2-7 3-3 72 17.8 22.2 5 12.0 15.0 39 3-1 3-9 73 18.2 22.8 6 ii. 6 14.4 40 3-6 4.4 74 18.7 23-3 7 ii. i 13-9 4i 4.0 5-0 75 19.1 23-9 8 10.7 13.3 42 4.4 5-6 76 19.6 24.4 9 IO. 2 12.8 43 4-9 6.1 77 20. o 2$. O 10 9.8 12. 2 44 5-3 6-7 78 20. 4 2 5 .6 ii 9-3 11,7 45 5.8 7.2 79 20.9 26.1 12 8.9 ii. i 46 6.2 7.8 So 21.3 26.7 13 8.4 10. 6 47 6.7 8-3 Si 21.8 27.2 14 8.0 IO. O 48 7-i 8.9 82 22.2 27.8 11 7.6 7- i 9-4 8.9 49 50 7.6 8.0 9-4 IO. O 83 84 22.7 23.1 28. 3 28.9 17 6-7 8 3 : 5i 8.4 10. 6 85 2 3 .6 29.4 18 6.2 7-8 5 2 8.9 . ii. i 86 24. o 3o.-o J 9 5.8 7.2 53 9-3 11.7 87 24.4 30.6 20 5-3 6-7 54 9.8 12.2 88 24.9 3I.I 21 4.9 6.1 i 55 10. 2 12.8 89 25-3 31-7 22 4-4 5-6 ; 56 10.7 13-3 90 25.8 32.2 2 3 4.0 5-o , 57 II. I 13-9 9i 26.2 ">-7 8 2. 5 24 3-6 4-4 ! 58 ii. 6 14.4 92 26.7 33-3 25 3- ! 3-9 59 12. O 15.0 93 27.1 33-9 26 2.7 3-3 60 12.4 I 5 .6 94 27.6 34-4 27 2 2 2.8 61 12.9 16. i | 95 28.0 35-0 28 1.8 2.2 62 13.3 16.7 96 28.4 35-6 2 9 1-3 i-7 63 13-8 17.2 97 28.9 36.1 30 0.9 1. 1 64 14.2 17.8 ; 98 29.3 36.7 31 32 0.4 O. O o. 6 0.0 3 14.7 + I5.I 18.3 +i8. 9 99 IOO 2 9 .8 +30.2 +?:5 x Reaumur = (32 + $x) Fah. = -| x Centes. x Centes. = (32 + f- x) Fah. = f x Reaum. x Fah. = J (x 32) Reau. = (. r 32) Cen. TABLES AND FORMULAE. PART III. ASTRONOMY. ASTRONOMY. XLVIL Of Sidereal and Solar Time. True or apparent solar time is that deduced from observations of the sun, or is the same as that shown by a well-adjusted sun dial. Mean solar time is derived from the time employed by the earth in revolving on its axis, as compared with the sun, supposed to move at a mean rate in its orbit, and to make 365.242218 revolutions in a mean Gregorian year. It cannot be immediately obtained from observation, but is al ways deduced from apparent time by the aid of the equation of time, which is the angular distance, in time, between the mean and true sun; or, mean solar time = apparent solar time i equa tion of time. Sidereal time is the portion of a sidereal day which has elapsed since the transit of the first point of Aries. Its point of origin cannot be determined by observation, but it is known at any moment by the right ascension of whatever object may be then in the meridian ; or, Sidereal time of a star s culmination = AR. of >K ; Sidereal time at mean noon = AR. mean % at mean noon ; and, generally, Sidereal time = sidereal time at mean noon i solar time from mean noon, (expressed in sidereal intervals;) Solar time = sidereal time sidereal time at mean noon, (the difference being reduced to a solar interval.) 1 86 ASTRONOMY. XL VI I. Of Sidereal and Solar Time Continued. EXAMPLE. To find the mean solar time of the passage of Altair over the meridian of Washington, on the loth July, 1849: h. m. s. AR. Altair, July 10, 1849 = 19 43 27.39 Sidereal time at mean noon at Washington 714 00.96 Sidereal interval past Washington mean noon. . . = 12 29 26.43 Retardation of mean on sidereal time = 02 02.77 Corresponding mean time interval past mean noon or mean time of culmination = 12 27 23.66 The nautical almanacs give the sidereal time at mean noon for each day of the year for a certain meridian. If the sidereal day be taken equal to 24 sidereal hours, the mean solar day will be equal to 24 11 3 m 56 s . 5 5 of those sidereal hours ; or the daily acceleration of sidereal on mean solar time (which is the mean motion of the earth in a mean solar day) is 3 m 56 s . 5 5 54 of sidereal time; hence the sidereal time at mean noon under any meridian other than that of the nautical alma nac used \vill be found by allowing the proportion of this quan tity due to the difference of longitude of the two places. If the mean solar day be taken equal to 24 mean solar hours, the sidereal day will be equal to 23^ 56 m 4 s .o9 of those solar hours, or the daily retardation of mean solar on sidereal time is 3 m 55 s -993 f s l ar time. The astronomical day begins at noon. In the civil or common method of reckoning, the day is supposed to commence at the preceding midnight. The civil reckoning is therefore 1 2 hours in advance of the astronomical reckoning, and in the above example, July loth, i2 h 27 m 23^.66 astronomical time, corresponds to July nth, o l1 27 m 23 s . 66 a. m. civil time. TIME. 187 XL VI II. To find the Time by an Altitude of the Sun or a Star. Sidereal time = AR. X * s hour-angle. Solar time = 24 h =J= & s hour-angle. 2 m = L + A + A cos L ."sin A where L = the latitude of the place of observation ; A = the. north polar distance of the sun or the star; A = the corrected altitude of the sun or star = observed altitude (refraction parallax) i semi-di ameter; and p = the hour-angle of the sun or star. The formula gives the arc in degrees, which must be converted into time, as in one of the following four cases: 1. When we have the corrected altitude of the sun s center, the hour-angle, /, in time, is the apparent time when the sun is in the west, or the complement of 24 hours when in the east. To reduce it to mean time apply the equation of time. 2. But should the sidereal time be required, transform the mean time thus obtained to sidereal time, as previously explained. 3. When the altitude is that of a star, the sidereal time is at once deduced from the hour-angle,/. 4. And if, in this last instance, solar time should be required, convert this sidereal time into solar time by means of the equa tion Solar time = AR. * AR. in which the sign -f- is used if the star is observed in the west, and the sign if in the east ; or, Mean solar time = the mean solar equivalent of (sidereal time of observation sidereal time of preceding mean noon at place.) i88 ASTRONOMY. The most common dial is that in which the plane of the dial is horizontal, and the style, placed in the meridian, is inclined to the plane of the dial at an angle equal to the latitude of the place. Hour-lines are drawn from the center, or point where the style intersects the plane, to the exterior limit of the surface of the dial. Their positions are calculated from the formula tan x = tan p sm L in which x = hour-angle on the horizontal plane ; /= 15. 30, 45, etc., the hour-angle on the equatorial plane \ and L = latitude of the place. The geometrical determination of these lines will be readily seen from the following figure. \ SUN-DIAL. 189 . Sun- Dial Continued. As the lines of IV, V, VIII, and VII cannot, generally, be directly drawn, owing to want of space upon the surface of the dial, draw, from any point of the line of IX hours, a line parallel to that of III hours, and take ab 1 = ab, and ac 1 = ac\ b 1 and c will be points in the lines VIII and VII. The lines IV and V will make the same angles on the opposite side. The line of VI is perpendicular to that of XII. To determine the Meridian Line. Take a point in the plane of the dial through which it is in tended the meridian plane shall pass. With this point as a cen ter describe several concentric circles. Fix a straight pin in the center, perpendicular to the plane of the dial, of such a length that the extremity of the shadow cast by it shall fall within the circles at XII M. Mark the points where the extremity of the shadow passes over these circles in the forenoon and again the same in tfye afternoon. Ihe line drawn from the middle of these arcs, contained between the points of passage, to the center of the circles will be the meridian. SUN-DIAL CORRECTION. Mean Time at Apparent Noon. 1 Day. January. February. March. April. May. June. . //. m. //. ;;/. //. m. //. Di. h. DI. h. DI. I 12 4 12 14 12 12 12 4 ii 57 ii 58 8 12 7 12 14 12 II 12 2 ii 56 ii 59 l6 12 10 12 14 1 12 9 12 O ii 56 12 O 24 12 12 12 I 3 12 6 II 58 ii- 57 12 2 Day. July. August. September. October. November December. ! h m // . 111 Ji. m. h. in. h. DI. //. ;;/. I 12 3 12 6 12 II 50 ii 44 II 50 8 12 5 12 II 58 II 48 II 44 ii 53 16 12 . 6 12 4 " 55 ii 46 ii 45 ii 56 24 12 6 12 2 ii 52 ii 45 ii 47 12 1 . 1 90 ASTRONOMY. For converting Intervals of SIDEREAL into Corresponding Intervals of MEAN SOLAR Time. Hours. Minutes. Seconds. h. in, s. m. S. m. s. s. S. s. s. i o 9.830 i 0.164 3i 5-079 i 0.003 31 0.085 2 o 19.659 2 0.328 32 5-242 2 0.005 32 0.087 3 o 29.489 3 0.491 33 5.406 3 O.OO8 33 0.090 4 o 39,318 4 0.655 34 5-570 4 O.OII 34 0.093 5 o 49.148 5 0.819 35 5-734 5 0.014 35 0.096 6 o 58.977 6 0.983 36 5.898 6 0.016 36 0.098 7 i 8.807 7 1.147 37 6.062 7 0.019 37 O. IOI 8 i 18.636 8 1.311 38 6.225 8 O.O22 38 0.104 9 i 28.466 9 1.474 39 6.389 9 O.O25 39 o. 1 06 10 i 38.296 10 1.638 40 6.553 10 0.027 40 o. 109 ii i 48.125 ii 1.802 4i 6.717 ii 0.030 4i O. 112 12 i 57-955 12 1.966 42 6.881 12 0.033 ^2 O.II5 13 2 7.784 13 2.130 43 7.044 13 0.036 43 O.IlS T4 2 17.614 14 2.294 44 7.208 14 0.038 4-4 O. 1 2O 15 2 27.443 15 2.457 45 7-372 15 0.041 45 o. 123 16 2 37.273 16 2.621 46 7.536 16 0.044 46 o. 126 17 2 47.103 17 2.785 47 7.700 17 0.047 47 0.128 18 2 56.932 18 2.949 48 7-864 18 0.049 48 0.131 19 3 6.762 T 9 3- "3 49 8.027 19 0.052 49 0.134 20 3 I6.59 1 20 3-277 50 8.191 20 0.055 50 0.137 21 3 26.421 21 3.440 5i 8.355 21 0.057 5i o. 140 22 3 36.250 22 3.604 52 8.519 22 O.O60 52 0.142 23 3 46.080 2 3 3.768 53 8.683 23 0.063 53 0.145 24 3 55.9 9 24 3-932 54 8.847 24 O.O66 54 0.148 25 4.096 55 9.010 25 0.068 55 0.150 26 4-259 56 9.174 26 O.O7I 56 0.153 27 4-423 57 9-338 27 0.074 57 0.156 28 4.587 58 9.502 28 0.076 58 0.159 29 4.751 59 9.666 2 9 o . 079 59 o. 161 30 4.915 60 9.830 30 0.082 60 0.164 The quantities taken from this table must be subtracted from a sidereal interval to obtain the corresponding interval in mean solar time. TIME. For converting Intervals of MEAN SOLAR Time into Corresponding Intervals of SIDEREAL Time. Hours. Minutes. Seconds. . h. m. s. tn. S. <jn. j. s. S. s. s. i o 9.856 i 0.164 3i 5.092 i 0.003 31 0.085 2 o 19.713 2 0.329 32 5-257 o O.OO5 32 o.oSS 3 o 29. 569 3 0.493 33 5.421 3 0.008 33 0.090 4 o 39.426 4 0.657 34 5.585 4 O.OII 34 0.093 5 o 49.282 5 0.821 35 5-750 5 0.014 35 0.096 6 o 59.!39 6 0.986 36 5.9M 6 0.016 36 0.098 7 i 8.995 7 1.150 37 6.078 7 0.019- 37 O. IOI 8 i 18.852 8 I.3I4 38 6.242 S 0.022 38 0.104 9 i 28.708 9 1.478 39 6.407 9 0.025 39 0.106 10 i 38.565 10 1.643 40 6.571 10 0.027 40 0.109 ii i 48.421 ii 1.807 4i 6.735 ii 0.030 4i O.II2 12 i 58*278 12 1.971 42 6.900 12 0.033 42 o. 115 13 2 8.134 13 2. 136 43 7.064 13 0.036 43 0.118 14 2 I7-99 1 14 2.300 44 7.228 14 0.038 44 O.T20 15 2 27.847 15 2.464 45 7.392 15 0.041 45 0.123 16 2 37.704 16 2.628 46 7-557 16 0.044 46 o. 126 17 2 47-560 17 2-793 47 7.721 17 0.047 47 0.129 18 2 57.416 18 2-957 48 7-885 iS 0.049 48 0.131 19 3 7-273 19 3- 121 49 8.050 19 0.052 49 0.134 20 3 17-129 20 3.285 50 8.214 20 0.055 50 0.137 21 3 26.986 21 3-450 51 8.378 21 0.057 51 o. 140 22 3 36.842 22 3.614 5 2 8-542 22 O.O6O 52 0.142 23 3 46.699 23 3-7/8 53 8.707 23 0.063 53 0.145 24 3 56.555 24 3-943 54 8.871 24 O.O66 54 0.148 25 4.107 55 9-35 25 0.068 55 0.151 26 4.271 56 9.199 26 0.071 56 0.153 27 4.436 57 9-364 27 0.074 57 o. 156 23 4.600 58 9-528 28 0.077 58 0.159 29 4.764 59 9.692 29 0.079 59 o. 161 . 30 4.928 60 9-856 30 0.082 60 o. 164 The quantities taken from this table must be added to a mean interval to obtain the corresponding interval in sidereal time. I 9 2 ASTRONOMY. To convert Parts of the Equator in Arc into Sidereal Time, or to convert terrestrial Longitude in Arc into Time. Arc. Time. Arc. Time. Arc. Time. Arc. ! Time. h. 111. o h. m. //. m. //. 7/7. I o 4 31 2 4 61 4 4 91 6 4 2 o 8 32 2 8 62 4 8 92 6 S 3 12 33 2 12 63 4 12 93 6 12 4 o 16 . 34 2 l6 64 4 16 94 6 16 5 20 35 2 20 65 4 20 95 6 20 6 o 24 ; 36 2 24 66 4 24 96 6 24 7 o 28 37 2 28 67 4 2S 97 6 28 8 o 32 33 2 32 68 4 32 98 6 32 9 o 36 39 2 3 6 69 4 36 99 6 36 10 o 40 ! 4 2 40 70 4 40 100 6 40 ii o 44 41 2 44 71 4 44 101 6 44 12 o 48 42 2 4 8 72 4 48 102 6 48 13 o 52 43 2 52 73 4 52 103 6 52 M o 56 44 2 56 74 4 56 104 6 56 15 I 45 3 o 75 5 o 105 7 o 16 i 4 46 3 4 76 5 4 1 06 7 4 17 i 8 47 3 8 77 5 8 107 7 8 18 I 12 48 3 12 78 5 12 108 7 12 IQ i 16 49 3 16 79 5 16 109 7 16 20 I 20 50 3 20 80 5 20 no 7 20 21 I 24 5* 3 24 Si 5 24 III 7 24 22 I 28 52 3 28 82 5 28 112 7 28 23 I 32 53 3 32 83 5 32 113 7 3 2 24 i 36 ! 54 3 36 84 5 36 114 7 36 25 I 40 55 3 40 85 5 40 H5 7 40 26 i 44 56 3 44 86 5 44 116 7 44 27 i 48 57 3 48 87 5 48 117 7 48 28 i 52 58 3 52 88 5 52 118 7 52 29 i 56 59 3 56 89 5 56 119 7 56 30 2 O 60 4 o go 6 o I2O 8 o SPACE INTO TIME. 193 To convert Parts of the Equator in Arc into Sidereal Time, or to convert 7errestrial Longitude in Arc into Time Continued. DEGREES. Arc. Time. Arc. Time. Arc. Time. Arc. Time. h. m. h. tn. h. in. h. DI. 121 8 4 151 10 4 181 12 4 211 14 4 122 8 8 152 10 8 182 12 8 212 14 8 123 8 12 153 10 12 183 12 12 213 14 12 124 8 16 154 10 16 184 12 l6 214 14 16 125 8 20 155 10 20 185 12 2O 215 14 20 126 8 24 156 10 24 1 86 12 24 216 14 24 127 8 28 157 10 28 187 12 28 217 14 28 128 8 32 158 10 32 iSS 12 32 218 14 32 129 S 36 159 10 36 189 12 36 219 14 S^ ,30 131 132 8 40 8 44 8 48 160 161 162 10 40 10 44 10 48 190 12 40 191 12 44 192 12 48 22O | 221 222 14 40 14 44 14 48 133 8 52 163 10 52 193. 1 12 52 223 14 52 134 8 56 164 10 56 194 12 56 224 M S^ 135 9 o 165 II 195 13 o 225 15 I 3 6 9 4 1 66 ii 4 196 13 4 1 226 15 4 137 9 8 167 IT 8 197 13 8 227 15 8 133 9 12 168 II 12 I 9 8 13 12 228 15 12 139 9 16 169 ii 16 199 13 16 229 15 16 140 9 20 170 II 20 2OO 13 20 230 15 20 141 9 24 171 II 24 201 13 24 231 15 24 142 9 28 172 II 28 202 13 28 232 15 28 143 9 32 173 II 32 203 13 32 233 15 32 144 9 36 174 II 36 2O4 13 36 234 15 36 145 9 . 40 175 II 40 205 13 40 235 15 40 I 4 6 9 4-1 176 ii 44 206 13 44 2 3 6 15 44 147 9 48 177 ii 48 207 13 48 237 15 48 148 9 52 178 ii 52 208 13 52 2 3 8 15 52- 149 9 56 179 ii 56 209 13 56 239 15 56 150 10 1 80 12 O 2IO 14 o 240 16 o 1 94 ASTRONOMY. To convert Parts of the Equator in Arc into Sidereal Time, or to convert Terrestrial Longitude in Arc into Time Continued. DEGREES. Arc. Time. Arc. Time. Arc. Time. Arc. Time. o //. m. h. in. //. ;;/. h. m. 241 16 4 271 18 4 301 20 4 331 22 4 242 16 8 272 18 8 302 20 8 332 22 8 243 16 12 273 IS 12 303 20 12 333 22 12 244 16 16 274 18 16 304 20 16 334 22 l6 245 16 20 275 IS 20 305 2O 2O 335 22 20 246 16 24 276 18 24 306 20 24 336 22 24 247 16 28 277 18 28 307 2O 28 337 22 28 2 4 S 16 32 2.78 18 32 ! 308 20 32 333 22 32 249 16 36 279 18 36 309 20 36 339 22 36 250 16 40 280 1 8 40 1 310 20 40 340 22 40 251 16 44 281 18 44 ! 3I1 20 44 34i 22 44 252 16 48 L 282 18 48 : 312 20 48 342 22 48 253 16 52 283 ia 52 313 20 52 343 22 52 254 16 56 284 18 56 3i4 20 56 344 22 56 255 17 ! 2S5 19 o 3i5 21 o 345 23 o 256 I 17 4 286 19 4 316 21 4 346 23 4 257 ; 17 8 287 19 8 ; 317 21 S 347 23 8 258 17 12 288 19 12 li 318 21 12 348 23 12 259 17 16 ! 289 19 16 319 21 l6 349 23 16 260 ; 17 20 290 19 20 320 21 2O 350 23 20 261 17 24 291 I 9 24 321 21 24 35i 23 24 262 17 28 292 19 28 322 21 28 352 23 28 263 17 32 293 19 32 323 21 32 353 23 32 264 17 36 294 19 36 324 21 36 354 23 36 265 17 40 295 19 40 325 21 40 355 23 40 266 17 44 296 19 44 326 21 44 356 23 44 267 17 48 297 19 48 327 21 48 357 23 48 268 17 52 298 19 52 328 21 52 358 23 52 269 17 56 299 19 56 329 21 56 359 23 56 270 18 o 300 20 i 330 22 360 24 o SPACE INTO TIME. To convert Parts of the Equator in Arc into Sidereal Time, or to convert Terrestrial Longitude in Arc into Time Continued. MINUTES. SECONDS. Arc. Time. Arc. Time. Arc. Time. Arc. Time. in. s. m. s. " s. " s. i o 4 3i 2 4 i 0.067 3i 2.067 2 o 8 32 2 S 2 0.133 32 2.133 3 12 33 2 12 3 0.200 33 2.200 4 o 16 34 2 l6 4 0.267 34 2.267 5 o 20 35 2 2O 5 0-333 35 2-333 6 o 24 36 2 24 6 O.40O 36 2.400 7 o 28 37 2 28 7 0.467 37 2.467 8 o 32 33 2 32 8 0-533 38 2-533 9 o 36 39 2 36 9 0.600 39 2 . 6()O 10 o 40 40 2 4 10 0.667 40 2.667 ii o 44 4i 2 44 ii 0.733 41 2-733 12 o 48 42 2 4 8 12 0.800 42 2.800 13 o 52 43 2 52 13 0.867 43 2.867 14 o 56 44 2 56 14 0-933 44 2-933 15 I 45 3 o 15 I. 000 45 3.000 16 17 i 4 I 8 46 47 3 4 3 8 16 17 1.067 1.133 46 47 3.067 3.133 18 I 12 48 3 12 18 I.2OO 48 3.20O 19 i 16 49 3 16 IQ 1.267 49 3.267 20 I 20 50 3 20 20 1-333 50 3o33 21 I 24 51 3 24 21 1.400 5i 3.400 22 I 28 52 3 28 22 1.467 52 3.467 23 I 32 53 3 32 23 1-533 53 3-533 24 I 36 54 3 36 24 i . 600 54 3 - 600 25 I 40 55 3 40 25 1.667 55 3.667 26 27 i 44 i 48 56 57 3 44 3 48 26 27 1-733 i . 800 56 57 3-733 3.800 23 i 52 53 3 52 28 1.867 58 3-867 2 9 i 56 59 3 56 29 1-933 59 3-933 30 2 O 60 4 o 30 2.OOO 60 4.000 196 ASTRONOMY. To convert Sidereal Time into Parts of the Equator in Arc, or to convert Time into Terrestrial Longitude in Arc. HOURS. MINUTES. SECONDS. Time. Arc. Time. Arc. Time. Arc. Time. Arc. Time. Arc. h. o m. , in. , s. , s. , i 15 i o 15 31 7 45 I o 15 31 7 4 5 2 30 2 o 30 32 8 o 2 o 30 32 8 o 3 45 3 o 45 33 8 15 3 o 45 33 8 15 4 60 4 I O 34 S 30 4 1 34 8 30 5 75 5 i 15 35 8 45 5 i 15 35 8 45 6 90 6 i 30 36 9 O 6 i 30 36 9 o 7 105 7 i 45 37 9 15 7 i 45 37 9 15 8 1 20 8 2 38 9 30 S 2 O 3S 9 30 . 9 135 9 2 15 39 9 45 9 2 15 39 9 45 10 150 10 2 30 40 10 o 10 2 3O 40 10 ii 165 ii 2 45 41 10 15 ii 2 45 4i 10 15 12 180 12 3 o 42 10 30 12 3 o 42 10 30 13 195 13 3 15 43 10 45 13 3 15 43 10 45 14 210 14 3 30 44 II O 14 3 30 44 II O 15 225 15 3 45 45 ii 15 15 3 45 45 ii 15 16 240 16 4 o 46 ii 30 16 4 o 46 ii 30 17 255 17 4 15 47 ii 45 17 4 15 47 ii 45 18 270 18 4 30 48 12 O 18 4 30 48 12 19 285 19 4 45 49 12 15 19 4 45 49 12 15 20 3OO 20 50 50 12 3O 20 5 o 50 12 30 21 315 21 5 15 5i 12 45 21 5 15 5i 12 45 22 330 22 | 5 30 52 13 o 22 5 30 52 13 o 23 345 23 5 45 53 13 15 23 5 45 53 13 15 24 360 24 6 o 54 13 30 24 6 o 54 13 30 25 6 15 55 13 45 25 6 15 55 13 45 26 6 30 56 14 o 26 6 30 56 14 o 27 6 45 57 14 15 27 6 45 57 14 15 28 7 o 58 14 30 28 7 o 58 14 30 29 7 15 59 14 45 29 7 15 59 14 45 30 7 30 60 15 o 30 7 30 60 15 o TIME INTO SPACE. I 9 7 To convert Sidereal Time into Parts of the Equator in Arc, or to convert lime into terrestrial Longitude in Arc Continued. TENTHS OF SECONDS. Time. Arc. Time. Arc. Time. Arc. Time. Arc. s. ,i s. s. " s. " o.oi 0.15 0.31 4.65 0.61 9-15 0.91 13-65 O.O2 0.30 | 0.32 4.80 0.62 9-30 0.92 13.80 0.03 0.45 0.33 4-95 0.63 9.45 0-93 13-95 0.04 0.60 0.34 5-10 o . 64 9 . 60 0.94 . 14.10 0.05 0.75 0.35 5-25 0.65 9.75 0-95 14-25 O.O6 0.90 0.36 5-40 0.66 ; 9.90 0.96 14.40 0.07 1.05 0.37 5-55 0.67 i 10.05 0.97 14-55 0.08 1.20 0.38 5-70 0.68 10.20 0.98 14.70 O.Og 1-35 0.39 5- 8 5 0.69 10.35 0-99 14.85 O. IO 1.50 0.40 6.00 , 0.70 i 10.50 i .00 1 5 . oo O. II 1.65. 0.41 6.15 0.71 i 10.65 _ 0.12 i. So 0.42 .6.30 0.72 10.80 1 0.13 1-95 0.43 6.45 0.73 , 10.95 c 0.14 2.10 O.44 6.6O ! O.74 II. IO T3 O 0.15 2.25 0.45 r -75 0.75 ; 11.25 s t-t-l Arc. 0.16 2.40 o . 46 6 . 90 0.76 11.40 o CO 0.17 2.55 0.47 7.05 o.77 ; H.55 1 o.iS 2.70 o . 48 7 . 20 0.78 i 11.70 o 0.19 2.85 0.49 7-35 0.79 11.85 H o . 20 3 . oo 0.50 7.50 O.8O ; 12. OO s. 0.21 3.15 0.51 7.65 0.81 ; 12.15 O.OOI 0.015 0.22 3.30 0.52 7.80 0.82 12.30 O.OO2 0.030 0.23 3.45 0.53 7-95 0.83 12.45 0.003 0.045 0.24 3 .60 0.54 8.10 0.84 12.60 O.OO4 0.060 0.25 3-75 0.55 ; 8.25 0.85 12.75 0.005 0.075 O.26 3-90 O.56 8.4O 0.86 12.90 O.OO6 o.ogo O.27 4-05 0.57 8.55 0.87 13-05 O.OO7 0.105 0.28 4.20 6.58 8.70 0.88 13.20 q.ooS O. I2O 0.29 4-35 0.59 8 - 8 5 0.89 ^3-35 o.oog 0.135 0.30 4-50 o . 60 9 . oo 1 0.90 13-50 O.OIO 0.150 198 ASTRONOMY. To convert Right Ascension in Arc into Mean Time. DEGREES. AR. in arc. Mean 1 time. AR. n arc. Mean 1 time. AR. in arc. Mean time. /, m. s. /, in. s. h. m. s. i o 3 59-345 | 31 2 3 39.686 61 4 , 3 20.027 2 i o 7 58,689 32 2 7 39-030 62 4 7 I9.37I 3 o ii 58.034 | 33 2 ii 38.375 63 4 ii 18.716 4 o T5 57-379 34 2 15 37.720 64 4 15 18.061 5 o T 9 56.724 35 2 19 37.064 65 4 19 17-405 6 o 23 56.068 36 2 23 36.409 66 4 23 16.750 7 : o 27 55-413 37 2 27 35-754 67 4 27 16.095 8 : o 3i 54.758 38 2 3i 35-099 68 4 3i 15-639 9 o 35 54.102 39 2 35 34-443 1 4 35 14.784 10 o 39 53-447 40 2 39 33.788 70 4 39 14.129 ii o 43 52.792 41 2 43 33-133 . 7i 4 43 13-474 12 o 47 52.136 42 2 47 32.477 72 4 47 12.818 13 o 51 51.481 43 2 5i 31.822 73 4 51 12.163 14 55 50.826 44 2 55 31.167 74 4 55 11.508 15 59 50.170 45 2 59 30.5H 75 4 59 10.852 16 I 3 49.515 46 3 3 29.856 76 5 3 10.197 17 I 7 48.860 47 3 7 29.201 77 5 7 9-542 18 I ii 48.205 48 3 ii 28.545 78 5 ii 8.886 19 I 15 47-549 49 3 15 27.890 79 5 15 8.231 20 I 19 46.894 50 3 T9 27-235 80 5 19 7.576 21 I 23 46.239 51 3 23 26.580 81 5 23 6.920 22 I 27 45.583 52 3 27 25.924 i 82 5 27 6.265 23 I 3i 44.928 53 3 3i 25-269 83 5 3i 5.610 24 I 35 44-273 54 3 35 24.614 84 5 35 4-955 25 1 39 43.617 55 3 39 23.958 85 5 39 4.299 26 I 43 42.962 56 3 43 23-303 86 5 43 3.644 27 I 47 42.307 57 3 47 22.648 87 5 47 2.989 28 I 51 41.652 ! 58 3 5i 21.992 88 5 51 2-333 2 9 I 55 40.996 .59 3 55 21-337 89 5 55 1.678 30 I 59 40.341 | 60 3 59 20.682 90 5 59 1.023 RIGHT ASCENSION IN ARC INTO MEAN TIME. I 99 To convert Right Ascension in Arc into Mean Time Continued. DEGREES. AR. in arc. Mean time. AR. in arc. Mean time. AR. in arc. Mean time. Ji. m. s. o h. m. s. o //. in. s. 9* 6 3 0.367 121 8 2 40.708 151 IO 2 2I.O49 92 6 6 59-712 122 8 6 40.053 152 10 6 20.394 93 6 10 59.057 123 8 10 39.398 153 10 10 19.738 94 6 14 58.401 124 S 14 38.742 154 10 14 19.083 95 6 is 57-746 : 125 8 18 38.087 155 10 18 18.428 96 6 22 57.091 126 8 22 37-432 156 10 22 17.773 97 6 26 56.436 127 8 26 36.7/6 157 10 26 17.117 98 6 30 55.730 128 8 30 36.121 158 10 30 16.462 99 6 34 55.125 129 8 34 35.466 ! 159 10 34 15.807 IOO 6 38 54-470 130 8 38 34.810 , 160 10 38 15.151 IOI 6 42 53-SI4 131 S 42 34.155 j| 161 10 42 14.496 102 6 46 53-159 132 8 46 33.500 162 j 10 46 13.841 103 6 50 52.504 133 8 50 32.845 ! 163 10 50 13.185 104 6 54 51-848 134 8 54 32.189 164 10 54 12.530 105 6 58 5LI93 135 8 58 31-534 165 10 58 11.875 I O6 7 2 50.538 136 9 2 30.879 166 II 2 11.220 107 7 6 49.883 137 9 6 30.223 167 ii 6 10.564 108 7 10 49.227 138 9 10 29.568 168 ii 10 9.909 109 7 14 48.572 139 9 14 28.913 169 II 14 9.254 no 7 18 47-9*7 140 9 18 28,257 170 ii 18 8.598 III 7 22 47.261 141 9 22 27.6O2 171 II 22 7.943 112 7 26 46.606 142 9 26 26.947 172 II 26 7.288 H3 7 30 45- 95i _ 143 9 30 26.292 173 II 30 6.632 114 7 34 45-295 144 9 34 25.636 J74 ii 34 5-977 H5 7 38 44.640 145 9 38 24.981 175 ii 38 5-322 116 7 42-43.985 I 4 6 9 42 24.326 176 n 42 4.666 117 7 46 43-329 147 9 46 23.670 177 IT 46 4.011 118 119 7 50 42.674 7 54 42.019 I 4 8 I 49 9 50 23.015 9 54 22. -360 178 179 ii 50 3.356 ii 54 2.701 1 20 7 58 41.364 il 150 9 58 21.704 i So ii 58 2.045 200 ASTRONOMY. To convert Right Ascension in Arc into Mean Time Continued. MINUTES. SECONDS. AR. Mean AR. Mean AR. Mean AR. Mean in arc. time. in arc. time. in arc. time. in arc. time. ;;/. s. m. s. s. a s. i o 3.9 8 9 3i 2 3.661 i 0.066 31 2.061 2 o 7.978 32 2 7-650 2 0.133 32 2.128 3 o 11.969 33 2 II.64O 3 0.199 33 2.194 4 o 15.956 34 2 15.629 4 0.266 34 2.261 5 o 19.945 35 2 19.618 5 0.332 35 2.327 6 o 23.935 36 2 23.607 6 0-399 36 2-393 7 o 27.924 37 2 27.596 7 0.465 37 2.460 8 31-9*3 38 2 3L585 8 0.532 38 2.526 9 o 35-902 39 2 35-574 9 0.598 39 2-593 10 o 39.891 40 2 39.563 10 0.665 40 2.659 j ii o 43.880 41 2 43-552 ii 0.731 4i 2.726 12 o 47.869 42 2 47-541 12 0.798 42 2.792 13 o 51.858 43 2 5L530 13 0.864 43 2.859 14 o 55.847 44 2 55.519 14 0.931 44 2.925 15 o 59-836 45 2 59.509 15 0.997 45 2.992 16 I 3.825 46 3 3.498 16 1.064 46 3.058 i? I 7.814 47 3 7.487 17 1.130 47 3- I2 5 18 I 11.803 48 3 H.476 18 1.197 48 3-191 19 I 15.793 49 3 15-465 19 1.263 49 3.258 20 I 19.782 50 3 19-454 20 1-330 50 3.324 21 I 23.771 5i 3 23.443 21 1.396 5i 3.391 22 I 27.760 52 3 27.432 22 1.463 52 3-457 23 I 31.749 53 3 3L42I 23 1-529 53 3-524 24 I 35.738 54 3 35-410 24 1.596 54 3-59 25 I 39.727 55 3 39-399 25 1.662 55 3-657 26 I 43.716 56 3 43.388 - 26 1.729 56 3.723 27 I 47.705 I 57 3 47.377 27 1-795 57 3.790 23 I 51.694 58 3 5L367 28 1.862 58 3.856 29 I 55.683 59 3 55.356 29 1.928 59 3-923 30 I 59.672 60 3 59-345 30 1-995 60 3.989 MEAN TIME INTO RIGHT ASCENSION IN ARC. 201 To convert Mean Time into Right Ascension in Arc. HOURS. MINUTES. I Mean time. AR. in arc. Mean time. AR. in arc. Mean time. AR. in arc. h. c m. , m. , i 15 2 27.85 i O 15 2 . 46 31 7 46 16.39 2 30 4 52.69 2 o 30 4.93 32 8 i 18.85 3 45 7 23.54 3 o 45 30.39 33 8 16 21.31 4 60 9 51-39 4 i o 9.86 34 8 31 23.78 5 75 12 19.24 5 I 15 12.32 35 8 46 26.24 6 90 14 47-oS 6 I 30 14.79. 36 9 i 28.71 7 105 17 14.93 7 i 45 17-25 37 9 16 31.17 8 120 19 42.78 8 2 O 19.71 33 9 3i 33-64 9 135 . 22 10.62 9 2 15 22.18 39 9 46 36.10 10 150 24 38.47 10 2 30 24.64 40 10 i 38.57 ii 165 27 6.32 ii 2 45 27.11 41 10 16 41.03 12 180 29 34.16 12 3 o 29.57 42 10 31 43-39 13 195 32 2.01 13 3 15 32.03 43 10 46 45.96 14 210 34 29.86 14 3 30 34-50 44 ii i 48.42 15 225 36 57.70 15 3 45 SM 6 45 ii 16 50.89 16 240 39 25.55 16 4 o 39.43 46 ii 3i 53-35 17 255 41 53.40 17 4 15 41.89 47 ii 46 55.81 18 270 44 21.24 18 4 30 44.35 48 12 I 58.38 19 285 46 49.09 19 4 45 46.82 49 12 17 0.74 20 300 49 16.94 20 5 o 49.28 50 12 32 3.21 21 315 51 44.78 21 5 15 5L75 5i 12 47 5-57 22 330 54 12.63 22 5 30 54-21 52 13 2 8.13 23 345 56 40.48 23 . 5 45 56.67 53 13 17 10.60 24 360 59 8.33 24 6 o 59.14 54 13 32 13.06 25 6 16 i. 60 55 13 47 15-53 26 6 31 4-7 56 14 2 17.99 27 6 46 6.53 57 14 17 20.45 28 7 i 9 . oo 58 14 32 22.92 2 9 7 16 11.46 59 M 47 25.38 30 7 3i 13.92 60 15 2 27.85 ASTRONOMY. To convert Mean Time into Right Ascension in Arc Continued. SECONDS. TENTHS OF SECONDS. Mean time. AR. in arc. Mean time. AR. in arc. Mean time. AR. in arc. Mean time. AR. in arc. s. ; // s. / a s. ,, s. ;/ i o 15.04 3i 7 46.27 O.OI 0.15 0.31 4.66 2 o 30.08 32 S 1.31 O.O2 0.30 0.32 4.81 3 o 45.12 33 8 16.36 O.O3 0.45 0.33 4.96 4 i o. 16 34 8 31.40 O.O4 0.60 0-34 5-12 5 i 15.21 35 8 46.44 O.O5 0-75 0-35 5-27 6 i 30.25 36 9 1.48 0.06 0.90 0.36 5.42 7 i 45.29 37 9 16.52 O.O7 1.05 0-37 5-57 8 2 0.33 3 8 9 31.56 0.08 1.20 0.38 5.72 9 2 15.37 39 9 46.60 O.Og 1-35 0.39 5.87 10 2 30.41 40 10 1.64. O. IO 1.50 0.40 6. 02 ii 2 45-45 4i 10 16.68 O.II 1.6 5 0.41 6.17 12 3 0.49 42 10 31.73 O. 12 i. Si 0.42 6.32 13 3 15-53 43 10 46.77 0.13 1.96 0-43 6.47 14 3 30.53 44 ii i. Si 0.14 2. II 0.44 6.62 15 3 45- 62 45 ii 16.85 0.15 2.26 0.45 6.77 16 4 0.66 46 ii 31.89 o. 16 2.41 0.46 6.92 17 4 15.70 47 ii 46.93 0.17 2.56 0.47 7-07 18 4 30.74 48 12 1.97 0.18 2.71 0.48 7.22 19 4 45.73 49 12 17.01 0.19 2.86 0.49 7-37 20 5 0.82 50 12 32.O5 O.2O 3.01 0.50 7-52 21 5 15- 86 5i 12 47.09 0.21 3-16 0.51 7.67 22 5 30.9 52 13 2.14 O.22 3-31 0.52 7.82 23 5 45-94 53 13 17.18 0.23 3.46 0-53 7-97 24 6 i. oo 54 13 32.22 O.24 3-61 0-54 8.. 1 2 2 5 6 16.03 55 13 47.26 0.25 3.76 0-55 8.27 26 6 31.07 56 14 2.30 0.26 3-91 0.56 8.43 27 6 46.11 57 14 17-34 0.27 4.06 0-57 8.58 23 7 i.i5 58 M 32.38 0.28 4.21 0.58 8.73 29 7 16.19 59 14 47.42 O.29 4.36 0-59 8.SS 30 7 31-23 60 15 2 . 46 0.30 4-51 0.60 9-3 MEAN TIME INTO RIGHT ASCENSION IN ARC. To convert Mean Time into Ri?ht Ascension in Arc Continued. TENTHS OF SECONDS. THOUSANDTHS OF SECONDS. Mean time. AR. in arc. Mean time. AR. in arc. Mean time. AR.in arc. Mean time. AR. in arc. s. " s. " s. s. 0.61 9.18 0.76 11-43 0.91 13.69 O.OOI 0.02 0.62 9-33 0.77 11.58 0.92 13.84 O.OO2 0.03 0.63 9.48 0.78 11.74 0-93 13.99 0.003 0.05 0.64 9-63 0.79 11.89 0.94 14.14 O.OO4 0.06 0.65 9.78 0.80 12.04 0.95 14.29 0.005 o.oS 0.66 9-93 o.Si 12. 19 0.96 14.44 O.OO6 0.09 0.67 10.08 0.82 12.34 0.97 14.59 0.007 O.II 0.68 10.23 0.83 12.49 0.98 14.74 O.OO8 O. 12 0.69 10.38 0.84 12.64 0.99 14.89 O.OOg 0.14 0.70 10.53 0.85 12.79 i .00 15-05 O.OIO 0.15 0.71 10.68 o.S6 12.94 0.72 10.83 0.87 13.09 0.73 10.98 0.88 13-24 0.74 11.13 0.89 13.39 0-75 11.28 0.90 13-54 CONSTANT LOGARITHMS. Logarithms. 12 hours, expressed in seconds 43200 4 6 / 3O8 37 Complement to the same 0000231 5 c 05 j.^ 16 ^ 24 hours, expressed in seconds = Complement to the same 86400. ooooi 157 4.9365137 c o6 3J.S6 3 360 degrees, expressed in seconds 1296000 6 1126050 To convert sidereal time into mean solar time 9 9988126 204 ASTROXOMY. FORM FOR SURVEY OF DETERMINATION or TIME, DATE AND STATION. 1843, October 13. Month of the Big Black River, C Sextant No. 2197, by Troitghton & Simms, and INSTRUMENTS ... Mean Solar Chronometer No. 76, by Charles 2 s *o ^ i ^ . o ^o o ^ in , o *" M u- & | * 55 o J 3 "o c *J* ~ .2 O -*- "^j Names of stars. <U "-3 ^ Hill u, i ! fj o .^ *5 ?-* *+J <L> i si <U "1> <j> V- O c/5 | ^ y c S o rt ^ S^ 8** ^ * H / // / // h. m. s. /. ;;/. J. 91 43 40 45 52 58. 8 7 5 47- 69 6 57 02.4 92 18 oo 46 10 09.3 7 07 28.67 6 58 43. 2 92 41 15 46 21 47.3 7 08 37. 15 6 59 52.8 a Andromeda, 93 04 05 46 33 12.6 7 9 44.37 7 oo 59. 6 (cast.} 93 45 20 94 13 45 46 53 50. 8 47 08 03. 7 7 ii 45.92 7 13 9- 73 7 03 01.2 7 04 25. 6 94 40 50 47 21 36.6 7 14 29.64 7 05 45- 95 07 2 5 47 34 54- 5 7 15 48. H 7 7 03- 6 Mean result of 8 observations on a Andromeda, in the # T 7 . 95 20 05 47 41 14. 7 //. ;;/, s. 8 55 32.36 n. in, s. 8 46 49. 2 95 oo oo 47 3i ii. 6 8 56 32.06 8 47 50.4 94 3 40 47 16 31-2 8 57 59.42 8 49 1 6. a Lyra 94 12 20 47 07 21. 8 58 54- 8 50 10.8 (-vest } 93 53 45 46 58 03. i 8 59 49.4 8 51 06.9 93 29 20 46 45 50. 2 9 01 02. i 8 52 19.4 93 07 35 46 34 57-3 9 02 07. 8 53 24.8 92 46 50 46 24 34. 5 9 3 9- 8 54 26. 92 28 45 46 15 3L7 9 04 02. 96 8 55 21.2 Mean result of nine observations on the star a Lyra, in the west Mean result of eight observations on the star a Andromeda, in the east, as above . CHRONOMETER ERROR. Slow of mean solar time at 8 h /. m. t by a mean of these results from east and west stars TIME BY OBSERVED ALTITUDES. RECORD AND COMPUTATION. by Observed Double Altitudes of East and West Stars. a tributary to the river Saint John, Maine. artificial horizon of Mercury. Younsr. Chronometer (C. Y. 76) slow of mean solar time by each observation. Remarks. Index error of sextant -f- 2 40" O 08 4^ 20 Error of eccentricity of sextant -1- I 32 8 45-47 8 44-35 8 44. 77 Thermometer, 31. 5 Fahrenheit. Barometer, 29.14 inches. Apparent right ascension of star o ll oo m 2i s . 72 8 44 <;6 Apparent declination of star 28 13 59". 5 X. 8 44.13 8 44. 64 8 44. 54 Apparent north polar distance of star . = 61 46 oo .5 = A Approximate latitude of this station .. 46 57 oo N. = L Approximate longitude of this station . = 4 h 37 m 47 s Sidereal time of mean noon at station . = 13 26 20 .83 o h o8 m 44 s . 74 //. ;;/. s. o 08 43. 16 8 41.66 8 43. 42 8 43. 20 8 42. 50 8 4.2 7O Thermometer, 29 Fahrenheit. Barometer, 29. 14 inches. Apparent right ascension of star = i8 b 31" 39". 16 Apparent declination of star north = 38 38 46". 5 Apparent north polar distance of star . = 5* 2I ! 3 -5 A Index error of sextant ...... ...... ~ {- 2 ; 40" 8 42 20 Error of eccentricity of sextant -f- I 32 8 43. oo 8 41. 76 o h o8 m 42 s .6 o 08 44.7 o h o8 m 43 s .6 Observer, Major J. D. Graham. Computer, Private F. Ilcrbst. 206 ASTRONOMY. Computation of the Fifth of the Preceding Altitudes of a. An dromeda, (formula, page 187.) Observed double altitude. Index error, sextant Eccentricity, sextant = 93 45 20" = + 02 40 = 4- 01 32 Double altitude corrected Altitude Refraction, (thermom., 31. 5 ;barom.. 29. i ).. = 93 49 32 46 54 4 6 56 .6 True altitude of * = A = 46 53 49 .4 cos = sin = cos L sin J = 9.8341894 9.9449899 9-779 I 793 L= 4 657 / J = 61 46 oo".5 A = 46 53 49.4 2 w = 155 3 6 49-9 ;;/ = 77 48 24.4 (;;/ A) = 30 54 35.0 sin- J p = \p 25o8 / oi // .5 p in arc = 50 16 03 .o (page 192) p in time = 3 21 04 .20 AR. * 24 oo 21 .72 Sidereal time of observation. . . = AR. / = 20 39 17. 52 Sidereal time, mean noon, at place, (nauti cal almanac) = 13 26 20 - 8 3 cos COS ;// sin m sin cos =. sin = (m - A) = (m A) 9.3247127 9.7106984 9.0354111 19.2562318 9.6281159 cos L sin A sin 4/ = Sidereal interval past mean noon . Retardation of mean on sidereal interval, (page 190) Mean solar interval past mean noon, or mean time p. m. of observation Time of observation by chronometer "Chronometer slow . . 7 I2 5 6 - 6 9 = 01 10 .93 7 II 45 -7 6 7 03 01 .20 8 44 .56 OBSERVATIONS FOR THE TIME. 207 L. To Find the Time by Equal Altitudes of the Sitn. Correction in time, to be applied as an equation to the mean of the times of observed equal altitudes of the sun, in order to obtain the time of its meridional passage : T T x = o tan D - T- d tan L 30 tan 7 J T " 30 sin 7^ T Make T T = A ; - ., rr == n 30 sin 7 T " 30 tan 7^ x = =p A d tan L + B d tan D Apparent noon = J (t -f / ) + x /, / = the times of observations ; T = (t /) = the interval of time between the observations, expressed in hours and decimals ; L = the latitude of the place of observation, (myius when south;) D = the sun s declination at apparent noon on the given day, (mums when south :) d = the hourly variation in the declination at noon, (minus when the sun is proceeding toward the south ;) and x = required correction in seconds, where A is to be minus- where the time of noon is required and plus where the time of midnight is required, i. e., when the first ob servation is made in the afternoon and the correspond ing one the morning following. Logarithmic values of A and B are given in the following tables. 208 ASTRONOMY. Equations to Equal Altitudes. Interval. Log A. LogB. Interval. Log A. Log B. h. in. h. in. 2 O 9.4109 9- 3959 3 o 9.4172 9- 3828 2 .4111 3955 2 .4174 .3822 4 4113 3952 4 .4177 .3817 6 .4114 .3948 6 .4179 .3811 8 .4116 3944 8 .4182 .3806 10 .4118 3941 10 .4184 . 3800 12 . 4120 3937 12 .4187 3794 14 . 4121 3933 H .4190 .3789 16 .4123 3929 16 4193 .3783 18 4125 .3925 18 .4195 3777 20 .4127 3921 20 .4198 3771 22 .4129 39 1 ; 22 .4201 .3765 24 4I3 1 .3913 24 .4204 3759 26 .4133 .3909 26 .4207 3752 28 .4135 .3905 28 .4209 .3746 3 .4137 .3900 30 . 4212 3740 32 .4139 .3896 32 .4215 3733 34 .4141 .3892 34 .4218 .3727 36 .4144 3887 36 .4221 .3720 38 .4146 .3882 ^o .4224 .3713 40 .4148 .3878 40 4227 .3707 42 .4150 .3873 42 .423 1 .3700 44 4152 .3868 44 .4234 3 6 93 46 4155 . 3863 46 .4237 .3686 48 .4157 .3859 48 .4240 3679 50 .4159 .3854 50 .4243 .3672 52 .4162 .3849 52 .4246 .3665 54 .4164 .3843 54 .4250 .3657 56 .4167 3838 56 .4253 3650 2 5 8 9.4169 9- 3833 3 58 9.4256 9- 3 6 43 x= =p A o tan L + B 5 tan D OBSERVATIONS FOR THE TIME. 209 Equations to Equal Altitudes Continued. Interval. Log A. LogB. Interval. Log A. LogB. h, m. h. m. 4 o 9. 4260 9- 3 6 35 5 o 9- 4374 9- 33 6 9 2 .4263 .3627 2 4378 3358 4 .4266 . 3620 4 .4383 .3348 6 .4270 3612 6 .4387 3337 8 4273 .3604 8 4391 .3327 10 .4277 3596 10 .4396 .33i6 12 .4280 .3588 12 .4400 335 H .4284 .358o H 4405 3294 16 .4288 3572 16 .4409 3283 18 .4291 35 6 4 18 .44H .3272 20 .4295 3555 20 .4418 3261 22 .4299 3547 22 4423 3249 24 .4302 3538 24 .4427 3238 26 .4306 3530 26 4432 .3226 28 .4310 3521 28 4437 3214 30 43 4 3512 30 .4441 3203 32 43 J 7 3503 32 . .4446 3!9i 34 .4321 3494 34 .4451 .3178 36 4325 .3485 36 4456 .3166 38 4329 3476 38 .4460 3 I 54 40 4333 3467 40 4465 . .3142 42 4337 3457 42 .4470 .3129 44 4341 .3448 44 4475 .3116 46 4345 .3438 46 .4480 3103 48 4349 3429 48 4485 .3091 5 4353 3419 5o .4490 .3078 52 4357 .3409 52 4494 .3064 54 .4361 3399 54 .4500 3051 56 .4366 3389 56 4505 3038 4 58 9-4370 9- 3379 5 58 9.4510 9- 324 x = ^ A d tan L -f B d tan D 210 ASTRONOMY. Equations to Equal Altitudes Continued. Interval. Log A. Log B. Interval. Log A. LogB. h. m. h. m. 6 o 9.4515 9. 3010 7 o 9.4685 9- 2530 2 .4521 .2996 2 .4691 . 2511 4 .4526 .2982 4 .4697 .2492 6 4531 .2968 6 .4704 2473 8 .4536 2954 , 8 .4710 2454 10 4542 .2940 10 .4716 2434 12 4547 .2925 12 .4723 .2415 H 4552 .2911 H .4729 2395 16 .4558 .2896 16 4735 2375 18 45 6 3 .2881 18 .4742 2355 20 .4569 .2866 20 .4748 2334 22 4574 .2850 22 4755 2313 2 4 .4580 2835 24 .4761 . 2292 26 .4585 . 2819 26 .4768 . 2271 28 4591 .2804 28 4774 . 2250 30 4597 .2788 3 .4781 .2228 32 . 4602. .2772 32 .4788 .2206 34 - . 4608 .2756 34 4794 .2184 36 .4614 2739 36 .4801 .2162 38 .4620 .2723 38 .4808 .2140 40 .4625 .2706 40 .4815 .2117 42 .4631 .2689 42 .4821 .2094 44 .4637 .2672 44 .4828 . 2070 46 .4643 2655 46 .4835 2047 48 .4649 .2638 48 .4842 . 2023 5 .4655 . 2620 50 .4849 .1999 5 2 .4661 . 2602 52 .4856 .1974 54 .4667 .2584 54 .4863 .1950 56 .4673 .2566 56 .4870 .1925 6 58 9. 4679 9- 2548 7 58 9. 4877 9. 1900 a? = ^p A d tan L + B d tan D OBSERVATIONS FOR THE TIME. 211 Equations to Equal Altitudes Continued. Interval. Log A. Log B. Interval. Log A. Log B. h. m. h. m. 8 o 9. 4884 9. 1874 9 9.5H5 9- 0943 2 . 4892 .1848 2 5 I2 3 . 0906 4 .4899 . 1822 4 5132 .0867 6 .4906 .1796 6 .5140 .0828 8 49 3 .1769 8 .5148 .0789 10 .4921 .1742 10 .5157 .0749 12 .4928 .1715 12 5 l6 5 .0708 14 4935 .1687 H .5174 .0667 16 4943 .1659 16 .5182 .0625 18 4950 . 1630 18 5 l 9 l 0583 20 .4958 . 1602 20 .5199 .0540 22 .4965 1573 22 .5208 .0496 24 4973 1543 2 4 5217 .0452 26 .4980 .1513 26 5225 . 0406 28 .4988 .1483 28 .5234 .0360 3 .4996 1453 30 5243 .0314 32 5003 .1422 32 5252 .0266 34 . 5011 .1390 34 .526! .0218 36 .5019 1359 36 .5269 .0169 38 .5027 .1327 38 .5278 . .0119 40 535 .1294 40 .5287 .0069 42 .5042 . 1261 42 .5296 . 0017 44 5050 . 1228 44 .5305 8.9965 46 .5058 .1194 46 .5315 .9911 48 .5066 .1159 48 .5324 .9857 50 574 .1125 50 5333 . 9802 52 .5082 .1089 52 5342 9745 54 .5091 .1054 54 5351 .9688 56 599 . 1017 56 .536i .9630 8 58 9.5107 9. 0981 9 58 9. 5370 8.9570 x if A d tan L -+- B d tan D = 2 I 2 ASTRONOMY. Equations to Equal Altitudes Continued. Interval. Log A. LogB. Interval. Log A. Log B. h. tn. /i. m. < 14 o 9. 6841 -9.0971 15 9- 7333 -9.3162 2 .6856 .1057 2 7351 3225 4 .6872 . 1141 4 73 6 9 .3287 6 .6887 . 1224 6 .7386 335 8 .6903 1 36 8 .7404 3411 10 .6919 .1387 10 .7422 3472 12 .6934 .1468 12 .7440 3533 H .6950 .1547 14 .7458 3593 16 .6966 . 1625 16 .7476 .3653 18 .6982 .1703 18 7494 .3713 20 .6998 .1779 20 7512 3772 22 .7014 .1855 22 7531 3831 2 4 .7030 .1930 24 7549 .3889 26 .7047 .2004 26 .7568 3947 28 .7063 .2078 28 .7586 .4005 3 .7079 .2150 30 .7605 .4062 32 .7096 .2222 32 .7624 .4119 34 .7112 .2293 34 . 7642 .4175 % 36 .7129 .2364 36 .7661 .4232 38 .7146 2434 38 .7680 .4288 40 . 7162 .2503 40 .7699 4343 42 .7179 .2571 42 .7718 4399 44 .7196 .2639 44 .7738 4454 46 .7213 .2706 46 7757 459 48 .7230 2773 48 .7776 4563 5 .7247 .2839 5 .7796 .4617 52 7264 .2 9 05 5 2 .7815 .4671 54 . 7281 .2970 54 .7835 .4725 56 .7299 3034 56 .7855 4778 H 58 9.7316 -9. 3098 15 58 9- 7875 -9.4831 x = =p A 8 tan L + B d tan D OBSERVATIONS FOR THE TIME. 213 Equations to Equal Altitudes Continued. Interval. Log A. Log B. Interval. Log A. Log B. h, m. h. m> 16 o 9- 7^95 - 9. 4884 17 o 9- 8539 -9.6383 2 79^5 4937 2 .8562 .6431 4 7935 .4990 4 .8585 .6478 6 7955 .5042 6 .8608 .6526 8 7975 594 8 .8632 .6573 10 .7996 .5146 10 .8655 .6621 12 .8016 .5197 12 .8679 .6668 H .8037 .5248 H .8703 6715 16 .8058 5300 16 .8727 .6762 18 .8078 5351 18 8751 .6809 20 .8099 .5401 20 .8775 .6856 22 .8120 5452 22 .8799 .6903 24 . 8141 5502 24 .8824 .6949 26 .8162 5553 26 . 8848 .6996 28 .8184 5603 28 .8873 7043 30 .8205 .5653 30 .8898 .7089. 32 .8227 .5702 32 .8923 7136 34 .8248 5752 34 .8948 .7182 36 .8270 .5801 36 .8973 .7228 38 .8292 5850 38 .8999 .7275 40 .8314 .5900 40 .9024 .73 2 i 42 .8336 .5948 42 .9050 .7367 44 .8358 5997 44 975 .7413 46 .8380 .6046 46 .9101 7459 48 . 8402 .6094 48 .9127 755 50 .8425 .6143 50 .9154 .7552 5 2 .8447 .6191 52 .9180 7598 54 .8470 .6239 54 . 9206 .7644 56 .8493 .6287 56 .9233 .7690 16 58 9.8516 -9-6335 17 58 9. 9260 -9.7736 x = =p A <5 tan L -f B <5 tan D 214 ASTRONOMY. Equations to Equal Altitudes Continued. Interval. Log A. Log B. Interval. Log A. LogB. h. in. h. m. 18 o 9. 9287 -9.7782 19 o 0.0172 -9.9167 2 93H .7827 2 .0204 .9213 4 9341 .7873 4 .0237 . 9260 6 .9368 .7919 "6 .0270 937 8 .9396 79 6 5 8 . 0303 9355 10 .9424 .8011 10 .0336 .9402 12 945 1 .8057 12 .0370 9449 H 9479 .8103 4 .0403 9497 16 .9508 .8149 16 0437 9544 18 953 6 .8195 18 .0472 9592 20 95 6 4 .8241 20 .0506 .9640 22 9593 .8287 22 .0541 .9687 24 .9622 .8333 24 .0576 9735 26 .9651 .8379 26 .0611 .9784 28 .9680 .8425 28 .0646 .9832 3 .9709 .8471 30 .0682 .9880 32 9739 8517 32 .0718 .9929 34 .9769 8563 34 0754 -9.9977 36 .9798 .8609 36 .0790 o. 0026 38 .9829 .8655 38 .0827 .0075 40 .9859 .8701 40 .0864 . 0124 42 .9889 .8748 42 .0901 .0173 44 .9920 .8794 44 0939 . 0223 46 .9951 .8840 46 .0976 .0272 48 9. 9982 . 8887 48 . 1015 .0322 50 o. 0013 8933 5o i53 .0372 52 .0044 .8980 52 . 1092 .0422 54 . 0076 .9026 54 .1131 0473 56 .0108 .9073 56 .1170 0523 18 58 o. 0140 9.9120 19 58 o. 1209 0.0574 x = =p A <5 tan L -f- B d tan D OBSERVATIONS FOR THE TIME. 215 Equations to Equal Altitudes Continued. Interval. Log A. LogB. Interval. Log A. LogB. //. m. h. m. 20 o o. 1249 o. 0625 21 o. 2623 o. 2279 2 . 1290 .0676 2 .2676 2339 4 .133 .0727 4 .2729 . 2401 6 .1371 .0779 6 .2783 . 2462 8 . 1412 . 0830 8 .2838 .2524 10 1454 .0882 10 .2893 .2587 12 .1496 0935 12 .2949 . 2650 14 .1538 .0987 14 35 .2714 16 .1581 . 1040 16 3 6 3 .2778 18 . 1623 .1093 18 .3120 .2843 20 .1667 . 1146 20 .3179 .2909 22 .1711 . 1 200 22 3238 2975 24 1755 1253 24 .3298 .3041 26 .1799 .1308 26 3359 .3109 28 .1844 . 1362 28 .3420 . .3177 3 .1889 .1417 30 .3482 .3 2 45 32 1935 .1472 S 2 3545 .3315 34 .1981 .1527 34 .3609 .3385 36 .2028 . 1582 36 .3 6 74 .3456 38 .2075 .1638 38 3739 .3527 40 . 2122 .1695 40 3805 3599 42 .2170 .1751 42 .3873 3673 44 .2218 . i8oS 44 3941 3747 46 .2267 .1866 46 . 4010 .3822 48 .2316 .1924 48 .4080 .3897 50 .2366 .1982 5 4151 3974 52 . 2416 .2040 52 .4223 .4052 54 . 2467 .2099 54 .4297 .4130 56 .2518 2159 56 4371 .4210 20 58 o. 2570 o. 2219 21 58 0.4446 0.4291 x = =f A 3 tan L + B d tan D 2l6 ASTRONOMY. Equations to Equal Altitudes Continued. Interval. Log A. LogB. Interval. Log A. LogB. 7z. m. h. m. 22 o- 4523 0.4372 23 o o. 7689 o. 7652 2 .4601 4455 2 .7842 .7807 4 .4680 4540 4 .8000 .7967 6 .4761 .4625 6 .8163 8i33 8 .4^42 .4711 8 8333 8305 10 .4926 4799 10 .8508 8483 12 . 5010 .4889 12 .8691 .8667 14 .5097 .4980 14 .8882 .8860 16 .5184 .5072 16 .9080 .9060 18 5274 5165 18 .9288 .9270 20 53 6 5 .5261 20 .9506 9489 22 545 8 5358 22 9734 .9719 24 5553 5457 2 4 o. 9975 o. 9961 26 .5 6 49 5557 26 i. 0228 I. 0216 28, .5748 .5660 28 .0497 .0487 30 .5848 .5764 30 .0783 .0774 32 5951 5871 32 . 1089 . 1081 34 . 6056 5979 34 . 1416 . 1409 36 . 6164 . 6090 36 .1770 .1764 38 .6273 .6204 38 2154 .2149 40 .6386 .6319 40 ?573 .2569 42 .6501 .6438 42 337 3033 44 .6619 .6559 44 3554 3552 46 .6740 .6684 46 .4140 .4138 48 .6865 .6811 48 4815 .4814 50 6993 .6942 50 5 6l 3 .5612 52 7 I2 4 .7076 52 .6588 .6587 54 .7259 .7214 54 .7844 .7843 56 .7398 7355 56 i. 9610 1.9610 22 58 o. 7541 o. 7501 . 23 58 2. 2627 2. 2627 x = =p A 3 tan L -f B d tan D EQUATION OF EQUAL ALTITUDES. 2 17 Computation of the Equation of Equal Altitudes to correct the Chro nometer for Noon, August 9, 1844, by the First of the following Equal Altitudes of the Sun s Limbs. x = ( A 3 tan L) + (B d tan D) T = 6 11 33 m log A = 9.4605 log B = 9.2764 d = 43".6 3 log d = 1.6397 log d = 1.6397 L = 45 48 log tan L = 0.0121 log tan D = 9.4493 ist term i2 s -95 = 1-1123 2 8 .32= 0.3654 2d term 2 .32 I -|- 10 .63 = equation of equal altitudes. Computation of the First Two of the following Pairs of Equal Alti tudes of the Sun s Limbs. ist pair. ad pair. A. M. / = i 11 28 m 2 3 3 .o i h 29 52 S .8 P.M. = / = 8 03 16.5 8 01 46.5 = 9 3 1 39-5 9 3 1 39 -3 __ = 4 45 49 .75 4 45 49 .65 Equat n of equal alts. = x = + I0 -63 10 .63 Time by chron. of appt. noon = 4 46 oo .38 4 46 oo .28 Correct mean time at apparent noon (Naut. Aim.) = o 05 09 .09 o 05 09 .09 Chron. fast of mean time at app t noon, August 9, 1844 = 4 40 51 .29 4 40 51 .19 2l8 ASTRONOMY. SURVEY OF ... . DETERMINATION OF THE TIME, Chronometer DATE AND STATION. 1844, August 9 American Camp, near Tasche INSTRUMENTS ^ Sextant No. 2197, by Trough ton &* Mean Solar Chronometer, No. 2440, Times, by chronometer, of ob Observed double altitudes of the sun s upper and lower served equal altitudes. 2" t = the elapsed time, =T. Equation of equal altitudes August gth. limbs. = X. A. M. = / P. M. = f Upper Limb. //. m. s. h. ;.Y. s. h. m. s. 78 50 oo" I 28 23 8 03 16. 5 i 6 33 +10.63 79 9 3 I 29 52. 8 8 01 46. 5 ) Lower Limb. 83 10 oo" i 45 01 7 46 40.5] 83 40 oo i 46 34. 5 7 45 6 - 2 j> 5 59i +10.24 84 oo oo i 47 38 7 44 04 J Upper Limb. 85 36 oo" i 49 23 7 42 18 j 5 48 + 10. I 87 02 10 i 53 55-5 7 37 46. 2 5 CHRONOMETER ERROR. Fastoi mean solar time at apparent noon of Au gust 9, 1844, by a mean of 7 pairs of equal altitudes of the sun TIME BY OBSERVED EQUAL ALTITUDES. 219 by Observed Equal Altitudes of the Sun s Limbs, to Correct the at Noon. reau s house, on the highland boundary between Maine and Canada. Simms, and Artificial Horizon of Mercury. by Parkinson 6 Frodsham. Chron o m e t e r No. 2440 fast of mean time at apparent noon by each pair of equal altitudes. h. m. s. 4 40 5i- 2 9 4 40 5 1 - *9 f4 40 5 T -9 <J4 40 51-5 U 40 52.15 4 40 5 T -5i 4 40 51. 86 Remarks. Index error of sextant Error of eccentricity of sextant Thermometer (a. m.) 70 Fahr. barom-. Thermometer (p. m.) 69 Fahr. barom. .. Sun s appar t declination at appar t noon(D)= 15 43 12" N. Hourly variation of sun s declination. . .(r5) = 43 // -63 Equation of time at apparent noon -+- 5 " 09*. 09 Latitude of station (approximate) -f- 45 48 = (L. ) 4 40 51.6 Observer, Major J. D. Graham. Computer, Do. 220 ASTRONOMY. Sun s Parallax in Altitude. Sun s Sun s horizontal parallax. Sun s Sun s horizontal parallax. altitude. altitude. 8". 7 8".8 8". 9 9".o 8". 7 8".8 8". 9 9 ".o H n n n u n n 8. 70 8.80 8.90 9.00 45 6. i S 6.22 6. 29 6.36 5 8.67 8.77 8.87 8.97 50 5-59 5.66- 5.72 5-79 10 8.57 8.67 8.76 8.86 55 4.99 5-05 5-12 5.16 15 8.40 8.50 8.60 8.70 60 4-35 4.40 4-45 4.5 20 8.18 8.27 8.36 8.46 65 3.68 3-7 2 3.76 3.80 25 7.88 7.98 8.06 8.16 70 2.98 3.01 3.04 3.08 30 7-53 7.62 7.70 7-79 75 2.25 2.28 2.31 2-33 35 7-13 7.21 7.29 7.37 80 1.51 i-53 1-55 1.56 40 6.66 6-74 6.82 6. 90 85 o. 76 o.77 0-77 0.78 45 6-15 6.22 6.29 6.36 90 o. oo o. oo o. oo o. oo Parallax in altitude ==. horizontal parallax X cosine of altitude. Decimals of an Hour. Minutes. Seconds. m. deem. m. deem. m. deem. s. deem. S. deem. s. deem. i . 01667 21 35oo 41 68333 i . 00028 21 . 00583 41 .01139 2 3333 22 .36667 42 . 70000 2 . 00056 22 . 00611 42 . 01167 3 . 05000 23 .38333 43 . 71667 3 . 00083 2 3 . 00639 43 .01194 4 . 06667 24 . 40000 44 73333 4 .00111 24 . 00667 44 .01222 5 08333 25 .41667 45 . 75000 5 . 00139 2 5 . 00694 45 . OI25O 6 . IOOOO 26 43333 46 . 76667 6 . 00167 26 . 00722 46 .01278 . 7 . 11667 27 . 45000 47 78333 7 .00194 27 . 00750 47 .01306 8 J 3333 28 .46667 48 . 80000 8 . OO222 28 . 00778 48: .01333 9 . 15000 29 .48333 49 .81667 9 . 00250 29 . 00806 49 .01361 10 . 16667 3 . 50000 50 .83333 10 . 00278 3" . 00833 50 .01389 ii 18333 3i .51667 5i . 85000 ii . 00306 3 1 . 00861 5i .01417 12 . 20000 32 . 53333 S 2 . 86667 12 00333 3 2 . 00889 52 .01444 13 .21667 33 . 55000 53 .88333 13 .00361 33 .00917 53 .01472 H 23333 34 .56667 54 . 90000 14 .00389 34 .00944 54 . 01500 15 . 25OOO 35 . 5 33o 55 .91667 15 .00417 35 . 00972 55 .01528 16 . 26667 36 i . 60000 5 6 .93333 16 . 00444 3 6 . OIOOO S 6 .01556 17 28333 37 .61667 57 .95000 17 . 00472 37 . 01028 57 .01583 -i 8 19 . 3OOOO .31667 38 j . 63333 39 . 65000 58 59 .96667 98333 18 19 .00500 . 00528 38 39 . 01056 . 01083 58 59 . Ol6ll . 01639 20 33333 40 . 66667 60 I. 00000 20 . 00556 40 . OIIII 60 .01667 TIME BY TRANSITS. 221 LI. The Transit Instrument, Knowing the apparent right ascension of a star, to compute the corrections to its observed transit on account of the three principal errors of the transit instrument in azimuth, in the inclination of the axis, and in collimation in order to obtain the correct clock error : E - T + a *Hk::J9 + b ^ (L ~ D) + ~ - AR. cos D cos D cos D where E denotes the error of the clock, minus when slow ; T, the observed time of transit ; L, the latitude of the place ; D, the declination of the star, plus when north, and minus when south, for the upper culminations, and vice versa for the lower culminations ; a, the deviation in the telescope in azimuth, plus when (pointing to the south) the vertical which it describes falls to the east, and minus when it falls to the west, and vice versa when pointing to the north ; b, the bias or inclination of the axis of the telescope, plus when the west end of the axis is too high ; c, the error in collimation, plus when the circle described by the line of collimation of the telescope falls to the east, and minus when it falls to the west, for upper cul minations, and vice versa for lower culminations ; and AR., the right ascension of the star.^ When the clock marks mean solar time, the mean time of transit of the object over the meridian must be substituted for AR. 222 ASTRONOMY. LI. The Transit Instrument Continued. i. To determine the value (in time) of the co -efficients 0, in the preceding formula : For inclination of the axis of the telescope : where w and e 1 denote respectively the values of w and e after reversing the level; d, the value of each division of the level in seconds of space; w, the inclination of the level to the west; and e, the inclination of the level to the east. for cottimation : c = -J (/ - t) cos D + J (b 1 - b) cos (L - D) where f and b denote respectively the values of / and b, after reversing the instrument , D, the declination of a circumpolar star; and /, the time of the transit of the circumpolar star deduced from an observation at a given side wire of the instrument. For the deviation -in azimuth : By observations of a circumpolar star : a = I2h ~ ( T/ ~ T ) + b CQS (L - D) - V cos (L + D) + 2 c 2 cos L tan D 2 cos L sin D where T 7 and V denote respectively the values of T and b at the lower culmination. TIME BY TRANSITS. 223 LI. The Iransit Instrument Continued. Deviation in azimuth by transits of a high and low star : where T , AR/, and D v denote respectively the values of T, AR., and D of the second star observed. Or - : cos L (tan D tan D ) If one of the stars is observed at its lower culmination, use 180 D and i2 h -f AR/ for its declination and right as cension. Or make sin (L - D) for cos D and sinJL_- DO for cos D - then _ (AR/ AR.) (T T) n 1 n n is negative for a star north of the zenith. 2. To find the equatorial interval of each wire from the central wire, observe the transit of a star of any decimation D ; then Equatorial interval = observed interval x cos D. 3. When the intervals on each side of the central. wire are equal, the mean of the times of transit over each wire will denote the transit over the middle wire. But should they not be equal, a correction must be applied to obtain a correct mean. Call I. II; IV. V, the equatorial intervals of each wire from the central wire, the instrument having say 5 wires ; then fl _L_ il\ (IV 4- V) Reduction to middle wire = ^ 5 cos D 224 ASTRONOMY. FORM FOR RECORD AN SURVEY OF Transits of Stars . 7c*i//i D COMPUTATION. STATION. Inch transit No ; Hardy No. 50. Sidereal Chronometer Illuminated end of axis, west. Date (1847). - October 6th. October 6th. October 6th. Observer T.J.L. T.J.L. T.J.L. Object TT Capricorni. 14 Capricorni. a Cygni. Level E. 3 2.2 W. 33 .o .32.2 W.33.o .32.7 W. 3 2.5 E-32-5 W.33-3 .32.7 W. 3 2.5 E. 33 .o W. 3 2.5 Value of i division of scale 7" . Z Wires I h. m, s. 20 17 33.0 17 53-5 18 12. 7 18 32.7 20 18 52. 5 h. m. s. 20 29 43.7 30 02.7 30 22. 30 4L7 20 31 oo. 7 h. m. s. 20 35 oo. o 35 26.0 35 52.0 36 18.7 20 36 45-5 II III IV V Sum 184,4 1 10. 8 142.2 Mean 20 18 12.88 .07 20 30 22. 1 6 .07 20 35 52.44 . 10 Reduc n to middle wire Transit on instrument. ^ ^ for collimation . "l: . )> for level .. 12. 8l + .10 + -17 22. 09 4- .04 + -IS 52.39 . 12 .01 tj } fordev n in az h Transit by chronom r. 20 1 8 13.08 20 3 22.31 20 35 52.21 AR. of star 20 1 8 36.66 20 30 45. 89 20 36 15. So Error of chronometer . 23.58 23.58 23-59 Chronometer at - slow of time - p.m., October 6th, 1847. TRANSIT INSTRUMENT. 22 5 Computation of the Corrections a and b, in the Preceding Transits. Declination of TT Capricorni = 18 42 S. 14 Capricorni = 15 29 S. a Cygni = 44 44 N. Latitude of Station = L = 43i3 / Level Correction of x Capricorni. = 43 3 E. 32.2 D = - 1 8 42 - D) = 6.0 55 cos < L ~ D) = o. 5 < cos D 32.2 33 33 64.4 66 66 64.4 = i.(> b ?~5. X 1.6 = o".2o 60 7 cos (L D) Level correction = o - - = o s .2o x o.c;o = o s .io cos D Deviation in Azimuth. (AR/ - AR.) - (T - T) n - n T and T being the times of transit corrected for level and col- limation. Combining - Capricorni and a. Cygni, h. in. s. AR/ = 20 36 15.80 AR. = 20 18 36.66 h. m. s. T = 20 35 52.22 T = 20 18 12.91 I 7 39-3 1 39-3 1 (AR/ - AR.) - (T - T) = - 0.17 (sinL - DO sin (-1 31 ) cos D k cos 4 444 r sin (L D) _ sin6i 55 cosD = cos 1 8 42 : + a = = oM8 0.03 0.93 0.96 Combining 14 Capricorni and a Cygni, a = + o s .i9 Correction for deviation in azimuth of JT Capricorni, 226 ASTRONOMY. Numerical Values of Factors >in (L D) cos (L D) cos D cos D For deviation. Star s declination D For level. Star s Z.-D. Star s Z.-D. . = (L-D) o 10 20 25 30 35 40 = (L D) I .02 . 02 .02 .02 .02 .02 .02 89 2 .04 .04 .04 .04 .04 .04 05 88 4 .07 .07 .07 .08 .08 .08 .09 86 6 . ii . II . II . II . 12 13 -H 84 8 .14 .14 15 15 .16 17 .18 82 10 17 .18 19 .19 .20 .21 23 80 12 .21 .21 .22 23 .24 25 .27 78 14 .24 25 .26 .27 .28 .29 32 76 16 .28 .28 .29 30 .32 34 .36 74 18 31 31 33 34 .36 .38 .40 72 20 34 35 .36 .38 .40 .42 45 70 22 37 .38 .40 .42 .44 .46 49 68 2 4 .41 .41 43 45 47 49 53 66 26 .44 45 47 .49 5i 54 57 64 28 47 48 50 52 54 57 .61 62 30 So 51 53 55 .58 .61 65 60 32 53 54 .56 .58 .61 .65 69 58 34 .56 57 59 .61 .65 -69 73 56 36 59 .60 63 .65 .68 .72 77 54 38 .62 63 .66 .68 71 75 .80 52 40 .64 .65 .68 71 74 .-78 .84 50 45 71 .72 75 .78 .82 .86 .92 45 So -77 .78 .82 .84 .89 93 I.OO 40 55 .82 83 .87 .90 95 .98 1.07 35 60 .87 .88 .92 95 I.OO i. 06 1. 13 30 65 .91 .92 .96 I.OO 1.05 I. 10 1. 18 25 70 .94 95 I.OO 1.04 1.09 1.14 1.23 20 75 97 .98 1.03 1.07 I. 12 1.17 1.26 15 80 .98 I.OO 1.05 1.09 I.I4 1.20 1.29 IO 89 I.OO i. 02 i. 06 I. 10 I-I5 1.22 1-31 I For colli- ? /- I mation ) I.OO i. 02 I. OO I. IO I. 15 I. 22 I -3 I cosD TRANSIT INSTRUMENT. 227 1 for 1 "acitttating the Reduction of Transit- Observations. cos D J J For deviation. Star s declination = D For level. Star s Z.-D. Star s Z.-D. = (L-D) 45 50 55 60 65 7 75 = <L-D) I .02 03 03 03 .04 05 .07 89 2 05 05 .06 .07 .08 . IO .13 88 4 . 10 . ii . 12 .14 17 .20 .27 86 6 15 .16 .18 .21 25 .31 .40 84 8 .20 . 22 .24 .28 33 .41 54 82 10 25 .27 30 35 .41 5 1 67 80 12 .29 3 2 36 42 .49 .61 .80 78 H 34 38 .42 48 57 .71 94 76 16 39 43 .48 55 .65 .81 i. 06 74 18 44 48 54 .62 73 .90 1.19 72 20 48 53 .60 .68 .81 I.OO 1.32 70 2Z 53 58 65 75 89 1.09 1-45 68 24 58 -63 7i .81 .96 1. 19 i-57 66 26 .62 .68 76 .88 1.04 1.28 1.69 64 28 .66 73 .82 94 i. ii 1.37 1.81 62 30 7i 78 87 I.OO i. 18 1.46 1.93 60 32 75 .82 .92 i. 06 1.25 i-55 2.05 58 34 79 87 97 I. 12 1.32 1.65 2.16 56 36 83 .91 1.03 1.18 1-39 1.74 2.27 54 38 87 .96 1.07 1.23 1.46 i. 80 2.38 52 40 .91 I.OO I. 12 1.29 1.52 1.88 2.48 5 45 I.OO I. 10 1.23 1.41 1.67 2.07 2-73 45 50 i. 08 1.19 1-34 r -53 1.81 2.24 2.96 40 55 1.16 1.27 1-43 1.64 1.94 2.40 3.16 35 60 1.22 1.35 I-5 1 1-73 2.05 2-53 3-35 30 65 1.28 1.41 1.58 1.81 2.14 2.65 3-50 2 5 70 I -33 1.46 i. 64 1.88 2.22 2-75 3-63 20 75 1-37 1.50 1.68 1-93 2. 29 2.82 3-67 15 80 1-39 r -53 1.72 i. 97 2-33 2.88 3-8i 10 89 1.41 1.56 i*74 2,00 2-37 2.92 3.86 I Forcolli- > T mation ) * " 1.41 1.56 1,74 2.00 2-37 2.92 3-86 ~~cos D 228 ASTRONOMY. LI I. Reduction of Transits by Least Squares. Let E be the error of chronometer at an assumed time T ; A? h) ^3> &c., the observed times of transit (corrected for rate and level error) of stars having the right ascensions AR. b AR. 2 , AR. 3 , &c.; a and c, the errors of azimuth and collimation ; and A!, A 2 , A 3 , c., Ci, C 2 , C 3 , &c., the factors of azimuth and of collimation for the several stars ; then /! -j- E + A! a + Ci c = AR t /2 + E + A 2 a + C 2 c AR 2 / 3 + E + A 3 a + C 3 c = AR 3 &c., &c. Let E = E + e where e is the unknown correction to an assumed chronometer error E; and let, also, ARi - /i = e, AR 2 / 2 = e- 2 : AR 3 - / 3 = ^ 3 &c., &c. then + e + A! ^ -f Ci c = c l E + s + A 2 + C 2 c = e 9 E + e + A 3 + C 3 c = ^ &c., &c. Let now ei E = tii ^2 E 7/2 3 E =. 7/3 &c., &c. then e + A! # + Ci = 7/! -j- A 2 + C 2 r = 77 2 e 4- A 3 a + C 3 ^ ;= 7/3 &c., &c. From which form the normal equations S +ZAa+ZCc=n ^Ae + ^A 2 ^ 4- -TAC^ = A;/ (i) JCe + AC0 + ,<: = <: from which e, ^, and c can be obtained. TRANSIT INSTRUMENT. 22 9 LII. Reduction of Transits, <5rV. Continued. If the errors of collimation are known, and the times /i, / 2 > ?3> &c., corrected for it, the azimuthal deviation and correction to assumed chronometer-error may be deduced from the equations 2 s + I a = n Equations (i) cannot be advantageously employed unless the nstrument be reversed. Example of the Computation of Equations (i). * Latitude, 36 38 N. April u, 1852 Assumed time, T = n h sidereal Chronometer losing i s .83 dailyAssume E = + 3 h 14 " 30^.0. Illum n star. n A C A C A 2 AC C 2 f a Urs. Maj h . in . s. 3 *4 2 9-77 29.91 29.87 28.50 30.56 30.19 30.06 -0.23 -0.09 -0.13 -1.50 + 0.56 + 0.19 + 0.06 -0-95 + 0.27 + 0.80 + 4.05 0.55 + 0.46 + 0.46 +2.17 + 1.07 + 1.04 + 4-39 -1.72 I. 01 i. 02 + 0.22 O.O2 -0.10 6.07 -0.31 +0.09 + 0.03 -6.16 0.50 O. IO 0.14 -6.58 0.96 0.19 0.06 + 0.90 0.07 0.64 16.40 0.30 O.2I O.2I 2.06 + 0.29 + 0.83 -17.78 + 0.95 0.46 -47 + 4.71 1.14 i. 08 19.27 2.96 i. 02 1.04 +31.22 (S Hydra. ... r y Cephei sub. polo U Urs. Maj 1 4052 B. A. C [4072 B. A. C 1.14 + 4-54 + 4.92 -8.53 + 18.73 + 16.86 Normal Equations. 7 + 4.54 a + 4.92 c = 1.14 4.54 e + J 8.73 a + 16.86 c = 6.16 4.92 e -|- 16.86 a -\- 31.22 c = 8.53 from which = -j- O 8 .C>9 a o s .i8 c =. o 8 . 19 hence, Azimuthal deviation of the instrument = o s .i8 W. of S. Error of collimation of mean of wires, illumi nation east = o s . 1 9 W. Error of chronometer, (slow) = 3 Tl 14 30". 06 230 ASTRONOMY. . Tables of Refraction. Table I gives the refraction when the barometer stands at 30 inches and the Fahrenheit thermometer at 50. Table II, to be used when greater accuracy is desired, gives the correction of the mean refraction depending upon the ob served height of the barometer and thermometer. In column A of this table, the refraction is regarded as a func tion of the apparent zenith-distance Z. The adopted form of this function is r = a,3 A ? x tan Z in which a varies slowly with the zenith-distance, and its loga rithm is therefore readily taken from the table with the argument Z. The exponents A and A differ sensibly from unity only for great zenith-distances, and also vary slowly; their values are therefore readily found from the table. The factor y? depends upon the barometer. The actual press ure indicated by the barometer depends not only upon the height of the column, but also upon its temperature. It is therefore put under the form /? = B T and log B and log T are given in the supplementary tables with the arguments "Height of the barometer" and "Height of the attached thermometer," respectively ; so that log ? = log B + log T Finally, log Y is given directly in the supplementary table with the argument " External thermometer." This thermometer should be so exposed as to indicate truly the temperature of the atmos phere at the place of observation. REFRACTION. 2 3 I L 1 1 1 . Tables of Refraction Continued. Example. Given the apparent zenith distance, Z = 783o / o // ; barometer, 29.770 inches; attached thermometer, o.4 F. ; external thermometer, 2.o F. From table II for 783o ; log a = 1.74981 A = 1.0032 ; A = 1.0328 and from the tables for barometer and thermometer log f = + 0.04545 log B = + 0.00253 log T = + 0.00127 log ft = -f- 0.00380 Hence the refraction is computed as follows : log a = 1.74981 A log /? = log /5 A = + 0.00381 A log Y lg V K = + 0.04694 log tan Z = 0.69154 r = 5 io".53 = 3io".53; log r = 2.49210 The true zenith-distance is, therefore, 78 30 + s io". S 3 = 7835 io".53 TABLE I. Mean Refraction. Barometer, 30 inches Fahrenheit thermometer, 50. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. / / // / / // / / // 5 30 9 7-0 6 30 7 53-9 36 29 35 9 o. i 35 7 48.7 I 24 54 40 8 53-4 40 7 43-5 2 18 26 45 8 46.8 45 7 38.4 *^ O 14 25 50 8 40.4 5 7 33-5 4 o ii 44 55 8 34-2 55 7 28.6 5 9 5 2 - 6 o 8 28.0 7 o 7 23.8 5 9 44.0 5 8 22.1 5 7 19.2 10 9 36.2 10 8 !6.2 10 7 14-6 15 9 28.6 15 8 10.5 15 7 10. i 20 9 21.2 20 8 4.8 20 7 5-7 25 9 14.0 25 7 59-3 25 7 1-4 2 3 2 ASTRONOMY. TABLE I. Mean Refraction Continued. Barometer, 30 inches Fahrenheit thermometer, 50. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. 7 30 6 57-1 10 30 5 4-6 13 30 3 58.1 35 6 53-0 35 5 2.3 35 3 56.6 40 6 48.9 40 5 o. o 40 3 55-2 45 6 44.9 45 4 57-8 45 3 53.7 50 6 41. o 50 4 55-6 5 3 52.3 55 6 37-1 55 4 53.4 55 3 5-9 8 o 6 33-3 II 4 51.2 14 o 3 49-5 5 6 29.6 5 4 49.i 5 3 48. i IO 6 25.9 IO 4 47.o 10 3 46.8 15 6 22.3 15 4 44.9 15 3 45-5 20 6 18.8 20 4 42.9 20 3 44-2 25 6 15.3 25 4- 4-O 9 25 3 42.9 30 6 ii. 9 30 4 38.9 30 3 41.6 35 6 8.5 35 4 36.9 35 3 40.3 40 6 5-2 40 4 35-0 40 3 39-0 45 6 2. 45 4 33-1 45 3 37-7 5 5 58.8 5 4 31.2 5 3 36.5 55 5 55-7 55 4 29.4 55 3 35-3 Q O 5 52.6 12 4 27.5 15 o 3 34.i 5 5 49-6 5 4 25.7 5 3 32.9 10 5 46.6 IO 4 23.9 10 3 31-7 15 5 43-6 15 4 22.2 15 3 30.5 20 5 40.7 20 4 20.4 20 3 29.4 25 5 37-9 25 4 18.7 25 3 28.2 30 5 35.i 3 4 17.0 30 3 27.1 35 5 32.4 35 4 15-3 35 3 25.9 40 5 29.6 40 4 13.6 40 3 24.8 45 C 27, O 45 4 12.0 45 3 23.7 50 5 24.3 5 4 10.4 5 3 22.6 55 5 21.7 55 4 8.8 55 3 21.5. 10 5 19.2 13 o 4 7-2 16 o 3 20.5 5 * 5 16.7 5 4 5- 6 5 3 !9-4 10 5 H.2 10 4 4. i IO 3 18.4 15 5 11.7 15 4 2.6 15 3 17.3 20 5 9-3 20 4 i.o 20 3 16.3 25 5 6 -9 25 3 59-6 25 3 15-2 TABLES OF REFRACTION. 233 TABLE I. Mean Refraction Continued, Barometer, 30 inches Fahrenheit thermometer, 50 Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction. 01 III , / // 16 30 3 14.2 20 o 2 38. 8 26 o I 58.9 35 3 13-2 10 2 37.4 10 i 58. I 40 3 I2 - 2 20 j 2 36. 20 i 57-2 45 30 2 34. 6 30 i 56.4 50 3 IO -3 40 2 33-3 4 i 55-5 55 3 9-3 50 2 32.0 50 i 54-7 17 o 3 8.3 i; 21 o ! 2 30.7 27 o i 53-9 5 3 7-3 10 2 29.4 IO i 53- 1 10 3 6.4 2O 2 2-8.1 20 i 52.3 15 3 5-5 30 2 26.9 30 i 51-5 20 3 4-6 40 2 25.7 40 i 5o.7 25 3 3-7 50 2 24.5 50 i 50.0 30 3 2.8 22 2 23.3 28 o i 49.2 35 3 1-9 10 2 22. I 10 i 48.4 4 3 l - 2O 2 20.9 20 i 47-7 45 3 o. i 30 2 19.8 30 i 46.9 50 2 59.2 40 2 18.7 40 i 46.2 55 2 58.3 50 2 17-5 50 i 45-5 18 o 2 57.5 23 o 2 16.4 29 o i 44.8 5 2 56.6 10 2 15.4 20 i 43-4 IO 2 55-8 j 20 2 14.3 40 i 42.0 15 2 54-9 30 2 13.3 30 o i 40. 6 20 2 54-1 40 2 12. 2 20 i 39.3 25 2 53-2 5 2 II. 2 4 i 38-0 3 2 52.4 j| 24 2 10.2 31 o i 36.7 35 2 51-6 | 10 2 9. 2 2O i 35-5 40 2 50.8 20 2 8.2 40 i 34-2 45 2 50.0 30 2 7.2 32 o i 33-0 2 49.2 4 2 6.2 20 i 31-8 55 2 48.4 50 2 5-3 40 i 30.7 19 o 2 47-7 2 5 2 4-4 33 o i 29.5 10 2 46. I 10 2 3.4 20 i 28.4 20 2 44. 6 20 2 2.5 40 i 27.3 3 2 43-1 3 2 1.6 34 o i 26.2 40 2 41.6 40 2 0. 7 20 i 25. i 50 2 40. 2 50 I i 59.8 ; 40 i 24. i 234 ASTRONOMY. TABLE I. Mean Refraction Continued. Barometer, 30 inches Fahrenheit thermometer, 50. Apparent altitude. Mean re fraction. Apparent altitude. Mean re fraction*. Apparent altitude. Mean re fraction. / / // o , / // o / / // 35 o I 23. i 47 o o 54-3 59 o o 35-0 20 I 22.0 20 o 53-7 20 34-5 40 I 21.0 40 o 53-1 40 o 34.1 36 o I 2O. I 48 o o 52.5 60 o o 33-6 20 I 19. I 20 o 51.9 20 o 33-2 40 I 18.2 40 o 51.2 40 o 32.7 37 o I 17.2 49 o o 50. 6 61 o o 32-3 20 I I6. 3 20 o 50. o i 62 o o 31.0 40 I 15.4 .40 o 49.4 63 o o 29!; 38 o I 14.5 50 o o 48. 9 64 o o 28.4 2O I 13-6 20 o 48.3 65 o o 27.2 40 i 12.7 40 o 47.8 66 o o 25.9 39 o I 11.9 5i o 47.2 67 o o 24.7 20 I II. 20 o 46.6 68 o o 23.6 4 I IO. 2 40 o 46.1 69 o o 22.4 40 o I 9.4 52 o o 45-5 70 o O 21. 2 20 i 8.6 20 o 45.0 71 o 20. I 40 i 7.8 40 o 44.4 72 o o 18.9 41 o i 7.0 53 o 43-9 73 o o 17.8 20 I 6.2 20 o 43-4 74 o o 16. 7 40 i 5-4 40 o 42.8 75 o o 15.6 42 o i 4.7 54 o o 42.3 ! 76 o o 14.5 2O i 3-9 20 o 41.8 77 o o 13-5 40 i 3.2 40 o 41.3 78 o o 12.4 43 i 2.4 55 o o 40.8 79 o o 11.3 20 i 1.7 20 o 40.3 80 o o 10.3 40 I I.O 40 o 39.8 81 o o 9.2 44 o i 0.3 56 o o 39-3 ! 82 o 8.2 20 o 59.6 20 o 38.8 83 o o 7.2 40 o 58.9 4 o 38.3 84 o o 6.1 45 o o 58.2 57 o o 37.8 85 o o 5.1 20 o 57.6 20 o 37-3 86 o o 4. i 40 o 56.9 40 o 36.9 87 o o 3.1 46 o o 56.2 58 o o 36.4 88 o 2.0 20 o 55-6 20 o 35-9 89 o O I.O 40 55- 40 o 35-5 90 o 0.0 REFRACTION. 235 TABLE II. BesseVs Refraction- Table. i 1 Arg.app. zen.-dist. 8 Arg. app. zen.-dist. T S Q N Log a. A S 4) N Log a. A / O O 10 20 o 1.76156 1.76154 . .76149 77 o IO 20 1-75229 * 75205 _^ 1-75180 y 1.0026 i .0026 1.0027 1.0252 1.0258 1.0264 30 o 35 1.76139 9 1.76130 * 3 40 1.75155 J i-75 I2 9 2 8 1.0027 i .0028 1.0272 I.02I 40 o 5 i.75ioi 2g 1.0029 1.0290 45 o 46 o 48 o 1.76104 1.76100 * 1.76096 J 1.76092 4 I.OOlS I .OOI9 1.0019 I .0020 78 o IO 20 30 1.75072 i 7543 ,o 1-75013 i.7498i H 1.0030 1.0030 1.0031 1.0032 1.0299 1.0308 1.0318 1.0328 1.76087 5 I.OO2I 40 1 74947 :. 1.0033 1.0338 50 o 1.76082 * 1.0023 50 1.749" ^g 1.0034 1-0347 52 o 53 o 54 o 55 o 56 o 1.76077 6 1.76071 6 i . 76065 1.76058 I 1.76050 g 1.76042 g 1.0025 I.OO26 1.0027 I .OO29 I.OO3I 1.0034 79 IO 20 30 40 5 1.74876 1.74839 * 1-74799 7 2 1-74757 J. I 747 I 4 . 1.74670 1.0035 1.0036 1.0037 1.0038 1.0039 1.0040 1.0357 1.0367 1-0377 1.0387 1.0398 1.0409 57 o 58 o 59 o 60 o 61 o 1.76033 I0 1.76023 1.76012 1.76001 1.75988 *3 1.0037 I .0040 1.0043 1 .0046 1.0049 80 o IO 20 30 40 1-74623 - 1-74573 * I-7452I | 1.74468 53 I-744I2 o 1.0041 1.0042 1.0043 1.0045 i .0046 I.O420 I.043I 1.0442 1.0454 i . 0466 62 o 1-75973 jg 1.0054 50 1-74352 64 1.0047 1.0479 63 o 64 o 65 o 66 o 67 o 68 o 1-75957 l8 1-75939 20 I.759I9 22 1.75897 2 6 I.7587 1 2Q 1-75842 33 1.0058 1.0063 1.0068 1.0075 1.0083 1.0092 81 o 10 20 30 40 50 1.74288 g 1.74223 6 I.74I55 72 1-74083 6 1.74007 7 1.73928 g 1.0049 1.0050 1.0052 1.0054 1.0056 1.0058 1.0493 1.0508 1.0523 1.0540 1-0559 1-0579 69 o 70 o 71 o 72 o 73 1-75809 g I-7577I ;L 1.75726 ^ I .OIOI I.OIII 1.0124 1.0139 1.0156 82 o 10 20 30 40 1.73845 gS I 73 W 94 1-73663 94 1.73564 J0 ^ 1-73459 H2 i. 0060 1.0062 1.0065 1.0067 1.0070 I. 0600 i . 0622 1.0646 1.0671 1.0697 74 o 1-75543 H 1.0175 5 1-73347 Il8 1.0073 1.0725 75 o 10 20 i -75457 l6 I-7544I l6 1.75425 I7 1.0197 I.O2OO I .O2O4 1. 0208 83 o IO 20 3 1-73229 I2 . I-73I05 4 1.72974 ^ 1.72832 3* 1.0075 1.0078 i. 0081 1.0084 1.0754 1.0784 1.0815 1.0846 40 1.75391 { I.O2I2 40 1.72681 *g i. 0088 1.0879 50 1.75373 Ig I. O2l6 50 I.725I9 I73 1.0092 1.0914 76 o IO 1-75355 IQ 1.75336 y I.O22O 1.0225 84 o 10 1.72346 l86 1.72160 1.0096 I.OIOO 1.0951 1.0992 20 f- 20 1.0230 20 1.71961 " 1.0105 I . 1036 3 1.75295 2 : 1.0235 30 I.7I749 227 I. OIIO 1.1082 40 50 I-75274 22 1.0241 I . 0246 40 50 I-7 I 522 * I.7I279 J 1.0115 I.OI2I 1.1130 1.1178 77 o 1.75229 1.0026 I.O252 85 o I.7IO2O I.OI27 1.1229 2 3 6 ASTRONOMY. TABLE 1 1 . BesseVs Refraction- Table. Factor depending upon the barometer. Eng. ins Log B. Factor depending upon the external thermometer. F. Log y. F. Logy. 2 7-5 0.03191 27.6 27.7 0.02876 27.8 27.9 . 28.0 0.02720 0.02564 0.02409 20 18 + 0.06279 o. 06181 o . 06083 +3 i 3.6 37 + 0.01185 0.01098 O.OIOII 28.2 28.3 28.4 28.5 28.6 28.7 28 8 0.02254 0.02099 0.01946 0.01793 0.01640 0.01488 0.01336 3 15 J 4 13 12 II 0.05985 0.05887 0.05790 0.05693 0.05596 0.05500 0.05403 38 39 40 4 1 42 43 44 0.00924 0.00837 0.00750 0.00664 0.00578 0.00492 0.00406 28.9 29.0 29.1 29.2 29-3 29.4 29-5 29.6 29-7 29.8 O.OIO35 0.00885 0.00735 0.00586 0.00438 0.00290 0.00142 + 0.00005 0.00151 0.00297 10 9 8 5 4 3 2 I 0.05307 0.05211 0.05115 0.05020 0.04924 0.04829 0.04734 o . 04640 0-04545 0.04451 0-04357 45 46 3 49 50 51 52 53 54 55 0.00320 0.00234 0.00149 + 0.00064 0.00021 O.OOIO6 O.OOigi 0.00275 o . 00360 0.00444 0.00528 29.9 30.0 30.1 3-2 30-3 30-4 3-5 30.6 3-7 30.8 3-9 31-0 0.00443 0.00588 0.00732 0.00876 O.OIO2O 0.01163 0.01306 0.01448 0.01589 0.01731 0.01871 + O.O2OI2 + 1 2 3 4 I 9 10 ii 12 0.04263 0.04169 0.04076 0.03982 0.03889 0.03796 0.03704 0.03611 0.03519 0.03427 0.03335 0.03243 5 6 11 g 61 62 63 64 65 66 67 0.00612 0.00696 0.00780 0.00863 0.00946 0.01029 O.OIII2 O.OII95 0.01278 0.01360 0.01443 0.01525 *3 0.03152 68 0.01607 Factor depending upon the attached ther mometer. 14 15 16 \l 0.03060 0.02969 0.02878 0.02787 69 70 7i 72 0.01689 0.01770 0.01852 0.01933 73 O.O2OI5 F. LogT. 19 20 0.02606 0.02516 74 75 0.02096 O.O2I77 22 0.02336 76 77 0.02338 -30 + 0.00242 23 24 0.02247 0.02157 78 79 0.02419 0.02499 10 0.00164 25 26 0.02068 0.01979 80 81 0.02579 0.02659 27 0.01890 82 0.02738 28 0.01801 83 0.02819 20 30 40 0.00047 + 0.00008 0.00031 29 3 31 0.01713 0.01624 0.01536 84 85 S6 0.02898 0.02978 0.03057 fin 32 0.01448 87 0.03136 33 0.01360 88 0.03216 80 90 0.00186 0.00225 34 + 35 0.01273 + 0.01185 89 + 90 0.03294 0.03373 0.00264 Log /? = log B + log T. LATITUDE. 237 LIV. To determine the Latitude from the Meridional Altitude of an Object whose Decimation is known. i. When the object observed is south of the zenith : L = 9o+D A = Z+D = 9o + Z A = 180 (A+ A) 2. When the star is between the zenith and the pole : L = A - A = D - Z = 90 - (Z + A) = A + D - 90 3. When the star is between the pole and the horizon to the north : L = A + A = 90 + A Z = 90 + A D = 180 (Z -f D) where L = the latitude sought ; D = the declination of the object, minus when south ; A = its north-polar distance ; A = its meridional altitude ; and Z = its meridional zenith-distance. A and Z must be corrected for refraction. When the sun is the object observed, A = observed altitude (refraction parallax) i semi-diameter. 238 ASTRONOMY. LV. Determination of the Latitude of a Place by the Method of Circum- Meridian A Ititudes. Reduction to meridian = . ( . cos / cos L) ) ( x = k \ i > ;// tan a < ( cos a J ( . cos / cos D ) ( . cos / cos D / cos a 2 sin 2 \ p sin i" 2 sin 4 ;;/ = sin i ^o a = 90 + D / Where A = a -f- * = the meridional altitude of the object ; a = its observed altitude (refraction parallax) i semi- diameter ; p = its correct hour-angle ; D = its declination ; / = the assumed latitude of the place ; and x = the required correction in seconds. When a star is the object observed and the chronometer marks mean time / = 1.005473; log / = 0.0023708 When the sun is observed and the chronometer marks sidereal time * = 0.99455418; log / = 9.9976285 and, generally, when the chronometer has a large losing rate, x must be multiplied by i + 0.00002315 r\ when it has a gaining rate it must be divided by i -f- 00002315 r\ r being the rate in 24 hours, which must be assumed minus when gaining, and plus when losing. LATITUDE. 239 LV. Determination of the Latitude, &c. Continued. The values of k and m for each value of p are given in the following tables. The meridian altitude, A = a 4- x for each observation ; for any number of observations, n, a 4. gn 4. . . . . x 1 + x" + = the mean, a, of all the observed altitudes -f- the mean, x, of all the corrections. Consequently, 1. Measure several successive altitudes of the object both before and after its meridional passage. 2. Note the times of each observation, and compute the time of the object s culmination ; the differences between this and the times of each successive observation are the values of / , p n , &c., in time, for which the corresponding values of k , k", &c., and m , m", &c., must be taken from the tables. 3. The means k and m of these results will be introduced into the equation for the value of the correction, x, to be applied to a to obtain the meridional altitude, A, of the object. 4. If the final latitude differ much from the assumed, the com putation should be repeated with the new value for /. 5. It is not necessary that the time of the object s culmination should be known with great precision, provided an equal number of altitudes be taken upon each side of the meridian, and at nearly equal distances from it. 6. The second correction, m, is seldom necessary, unless great accuracy is desired, and the object is observed more than ten minutes of time from the meridian. - . 240 ASTRONOMY. t Reduction to the Meridian ; Values of k , f-^_ sin i Sec. o m I m 2 m 3 m 4 5 m 6 7 m u // // u u // u II 0.00 I. 9 6 7.8 17.7 3i-4 49.1 70.7 96.2 i 0. CO 2.03 8.0 17.9 31.7 49-4 71.1 96.7 2 0.00 2. 10 8.1 1 8. i 31.9 49.7 7L5 97.1 3 0.00 2.16 8.2 18.3 32.2 50.1 7L9 97.6 4 0. 01 2.2 3 8.4 18.5 32.5 54 72.3 98.0 5 0. 01 2. 3 I 8.5 18.7 32.7 50-7 72.7 98.5 6 0.02 2.38 8.7 18.9 33-0 5i.i 73-1 99.0 7 0.02 2-45 8.8 19.1 33-3 5L4 73-5 99-4 8 0.03 2.52 8.9 19.3 33-5 51-7 73-9 99-9 9 0.04 2.60 9-1 19.5 33-8 52-1 74-3 100.4 10 o. 05 2.67 9.2 19.7 34-1 5 2 - 4 74- 7 100.8 ii o. 06 2-75 9-4 19.9 34-4 S 2 . 7 75- 1 101.3 12 0.08 2.83 9-5 20. i 34-6 53- 1 75- 5 101.8 13 o. 09 2.91 9.6 20.3 34-9 53-4 75.9 102.3 H 0. II 2.99 9.8 20.5 35-2 53-8 76.3 102. 7 15 O. 12 3-07 9-9 20.7 35-5 54-1 76.7 103.2 16 o. 14 3.15 10. I 20. 9 35.7 54-5 77.1 103.7 17 o. 16 3-23 10.2 21.2 36. o 54-8 77-5 104.2 18 o. 18 3-32 10.4 21.4 36.3 55-1 77-9 104.6 19 o. 20 3-40 10.5 21.6 36.6 55.5 78.3 105.1 20 0.22 3-49 10.7 21.8 36.9 55.8 1 78.8 105.6 21 o. 24 3-58 10.8 22.0 37-2 56.2 79.2 106. i 22 0.26 3-67 II. 22. 3 37-4 56.5 79.6 106.6 2 3 0.28 3.76 1 1.2 22.5 37-7 56.9 80.0 107.0 24 0. 3 I 3-85 ii. 3 22.7 38.0 57.3 80.4 107-5 25 0-34 3-94 ii. 5 22.9 38.3 57.6 80.8 1 08. o 26 -37 4.03 11. 6 23.1 38.6 58.0 81.3 108.5 27 0.40 4.12 ii. 8 23.4 38.9 58.3 81.7 109.0 28 0-43 4.22 11.9 23.6 39-2 58. 7 ! 82. i 109.5 29 0.46 4.32 ; 12. i 23-8 39.5 59- o 82. 5 1 10. t LATITUDE. 241 Reduction to the Meridian; Values of k = 2 sm sin i Sec. O m ,m 2 m 3" 5 m 6m r " n n II n // n n 3 0.49 4.42 I2. 3 24.O 39-8 59.4 83.0 110.4 3i 0.52 4.52 I2. 4 24.3 40. i 59.8 83.4 no. 9 32 0.56 4.62 12.6 24-5 40.3 60. i 83.8 in. 4 33 0-59 4.72 12.8 24.7 40.6 60. 5 84.2 111.9 34 0.63 4.82 12.9 25.0 40.9 60.8 84.7 112.4 35 o. 67 4.92 13.1 25.2 41.2 61.2 85.1 112. 9 36 0.71 5.03 13.3 25.4 4L5 61.6 85.5 H3-4 37 o.75 5-13 13.4 25-7 41.8 61.9 86.0 H3-9 38 0.80 5-24 13.6 25.9 42.1 62.3 86.4 114.4 39 0.83 5-34 13.8 26.2 42.5 62.7 86.8 114.9 i 40 0.87 5-45 14.0 26.4 42.8 63. o 87.3 4i 0.91 5.56 14. 1 26.6 43-1 63.4 87.7 115-9 42 0.96 5.67 14.3 26. 9 43-4 63.8 88.1 116.4 43 I. 01 5-78 14.5 27.1 43.7 64.2 88.6 116.9 44 i. 06 5-90 14.7 27.4 44.0 64-5 89.0 117.4 45 I. 10 6.01 14.8 27.6 44-3 64.9 89.5 117.9 46 I - r 5 6.13 15.0 27-9 44.6 65-3 89.9 118.4 47 1.20 6. 24 15.2 28.1 44-9 65-7 90.3 118.9 48 1.26 6. 36 15.4 28.3 45-2 66.0 90.8 H9.5 49 SI 6.48 15.6 28.6 45-5 66.4 91.2 120.0 50 I. 3 6 6.60 15.8 28.8 45.9 66;8 91.7 120.5 5i 1.42 6.72 15.9 29.1 46.2 67.2 92.1 121. O S 2 1.48 6.84 1 6. i 29.4 46.5 67.6 92.6 I2I.5 53 L53 6.96 16.3 29. 6 46.8 68.0 93-0 122.0 54 i-59 7.09 16.5 29.9 47-1 68.3 93-5 122.5 55 1.65 7.21 16.7 30.1 47.5 68.7 93-9 I23.I 56 1.71 7-34 16. 9 3 4 47-8 69.1 94-4 123.6 57 1.77 7.46 17.1 30. 6 48. i 69.5 94-8 124. I 58 1-83 7.60 17.3 3 .9 48.4 69.9 95-3 124.6 59 1.89 7-72 17-5 3i. i 48.8 70.3 95-7 I25.I 16 242 ASTRONOMY. 2 sin ^ i) Reduction to the Meridian ; Values of k = . sin i" Sec. 8" 9 m I0 m u m 12 i3 m 14 " n n n n n // n o 125.7 159.0 196.3 237.5 282. 7 33L8 384- 7 i 126.2 159.6 197.0 238.3 283.5 332.6 385.6 2 126. 7 160.2 197.6 239.0 284.2 333-4 386.6 3 127.2 160.8 198.3 239.7 285.0 334-3 387. 5 4 127.8 161.4 198.9 240.4 285.8 335-2 388.4 5 128.3 162. o 199.6 241.2 286.6 336.0 389-3 6 128.8 162.6 200. 3 241.9 287.4 336.9 390.2 7 129.3 163. 2 200. 9 242.6 288.2 337-7 391-1 8 129.9 163.8 201.6 243-3 289.0 338.6 392.1 9 130.4 164.4 202.2 244.1 289.8 339-4 393- 10 131.0 165.0 202.9 244.8 290. 6 340.3 393-9 ii 131-5 165.6 203.6 245-5 291.4 341-2 394.8 12 132.0 166.2 204.2 246.3 292.2 342.0 395-8 13 132.6 166.8 204.9 247.0 293.0 342.9 396.7 14 i33.i 167.4 205.6 247.7 293.8 343-7 397-6 15 !33.6 168.0 206. 3 248.5 294.6 344-6 398.6 16 134.2 168.6 206. 9 249.2 295-4 345-5 399- 5 17 134.7 169.2 207. 6 249.9 296.2 346.4 400.5 18 135.3 169.8 208.3 250.7 297.0 347-2 401.4 19 135.8 170.4 208. 9 251.4 297.8 348.1 402.3 20 136.3 I7I.O 209.6 252.2 298.6 349-0 403-3 21 136.9 I7I.6 210.3 253. 299.4 349-8 404.2 22 137.4 172.2 211. 253.6 3OO. 2 350.7 405.1 23 138.0 172.9 211. 7 254-4 301. o 351.6 406. o 24 138.5 173.5 212.3 255-1 301.8 352.5 407.0 25 I39. 1 I74.I 213. o 255.9 302. 6 353-3 408. o 26 139.6 174- 7 213.7 256.6 303.5 354-2 408.9 27 140.2 175-3 214.4 257.4 34-3 355-1 409.9 28 140.7 175-9 215. I 258. I 305-1 356.o 410. 8 29 141-3 176.6 215.8 258.9 305.9 35 6 -9 411. 7 1 LATITUDE. 243 Reduction to the Meridian Values of k = J % P sin i Sec gm 9 m I0 m II I2 ln I3 m 14 u // n // // n // 3 141.8 177.2 216.4 259.6 306.7 357.7 412.7 3 1 142.4 177.8 217. i 260.4 307.5 358.6 413.6 32 143.0 178.4 217.8 261. i 308.4 359-5 414.6 33 143-5 179.0 218.5 261. 9 309.2 360.4 415.5 34 144.1 179.7 219.2 262.6 310.0 36L3 416.5 35 144.6 180.3 219.9 263,4 310.8 362.2 417-5 36 145.2 180.9 220. 6 264. i 3".6 363.1 418.4 37 145.8 181.6 221.3 264.9 312.5 364.0 419.4 38 146.3 182.2 222.0 265.7 3I3.3 364.8 420.3 39 146.9 182.8 222. 7 266.4 3H.I 36<5.7 421.3 * - 40 147-5 183.5 223.4 267. 2 3 r 5-o 366.6 422.2 4i 148.0 184. i 224. I 267.9 315.8 367.5 423.2 42 148.6 184.7 224.8 268.7 316.6 368.4 424.2 43 149.2 185.4 225.5 269.5 317.4 369.3 425- i 44 149-7 1 86. o 226. 2 270.3 318.3 370.2 426. i 45 150.3 186.6 226. 9 271.0 3i9.i 37LI 427.0 46 IS ^ 187.3 227. 6 271.8 319.9 372.0 428. o 47 I5L5 187.9 228.3 272.6 320.8 372.9 429.0 48 152. o 188.5 229. o 273-3 321.6 373-8 429.9 49 152. 6 189.2 229; 7 274.1 322.4 374-7 430.9 50 153-2 189.8 230.4 274.9 323-3 375-6 43L9 5i 153.8 190.5 231.1 275.6 324-1 376.5 432.8 52 154.4 191. i 231.8 276.4 325-0 377-4 433-8 53 154.9 191.8 232.5 277.2 325-8 378.3 434-8 54 155-5 192.4 233.2 278.0 326.7 379.3 435-8 55 156. i I93.I 234.0 278.8 327.5 380.2 436.7 56 i5 6 .7 193.7 234.7 279.5 328.4 381.1 437-7 57 157.3 194.4 235-4 280.3 329-2 382.0 438.7 58 157.8 195.0 236. i 28I.I 33.o 382.9 439-7 59 158.4 195.7 236.8 281.9 330.9 383-8 440.6 244 ASTRONOMY. 2 sin 2 A -h Reduction to the Meridian ; Values of k -L sin i" Sec. i5 m i6 m 17 i8" 19 20 m 2I m n n n n // a 441. 6 502.5 567-2 635.9 708.4 784.9 865-3 i 442. 6 53-5 568.3 637-0 709.7 786. 2 866. 6 2 443- 6 504.6 569-4 638.2 710.9 787. 5 868. o 3 444.6 505-6 57o:5 639-4 712.1 788.8 ^69.4 4 445- 6 506.7 571-6 640. 6 7I3-4 790.1 870.8 5 446.5 5 n 7- 7 572.8 641.7 714.6 791.4 872. i 6 447-5 508.8 573-9 642.9 7I5-9 792.7 873-5 7 448.5 509.8 575-0 644.1 717.1 794- o , 874.9 8 449-5 5io-9 576.1 645.3 718.4 795-4 876.3 9 450-5 5-9 577-2 646.5 719.6 796.7 877.6 10 451-5 S 1 3> 578.4 647.7 720.9 798. o 879. o ii 452-5 514.0 579-5 648.9 722. i 799- 3 880. 4 12 453-5 5i5.i 580.6 650. o 723.4 800.7 881.8 13 454-5 516. i 581.7 651. 2 724.6 802.0 883.2 14 455-5 5*7.2 582.9 652.4 725.9 803-3 884.6 5 456.5 518-3 584.0 653-6 727.2 804.6 886.0 16 457-5 519-3 585-1 654.8 728.4 806.0 887.4 17 458.5 520.4 586.2 656.0 729.7 807.3 888 . 8 18 459-5 521.5 587.4 657.2 . 730. 9 808.6 890.2 19 460.5 522.5 588.5 658.4 732.2 809.9 891.6 20 461.5 523- 6 589.6 659.6 733-5 811.3 893.0 21 462.5 524.6 590.8 660.8 734-7 812.6 894.4 22 463-5 525.7 59i-9 662.0 736-0 813.9 895.8 23 464-5 526.8 593-0 663.2 737-3 815.2 897.2 24 465-5 527.9 594-2 664.4 738.5 816.6 898.6 25 466.5 528.9 595-3 665.6 739-8 817.9 900. o 26 467-5 53o.o 596.5 666.8 741. i 819.2 901.4 27 468.5 53I-I 597-6 668.0 742.3 820.5 902.8 28 469-5 532.2 598.7 669.2 743-6 821.9 904.2 29 470.5 533-2 599-9 670. 4 744-9 823.2 905.6 LATITUDE. 245 Reduction to the Meridian ; Values of k g(^~pr~ Sec. i5 m i6 m i7 m i8 m I9 m 20 m 2I m // a a a ii u u 30 471-5 534-3 601. o 671.6 746.2 824.6 907.0 31 472.6 535-4 6O2. 2 672.8 747-4 825.9 908.4 32 473- 6 536.5 603.3 674.1 748.7 827.3 909.8 33 474.6 537-6 604.5 675.3 750.0 828.6 9II.2 34 475-6 538.7 605.6 676.5 751-3 829.9 912. 6 35 476.6 539.7 606.8 677.7 752.6 831.2 914.0 36 477.6 540.8 607.9 678.9 753-8 832.6 915.5 37 478.7 541-9 609. I 680.1 755-1 833.9 916.9 38 479-7 543-0 610. 2 681.3 756.4 835.3 918.3 39 480.7 544-1 611.4 682.6 757.7 836.6 919.7 40 481. 7 545-2 612.5 683.8 759.o 838.0 921.1 4i 482.8 546.3 613.7 685.0 760.2 839.3 922.5 42 483.8 547 4 614.8 686.2 761.5 840.7 923-9 43 484.8 548.4 616. o 687.4 762.8 842.0 925-3 44 485-8 549-5 617.2 688.7 764.1 843.4 926.8 45 486.9 550.6 618.3 689.9 765-4 844-7 928.2 46 487.9 55L7 619.5 691. i 766.7 846. I 929.6 47 488.9 552.8 620.6 692.4 768.0 847.5 931 o 48 490.0 553-9 621.8 693.6 769-3 848.9 932-4 49 491-0 555.0 623.0 694.8 770.6 850.2 933-8 50 492.0 5 56. i 624. i 696.0 771.9 851.6 935-2 5i 493- 1 557-2 625.3 697.3 773-1 852.9 936.6 52 494.1 558.3 626.5 698.5 774-5 854.3 938.1 53 495-2 559.4 627.6 699.7 775-8 855.7 939-5 54 496.2 560.5 628.8 701. o 777-1 857.1 940.9 55 497.2 561.6 630. o 7O2. 2 778.4 858.4 942.3 56 498 3 562.7 631.2 703.5 779-7 859.8 943.8 57 499-3 563-9 632.3 704.7 781.0 86I.I 945.2 58 500.3 565-0 633.5 705.9 782.3 862.5 946.6 59 501.4 566.1 634.7 707.1 783-6 863.9 948.1 ! 246 ASTRONOMY. Reduction to the Meridian; Values , 7 2 sin 2 J P V /v ii sm i" Seconds. 22 ra 23 m 24 Seconds. 22 m 2 3 m 24 ii ii n n // // o 949.6 1037.8 1129.9 30 993-2 1083. 3 1177.5 i 95 I -o 1039- 3 1131.4 31 994-7 1084. 8 1179.1 2 952.4 1040. 8 1133-0 32 996.2 1086. 4 1180.7 3 953-8 1042. 3 1134.6 33 997.6 1087. 9 1182.3 4 955-3 1043. 8 1136.2 34 999.1 1089. 5 1183.9 5 956.7 1045.3 H37-8 35 1000. 6 1091. o 1185.5 6 958.2 1046. 8 H39.3 36 1 002. i 1092. 6 1187. 1 7 959-6 1048. 3 1140.9 37 1003. 5 1094. i 1188.7 8 961. i 1049. 8 1142.5 38 1005.0 1095- 7 1190.3 9 962.5 1051-3 1144. o 39 1006. 5 1097. 2 1191.9 10 963-9 1052. 8 1145.6 40 1008. o 1098. 8 "93-5 ii 965.4 1054.3 1147.2 4i 1009. 4 1100.3 1195.1 12 966.9 1055-9 1148.8 42 1010. 9 noi. 9 1196.7 13 968.3 1057.4 1150.4 43 1012.4 1103.4 1198.3 14 969.8 1058. 9 1152.0 44 1013.9 1105. o 1199.9 15 971.2 1060. 4 U53-6 45 1015.4 1106. 5 1201. 5 16 972.7 1062. o II55-2 46 1016. 9 1108. i 1203. I 17 974.1 1063. 5 1156.8 47 1018. 4 1109. 6 1204. 7 18 975-5 1065.0 H58.3 48 1019. 9 IIII. 2 1206. 4 19 977.0 1066. 5 H59.9 49 1021.4 1112. 7 1208. o 20 978.5 1068. i 1161. 5 50 1022. 8 1114.3 1209. 6 21 979-9 1069. 6 1163. i 5i 1024. 3 1115.8 I2II. 2 22 981.4 1071. i 1164.7 52 1025. 8 1117.4 1212. 9 23 982.9 1072.6 1166. 3 53 1027.3 1118.9 I2I4.5 2 4 984.4 1074. 2 1167.9 54 1028. 8 1120.5 1216. i 2 5 985.8 1075. 7 1169.5 55 1030. 3 1 122. 1217.7 26 987.3 1077.2 1171. i 56 1031. 8 II23.6 1219.4 27 988.8 1078. 7 1172.7 57 1033-3 1125. I 1221. O 28 990.3 1080. 3 1 1 74- 3 58" 1034. 8 1126. 7 1222. 6 29 991.8 1081.8 II75-9 59 1036. 3 1128.3 1224. 2 LATITUDE. 247 Second Part of the Reduction to the Meridian. r 2 sin 4 ^ p Values of m = . -^-- sm i" Minutes. s 10 s 20 s 30 s 40* jo- 5 O. 01 0. 01 ii O. OI O. OI 0. OI O. OI 6 O. OI 0. 01 O. OI o. 02 o. 02 o. 02 7 o. 02 0.02 o. 03 0.03 o. 03 o. 04 8 o. 04 o. 04 o. 05 0.05 o. 05 o. 06 9 o. 06 0.07 0.08 0.08 0.08 o. 09 10 0.09 O. IO 0. II O. II 0. 12 0.13 ii 0.14 0.15 0.15 o. 16 0.17 o. 18 12 o. 19 o. 20 0.22 0.23 0.24 0.25 13 0.27 0.28 0.30 0.31 0-33 o.34 H o. 36 0.38 o 39 .0.41 0.43 o.45 15 0.47 0.49 0.52 0-54 0.56 o-59 16 o. 61 o. 64 o. 67 0.69 0.72 o.75 17 0.78 o. Si 0.84 0.88 o. 91 0-95 18 0.98 I. 02 1. 06 1.09 1.13 1. 18 19 1.22 1.26 1.30 1-35 I. 40 1-44 20 1.49 1.54 1. 60 1.65 1.70 1.76 21 1.82 1.87 1-93 1.99 2.06 2. 12 22 2.19 2.25 2.32 2.39 2. 46 2-54 23 2.61 2.69 2-77 2.85 2-93 3.01 24 3.10 3.18 3.27 3-36 3-45 3-55 25 364 3-74 3.84 3-94 4.05 4.i5 26 4.26 4-37 4.48 4. 60 4.72 4.83 27 4.96 5.08 5. 20 5-33 5.46 5.60 28 5-73 5-87 6. 01 6.15 6. 30 6.44 29 6-59 6.75 6. 90 7.06 7.22 7.38 3 7-55 7.72 7.89 8.06 8.24 8.42 31 8.61 8-79 8.98 9.17 9-37 9-57 32 9-77 9-97 10. 18 10.39 10. 61 10.82 33 ii. 04 11.27 11.50 u-73 11.96 12. 2O 34 12.44 12. 69 12.94 13.19 13-45 I3.7I 35 13-97 14.24 14-51 14.78 15. 06 15.35 248 ASTRONOMY. FORM FOR SURVEY OF v DETERMINATION OF THE LATITUDE, and South of DATE AND STATION. 1843, October 13. Month of the Big Black River, NAME OF STAR, y Pegasi, South of the Zenith. ( Sextant No. 2197, by Troughton 6 Simms, and INSTRUMENTS . . . ? ( Mean Solar Chronometer No. 76, by Charles ^o; Times of ob MERIDIAN DISTANCES, Q ^ !~ ^ servation = p. o v ~ fl h * * Q .2 || S-i O g by chro In mean In sidereal sin i" -> o ~ ^ J-s 6 nometer. solar time. time. 8 1 ^ ^ h. m. s. m. s. m. s. n I 10 18 40.4 9 44-2 9 45-8 187.3 3 49-8 2 19 44.4 8 40.2 8 41.6 148.3 2 5^9 3 20 48 7 36.5 7 37-7 114.2 2 27.3 4 21 46.4 6 38.2 6 39-3 86.9 I 47.6 5 22 44.4 5 40-2 5 4i. i 63-4 r I I 7 .8 6 23 54 4 30-5 4 3i- 2 40.1 M o 49.2 7 25 12 3 12.6 3 13- ! 20.3 QJ D-, o 24.9 8 26 46 i 38.6 i 38.8 5-2 o 06.3 9 28 16.4 o 08.2 o 08.2 o. o a 00.0 10 29 42 i 17-4 i 17. 6 3.2 c o 03. 9 ii 31 42 3 17.4 3 17.9 21.4 1/5 o 26.2 12 32 54.4 4 29.8 4 3-5 40.0 6 o 49 13 34 18 5 53-4 5 54-3 68.5 i 24 J 4 36 14.2 7 49-6 7 50.9 123.5 2 31-5 15 3832.2 10 07.6 10 09.2 202.3 4 08.2 16 40 06 ii 41-4 ii 43-3 269.9 . 5 3 1 - 1 Observer, Major J. D. Graham. Computer, do. do. LATITUDE. 2 49 RECORD AND COMPUTATION. from Observed Double Circum- Meridian Altitudes of Stars, North the Zenith. a tributary to the river Saint JoJin, Maine. Artificial Horizon of Mercury. Young. True circum-meri Observed double circum-meridi an altitudes of dian altitude of star, as corrected for refraction and True meridian al titudes deduced, Latitude deduced from each ob servation, = L = star 1 . errors of instru =(a + *) = A. (90 -f- D A). ment, = a. o / // , / // o / // H4 34 15 57 1838.5 57 22 28. 3 46 56 42. 55 36 15 57 19 38.5 57 22 30.4 56 40. 45 37 10 57 20 06 57 22 33.3 5 6 37-55 38 10 57 20 36 57 22 23.6 56 47- 25 39 30 57 21 16 57 22 33.8 56 37.05 40 30 57 21 46 57 22 35.2 56 35. 65 41 05 57 22 03.5 57 22 28.4 56 42. 45 4 1 5 57 22 26 57 22 32.3 56 34- 55 4i 5 57 22 26 57 22 26 56 40. 85 41 50 57 22 26 57 22 29.9 56 36. 95 41 oo 57 22 oi 57 22 27.2 56 39. 65 39 45 57 21 23.5 57 22 12.5 56 58.35 38 40 57 20 51 57 22 15 56 55- 85 36 30 57 19 46 57 22 17.5 56 53- 55 33 20 57 18 ii 57 22 19.2 56 51.85 3 5 57 16 56 57 22 27. i 46 56 39- 75 LATITUDE Deduced from a mean of 16 altitudes of star y Pegasi 46 56 43".4 Deduced from a mean of 10 altitudes of star y Cephei, observed this night with same sextant .. 46 57 10 . 7 Mean, or latitude adopted 46 56 5 7" 2 5 ASTRONOMY. Form for Record and Computation Continued. D=apparentdeclinat nofstar==i4i9 / io // .85 N. logcos 9. 98629 /= approximate latitude of place = 4657 / . .logcos 9. 83418 Sum ... 19.82048 a = approximate merid. alt. of star=5722 / io // .logcos 9. 73176 cos / cos D cos a = constant multiple = 1.227 ... log o. 08872 Refraction (ther. 28, bar. 29.14 in.) for mean obs d alts. o 39" Index-error of sextant _|_ 2 40 * Error of eccentricity, &c., of sextant -f i 40 Apparent AR. of the star f Pegasi o h c>5 m i4 s . 09 Sidereal time at mean noon at this station 13 26 20. 83 Sidereal interval from mean noon of star s culmi nation I0 3 8 53 . 16 Retardation of mean on sidereal time i 44 . 96 Mean time of culmination of star f Pegasi 10 37 08 . 2 Chronometer (C. Y. 76) slow of mean time at time of observation 08 43 . 6 Time by chronometer of culmination of star ^ Pegasi io ll 28 m 24 s . 6 On this night, October 13, 1843, Major Graham obtained for the latitude of this station, from 75 observations on 5 stars south of the zenith, com bined with 21 observations on f Cephei and Polaris, to the north 46 56 56".3 On the night of October 24, by 43 observations on 4 southern stars, combined with 2 observations on Y Cephei, the latitude deduced was 46 56 57. 2 On September 17, 1844, 66 observations on north and south stars gave for the latitude of this station 46 56 60.4 * The error of eccentricity is approximately ascertained by comparing latitudes, well determined by observations on north and south stars, with that which will result from north or south stars individually of various me ridional altitudes. It varies with the altitudes observed; that is to say, it is different for different parts of the limb of the instrument. LATITUDE. 251 LVI. To Determine the Latitude by an Altitude of a Star Near the Pole, at any hour. L = A (A cos/) + (A sin/) 2 tan A /5 1 (A sin/) 2 (A cos/) where A = the observed altitude, corrected for refraction, &c. ; A = the polar distance of the star in seconds of arc; a = J sin i"; log a = 4.3845449; /9 = sin 2 i"; log ft = 8.89403; and / = the hour-angle of the star. i / = sidereal time AR. * = solar time -j- AR. AR. * / is plus when the star is west, and minus when it is east of the meridian. The sign of cos / should also be attended to, for when / is greater than 6 h , or 90, the cosine is negative, and the second and fourth terms change the sign minus to plus. . The fourth term may be generally omitted ; its greatest value being only o"^. This formula is only applicable to stars within a very few degrees of the pole. For other circumpolar stars tan x = tan A cos / s cos A in which the upper sign is used when the star is above the pole; the under when below the pole. 2 5 2 ASTRONOMY. FORM FOR RECORD SURVEY OF DETERMINATION OF THE DATE AND STATION. 1843, September 6 Woodstock, NAME OF STAR. Polaris, (a Ursa Minoris,) observed on ( Sextant No. 2197, by Tmtghton 6 INSTRUMENTS ...\ Mean Solar Chronometer, No. 2440, Times of ob- MERIDIAN DISTANCES. o servation True sidereal 1 by mean solar chro- times of ob A cos/ JE> nome t e r servation. In sid l time, In arc, 6 No. 2440. P. h. m. s. h. m. s. h. m. s. o / // / // i i 33 02. 5 20 05 34. i 4 5 8 23- 2 74 35 48 24 18. i 2 i 34 28 20 06 59. 8 4 56 57-5 74 14 22. 5 24 54- 5 3 1 35 42. 7 20 08 14. 7 4 55 42. 6 73 55 39 25 19.8 4 i 36 38. 2 20 09 10. 4 4 54 46. 9 73 4i 43- 5 25 41-4 5 1 39 07. 5 20 ii 40. i 4 S 2 17-2 73 4 18 -26 34.7 6 I 41 II. 2 20 13 44. I 4 5 r 3-2 72 33 22. 5 27 27.1 7 i 44 28. 2 20 17 01. 7 4 4 6 55- 6 7i 43 54 28 40. 8 Observer, Major J. D. Graham. Computer, Do. LATITUDE. 253 AND COMPUTATION. t LATITUDE, from Observed Double Altitudes of Polaris. New Brunswick, (Grovels Inn.) between four and five hours before its upper meridian passage. Simms, and Artificial Horizon of Mercury. by Parkinson d> Frodsham. -f- a( A sin/) 2 . tan A . (A cos/) Observed double alti tudes of Po laris out of the merid ian. True altitudes of star, as cor rected for re fraction and errors of in strument, = A. Latitude de duced from each obser vation, = L. i n , i n / // / // + ! ! ! 63 + I 11.41 - o. 32 0.33 93 01.30 93 02-45 46 31 58.6 46 32 36 46 08 51.8 46 08 52. 6 -f- I 1 1. 2O 0.33 93 03. 50 46 33 08. 6 46 08 59. 7 + I 11.04 0.33 93 04.40 46 33 33- 6 46 09 02. 9 -{- I IO. 63 - 0.34 93 06. 15 ; 46 34 21 46 08 56. 6 -f I 10.28 o.35 93 08. 20 46 35 23. 5 46 09 06. 3 -f i 09.68 0.37 93 I0 - 5 46 36 38. 5 46 09 07 LATITUDE, deduced from a mean of 7 altitudes of star Polaris, 46 D 08 59 // -4* 254 ASTRONOMY. Form for Record and Computation Continued. Apparent declination of star = 88 28 3o".5. Apparent N. P. D. of star = 1 31 29".$ = 5489". 5 = A Refraction (ther. 57; bar. 30.013 inches) ......... . * Index-error of sextant _ // . * Errors of eccentricity, &c., of sextant _j_ z / 2 g// Apparent AR. of the star Polaris (a Ursa; Minoris) i h O3 m 5 ys. Sidereal time at mean noon at this station : j oo 2 j \ Sidereal interval from mean noon of star s culmination ... Retardation of mean on sidereal time . . , 10 03 30 .2 2 18.2 14 OI 12 Mean time of culmination of star Polaris ......... Chronometer No. 2440 fast of mean time at time of observation . 4 29 24 .8 Time by chronometer of culmination of star Polaris .......... 6 h 30 36 s . 8 The reduction of the mean time of observation to sidereal time, in the pre ceding example, might have been omitted by using table of AR. in Arc into Mean Time, pages 198, &c. Thus : Mean time of observation Mean time of culmination of Polaris .................... . 6 O2 s -, o 4)1 57111 Hour-angle, /, in intervals of mean time Sidereal equivalents, in arc ........................ 4 h = 60 57 m = 14 i? 20 .45 34 s = 8 31 .40 o".3 = 4.51 Computation, First - 74 35 47-75 Observation. ist term logcos/(+) = log A = A cos p 9.4242480 3-7395327 2d sin/ A A sin/ (A sin/) log a tan A term. = 9.984" = 3-73953 3d term. . . . =3-16378 . . . = 7-44728 log/9 =8.89403 3.1637807 2 4 18".! 31 58".6 = 3-72364 ist term = A = 46 2 = 7.44728 = 4-38454 = 0.02325 46 2d term = -f- 07 40 .5 i ii .63 2d term = 1-85507 = 7i".6 3 + i 11". 63 3d term= o /7 .32 46 3d term = 08 52 .13 o .32 Latitude = 46 o8 5i".8i LATITUDE. 255 LVI. Determination of the Latitude by Transits over the Prime Vertical. Suppose a transit-instrument so placed that the transit- axis is on the meridian, or very nearly so, and that the axis is horizontal, and the collimation nothing : i. Call the time T at which a star whose declination is D passes the middle wire of the instrument on the eastern side of the meridian, the clock-correction to reduce the observed time to the true E, and the right ascension of the star AR. ; and let T and E denote the corresponding quantities for the western transit. Then the two hour angles, in sidereal time, will be, the eastern negative, / = T + E - AR.; / = T + E - AR. Let the unknown latitude of the place be L, and the azimuth of the line of collimation a. The spherical triangle, formed by great circles connecting the zenith, the pole, and the place ot the star, gives the following relations : cos t cos D sin L sin D cos L cos D sin / cos t 1 cos D sin L sin D cos L cos D sin / Whence cos 4 / / If the instrument is very nearly on the prime vertical, cos J (f + /) = cos o = i and tan L = tan D sec J (t 1 t) for the passage over the middle wire of the instrument. 2. Call the time of passage of the star, from a side wire to the middle wire, r. Let the distance, in arc, of one of the lateral wires from the middle wire, measured on a great circle, be 15 // ./ being the equatorial interval of the wire, in time. 2 5 6 ASTRONOMY. LVI. Prime Vertical Transits Continued. Then, to reduce the transit over a side wire to the center wire, = [sin (L + D) . sin (L - D) - 1 //] 1 The upper sign of the term - 1 / / is to be used for wires crossed by the star earlier than the middle wire in the eastern transit, and later in the western transit, and the lower sign in the opposite cases. An approximate latitude may be used for L. 3. Should the optical axis not coincide with the middle wire, substitute / i c for / in the above, according as the error of collimation, c, lies on the same or opposite sides of/. 4. The preceding formula gives the latitude on the supposition that the axis of the instrument is parallel to the horizon. If the instrument is on the prime vertical, but the north end of the axis is, for instance, n seconds too high, the axis is parallel to the horizon of a place whose latitude is n seconds less than where the instrument is placed, and the true latitude is, therefore, L + n 5. But should the instrument not be on the prime vertical, the true latitude becomes L -[- n sin a a being the azimuth of the center wire of the telescope, supposed in collimation. This may be found from the time elapsed between the east and west transits of the same star : thus cot u = tan ( t t) sin D sin u sin a = cos D cos L a is taken between o and 90 when the north end of the transit- axis is between the north and west, and between 90 and 180 when the same end is between the north and east. If n is called plus when the north end of the axis is too high, and vice versa, the signs of the corrections are indicated by those of the quantities resulting from the formula. When a is nearly 90, the correction is exceedingly small ; so that, when the instrument is placed nearly east and west, we may proceed in all the computations as if it were exactly so. LATITUDE. 2 cj7 LVI. Prime Vertical Transits Continued. 6. The instrument should be set up in the firmest manner. A change of azimuth between the east and west transits of a star will affect the result much less than an equal change of level. It is better, in order to obtain a close result in the shortest time, to observe several stars on the same evening, and between the first and last observations to determine with the level the inclination of the axis several times, and then to interpolate for transits between the times of observation of the level. It is of course understood that the changes of inclination must be small, which will be the case if the instrument is properly placed. 7. In order to point the telescope rightly, the hour-angles and zenith-distances of the stars to be observed must be computed for the time of transit. When the telescope is on the prime vertical, calling p the hour- angle, and z the zenith-distance of the star, then cos/ = tan D cot L cos z ^= -r An allowance must be made for the time of crossing the first wire, and for change of zenith-distance from the first to the middle wire. 8. To correct, for errors of collimation, irregularity in the pivots, &c., the instrument may be reversed between the transits over each vertical; /. e., the wires on one side of the center wire are observed, the instrument reversed in its Y s, and the transit over the same wires continued, but in an inverse order; so that, in each vertical the same wire is at one time as far north as it is at another south of the optical axis. Then let L = the latitude sought ; D = the apparent declination of the star; t = the hour-angle, illuminated axis north; = J diff. of sidereal time of transit over the same wire, for same position of axis ; and t 1 hour-angle, illuminated axis south. tan D tan L = cos J (if + /) . cos % (t 1 258 ASTRONOMY LVII. To determine the Latitude of a Place by observing the Difference of the Meridional Zenith-Distances of Two Stars on Opposite Sides of the Zenith, with the Zenith and Equal-Alti tude Telescope. Compute an approximate latitude by the formula L = J [180 - ( A + A )] + i (* - * ) where A and A 7 are the polar distances of the south and north stars, respectively, and (s z ) the quantity measured by the micrometer. Then 1. The correction for level is applied by adding the angle which the vertical axis of the instrument makes with the zenith when the inclination is southward, or subtracting it when to the northward. This correction is found by multiplying the value of one division of the level-scale, in arc, by one-half the mean change, in level-divisions, which any one end of the bubble undergoes by reversing the instrument on the meridian ; or, if o and e, o and <? , denote the readings of the object and eye-ends of the bubble, for south and north stars, respectively ; corrections for level = ^ (o e } J (o c) x the value of one division of the level-scale in arc. 2. The correction for error of meridional position of the central vertical wire is found by computing the usual "reduction to the meridian " for each star ; then the difference between the reduc tions for the northern and southern stars is taken, and one-half that difference added or subtracted, according as the reduction for the northern star is greater or less than that for the southern ; or, correction for position, = ;// being the reduction for stars south, and m 1 for stars north of the zenith. * LATITUDE. 259 L VI I. Zenith Telescope Continued. 3 . When the star is obser ved off the line of collimation, the instrument remaining in the plane of the meridian, 1 2 sin 2 ^ i> . . f J. X $ sin 2 D sin i" The correction to the latitude is one-half of this quantity, whether the star be north or south ; and if the two stars forming a pair are observed off the line of collimation, two such corrections, separately computed, must be added to the latitude. D is minus wJien south. Values of m are given in the following table : D. 10* 5 20 s 25 s 30- 35 s 40 45 s 50* 55 6 o" D. Q 5 .00 .01 .02 03 .04 .06 .08 . 10 . 12 .14 .1 " O 7 85 10 .01 .02 .04 .06 .08 . ii J 5 .19 23 .28 .; !4 80 15 .01 03 05 .09 . 12 17 .22 .28 34 .41 .4 ^9 75 20 .02 .04 .07 . ii .16 .22 .28 36 44 53 -< >3 7o 25 .02 05 .08 13 .19 .26 34 .42 52 63 -3 r 5 65 30 .02 05 .09 15 .21 .29 38 .48 59 .71 .* 5 60 35 03 .06 . IO .16 23 31 .41 52 .64 77 ><> 2 55 40 03 .06 . II i7 .24 33 43 54 .67 .81 .5 7 5 45 03 .06 . II 17 25 33 44 55 .68 .82 .5 8 45 4. The correction for refraction is applied similarly to reduc- tion to meridian (2) but with a contrary sign ; or, Prm f*C*\"\ f\ r r 1 v-xUl J CL llU 2 - - r r 1 being small, no note need be taken of the state of the barometer and thermometer at the time of observation. 260 ASTRONOMY. L VI I. Zenith Telescope Continued. The following table gives the correction to the latitude for differential refraction; arguments, one-half difference of zenith- distances on one side and zenith-distance on the top : Zenith-distance. SS? 2 o 10 2O 25 30 35 / // ,, ii ii u tl . OO . OO . OO . OO . OO .00 o 30 . 01 . OI . OI . OI .01 . OI I . 02 . 02 .02 .02 .02 . 02 I 30 . 02 ^3 .03 .03 03 03 2 .03 3 .04 .04 .04 05 2 30 .04 .04 .05 .05 05 .06 3 05 05 .06 .06 .07 .08 3 30 .06 .06 .07 .07 .08 .09 4 .07 .07 .08 .08 .09 . 10 4 3 .08 .08 .09 .09 . IO . II 5 .08 .09 . 10 . 10 . II 3 5 30 .09 . 10 . 10 . II . 12 .14 6 . 10 . 10 . II . 12 13 .15 6 30 . II . II . 12 *3 .14 .16 7 . 12 . 12 13 .14 15 .18 7 30 !3 13 H .15 .16 .19 8 3 .14 !5 .16 .18 .21 8 30 .14 15 .16 .17 .19 .22 9 .15 .16 .17 .18 .20 23 9 30 .16 17 .18 . 20 .21 .24 10 .17 .18 .19 . 21 23 .26 10 30 .18 19 . 20 . 22 .24 .27 II .18 .19 . 21 23 25 .28 II 30 .19 .20 .22 .24 .26 30 12 .20 .21 23 25 .27 31 The sign of the correction is the same as that of the micro meter-difference. LATITUDE. 261 L VI I. Zenith Telescope Continued. 5. To find the value a of one division of the micrometer, note the time by chronometer of the transit of Polaris or other close circumpolar star over the movable wire placed vertically and set successively before the star for each turn or half-turn of the screw. Then let x be the arigular distance from the meridian at any reading of the screw; /, the hour-angle of the star at the same instant ; and D, its declination : sin x = cos D sin / The value of x is computed for each reading, and the differences of these values divided by the differences of the corresponding micrometer-readings give values for the screw. A better method, as it avoids displacing the micrometer, is to observe a close circumpolar star near its elongation, when rapidly rising or falling, with but slight motion in azimuth. The level should be carefully noted in order to allow for possible changes, and a correction applied for differential refraction. The sidereal time of elongation and the azimuth of the star can be determined from LXI. About 40 or more minutes before elongation, transits are noted, the micrometer being set in advance consecutively by whole or half turns of the screw throughout its length. A correction for rate of chronometer should be applied if sensible. Let / = the difference between the time of observation and the time of elongation of the star; and s" = the number of seconds of arc from elongation in thte direction of the vertical. s" = 15 cos D [t 1 (15 sin i") 2 t ? >] where / is expressed in seconds of time. 262 ASTRONOMY. LVII. Zenith Telescope Continued. Values of i (15 sin i") 2 / 3 for minutes of time from elongation are given in the following table : /. Term. t. Term. || /. Term. /. Term. m. s. VI. \ S. in. s. in. s. 5 o. o 15 0.6 25 3-o 35 8.2 6 o. o 16 0.8 26 3-3 36 8.9 7 o. o 17 0.9 27 3-7 37 9.6 8 O. I is ; i.i 28 4.2 38 10.4 9 0. I 19 i-3 29 4-6 39 ii. 3 10 0.2 20 i-5 30 5-i 40 12.2 ii 0.2 21 T 8 I. o 31 5-7 4i I3.I 12 0-3 22 2. O 3 2 6.2 42 14,1 13 0.4 2 3 2-3 33 6.8 43 I5.I 14 o-5 24 2.6 34 . 7-5 44 16.2 It is convenient to apply these values to the observed time of noting, additive to the observed time before, and subtractive after, either elongation. The correction to be applied to the observed times of noting for change of level is given by the formula where N , S , the north and south readings for a selected state of level, N, S, the readings for any other state, and b the value of one division of level-scale in seconds of arc ; the upper sign to be used for western, the lower sign for eastern elongation. After these two corrections have been applied we have in one column the readings of the micrometer, and in another the cor responding times, such as would have been obtained if the star had moved uniformly in a vertical line, leaving out of considera tion for the present the change in refraction and the rate of the chronometer. Various methods of combination might be adopted for the determination of the turn of the screw. That of subtracting the values resulting from the first operation from those of the middle one; next, those of the second from those of the middle one, plus one, and so on, is recommended for its simplicity. LATITUDE. 263 L VI I . Zenith Telescope Continued. A number of values are thus obtained for the time of a given number of turns or half-turns, from which is deduced the value of one turn : thus Mean time for one turn, ii6 s -774 lg 2 -6735 cos D log = 8.40750 15 log =1.1 7609 One turn 44"- 765 1-65094 Correction for refraction .025 Correction for rate .003 Resulting value ....... 44 .737 The correction for refraction, in seconds of arc, is negative for either eastern or western elongation, and equals the change ot refraction for the space equal to one turn ; equal to the value ot one turn times the difference of refraction for i at star s altitude divided by 60. 6. The value b of i division of the level-scale will be best found by using, in conjunction with the micrometer, a distant point as a mark, or the central wire of another instrument used as a collimator ; for the space above or below the mark, passed over by the horizontal wire of the micrometer, during the bubble s run over the scale, as the telescope s elevation is gradually altered, may afterward be measured by the micrometer-screw. The temperature should be noted, since the result may change with a change of temperature. Including all corrections, the general expression for latitude is __ . ., , . , 2 ~ 2 4 2 " 2 a and b being the arc values of one division of the micrometer and level-scale respectively. To correct as much as possible an erroneous determination of the value of the micrometer-screw, select stars for observation, such, if practicable, that the greatest zenith-distance of a pair will belong as often to the north star as to the south star ; be cause if the zenith-distance of the north star is the greatest, the observed quantity is subtractive; if least, additive. For, as a general rule, the error of latitude arising from an erroneous value to the micrometer-screw will be the least when in a set of stars 264 ASTRONOMY. b -a cfl oi 1 1 1 S" "* ^ o oo ^ . oo oo oo o v q q S S 0066 cr! G .2 2 000 o o o 6 III? 1 "3 ro r^ \o r^ i- 1 o 0* M + + 1 + 1 JJ 0; co M G O ^ 5 T~i~s ? s 3" s a? i i i i i 3 S 3 3 i 5 o> <* c o , i | ! ? 1 ? s ? c "o Q ^ m oo 00 O> 10 n ^ f? H" 2 ON 00 ON *l vo M m + + 1 + 00 ro q M M O_N * vo ON w 00 VO vo ON oo ro 4- * t--vo woo ooro t^ro ONm t^c^ t^O ONCTN H W MM Mro COM ometer. Q N 5 ^ ON ro M ^ oo oo ON f^ o 1 o i be q 1 j-^NM fOM -^-00 rOM ^ONvo^ oo^rl * roin Si 2 JJ 2S M" M" fc c* S5 oi ^ c/i ^ c/5 <H i/J 5 O o 5? ^ S3 3 ON VO vo AZIMUTHS. 265 LVIII. Knowing the Time and the Latitude of the Place, to find the Azimuth of the Sun or a Star. A - J (A + S) T J (A - S) the upper or negative sign is used when A is greater than A ; where A = the azimuth counted from the north, which must be subtracted from 180 if counted from the south; S = the angle at the star, called the angle of variation ; X = the co-latitude of the place ; A = the north-polar distance of the sun or star ; and p = the hour-angle at the pole. Without the Use of a Chronometer, by observing the Altitude of the Sun or Star at the Same Instant with the Observation of the Azimuth. Let Z = the zenith-distance, corrected for refraction, parallax, and semi-diameter: then sin k . sin (k A) cos 2 A A = : ^-A . sin Z sin X 2 k = Z + A + A 266 ASTRONOMY. LIX. To find the AMPLITUDE of a Celestial Object at its Rising or Setting; by " Amplitude" is meant the complement of the azimuth, or distance from the east or west points of the horizon. This is a particular case of the preceding problem. When the object appears to be in the horizon, its zenith-d.istance, instead of being 90, is, on account of refraction and parallax, 90 -f / , where k horizontal refraction horizontal parallax = 36 29" horizontal parallax. For stars, the* horizontal parallax = o and k = 9036 / 29" ; for the sun, k = 36 / 29 // 8".8 = 90 36 2o".2. The mean refraction and mean horizontal parallax are here used, as these observations are not susceptible of much accuracy. LX. To find the True Meridian by the Method of Equal Alti tudes of the Sun. The instrument remaining stationary, observe the readings of the horizontal limb when the altitude of the sun s center is the" same in the forenoon and afternoon. Then, the correction to the mean of these two readings for the change in the sun s declination in the interval is 4 (D - D ) cos L . sin J (/ f) where D D = the change in the sun s declination in the interval of the observations ; (t / ) = this interval of time expressed in arc ; and L = the latitude of the place. AZIMUTHS. 267 LXI. To find the Azimuth of POLARIS, or other Close Circumpolar Star, at its Greatest Eastern or Western Elongation. cos/ = tan A cot / = cot D tan L = tan L tan A cos L sin A = sin A = cos D where p = the hour-angle of the star ; D = its declination ; A = its polar distance ; A = the required azimuth ; L = the latitude of the place; and I = the co-latitude. The first equations give the hour-angle of the star at its greatest elongation; hence the sidereal time of elongation; The second, the azimuth of the star at its greatest elongation. The azimuth at any hour-angle is found by the methods in LVIII, or by the formula A (in seconds) = ^~ < A -f- A 2 sin i" cos/ tan L + J A 3 sin 2 i" [(i + 4 tan 2 L) cos 2 / tan 2 L] \ If the hour-angle is counted from the lower culmination, change the sign of the second term. The most approved method is to observe a series of azimuths of the star about the elongation, say for not more than 30 minutes before and after, and to reduce them to the elongation. To do this, compute, from the known latitude, the azimuth of the star at its greatest elongation = A, and call the sidereal time from elongation /; the correction to the azimuth will be c = (112.5) f 1 sm I " * an A log (112.5) sin *" = 6 -73 6 7 2 74 The quantities found in the table for "Reduction to the meridian" /smH/\ V sin L" ) correspond very nearly to (112.5) ^ s ^ n **-" so, by entering the table with the time from elongation, and 268 ASTRONOMY. LXI. To find the Azimuth of Polaris, &c. Continued. multiplying the tabular quantities by tan A, we obtain the re quired correction in seconds of arc. This will be found a con venient substitute for the more rigorous method. The formula may be separately applied to each observation, or the work may be shortened by computing only the azimuth corresponding to the mean hour angle and applying to it a correction to mean azimuth. Let 11 be the number of observations on the star ; A b the azi muth corresponding to the mean hour-angle; and let, also, r = the difference between the time of any observation and the mean of the times; then, for a circumpolar star, Correction to mean azimuth = A : tan A r -- zL!^lil n sin i" Example. Means of the times of observations by chronometer = 3 h 48 i2 s .3. Chronometer time of observation. T Tab. quantities, (page 240.) log 27". 4 = 1.4380 log tan A 8. 5 144 //. m. s. 3 42 3- 5 44 08. o 45 52.o 47 15- 48 59. o 5o 34- o 52 28. o 3 53 5i-5 1)1. S. 4 04.3 2 30-3 57.3 46.7 2 21. 7 4 15.7 5 39-2 63.7 32-5 10.7 1.8 I. 2 II. 35-7 62.8 9- 9524 Correction. = o". 90 IS = 27.4 Azimuth of star at elongation..! 52 42". 8 Chronometer time of elongation. 4 h o6 m 2O S . 5 Mean time of obs n by chron.-3 h 48 12 s . 3 log log log t^ . 6.0774174 (112.5) sin i". -6. 7367274 tan A 8.5162425 t i 088". 2 = i8 m 8". 2 Cor. = 2i".2 =.. 1.3263873 Azimuth corresponding to mean hour-angle = i 52 2i".6 Correction to mean azimuth = 0.9 Mean azimuth = 1" 5 2 20". 7 AZIMUTHS. 269 LXI. To find the Azimuth of Polaris, &c. Continued. Azimuths are usually reckoned from the south and in the direction of south to west. When circumpolar stars are observed, it is more convenient to reckon from the north meridian. The determination of primary azimuths supposes the local time to be known; for secondary azimuths observations for time and azimuth may be made together. The sun is only employed in connection with the inferior class of azimuths. For the purpose of referring azimuths observed at night to the direction of any geodetic signal, a mark is set up, consisting of a perforated box, (about 3^ of a foot cube,) through the front face of which the light of a bull s-eye lantern is shown, appearing of about the size and brilliancy of the star observed upon. The distance of this mark from the place of observation is generally determined by local circumstances, but should not, if possible, be nearer than about a mile, in order that the sidereal focus of the telescope may not require changing. For day observations a vertical black stripe, of the same width as the aperture, is painted upon a white wand placed vertically above it. If the diameter of the aperture is a quarter of an inch, it will subtend at the distance of a mile an angle of a little more than o /7 .8. The horizontal angle between the mark and any trigonometrical station is measured in connection with the triangulation by com bining it with all other directions radiating from the station observed from. Observations for azimuth are usually made in sets, commencing with a number on the mark, followed by about an equal number of readings on the star preceded and followed by level-readings. The instrument is then reversed, and the preceding operations are repeated in the reverse order, the number of readings on the star and mark being as before. In these observations the optical axis of the telescope of the theodolite must be made to describe a truly vertical plane. If the axis of the telescope is not horizontal, the correction to the azimuth will be i - | ( W + w >) _ ( e _ e f ) \ tan L where d = the value of one division of the level-scale in seconds 270 ASTRONOMY. LXI. To find the Azimuth of Polaris, &c. Continued. of arc, w, e, and / , e , the west and east readings of the level before and after reversal. The circumpolar stars a, 3, A Ursae Minoris and 51 Cephei are those almost exclusively used. When d Ursae Minoris and 51 Cephei culminate on either side of the pole, Polaris is not far from its elongation; and, on the contrary, when the pole-star culminates, the other two are not far from their elongations on either side of the meridian. A Ursae Minoris, from its greater proximity to the pole, and its small size, presents to the larger instruments a finer and steadier object than Polaris. LXII. Correction for RUN in reading Microscopes. As it is difficult to adjust the microscopes so that five revolu tions of the micrometer-screw shall carry the wire exactly over one of the five-minute spaces on the limb of the instrument, (if it be so graduated,) it is preferred to observe the number of revolutions and the part of a revolution made by the screw while the wire passes over the space ; then, let ;;/ = the mean of first readings ; that is, the readings obtained by turning the screw in the direction of increasing numbers from the zero of the comb ; and m 1 = the mean of second, or reverse, readings ; then (mean) run = r = m m -f- 300 and 300 . /;/ 700 (r 4- m 300) true (mean) readme = ^ - = ^ ^ r r = the number of minutes and seconds to be added to the degrees and minutes of the limb. LONGITUDE. 271 LXIII. Longitude by Lunar Culminations. i. Make / = true longitude sought ; / = approximate longitude ; m = observed change, in right ascension, of the moon s bright limb between the first meridian and that sought; ;;/ = computed change in same, by interpolation ; V = rate of motion, in right ascension, of the moon s bright limb, when on the meridian / ; and I = the constant difference between the values of the inde pendent variable, or arguments, corresponding to the consecutive tabulated values of the right ascension of the moon ; then 2. Interpolation. Take the following scheme, viz I F A, A 2 A 3 A 4 ^ / " a" 1 b" /" a" c" b d / a c 1 e b d f f t <*, c i e , t u ^ c n d, *, a in b " In which the column I contains the independent variable, or argument, as time, terrestial longitude, degrees, and the like ; F, the value of the function of this variable, as found in any set of tables; A b A 2 , &c., the first, second, &c., order of differences of these functions. 272 ASTRONOMY. LXIII. Longitude by Lunar Culminations Continued. Make s = the value of the function corresponding to any value t s between t 1 and //; t = t // 1 1 a e = a + a, Then, according to Bessel, Ast. Nach. No. 30 i) ./.(/- i) (/ 2 ] 1.2.3.4" . ^ _i_ c . Or, making A! = b A, = d A, = s = a + / (3) A .2 1.2.3 or, by the ascending powers of /, s = a + A/ + B/ 2 + C/ 3 + D/ 4 + &c. . . (4) LONGITUDE. 273 LXIII. Longitude by Lunar Culminations Continued, in which, stopping at the fourth differences, A = A, AA 2 4- T LAo - . . . . (5) B = JA 2 - JA 3 - -J T A 4 C = 1A 3 - ,VA 4 D = J T A 4 3 Also, V = ~ = A + 2 B / + 3 C / 2 + 4 D / 3 + . . (6) ; = s - a 1 = A t + B / 2 + C / 3 + D / 4 -f . . (7) The value of m may be obtained either from observations on the two meridians, or by observations on one and the tabulated results under the head of Moon Culminations in the Nautical Almanac, which may be used as actual observations on the meridian of the ephemeris. NOTE. If the lunar tables were perfectly accurate, the true lon gitude given by the observation would be found at once by com paring the observed right ascension with that of the ephemeris. There are two methods of avoiding or eliminating the errors of the ephemeris. In the first, the observation is compared with a corresponding one on the same day at the first meridian, or at some meridian the longitude of which is well established. In this method the increase of the right ascension in passing from one meridian to the other is directly observed, and the error of the ephemeris on the day of observation is consequently avoided; but observations at the unknown meridian are frequently rendered useless by a failure to obtain the corresponding observations at the first meridian. In the second method, the ephemeris is first corrected by means of all the observations taken at the fixed observatories during the semi-lunation within which the observation for longitude falls. The corrected ephemeris then takes the place of the correspond ing observation, and is even better than the single corresponding observation, since it has been corrected by means of all the ob servations at the fixed observatories during the semi-lunation. Chauvenefs Practical Astronomy. 18 274 ASTRONOMY. LXIII. Longitude by Lunar Culminations Continued. 3. Example. Let / = 4 h 55 m 5o s west from Greenwich, and suppose the following transits with a chronometer marking sidereal time. The error of the time-keeper is not material, and the transit is very nearly in the meridian, viz : Feb. 18. Geminorum 6 h 54 4i s -75 d Geminorum 7 10 38 .97 ]) s first limb .. 7 38 " o6 s .76 C Cancri 8 03 06. u 3)22 08 26.83 7 22 4 8 -943 o 15 17.817 Chronometer rate + 3 s daily o .0318 o h 15 I7 S .7S5 The corresponding observations at Greenwich, as given by the Nautical Almanac, are : Feb. 18. Geminorum 6 h 54 57 8 .4i 6 Geminorum 7 10 54 .36 D s first limb .. ;b 27 47 3 .66 Cancri 8 03 21.44 3)22 09 13.21 7 23 04.403 o 1 04 433.257 10 34.528 Next compute this increase from Nautical Almanac. The right ascensions of the moon are given in that work for the upper and lower passages over the meridian of Greenwich. The independent variable is, therefore, terrestrial longitude, of which the unit is one hour, and the intervals between the consecutive tabulated values of its function, 12. The increase to be com puted is for the interval of passage from the upper meridian of Greenwich to that 4 h 55 5o a west. Now, according to the scheme and equations (2), (3), and (5), LONGITUDE. 2 75 VO "M O VO vo VD 1O *"* **""* O vo O vo oo oo vo TJ- ^ oo ro vo r i* t^ d vO "^J. ^ CO N 00 ^O CO ^ C II ^ 11. OO HH co ON w t>. O vo x> -. g fO 1O ION OO* I """ 1? 1 + II II II II ^ >-~- ^1! II II II II <J M Q II ^sk , ( 1-1 C* CO T <] O O O w ta fcjo w ,3 2 r2 r2 OO 00 vO^ O vo b o M 1-1 O <3 it <3 >Q vo ON Q CJ ^-O T ? + r-^ M oo O ON ON 1 1 1 TJ- ON ON TJ- ON to O ^O II j> n o d ^ Tl- + 1 1 | - CJ rj- CXD ON ^J" ^O vq vo oq 1 1 T II II II II ON O O ON ON to O N "q q "q q ^ 1 1 1 II II ^ JT b o o o +1 I "t" W N HH C4 VO rj- b o o vo ro ^ ^ 00* OO fO * ? *? t 1 + 1 b ^ 3. o^ o C4 vo N vo vo <i S vo 10 vo vo c 00 vo T}- xo vo + + + + + + 1 II N VO OO o ft oo Z i> \f) \o co vo T}* C N* n- ^ s vo vO t^* OO ON O fl s VO cS C/3 S ^ O ?o ** t- ro 00 rl- ro O N vo HH TJ- o II II II II <! pq u a ^ vO t>. t>. t-. 00 00 ^ (^ -3 o CS s W " S ii C* 3 o OTO & J & J ^ J 2J "o u ^3 z X t^ 00 ON ^11 VJ- 1 t NH ^H j Q rj c/5 H U 276 ASTRONOMY. LXIII. Longitude by Lunar Culminations Continued. Then equation 7 A log 3.1893180 t log 9.6137147 2.8030327 Nos.. -f 635 S .337 B log 0.7108786 f z log 9.2274294 9.9383080 Nos.. 0.867 C log 9.0374265 / 3 log 8.8411441 7.8785706 Nos.. 0.007 D log 8.5910646 / 4 log 8.4548588 7.0459234 Nos.. -j- o .001 634 .464 And equation 6 A ---- B log 0.7108786 t log 9.6137147 2 log 0.3010300 0.6256233 Nos.. 4.222 C log 9.0374265 / 2 log 9.2274294 3 log 0.4771213 8.7419772 Nos.. 0.055 D log 8.5910646 / 3 log 8.8411441 4 log 0.6020600 8.0342687 Nos.. + o.on V = 1542.128 LONGITUDE. 277 LXIII. Longititde by Lunar Culminations Continued. And equation (i) I = I2 11 log 4-6354837 MI fji 1 = o s .o64 log 8.8061800 V = i542 8 .i28 log ac 6.8118797 i 8 -792 . . . 0.2535434 Whence / = 4 11 55 5 o s + i s -79 2 = 4 11 S5 m 5 l3 -79 2 If m be the observed increase of right ascension between any meridian not the first, (but of which the longitude is well known,) and the meridian sought, interpolate the increase m, for the known meridian as well as m 1 for that sought. Then, for m m , in equation (i), substitute ;;/ (m 1 m,), and the result will be the corrected longitude from the first meridian, as before. It often happens that two observers do not use the same number of wires, or do not observe the same number of stars at the two places. In such cases the observed increase of the right ascension of the moon s limb requires a correction, which Mr. Walker deduces as follows, from Gauss s method : For the eastern observatory and western station respectively, let A and A = the observed AR. of a star ; E = A A for the same star ; E = a similar value for another star; / and / = the number of wires on which each limb was observed ; a and a = similar values for a star; / 1 1 A = for the moon s limb ; - for one star ; a + a u = a similar value for another star ; -T = symbol to denote the aggregate of similar quantities; and s = the correction required; ASTRONOMY. LXIII. Longitude by Lunar Culminations Continued, then A u /> + u m - m> + e J. _ / J ~~V~~ Also, calling W the weight of each day s comparison, W = 7 ^4r-i in which s is the same as as y and <r = u -f- u -f- 7/ 4- &c. For the weight of the result of all the comparisons, we have Let e denote the probable error of observation, and E the probable error of the final result; then, (<r + A) z 2 It also frequently happens that the moon cannot be observed on the middle wire, in which case she is far enough from the meridian to have a sensible parallax in right ascension ; and, as it may be very desirable not to lose the observation, this parallax must be computed and applied to the hour-angle from the middle wire, which is supposed to be nearly coincident with the meridian. Denoting this parallax in right ascension by /, the horizontal parallax by w, the latitude of the place of observation by p, and the true declination of the moon by <?, we have from the ordinary series for the parallax in right ascension, neglecting the terms after the first, which would in this case be insignificant, p = sin w cos <p sec d in which is the hour-angle, or equatorial interval in sidereal LONGITUDE. 279 LXIII. Longitude by Lunar Culminations Continued. time from the lateral wire on which the moon is observed to the central wire; so that, at the instant of observation, the actual distance of the moon s limb from the central wire is sin w cos <p sec <5 and the reduction to meridian or middle wire will be , i sin w cos <p sec o cos d i 0.00277;;? in which ;;/ is the motion of the moon in right ascension in one day, expressed in degrees. The upper sign is to be used when the observation is on a wire before and the lower after the middle wire. In what precedes the approximate longitude I 1 is supposed to be known. When this is not the case, it may be found from and the interpolation is then to be made for this value of I to obtain the value of m . 280 ASTRONOMY. LXIV. Longitude by the Electric Telegraph. [From Cliauvenet s Practical Astronomy.] It is evident that the clocks at two stations, A and B, may be compared by means of signals communicated through an electro- telegraphic wire which connects the stations. Suppose at a time T by the clock at A, a signal is made which is perceived at B at the time T by the clock at that station. Let JT and JT be the clock-corrections on the times at these stations respectively, (both being solar or both sidereal.) Let x be the time required by the electric current to pass over the wire, then, A being the more easterly station, we have the difference of longitude A by the formula I = (T + JT) - (T + JT ) + x = *i + x Since x is unknown we must endeavor to eliminate it. For this purpose let a signal be made at B at the clock-time T", which is perceived at A at the clock-time T x// , then we have A = (T " + AT") - (T" + JT") - x = A 2 - x In these formulae ^ and J 2 denote the approximate values of the difference of longitude, found by signals east-west and west- east, respectively, when the transmission-time x is disregarded, and the true value is Such is the simple and obvious application of the telegraph to the determination of longitudes ; but the degree of accuracy of the result depends greatly more than at first appears upon the manner in which the signals are communicated and received. Suppose the observer at A taps upon a signal-key at an exact second by his clock, thereby producing an audible click of the armature of the electro-magnet at B. The observer at B may not only determine the nearest second by his clock when he hears this click, but may also estimate the fraction of a second ; and it would seem that we ought in this way to be able to determine a longitude within one-tenth of a second. But before even this LONGITUDE. 281 LXI V. Longitude by the Electric Telegraph Continued. degree of accuracy can be secured, we have yet to eliminate, or reduce to a minimum, the following sources of error : 1. The personal error of the observer who gives the signal; 2. The personal error of the observer who receives the signal and estimates the fraction of a second by the ear; 3. The small fraction of time required to complete the galvanic circuit after the finger touches the signal-key ; 4. The armature time, or the time required by the armature at the station where the signal is received, to move through the space in which it plays and to give the audible click ; 5. The errors of the supposed clock-corrections, which involve errors of observation and errors in the right ascensions of the stars employed. For the means of contending successfully with these sources of error we are indebted to our Coast Survey, which, under the superintendence of Professor Bache, not only called into exist ence the chronographic instruments, but has given us the most efficient method of using them. The " Method of Star-Signals," as it is called, was originally suggested by the distinguished astronomer, Mr. S. C. Walker, but its full development in the form now employed by the Coast Survey is due to Dr. B. A. Gould. Method of Star- Signals. The difference of longitude between the two stations is merely the time required by a star to pass from one meridian to the other, and this interval may be measured by means of a single clock placed at either station, but in the main galvanic circuit extending from one station to the other. Two chronographs, one at each station, are also in the circuit, and, when the wires are suitably connected, the clock-seconds are recorded upon both. A good transit-instrument is carefully mounted at each station. When the star enters the field of the transit-instrument at A, (the eastern station,) the observer, by a preconcerted signal with his signal-key, gives notice to the assistants at both A and B, who at once set the chronographs in motion, and the clock then records its seconds upon both. The instants of the star s transits 282 ASTRONOMY. LXIV. Longitude by the Electric Telegraph Continued. over the several threads of the reticule are also recorded upon both chronographs by the taps of the observer upon his signal- key. When the star has passed all the threads the observer indi cates it by another preconcerted signal, the chronographs are stopped, and the record is suitably marked with date, name of the star, and place of observation, to be subsequently identified and read off accurately by a scale. When the star arrives at the me ridian of B, the transit is recorded in the same manner upon both chronographs. Suitable observations having been made by each observer to determine the errors of his transit-instrument and the rate of the clock, let us put T t = the mean of the clock-times of the eastern transit of the star over all the threads, as read from the chronograph at A; T 2 = the same, as read from the chronograph at B; T/ = the mean of the clock-times of the western transit of the star over all the threads, as read from the chrono graph at A; T 2 = the same, as read from the chronograph at B; ^, e = the personal equations of the observers at A and B, re spectively; T, - = the corrections of T { and T/ (or of T 2 and T/) for the state of the transit-instruments at A and B, or the re spective "reductions to the meridian;" o T = the correction for clock-rate in the interval T/ TI ; x = the transmission-time of the electric current between A and B ; and > = the difference of longitude; then it is easily seen that we have, from the chronographic records at A, ;. = T/ + ST -f r + c >- x - (T, + r + e) and from the chronographic records at B, I = IV + IT + r + e + x - (T 2 + r + ) LONGITUDE. 283 LXIV. Longitude by the Electric Telegraph Continued. and the mean of these values is X = [ J (TV + T 2 ) + r ] - [ J (Ti + T 2 ) + r] + flT + ^ - <? which we may briefly express thus: A = A! + ^ - ^ in which A! = the approximate difference of longitude found by the exchange of star-signals, when the personal equations of the observers are neglected. This equation would be final if e? e, or the relative personal equation of the observers, were known ; however, if the observers now exchange stations and repeat the above process, we shall have, provided the relative personal equation is constant, X X 2 -f e e 1 in which ^ 2 is the approximate difference of longitude found as before ; and hence the final value is I have not here introduced any consideration of the armature- time, because it affects clock-signals and star-signals in the same manner; and therefore the time read from the chronographic fillet or sheet is the same as if the armature acted instantaneously. It is necessary, however, that this time should be constant from the first observation at the first station to the last observation at the second, and therefore it is important that no changes should be made in the adjustment of the apparatus during the interval. As the observer has only to tap the transit of the star over the threads, the latter may be placed very close together. The reticules prepared by Mr. W. Wiirdemann for the Coast Survey have generally contained twenty-five threads, in groups or " tallies " of five, the equatorial intervals between the threads of a group being 2 8 .5, and those between the groups, 5 s ; with an additional thread on each side at the distance of io 8 , for use in observations by "eye and ear." Except when clouds intervene and render it necessary to take whatever threads may be available, only the 284 ASTRONOMY. LXIV. Longitude by the Electric Telegraph Continued. three middle tallies, or fifteen threads, are used. The use of more has been found to add less to the accuracy of a determina- tion than is lost in consequence of the greater fatigue from concentrating the attention for nearly twice as long. A large number of stars may thus be observed on the same night ; and it will be well to record half of them by the clock at one station and the other half by the clock at the other station, upon the general principle of varying the circumstances under which several determinations are made, whenever practicable, without a sacrifice of the integrity of the method. For this reason, also, the transit-instrument should be reversed during a night s work at least once, an equal number of stars being ob served in each position, whereby the results will be freed from any undetermined errors of collimation and inequality of pivots. Before and after the exchange of the star-signals, each observer should take at least two circumpolar stars to determine the instru mental constants, upon which r and T depend. This part of the work must be carried out with the greatest precision, employing only standard stars, as the errors of r and r come directly into the difference of longitude. The right ascensions of the " signal- stars" do not enter into the computation, and the result is, there fore, wholly free from any error in their tabular places ; hence, any of the stars of the larger catalogues may be used as signal- stars, and it will always be possible to select a sufficient number . which culminate at moderate zenith-distances at both stations, (unless the difference of latitude is unusually great,) so that instrumental errors will have the minimum effect. A single night s work, however, is not to be regarded as con clusive, although a large number of stars may have been observed and the results appear very accordant; for experience shows that there are always errors which are constant, or nearly so, for the same night, and which do not appear to be represented in the corrections computed and applied. Their existence is proved when the mean results of different nights are compared. More over, it is necessary to interchange the observers in order to eliminate their personal equations. The rule of the Coast Survey has been that when fifty stars have been exchanged on not less than three nights, the observers exchange stations, and fifty stars LONGITUDE. 28 5 LXIV. Longitude by the Electric Telegraph Continued. are again exchanged on not less than three nights. The ob servers should also meet and determine their relative personal equation, if possible, before and after each series, as it may prove that this equation is not absolutely constant. Before entering upon a series of star-signals, each observer will be provided with a list of the stars to be employed. The prep aration of this list requires a knowledge of the approximate difference of longitude, in order that the stars may be so selected that transits at the two stations may not occur simultaneously. Example. For the purpose of finding the difference of longi tude between the Seaton station of the United States Coast Survey and Raleigh, a list of stars was prepared, from which I extract the following for illustration. The latitudes are : Seaton station, (Washington) <p = -f- 38 53 7 -4 Raleigh station, (North Carolina) y = -f 35 47 .o and Raleigh is assumed to be west from Washington 6 m 30 s . Seaton sidereal Star. Mag. a 6 time of Raleigh transit. * h, m. s. 1 h. m. s. No. 5036, B. A. C. 3 15 09 36 + 33 52 15 16 06 5084 4-3 18 58 37 54 25 28 5i3i 4/ 2 27 02 3i 5i 33 32 5*92 5 36 35 26 46 43 05 5259 5 45 43 36 07 52 13 5322 4/2 55 59 23 12 16 02 29 5388 5 16 04 09 45 J 9 10 39 5463 3-4 15 21 46 40 21 51 The following table contains the observations made on one of these stars at the above-named stations by the United States Coast-Survey telegraphic party in 1853 April 28 under the direction of Dr. B. A. Gould. In this table " Lamp W " expresses the position of the rotation- axes of the transit-instruments. The first column contains the symbols by which the fifteen threads of the three middle tallies 286 ASTRONOMY. LXIV. Longitude by the Electric Telegraph Continued. were denoted ; the second column, the times of transit of the star over each thread at Seaton, as read from the chronographs at Seaton ; the third column, the times of these transits as read from the chronographs at Raleigh ; the fourth column, the mean of the second and third columns ; the fifth column, the reduction of each thread to the mean of all, computed from the known equatorial intervals of the threads ; the sixth column, the time of the star s transit over the mean of the threads, being the algebraic sum of the numbers in the fourth and fifth columns ; and the remaining columns, the Raleigh observations similarly recorded and reduced. Seaton-Raleigh, 1853, April 28. Star No. 5259, B. A. C. Thread. Seaton observations, lamp W. Raleigh observations, lamp W. T, T 2 Mean Red. T, + T, T/ T, Mean Red. T/ + T./ 2 2 h. m. s. s. s~ s. J. h, 711. S. j. j. j. J. Ci 37-97 38.00 37.98 + 2 5.49 3-47 11.00 11.00 + 25.45 36.45 C 2 4i-37 4i-34 41.36 22.21 3-57 14.58 14.50 14-54 22.25 36.79 C 3 44-03 44.21 44.12 19.06 3-i8 17.60 17.55 17.58 19.05 36.63 C 4 47.81 47-74 47.78 I5-7I 3-49 20.88 20.79 20.84 15-85 36.69 C 5 50.76 50.70 50.73 12.71 3-44 23.90 23.87 23.89 12.70 36.59 D! 56.96 57-10 57-03 6.21 3-24 30.19 30.05 30.12 6.32 36.44 D a 0.06 0.04 0.05 3-25 3-30 33-34 33.25 33-3 3.18 36.48 D 3 15 46 3.40 3.38 3-39 4- 0.05 3-44 i5 S 2 36.40 36.30 36.35 + 0.07 36.42 D 4 6.70 6.70 6.70 - 3-03 [3-67] 39.61 39-53 39-57 - 3-16 36.41 D 5 9-58 9-58 9-58 6.28 3-30 43.00 43.00 43.00 6.36 36.6 4 EI 16.03 15-93 15.98 12.54 3-44 49.04 48.81 48.92 12.75 [36.17] E 2 19.26 19.30 19.28 15-83 3-45 52-30 52.33 52-32 15.90 36.42 E 3 22.47 22.45 22.46 18.99 3-47 55-50 55.41 55.46 19.10 36.36 E 4 25.60 25.60 25.60 22.23 3-38 58.73 58.60 58.67 22.20 36.47 E 5 28.60 28.70 28.65 25-33 3-32 2.08 2.08 2.08 25-38 36.70 Mean . 3-392 Mean . 36.535 The numbers in the last column for each station would be equal if the observations and chronographic apparatus were per fect; and by carrying them out thus individually we can estimate their accuracy. The numbers 3.67 at Seaton and 36.17 at Ra leigh are rejected by the application of Peirce s Criterion, (Method of Least Squares,) and the given means are found from the re maining numbers. LONGITUDE. 28 7 LXIV. Longitude by the Electric Telegraph Continued. The corrections of the transit-instruments for this star (d = + 36 6 .$) were for the Seaton instrument., r = 0.028 for the Raleigh instrument T = 0.193 The rate of the clock was insensible in the brief interval TV T. Hence, neglecting the personal equations of the ob servers, the difference of longitude is found as follows : 4 (TV + T a ) + T = 15* S2 m 36 8 -342 i(Ti + T 2 ) + r = 15 46 3-364 6 32 .978 In this manner seven other stars were observed on the same night, and the results were as follows : Star. fa Diff. from mean, = v. m. s. s. 5036, B. A. C. 6 33-03 -f 0.04 5084, B. A. C. 33-9 -f- 0. 10 5131,6. A. C. 32.91 0.08 5192, B. A. C. 33-00 -f- o.oi 5259, B. A. C. 32.98 o. 01 5322, B. A. C. 33-00 -f- O.OI 5388, B. A. C. 33-02 + 0.03 5463, B. A. C. 32.91 0.08 Mean, fa = 6 32. 99 From the residuals, v, we deduce the mean error of a single de termination by one star, and hence the mean error of the value 6 m 32^99 is , o s .o6 = j= = O s .02 But this error will be somewhat increased by those errors of the instruments which are constant for the night, and not repre sented in T and r 7 , and by the errors of the personal equations yet to be applied. Moreover, a greater number of determina tions should be compared in order to arrive at a just evaluation of the mean error. 288 ASTRONOMY. LXV. Formula for Probable Error and Precision. [Contributed by Lieutenant Mercur, Corps of Engineers.] i. Let ;;/ = the number of observations; ;/, n , &c. = results found by observation ; x = their arithmetical mean ; i , z> , &c. = (#;/), (x ;/ ), &:c. = the residual errors of ob servation; e = the mean error of ;/, n , &c. ; 7] = the mean of errors, (arithmetical;) E = the mean error of final result; E! = the mean error of the observation assumed as the standard of excellence; r = the probable error of a single observation, ;/, n , &c. ; R = the probable error of the final result, x; h = the measure of exactness of a single observation, X , &c.; H = the measure of exactness of the final result, x ; /, / , c. = the weights of different observations or sets of observations; e f , e", &c. = the mean errors of observations corresponding to/,/, &c.; I = symbol representing the sum; r 0.6745 e = 0.8453 >j = 0,8453 ~: Iv (m i) /- T- / / 2v* /-4S49S^ 7;2 r R = 0.6741; E 0.6745 / - r- = / Y . = = 9 \m(m*-i) V m (m i) -/ m 7,^0.46936 H = //V^ r 2. By "the weight of any determination" is meant its relative approximation to the true value. It may be measured by the number of observations (each of which may be considered as good as the other, and of which one is assumed to represent the unit of excellence) necessary to give a result equally near the true value. ERROR AND PRECISION. 289 LXV. Formula for Probable Error, &c. Continued. Then, since the weights of observations are inversely as the squares of their mean or probable errors, and when we arbitrarily assign weights to each observation or set of observations, -i) in which ;// = the number of observations or sets of observations whose weights are/,/ , &c. When observations are combined by weights, the probable error is given by the formula R = 0.6741; 745 3. PEIRCE S Criterion for the rejection of doubtful observations. To apply this to any set of observations involving but one unknown quantity : * Let m = the number of observations taken; ;/ = the number of doubtful observations to be re jected, (to be found by trial;) s = the mean error of one observation in the set of /// observations; i , v j &c. = the residual errors of the observations; and -/. = the ratio of the required limit of error for the re jection of n observations to the mean error -, so that xs is the limiting error. Find from the table the value of x 2 for n i, n = 2, n = 3, &c., in succession, and reject all observations in which xe>z ; stopping, however, when the value of xs found for any particular value of n does not reject any observations (not already rejected) for a value of n numerically one less. * For rules for determining mean and probable errors, and for applying the Criterion to cases involving more than one unknown quantity, see Chauvenet s Manual of Spherical and Practical Astronomy, vol. II, pages 469 to 566. 290 ASTRONOMY. LXV. Formula for Probable Error, &c. Continued. Example. To determine the value of one turn of the microm eter of a zenith-telescope, the following "reduced intervals" were obtained, each corresponding to the ten turns of the micrometer : Between observa Reduced intervals. V tions. 111. S. i and 10 20 35.1 II .0 121. OO 2 and 12 3 and 13 43-9 41.2 2.2 4-9 4.84 24.01 when n i K* 3.707 K* e a = 204.148 4 and 14 51-2 5-i 26.OI K e = 14.28 5 and 15 46.6 -5 00.25 which rejects (n and 21) 6 and 16 44.0 2. I 4.41 7 and 17 38.2 7-9 62.41 in ~ 1 1 8 and 18 54-2 8.1 65.61 when n = 2 K* 2.621 9 and 19 47-9 1.8 3-24 K* e 2 == 144.341 10 and 20 43-9 2.2 4.84 which rejects none other. ii and 21 21 OI .4 15-3 234.09 Sum = 228 27.6 v l ---= 550.71 x = 20 46.1 * l = 55-Q7 1 After rejecting the interval between (n and 21) the probable error is found as follows : Between observa tions. Reduced intervals. V V* /;/. j. i and ii 20 35.1 9-5 90.25 2 and 12 43-9 0.7 49 3 and 13 41.2 3-4 11.56 4 and 14 51.2 6.6 43-56 5 an d 15 46.6 2.0 4.00 6 and 16 44.0 0.6 .36 7 and 17 38.2 6.4 40.96 8 and 18 54-2 9.6 92. 16 9 and 19 47-9 3-3 10.89 10 and 20 43-9 0.7 49 Sum = 207 26.2 ,..= 294.72 X 20 44.62 * = 32-749 / 2 94 -.72 _ , 9- 2 5 | p ro 5 a bi e error of single set = 0.6745 x 5.72 or r = 3 s -86 Probable error of final result = or This is for ten revolutions ; for one revolution the probable error is o .122. PEIRCES CRITERION. Peirce s Criterion. Values of K" for // = i. 291 n i 2 3 4 5 6 7 8 9 1.480 j 4 1.912 I.l6 3 5 2.278 !-439 6 2.592 1.687 i. 208 7 2.866 i .910 1.409 : I-45 8 3.109 2. 112 1.589 1.229 9 3.327 2.2 9 5 L753 i 1.388 I.OQI IO 3-526 2.464 1.904 1.242 ii 3-707 2.621 2.045 ; 1.662 J-373 I. 122 12 3.875 2. 7 66 2.176 i 1.785 1.492 1.249 1.018 13 4.029 2.902 2.299 ; 1.901 i .604 1.362 I.I45 14 4-173 3.030 2.416 ! 2.009 1.709 | 1.465 1-255 1 1.053 15 4-309 3- I 5 I 2.526 2. Ill 1.807 I I o6i J-354 1.163 16 4-436 3-264 2.630 2.207 1.898 ; 1.651 445 1-259 1.080 J 7 4* 555 3-371 2.729 ; 2.300 1.985 | 1.736 529 1-347 i . 176 18 4 668 3-475 2.824 2.389 2.069 i 1.817 .609 . 1.428 1.261 J 9 4.776 3-57T 2.914 2.474 2.150 1.895 .685 i 504 r-34 1 20 4.878 3.664 3.001 : 2.556 2.227 : J -970 757 i 1-576 21 4.9/5 3-755 3.084 1 2.634 2.301 ! 2.041 .827 i 1.644 1.483 22 5.068 3.840 3.164 ; 2.709 2.373 ! 2.109 .893 1.710 1 549 23 5- 1 57 3.923 3.240 ; 2.782 2.442 2.176 957 T -773 i .612 24 5-242 4.002 3.3I5 2.852 2.509 ! 2.240 .019 1-833 1.671 25 5-324 4.078 3-387 ; 2.920 2.573 2.302 .079 1.892 1.729 26 5-403 4- I 5 I 3.456 2.986 2.636 : 2.362 T 37 1.948 1.784 27 28 5-479 4.222 ^s88 : 3.049 2.697 i 2.420 .194 2.003 1.838 29 5.622 4.291 4-358 3-651 3.111 2.756 ; 2.477 3.171 ; 2.813 ; 2.532 .249 .302 2.056 2.108 1.941 30 5-690 4.422 3-712 3.229 2.869 ; 2.586 354 2.158 1.990 31 5 756 4-484 3-772 3-285 2.923 2.638 .404 2.207 2.038 3 2 5.820 4-545 3.829 3-34 2.976 2.689 454 2-255 2.085 33 5.882 4.604 3-884 3-394 3.028 2.738 .502 2.302 2.130 34 5-942 4.661 3-939 3-446 3.078 2.787 -549 2-347 2-175 35 6.001 4-7 I 7 3-992 3-497 3.127 f 2.834 594 2.392 2.218 36 6.058 4-77* 4.044 3-547 3-174 2.880 639 2.436 2.261 37 38 6. 113 6.167 4^874 4-095 4.144 3-595 3-643 3.221 3.267 2.926 2.970 .683 .726 2.478 2.520 2.302 2-343 39 6.2IQ 4-925 4.192 3.689 3-312 3.013 .768 2.561 2-383 40 6.270 4-974 4-239 3-734 3-055 .809 2.601 2.422 4 1 4 2 6.320 6.360 5.022 5.069 4.285 3-779 3.822 3.398 3-440 3-97 3.138 . .849 .888 2.640 2.678 2 . 460 2-497 43 6.416 5.H4 4-375 3-865 3.481 3-178 i 927 2. 716 2-534 44 45 6.463 6.508 5-159 5.202 4.418 4.460 3.906 3.521 3.947 3o6i 3-217 ; 3-255 -965 3.002 2-753 2.789 2.570 2.606 46 47 48 49 6-552 6.596 6.639 6.681 5-245 5-287 5-328 5.368 4.501 4-542 4.581 4.620 3-987 4.026 4-065 4.103 3.600 3-638 3-675 3-7 12 3-293 3.33 1 3-366 j 3.401 3-39 3-075 3.110 3- I 45 2.825 2.860 2.894 2.928 2.641 2.675 2.708 2.741 50 6.720 5-408 4-657 4.140 3.748 3-436 3-1.79 2.962 2-774 5i 52 53 54 55 56 57 6.761 6.800 6.838 6.876 6.913 6.950 6.986 5-447 5-484 5-522 5-559 5-595 5.630 5-665 4-695 4-732 4.768 4.804 4-839 4.873 4.907 4.176 4.212 4.247 4.282 4.316 4-349 4-382 3-784 3.819 iisj 3.920 3.952 3.984 3-471 3-505 i 3-538 i 3-571 ! 3-603 ! 3-635 | 3-213 3.246 3-279 3-3 11 3-342 3-373 3-44 2-994 3.027 3-059 3.090 3.121 3^181 2.806 2.838 2.869 2.899 2.929 2-959 2.988 58 7.021 5-699 4.941 4-4*5 4.016 3.697 1 3-434 3.210 3- OI 7 59 60 7.056 7.090 5-733 5.766 4-974 5.006 4-447 4.478 4.047 3.728 4.078 3.758 : 3-463 3-492 3-239 3.268 3-046 3-074 Geographical Positions. Latitude. Cambridge, Observatory 42 22 48. I Quebec, Citadel 46 48 1 7. 3 New York, Observatory 40 43 48. 5 Oswego, Court-House 43 27 49.1 WASHINGTON, Observatory 38 53 38. 8 Buffalo, Michigan and Exchange streets 42 52 41.8 Detroit, New L. S. Observatory 42 19 58. 6 Chicago, City Hall 41 53 06. 2 Saint Louis, Washington University 38 37 Saint Paul, Custom-House 44 53 Fort Leaven worth, Engineer Observatory 39 21 Omaha, Coast- Survey Observatory 41 16 Denver, Mint 39 45 01.8 Salt Lake, Coast-Survey Observatory 40 46 San Francisco, Washington Square 374755-3 Longitude west from Greenwich. Ji. in. s. 4 44 31.04 4 44 49-42 4 55 5 6 - 6 5 5 06 01. 05 5 08 12. 12 5 15 27.58 5 3 2 12-24 5 50 32. 08 6 oo 49. 02 6 12 21. 84 6 19 39-35 6 23 46. 33 6 59 5 8 - 7 2 7 27 35-45 8 09 38.23 TABLES AND FORMULAE PAR T I V. APPENDIX. APPE N DIX. LXVI. Field Magnetic Observations. [By Captain CHAS. W. RAYMOND, Corps of Engineers.] These observations have for their object the determination of the magnetic declination, dip, and intensity, at any given time and place. The following instructions have reference to the determination of the magnetic elements on land. It will be supposed that the theodolite magnetometer and dip- circle are the instruments em ployed ; the first to determine the declination and the horizontal component of the intensity, and the second to determine the in clination or dip. Magnetic Declination, The magnetometer having been mounted upon its tripod, or upon a sound post firmly imbedded in the ground, the horizontal limb and the rotation-axis of the telescope must be leveled, the vertical wire of the telescope made truly vertical, and its collima- tion-error reduced. The magnet must then be suspended by as few filaments as possible ; four or five are usually required. The magnet is then made horizontal by adjusting the balancing-ring, the position of which should be carefully preserved throughout the experiments. In order to adjust the magnet-scale to the stellar focus of its lens, the telescope must be turned upon the sun or a star, and adjusted to perfectly distinct vision. The tele scope must then be turned upon the suspended magnet, and the scale-ring screwed in or out until the scale is seen with perfect distinctness. The lines of detorsion and collimation must now be brought into the plane of the magnetic meridian by the following method : The magnet being suspended, turn the instrument in azimuth until the scale is seen through the telescope. Remove the magnet and suspend the brass detorsion-cylinder. Bring the axis of the cylinder, by estimation, into the plane of the magnetic meridian by turning the torsion-circle. Remove the cylinder and suspend the declination-magnet. Turn the instrument in azimuth until the vertical wire of the telescope bisects the scale- zero. Remove the declination and suspend the short magnet. 296 APPENDIX. LXVL Field Magnetic Observations, &c. Continued. Turn the torsion-circle until the vertical wire of the telescope coincides with the scale-zero. Exchange the magnets and adjust as before. The time, temperature, and readings of the verniers and torsion-circle should be recorded. This method requires that the magnets and detorsion-cylinder should be of equal weight. With the cylinder alone, the plane of detorsion may be determined to within about one degree, an error which does not seriously affect the accuracy of declinations observed in ordinary field- work. The instrument is now in adjustment for observations of dec lination. The angular value of one scale-interval and the scale-zero of each magnet employed must be determined at some convenient time. The former is unchangeable. The latter .must be occa sionally redetermined, as it is liable to change through accident. To determine the angular scale-value, fix the magnet in the position which it occupies when suspended, in such a way that the instrument may be moved in azimuth without disturbing it. Turn the instrument in azimuth until the vertical wire of the telescope coincides with a scale-division. Record the vernier and scale readings. Turn the instrument until the vertical wire coincides with another division, and record as before. Repeat over different parts of the scale until a sufficient number of ob servations have been obtained. Each pair of observations furnishes a single determination of the required value. The probable error of the mean value may be determined by the method of least squares. To determine the scale-zero, or reading of the magnetic axis, suspend the magnet, turn the instrument in azimuth until the vertical wire of the telescope coincides with a division near the middle of the scale, and record the scale-reading. Invert the magnet, and record the reading corresponding to this position. Move the instrument slightly in azimuth, and repeat this operation until three or four readings with the scale erect, and as many with the scale inverted, have been obtained. The scale-zero may then be determined by the method of alternate means, (see Form A.) These observations should, if practicable, be made about the epoch of the day when the magnet is stationary. FIELD MAGNETIC OBSERVATIONS. 297 LXVI. Field Magnetic Obsewations, <5rv. Continued. The co-efficient of torsion may be determined as follows : The declination-magnet being suspended and the instrument in ad justment for observations of declination, record the readings of the scale and torsion-circle. Turn the torsion-circle through an angle of 90, and record this difference of arc and the corre sponding scale-reading. Turn the torsion-circle 180 in the reverse direction, and record as before. Finally, turn the torsion- circle back to its original position, and repeat the readings and record. Computation. H u__ F " 90 u TT i + -^ = co-efficient of torsion; and u = difference in scale-readings (reduced to arc) cor responding to a change of direction of the magnet caused by twisting the suspension- thread through an angle of 90. The mean value of u deduced from the observations is em ployed. For convenience the co-efficient of torsion is usually applied to the angular value [a] of the scale-interval, (see Form B.) At some convenient time, while the instrument is in position, the vernier-readings corresponding to the astronomical meridian must be determined, either by turning the telescope upon a point of which the azimuth is known, or directly, by any suitable method. * The preliminary adjustments and determinations having been made, the instrument is left in position for observations of the diurnal variations in declination. At some time early in the morning, the north end of the magnet attains its most easterly position, which is called the morning eastern elongation. The record of scale-readings must be commenced early enough to include this elongation. When this point is fairly passed, or the north end of the magnet has fully commenced its westerly motion, the readings may be discontinued until about noon, when they must be resumed and continued until the western elongation has been observed, and the easterly motion has fairly set in. 298 APPENDIX. LXVI. Field Magnetic Observations, &c. Continued. The telescope should be reversed at each observation, and a mean of the readings in the two positions should be taken. The temperature should be noted. The readings should be made half or quarter hourly during the periods of observation. The observations of the first day will determine the approximate times of elongation, or turning-hours, in accordance with which the periods of observation are to be subsequently regulated. Computation. / H \ * = * (* +p )(<-*+ 3 == value of declination for the day ; o = declination at instant of final instrumental ad justment, which is -J 1 when the magnetic meridian is { we ^ 1 of the true north meridian ; ( east ) a = angular scale-value ; TT i + u = co-efficient of torsion ; e = mean of scale-readings at the elongations, which s = scale-reading of magnetic axis, (zero of magnet- scale,) which is { + | when { jf^^ } than ,; ^ r = difference between scale-readings at elongations, or daily range. j = factor tor reduction to the mean ol no vations. It may be taken, with its the following table : 7 urly obser- sign, from i January. February . . . March o. 089 0.040 O OIQ May . o. 01^ j September . i October - o. 044 o 006 " June ( O OIO Tulv + o ooc November . December.. . . o. 096 O I ^4. April o. 068 j u v August .. .. 0.023 1 FIELD MAGNETIC OBSERVATIONS. 299 LXVI. Field Magnetic Observations, crv. Continued. The correction to d* is positive or negative according as it in dicates a motion from or toward the true meridian. (See Form C. ) Magnetic Intensify. For the determination of the horizontal intensity two distinct series of experiments are required experiments of deflection and experiments of oscillation. The experiments of deflection are made as follows : The de flection-bar is made fast in its position, and the copper damper placed within the box. The instrument is then adjusted as for observations of declination, the short magnet being suspended. The experiments should be made, when practicable, at three distances. The first position of the deflector should be at about three times its length from the suspended magnet ; the third, at a distance about one-third greater ; and the second midway be tween the other two. These distances are measured from center to center. At the beginning, the verniers are read and time noted, in order to follow changes of declination. The tempera ture is recorded at each observation. The long magnet is placed upon its carriage on the deflection-bar, on either side of the sus pended magnet, and at the nearest distance. The instrument is then turned in azimuth until the vertical wire of the telescope coincides with the scale-zero. The time is then noted and the verniers read. The carriage is then moved to the next distance, and finally to the greatest distance, and the observation is re peated at each position. The deflector is then reversed on its carriage, and the observa tions are repeated at the three distances, beginning with the greatest and ending with the least. At the nearest distance the magnet is again reversed, and the complete set of observations is repeated, in order to obtain a double set of results. The deflector is now placed on the opposite side of the suspended magnet at the nearest distance. A double set of experiments similar to that already described is then made. At suitable intervals special observations should be made to measure changes of declination. For this purpose the deflector is removed, and the instrument is turned in azimuth until the vertical wire of the telescope coincides with the scale-zero. The 3 APPENDIX. LXVI. Field Magnetic Observations, &c. Continued. verniers are then read and the temperature and time noted. From these observations we may, by simple interpolation, deter mine with sufficient accuracy corrections for the reduction of the observed angles of deflection to the same declination. Computation. m m = magnetic moment of the deflector ; X = horizontal intensity ; ; = distance between the centers of the magnets ; and P = constant depending on the distribution of magnetism in the magnets. 1 he value of -^ must be computed for each distance separately and a mean adopted. (See Form D.) The correction depending upon the constant P may be neg lected when there is a considerable difference between the lengths of the magnets. Its value is greatest when the magnets are equal in length, and zero when the lengths are in the proportion of i to 1.224. To determine the value of P, deflections are made alternately at two different distances, which should be in the proportion of i to 1.32. About twenty-five corresponding sets should be obtained. Computation. A A, P = A . ? > = value of - Y for *M j -A. ^ a.^iigv^i j { 1 To reduce the angles of deflection determined from different sets to the same temperature : sin 7/ sin = - (4 _- {]q 7/ = observed angle at temperature / ; u = angle reduced to standard temperature /; and q = temperature-constant, determined as explained here after. FIELD MAGNETIC OBSERVATIONS. 301 LXVI. Field Magnetic Observations, erv. Continued. The experiments of oscillation are made as follows : The in strument having been adjusted as for observations of declination, with the long magnet suspended, and the co-efficient of torsion having been determined, the magnet is made to oscillate hori zontally by attracting or repelling one of its poles. The impulse should be sufficient to make it oscillate beyond the limits of the scale for at least ten minutes, as steadiness of motion is thus acquired. All vertical oscillations must be carefully checked. When the amplitude is sufficiently reduced, the scale is read at the limits of an oscillation, and a division near the mean of these readings is selected as the zero or point at the passage of which the times are to be noted. The intervals of oscillation may now be noted in the following way : The approximate interval corre sponding to six oscillations is noted for convenience. The instant of passage is then noted at every sixth oscillation up to the sixtieth; then at the hundredth, two hundreth, and three hundreth ; then at every sixth oscillation up to the three hundred and sixtieth ; and then at the four hundredth, if it be desirable to prolong the observations to this extent. The number of oscilla tions timed should depend on the length of the interval. The entire time of observation should not exceed a quarter of an hour. The approximate time of ten oscillations is computed for convenience at the sixtieth. For the semi-oscillations timed the magnet will always move in the same direction. The amplitude of oscillation at the beginning and end of the experiments should be noted. At suitable intervals the mean reading of the scale should be observed, since the zero is liable to changes due to variations of declination. The temperature should be observed at intervals, the bulb of the thermometer being placed within the box. Computation. * 2 K mX = -7p- ;;/ = magnetic moment of the magnet; X = horizontal intensity ; * = 3-M-I59; K = moment of inertia of magnet -and stirrup ; 302 APPENDIX. LXVI. Field Magnetic Observations, 6-V. Continued. T = corrected time of oscillation ; T* ^ T*(. + 5) (i -( - ) T = observed time of oscillation ; ;} = teperat n re of magnet when tj temperature-constant, or change in magnetic moment for a change in temperature of i Fahrenheit. (See Form E.) To determine the moment of inertia of the magnet and stirrup, [K], the moment of inertia of the inertia-ring must first be com puted. For this purpose accurate measurements of its outer and inner radii (in decimals of a foot) and the weight of the ring (in grains) are required. These data are usually furnished by the maker. The instrument being in adjustment with the long magnet suspended, the inertia-ring is balanced upon the magnet by means of the balancing-blocks. The time of a single oscillation is then determined as before described. The load is then removed and the time of oscillation is again determined. At least twelve sets of these experiments should be made. A separate determi nation of the co-efficient of torsion must be made for the loaded magnet. The temperature must be recorded throughout the experiments. Computation. K" = 4 (r* + r 2 ) w K = moment of inertia of the suspended mass ; K" = moment of inertia of the ring ; T ) , r i -11 s r ( loaded ) T J = corrected time of single oscillation of | unloaded f magnet ; radius of rin S> in feet ; and ft = weight of ring, in grains. FIELD MAGNETIC OBSERVATIONS. 303 LXVI. Field Magnetic Observations, crV. Continued. The values of - 2 K for different temperatures should be tabu lated. The co -efficient of expansion for brass (o.ooooi) may be employed for their computation. The change in magnetic moment for a change in temperature of i Fahrenheit, \q\, is best determined by the method of deflections. The magnet for which q is to be determined is the deflector. At least three consecutive sets of deflections should be made, the first and third being at about the same temperature, and the intermediate set at a very different one. A mean of the results from the first and third sets must be compared with the result from the intermediate set. The required difference of tempera ture may be produced by a jacket of ice and hot water, or advantage may be taken of extreme natural temperatures. Computation, a 11 cot // q = temperature-constant ; / \ ( higher ) /!,} = ! lower } temperature ; // = difference of scale-readings corresponding to / / ; a = angular value of one scale-interval ; and ;/ = angle of deflection corresponding to /. Computation of the Horizontal Intensity and Magnetic Moment. X= / V a X := absolute horizontal intensity; ;//.. = magnetic moment of deflecting and oscillating magnet ; = ;;/X, determined by experiments of oscillation ; and t = -^ - , determined by experiments of deflection. Strictly, the values of a and /? should be corrected for the effect of induction. These corrections are very small, and require 34 APPENDIX. LXVI. Field Magnetic Observations, &c. Continued. special apparatus for their determination. They are therefore neglected in field-work. (See Form E.) To reduce m to a Standard Temperature. MO = m (i + (/- t^q) m$ = value of /// at standard temperature / ; / = temperature at time of experiments ; and q = temperature-constant. Computation of the Total Intensity. = X sec = total intensity ; X = horizontal intensity ; and = inclination or dip. To convert measures of intensity expressed in English units into their equivalents expressed in the metric system, multiply by 0.46108, (log = 9.66378.) To convert measures of intensity expressed in metric units into their equivalents expressed in English units, multiply by 2.1688, (log = 0.33622.) Magnetic Inclination. The dip-circle having been mounted on its tripod or post, the horizontal limb must be leveled. The needle is charged by means of the magnetizing-bars, and then suspended as follows : Raise the Y s, and placing the needle in them, lower it gently upon the agate supports. Turn the vertical circle slowly in azimuth around the entire circle, and see whether the needle plays freely, and whether its face lies in the plane of the face of the vertical circle in all azimuths. If necessary, the agate supports must be re-adjusted. The face of the vertical circle is then brought into the plane of the magnetic meridian by the following method: Turn the vertical circle in azimuth until the needle is vertical. Record the reading of the azimuth-circle. Reverse the needle on its supports ; make it again vertical by a slight movement in azimuth, and record as before. Turn the vertical circle 180 in azimuth, and repeat the double observation. A mean of the four readings is the reading of the magnetic prime-vertical, from which the settings of the magnetic meridian are obtained by FIELD MAGNETIC OBSERVATIONS. 305 LXVI. Field Magnetic Observations, &c. Continued. adding and subtracting 90. Set the vertical circle at one of these readings. (See Form F.) The observations for the determination of the inclination are made as follows : Record the reading of the azimuth-circle. Record the polarity of the needle, (marked end north or south?) the position of the vertical-circle, (face east or west,) a mean of the readings of the vertical circle at the ends of the needle, and the temperature Fahrenheit. Raise the needle, reverse it on its supports, and repeat the observations. Bring the needle back to its original position, and observe as before. Reverse it again, and repeat the observations. Turn the vertical circle 180 in azimuth, and repeat the observations. Remove the needle, and reverse its polarity. Suspend it again, and repeat the observations with the circle and needle in both positions, as before. Computation. = magnetic inclination ; n | = mean of observed values of j J^ 6 | reversal of polarity ; c = constant correction for errors of axle and limb. The constant correction (c) is determined as follows : A complete set of experiments must be made in the plane of the magnetic meridian. The vertical circle is then turned in either direction about 45 in azimuth, and a similar series of experiments is made. The vertical circle is then turned 90 in azimuth in the opposite direction, and the experiments are again repeated. Computation. c = 6 0i- t cot 2 = cot 2 + cot 2 0" 6 ) \ = observed inclinations in planes at right angles to each other; and 0! = observed inclination in the plane of the magnetic meridian. 20 306 APPENDIX. FORM A. Determination of the Zero of the Magnet- Scale. Station, Fort Yukon, Alaska. Date, August 12, (p. m.,) 1869. Observer, C. W. R. Recorder, C. W. R. Scale-zero of declination-magnet. Position of scale. Reading of scale. Alternate means. Zeros. Mean zero. Erect 4C. OO Inverted $7 OO 4^ .0 CT 2^ Erect 46. oo CC. CO co. 7C, Inverted .4 OO A A 7 C 40 ^ 7 50.18 Erect 4^ ^O CC 2C. 4.0-27 Inverted ^6. c,o FORM B. Determination of the Co-efficient of Torsion. Station, Fort Yukon, Alaska. Date, August 13, 1869. Observer, C. W. R. Recorder, C. W. R. Declination-magnet suspended. Circle-readings. Scale-readings. DifT. of arc. Diff. of scale. Mean for 90. 3-95 5I-25 3-95 12.95 3-5 79-5 90 1 80 20.75 49.00 23-56 3-95 55-oo 90 24.50 Compu ta tion. 0.0 = tt 23 d . 56 == 90 It = H_ F . , H 699 .7 log log log 2.84491 5.50961 32330 -3 0.00216 1.00216 29". 76 7-33530 h F *( +?)- H_ u F~ 90 u *a= 29". 70 FIELD MAGNETIC OBSERVATIONS. 307 FORM C. Declination Record of Observations. Station, Fort Porter, Buffalo, N. Y., 222 feet north of flag-staff. Date, June 14, 1872. Observer, A. N. L. Azimuth-reading at adjustment, ver. B 158 09 oo" ver. A 338 09 30 Reading of mark on flag-staff, .ver. B : 166 21 oo ver. A 346 21 oo Azimuth of flag-staff, west of north 175 35 53 Magnet C 2 suspended; scale-zero, 21.7. Time. Scale. Remarks. h. m. 6 oo a. m. 21.7 6 20 22.3 6 45 22.6 7 oo 22.7 Maximum. 7 30 22.4 8 oo 22. ii 30 15.6 12 oo m. 14.75 12 30 p. m. 14.35 Minimum. 12 45 14.65 I OO I 5-5 1 15 15-7 Computation. Correction to (5 . Astr. merid .. 161 56 53" Az. at adjust ment . 158 09 15 e = 18 S = 21. 5 7 e s = T > .2 <* = + 3 47 38 Corr. to d = -}- 9 4 5 r 8. 35 01 e s +f> 3-2S / 4- . fr=-\-o. 08 T + F)-" 56 ^ = + 3 5 6 42. 5 308 APPENDIX. FORM D. Horizontal Intensity Experiments of Deflection. Station, Fort Yukon, Alaska. Date, August 14, 1869. Observer, C. W. R. Recorder, C. W. R. Magnets at right angles to each other ; long magnet deflecting ; short magnet suspended. Magnet at i .O9 east and west. W. 3 4 9 10 15 16 21 22 Mean E. W. E. W. E. W. for E. 81.0 82.0 81.0 82.0 85.5 87.0 84.0 85.0 and H 71 18 251 18 55 33 235 33 71 21 251 21 55 35 235 35 7i 34 251 34 55 3 235 30 71 36 251 3 6 55 42 2 35 4 2 161 18 H5 33 161 21 H5 35 161 34 145 3 161 36 145 42 756.6 h. 111. 2 IO 2 18 3 02 ? OO 4 13 4 22 5 22 5 35 161 17.5 145 33- 7 161 19. 7 145 3 6 -4 Mean 161 34. i 145 29.8 161 36.5 145 41.4 Mean 15 43- 8 7 5 1 - 9 15 46.0 7 53-0 15 43-3 7 51-6 7 52-2 16 04. 3 8 02. i 16 06. 7 8 03.3 15 55-1 7 57-5 8 01. o Remarks. Computation. . jh 34111 p. m. Turned instrument on 50.18 declination-scale. Verniers, E., 63 51 ; W., 2 435 I/ - Temperature, 78.5 F., (attached.) 3 h 45 m P- ni - Deflector away to show changes of declination. Temperature, 87 F. Verniers, E., 63 49 ; W., 243 49 . 6 h oo m p.m. End of experiments. Tem perature, 82 F.; scale, 58.50. Verniers, E., 63 54 ; W., 243 54. "X log. 9- H049 o. 11228 9- 69897 > 95 i 74 FIELD MAGNETIC OBSERVATIONS. 39 FORM E. Horizontal Intensity Experiments of Oscillation. Station, Fort Yukon, Alaska. Date, August, 16, 1869. Chronometer, Bliss & Creighton, 1609, M. T. Daily rate, unknown. Observer, C. \V. R. Recorder, J. J. M. Long magnet suspended without load. , t/5 o c ^* *-> r* g g u-, -2 o |l 2 ^ 1 Remarks. . r^ .3 O a rH *O K O 5 QJ VH .^5 f/5 O H Q H //. ;;/. s. s. s. o i 16 05.0 82.5 6 16 40. 8 Approximate time of 6 oscilla 12 17 16.5 tions at the beginning 35 s . 18 17 52.3 24 18 28.3 Scale reading noted just be 30 19 04.0 179.0 59.67 fore 2OOth oscillation to 36 19 39. 7 178.9 59.63 detect changes of declina 42 20 15.5 179.0 59.67 tion. Reading, 65. oo; lo fc 48 20 51.3 179.0 59.67 cal time, I2 h I5 m p. m. 54 21 27. O 178.7 59-57 60 22 02.8 178.8 59.60 IOO 26 01.5 Si.o 238.7 59.67 200 35 57-5 81.5 596.0 59-60 Time of 10 oscillations, 59 S .64; time of I oscillation, 5 s -964. Computation. q 0.00015 T Logarithms. f t o.o6 T /2 o. 77554 (f t)q o. 000009 J + F l 1.55108 I -(/ -/) / 0.99991 l -(ff)q o. 00103 Logarithms. T , 9. 99996 m* X mX 8.95174 9. 83 2 42 7T 2 K niX. 1.55207 1.38449 up 8. 78416 m 9- 83242 X 9- 39208 o. 44034 Experiments of deflection. APPENDIX. FORM F. Inclination Determination of the Dip. Station, Willet s Point, N. Y. Date, August 5, 1872. Observer, C. W. R. Recorder, C. W. R. Dip -circle by Wurdemann ; Lloyd s needle. Meridian observations. Settings for magnetic meridian. Face of circle. Face of needle. Readings of hor. limb. S. S. N. N. S. N. S. N. 157 27 158 42 337 oi 338 12 Magnetic prime-vertical, 157 50 ; settings, 247 50 , 67 50 . Marked end. Face of circle. Face of needle. Means of N. and S. ends. Means. Means. N. E. E. E. W. 73 04 73 oo 72 12 73 02 oo" 72 36 30" W. 72 10 72 ii oo W. E. E. 72 3 72 28 72 29 oo W. W. 72 49 72 49 72 49 oo 72 39 oo Mean ... N 72 37 45 S. E. E. E. W. W. 72 43 72 45 73 oo 73 12 72 44 oo" 73 06 oo 72 55 oo" E. E. W. 72 49 72 47 73 : 4 72 48 oo 73 oi oo W. 73 J 4 73 H oo Mean . . . .[ ] 72 58 oo Resulting inclination a ~r P 72 47 5 2 Computation. = 72 47 52"; c = + 18"; 6 = 72 48 . YC (35S2