No. 3. Pap er s relating to the duties of the Corps of Topographical Engineers. A COLLECTION oil TABLES AND FORMULAE USEFUL IN SURVEYING, GEODESY, AND PRACTICAL ASTRONOMY, INCLUDING ELEMENTS FOR THE PROJECTION OF MAPS. PREPARED FOR THE USE OF THIE CORPS OF TOPOGRAPHICAL ENGINEERS, BY CAPTAIN T' JT. LEE, Topographical Engineers, U. S. Army. SECOND EDITION) WITH ADDITIONS, WASHINGTON TAYLOR & MAURY. GIDEON AND CO., PRINTERS, 1853. BUREAU OF TOPOGRAPHICAL ENGINEERS, Washington,. pril 4, 1853. SIR: The edition of Topographical Papers, No. 3, a colb lection of Tables and Formulae, etc., prepared by you in 1849, having become exhausted, and the great use of the collection being fully proved, the Hon. Secretary of War, appreciating its value, has authorized the printing of a new edition, with the corrections and additions which have been suggested by experience. You will give this your immediate attention. Respectfully, sir, Your obedient servant, J. J. ABERT, Col. Corps T. E. CAPT. T. J. LEE, Corps Top'l DEig'rs, Washington. TO COL. J. J. ABERT, Chief Corps of Topographical Engineers. SIR: I have endeavored, in the following pages, to comply with your instructions by presenting, in as condensed a form as practicable, such Tables and Formulae as may prove most useful to an officer engaged in the active duties of a survey. In the selection of the matter it has been my aim. to present the best methods, as far as they have been practised by us, or may be applicable to the nature of our duties, in such forms as to be convenient for reference, and still secure a high degree of' accuracy in the reduction of such observations as may be requisite for the minute survey of a limited extent of country, as'well as for the exact determinamtion of Geographical Positions or for distant Explorations. With such a subject I call lay claim to but little that is original, and although aware of the many imperfections in this Collection, I still trust that it may not be without its utility, and that as a JManual of easy reference it may meet the wants of my brother officers. Although every precaution has been taken to ensure accuracy of print, it is not improbable that some errors may have escaped correction. A table of errata is appended, and I would be obliged by the communication of any others that may be detected. Very respectfully, Your obedient servant, THOMAS J. LEE, Cap. Top. Engineers WASHINGTON, l4ugust 8, 1849. E R AT A. Page 9; 27878400 square feet =1 square mile. Page 64; Dp = (5.0857556) Cos P -- (2.00835) Cos 3 i -L etc.; Page 145; opposite 24 hours, read 3Tn 56F.555. PREFATORY. The following explanations of the sources from which the several portions of this Collection were derived, will serve to establish the degree of confidence with which each may be received. PART I. Pp. 1 —3. Baily, Astronomical Tables and Formule. Francceur, Geodesie, Paris, 1840. 4. Francaeur, Geodesie. 5 —7. Begat, Traite de Geodesie, Paris, 1839. 8-14. Weights and measures. Those of the United States will be found in the Report of Professor Bache, Superintendent of Weights and Measures, July 30, 1848, Ex. Doe., No. 84, 30th Congress, 1st session. The remaining part of the article is from Brande's Dictionary of Science and Art, American edition, the quantities having been compared, when practicable, with Alexander's Universal Dictionary of Weights and Measures, Baltimore, 1850, and with the tables in Appleton's Dictionary of Mechanics and Engineering. 14-15. These tables are from the Edinburg Philosophical Journal of October, 1837. 16-17. Abridged from tables in Hulsse's Sammnlung Mathematischer Tafeln, Leipsic, 1840, with the addition of the relation of Spanish and Mexican measures. 18. Claudel, Aide Memoire des Ingenieurs, etc., Paris, 1849. The length of the Spanish vara was compared with its value given in Francceur, Begat, Brande, and Hulsse's works. 19. Ordnance Manuel, 1850. 20-22. BEat, GEodEsie. VIII PREFATORY. Pp. 23. McNiel's]Railway Tables, London, 1833. Beardmore, Hydraulic and other Tables, London, 1852. 24-25. Storrow on Water Works, Boston, 1833. 26-30. Railroad Manual, by Brevet Lieut. Col. Long, Corps Topographical Engineers, Baltimore, 1829. 31-32. Davies' Surveying. 33-47. Abridged from the Traverse Tables of Captain J. T. Boileau, Bengal Engineers. 48-49. Beardmore, Hydraulic and other tables, London, 1852. 50. Regulations of the Subsistence Department. PART II. 53. Begat Ghodgsie. 54-55. Galbraith, Mathematical Tables and Formule, Edinburg, 1834. 56. Francour, Geodesie. 57. Baily, Astronomical Tables and Formula, London, 1827. 58-59. Galbraith, Mathematical Tables, etc. 60. Baily, Astronomical Tables.-Begat, Ghodasie. 61. These values were carefully compared with the original in the Astronomische Nachrichten, No. 438. 62-64. I have adopted the yard as a unit, it being the unit of our lineal measures, and have, in the text, given the reasons for adhering to Kater's value of the metre. I have also preferred the established ratio of the metre to the toise, to that derived from Mr. EIassler's comparisons. (See Hassler's report of 1832, Doc. No. 299, 22d Congress, 1st session.) In reducing Bessel's Terrestrial Elements to English yards, and in other computations hereafter to be noticed, I was fortunate in securing the services of Mr. John Downes, now attached to the American Nautical Almanac establishment, whose well known reputation is the surest evidence of their accuracy. These reductions were also compared with a computation of my own. PREFATORY. IX Pp. 65-69. These are in the form given in Begat, Gdodesie. They are sufficiently accurate for our ordinary wants; for very extended operations, it is not to be presumed that the officer would make this collection his only guide. 70 —71. At the solicitation of several officers I have introduced into this edition examples to explain the application of many of the formulae. 72-77. The values of N and R, etc., within the limiting parallels of the territory of the United States, were computed by Lieut. Thom, Corps Topographical Engineers; afterwards by Mr. Downes, and the two carefully compared. 78-80. Begat, G6odesie. 81-82. Trigonometrical surveying-Lieutenant Frome, Royal Engineers. 83-86. Baily, Astronomical Tables and Formulae. 87-94. Abridged from Guyot's Meteorological Tables-prepared for the Smithsonian Institution. 1852. 95. Adapted, from Guyot's tables, to English inches and Fahrenheit's Thermometer scale. 96-98. The first method is from a manuscript of the late J. N. Nicollet, who probably obtained it from Mr. Hassler, as it is the projection in use at the Coast Survey office. The remaining methods will be found in Francceur, G6odesie. 99-128. The whole of these tables were computed, under my direction, for the Bureau of Topographical Engineers, by Mr. Downes. They were, occasionally, compared with similar quantities (in metres) in the manuscript tables in the Coast Survey office. Appendix. Magnetical observations-from the Magnetical Instruc129-137. tions prepared by order of the British Government, by Lieut. J. C. B. Riddel, Royal Artillery. 1844. 138. Eighth report of the British Association, 1838, page 91. B X PREFATORY. PART II1. Pp. 141-143. Francceur, Astronomie Pratique-Baily, Ast. Tables and Formulae. 144-145. Baily, Ast. Tables and Formulae. 146-157. Downes, U. S. Almanac, 1845. Compared, also, with Baily, Ast. Tables whenever practicable. 158-160. This, with subsequent examples of Sextant observations, was obtained through the kindness of Brevet Lieut. Col. J. D. Graham, Corps Topographical Engineers, from the records of the Northeastern Boundary Survey. 161-169. Baily, Ast. Tables. 170-172. Lieut. Col. Graham. 173. American Almanac; Downes's U. S. Almanac. 174-181. Ivory's Refractions, from Galbraith, Math. Tables and Formulae. The zenith distances are changed to altitudes, as more convenient for our purposes. 182-184. Baily, Ast. Tables and Form.; Simms on Math. Instruments, London, 1836. 185. Original. This table, and the one on page 188, will be found convenient in setting up a Transit Instrument. 186-187. Extracted from some of my own observations whilst attached to the Coast Survey. 189-191. Francceur, Astronomie Pratique. 192-199. Baily, Ast. Tables and Formulae. 200-202. Lieut. Col. J. D. Graham. 203. Francceur, Astronomie Pratique. 204-206. Lieut. Col. J. D. Graham. 207-210. Fram a manuscript translation of an article by Prof. Hanson, Ast. Nach., No. 143. The method of reversals, described by Struve in his notice of the Rhepsold Instrument, Ast. Nach., Vol. 20, is undoubtedly the best; but, for the want of a reversing apparatus, is ill suited to such Transit Instruments as are usually carried into the field. PREFATORY. X1 Pp. 211-214. From a description, by myself, of the use of the zenith and equal altitude Telescope, printed for the Bureau of Topographical Engineers in 1848. 215-218. Francceur, Ast. Prat.; Simms on Math. Instruments. Reduction to elongation and corrections for level, R. H. Fauntleroy, U. S. Coast Survey. 218. Correction for Run. Henderson's Edinb. Ast. Observations. 219-221. Downes, U. S. Almanac. Francccur, Ast. Prat. 222-223. Frome, Trigonomerical Surveying. 224-231. Downes, U. S. Almanac; Walker, Trans. Am. Phil. Society, Vol. VI; Prof. Bartlett on Longitude by lunar culminations, printed for the Bureau of Topographical Engineers. 232-237. Gummere's Astronomy. 238-240. From a manuscript explanation, by Prof. Bartlett, of an article by Encke, translated in Taylor's Scientific Memoirs, part VII. 241-242. The authorities are given in the text. T. J. L. CONTENTS. PART 1.-Miscellaneous. Page. Trigonometry-equivalent expressions.. 1 Trigonometrical Series..... 3 Signs of Trigonometrical Lines... 3 Arcs in parts of Radius.. 4 Solution of Plane Triangles.. 5 Solution of Spherical Triangles 6 Weights and Measures of the United States. 8 English System of Measures.. 9 Tables of British Weights. 11 Miscellaneous Measures..12 French System of Measures. 13 Table for converting Metres into French and English feet, and vice versa. 14 Foreign Measures of Length.. 16 Foreign Itinerary Measures.......17 Comparisons of French and English Measures 18 Spanish and Mexican Measures of Length. 18 Specific Gravities... 19 Analytical expressions for Lines, Surfaces and Solids 20 Lengths of Circular Arcs.23 Measurement of flowing Water.24 Table of Surface, Bottom, and Mean Velocities.. 25 To trace Railroad curves by means of deflections.. 26 XIV CONTENTS. Page. Ordinates to Circular Arcs.... 30 Land Surveying with Compass and Chain... 31 Calculation of the area of a tract of land... 32 Traverse Table....... 33 Chains, Yards, and Feet, with their reciprocal equivalents........... 48 Component parts of the Army Ration... 50 PART II.- -Geodesy. Reduction to Centre of Station.. 53 Correction for Phase..... 53 Spherical Excess. 54 Reduction of Bases.. v e X 55 Correction for Temperature in metallic rods 57 Measurement of distances by Sound... 58 Velocity and Force of Winds..... 58 Problem of the three Points 59 Formulae for computing the principal Geodetic quantities depending on the Spheroidal figure of the Earth........ v 60 Bessel's magnitude and figure of the Earth.. 61 Ratio of the metre to the English yard... 62 Numerical values of Bessel's terrestrial elements in English yards. 63 Constant Logs. useful in Geodetic computations. 64 Formule for Geodetic Latitudes, Longitudes, and Azimuths..............65 Measurement of distances by Astronomical observations..67 Computation of the sides of a triangle..... 70 Computation of the Geodetic determination of positions ~ 70 CONTENiTS. XV Page. Log. values of the Normal, or radius of curvature of the perpendicular to the Meridian, in different Latitudes....72 Log. values of the radius of curvature of the Meridian in different Latitudes. 75 Trigonometrical Levelling. 78 Table of corrections for curvature and refraction.. 81 Table for reducing inclined measures to horizontal. 82 Table of the ratio of Slopes........ 82 Barometrical measurement of Heights... 83 Table for converting Fahrenheit's scale of the Thermometer to Reaumur's and the centesimal.. 86 Tables for comparing French and English Barometers 87 Table of corrections for Capillary action.... 94 Measurement of Heights with the Thermometer. 95 Formule for the Projection of Maps.. 96 Co-ordinates for the Projection of Maps, in yards. 99 Values of arcs of the Parallel, in yards.. 118 Values of Meridional arcs, in yards..118 Lengths of Degrees of Latitude and Longitude in different Latitudes, in nautical and statute miles. 128 APPENDIX TO PART II: On the use of the Portable Declinometer in the determination of magnetic variation and horizontal intensity. 129 To compute the variations in the magnetic variation due to changes of Latitude and Longitude.. 138 PART III. —,dstronomy. Of Sidereal and Solar time.. 141 To find the time by an altitude of the Sun, or a Star -143 Table for converting Sidereal into Mean Solar time. 144 Table for converting Mean Solar into Sidereal time. 145 XVI CONTENTS. Page. Table for converting Space into time.... 146 Table for converting Time into Space.. 150 Table for converting AR. in arc into mean time. 152 Table for converting mean time into AR. in arc. 155 Form for record and computation of the determination of the time by altitudes of Stars..... 158 Computation of an example of the method of finding the time by altitudes of a Star. (Formulae page 143).160 To find the time by equal altitudes of the Sun. 161 Table of Equations to equal altitudes.... 162 Form for record and computation of the determination of the Time by equal altitudes of the Sun. 170 Computation of an example of the method of finding the time by equal altitudes of the Sun, (form. page 161)......... 172 Table of the Sun's Parallax in altitude. 173 Table of decimals of an hour..173 Table of mean Refractions..174 Table of the corrections to the tabular Refractions for variations in the Thermometer and Barometer 181 The Transit Instrument-corrections to observed Transits....182 Table to facilitate the reduction of Transit observations 185 Form for record and computation of observed Transits 186 Example of the method of computing Transit corrections............. 187 Rules for determining the direction of the deviation of the Transit Instrument in azimuth.... 188 To determine the Latitude by meridional altitudes. 189 To determine the Latitude by circum-meridional alts. 190 CONTENTS. XVII Page. Tables for reduction to the meridian, values of k. 192 Do. do. do. values of m 199 Form for record and computation of the determination of the Latitude by circum-meridian altitudes of Stars..200 Computation of some of the quantities in the preceding method 202 To determine the Latitude by altitudes of circumpolar Stars............ 203 Form for record and computation of the method of determining the Latitude by altitudes of Polaris 204 Computation of an example of this method... 206 To determine the Latitude with the Transit Instrument by transits of Stars over the Prime Vertical 207 To determine the Latitude with the zenith and equal altitude Telescope.......... 211 Form for record and computation, of this method. 214 To find the Azimuth of the Sun or a Star.. 215 To find the Amplitude of the Sun or a Star.. 216 To determine the true meridian by equal altitudes of the Sun. o. o... 216 To find the Azimuth of Polaris at its greatest elongation.......... 217 Corrections to observed Azimuths for errors of level 218 Corrections for Run in reading Microscopes... 218 To determine the Longitude by Lunar Distances. 219 Table for correcting the Moon's parallax.... 222 Table of the augmentation of the Moon's semidiameter...a o.... 223 To determine the Longitude by Lunar culminations 224 XVIII CONTENTS. Page. The value of a quantity at three consecutive whole hours being given, to find its value at an intermediate time. 232 To find the Longitude by occultations of Stars by the Moon. 233 Formula for probable error and precision.. 238 Geographical positions of some of the principal Observatories, and of notable points in the western part of the United States... 241 TABLES AND FORMULatE. P ART I. MISCELLANE OUS. TRIGONO METRY. I. Equivalent Expressions. Sin 2X + cos 29x - 1. Sin x = cos x. tang x cos x cot x, 1 — cos 2x =-2 sin x. cos x tang x -- 1 + tang 2x 1 cosecant x sin x Cos x -- tang x = sin x. cot x = V 1 -- sin 2X = cos 2~x - sin x secant x sin x Tang x -c COS X cot x sin x Vi- sin 2x sin 2 x 1 + cos 2 x Cotang x = tang x 2 TRIGONOMETRY. Secant x = cos X Cosecant x =. sin x Versed sin x -- 1 — cos x =2 sin 2~ x Co-versed sin x - 1 - sin x Chord x _ 2 sin ~ x Sin (A B) = sin A cos B i sin B cos A Cos(A JZ B) = cos A cos B = sin A sin B Sin 2 A 2 sin A cos A Cos 2 A = 2 cos 2A - I 1 -- 2 sin 2A - cos 2A - sin 2A 2 cos 2 A 1 + cos A 2 sin 21 A 1 -- cos A Tang~(A B) -tang A 4- tang B 1 =F tang A tang B Tang A 1 -cosA 1- cosA T1 + cos A sin A Sin A A2 sin B = 2 sin ~ (A ~= B) cos 2 (A =F B) Cos A + cos B = 2 cos ~ (A + B) cos I (A - B) CosA - cosB =2sin - (A +B) sin - (B-A) Sin2A - sin2B = sin (A + B ) sin (A - B ) Cos2A- sin2B = cos (A + B ) cos (A- B ) sin ( A4- B) Tang A t: tang B sn cos A cos B Cot A colt B sin( A i: B) sin A sin B TRIGONOMETRY. 3 Sin A + sin B tang 2 (A + B) Sin A - sin B- tang ~ (A- B) 1 sin A _ tang 2( 450~:t A) 1:~ sin A I o sin A _ tang (450~: ~ A) Cos A II. Trigonometrical Series. A3 A5 A7 Sin A = A - + 234 237+ etc. Sin A -- A -- 2.3 - 12.3.4.5 -2.3..-~.7 - etc. A2 A4 A6 Cos A -1 —+ -~ - 2+etc. C2o + 2.3.4 2.....6 + etc. sin 3A 3 sin5A 3.5 sin 7A Arc A - sin A+ - 2,3 + 2.4.5 + 2.4.6.7 etc. = tang A -- tang3A + I tang 5A-+-tang 7A.. X2 X4 Log sin A = log (A - + 1-0 etc.) =log A - M +4 + X4 - log A -- M - + 180 2835) M = logarithmic modulus = 0.4342945..... Log M - 9.6377843113..o. III. Table of signs of Trigonometrical lines. Quadrants. Sine. Cosine. Tang. Cct. Secant. Cosecant. 1.5. 9. + + + + + + 2. 6. 10. 5+ - - + 3. 7. 11. -- - + - - 4. 80. 12 &c. t- + - F- -&- - 4 TRIGONOMETRY. IV. Ratio of the circumference of a circle to its diameter. — = 3.14159 26535 898..... Log X- = 0.49714 98726 941.... The radius being unity, the number of degrees in an 180~ 1 arc equal to radius = rO _ - - = a 570.29578 57~. 171.441t.8. 10800' 1 The number of minutes - r=t a or - are It sn1 -3437t.74677. sin 11 648000" 1 The number of seconds = rl -sin - - sin 1!206264".80625. Log r~ = 1.75812 26324 09172 Comp log rO 8.24187 73675 90828 Log rt = 3.53627 38827 92816 Comp log rl _ 6.46372 61172 07184 = log sin 1t Log r"1 = 5.31442 51331 76459 Comp log rt/ = 4.68557 48668 23541 = log sill 1"I Let a be the length of an arc of a circle whose radius is 1, and atl the number of seconds in that arc, as 1 r.si n and R: rt:: a: all or atl = r/ a; a — =atl sin 1// In an equation, therefore, any arc a of a circle whose radius is 1, is expressed in seconds by changing a into all sin 1,,. TRIGONOMETRY. 5 V. Solution of Plane Triangles. In the following formulae A, B, C, represent the angles, and a, b, c, the sides opposite, respectively. 1. Any plane triangle, a2 b2 + c- - 2 b c cos A sin A sin B sin C a b c tang ~ (A + B) cot ~ C a + b tang ~ (A - B) tang ~ (A - B) a - b sinA — (s-b) (s —c)~ cos ~ A (s —a } a+b + c 2. Right angled triangles, making A = 90~ in the preceding, they become a2 = b6 + c2 b = a sin B a cos C, c a sin C =a cos B b c tang B - tang C - C b 6 TRIGONOMETRY. VI. Solution of Spherical Triangles. a, b, c, represent the arcs, and A, B, C, the angles opposite. 1. Oblique spherical triangles sin A sin B sin C silla sin b sin c rco a=cos b sin (c+,) sin p cot q = tang b cos A cos B sin (C -) cos A = sin p'cot p = tang B cos a cot a tang b sin (C -,) sin q cot cot A cos b JVapier's /lnalogies. tang (a — b) = tang c c (A - B) C os (A +- B) tang ~ (a-b6) tang c sin (A-B) 2~ ~~sin ( A - B) cos g ( a - b ) ltang (A + B) cot C s 2f acos (a + b) sin 3 (a -1-) tang ~ (A - B) =cot C s sin ~ (a — b) TRIGONOMETRY. 7 VI. Solution of Spherical Triangles-Continued. sin S. sin (A - S) sin "~ a =.2 sin "$ a sin B. sin C sin (B-S) sin (C-S) 2a sin B. sin C tn Ia sin S. sin(A - S) tang, a - sin ( B - S ). sin ( C - S) sin (s-b). sin (s- c) sin A- Asin b. sin c sin s. sin (s-a) sin b. sin c tang I A sin (s - b). sin (s - c) tang ~~ Asin s. sin (s — a) In which S and s represent the half sum of the three angles diminished by 90~ and the half sum of the three sides, respectively. 2. Right angled spherical triangles, a, being the hypo. thenuse. 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 tang b - tang B. sin c cos C - sin B. cos c tang c = tang C. sin b tang b tang a. cos C sin b6 sin a. sin B tangc — tanga. cos B sin c - sin a. sin C 8 WEIGHTS AND MEASURES. I. Weights and JJfeasures of the United States. The actual standard of length is a brass scale of 82 inches in length, made by Troughton, of London, and now in the possession of the Treasury Department. The standard of weight is the troy pound, copied in 1827, by Captain Kater, 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 Thermometer. 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 distilled water, at the temperature of maximum density of water, the vessel being weighed in air in which the Barometer is 30 inches at 620 Fahrenheit. The bushel is a measure containing 543391.89 standard grains (77.6274 pounds avoirdupois) of distilled 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. The avoirdupois pound 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 AND MEASURES. 9 II. English System of Measures. The unit of lineal measure is the yard. The yard is divided into 3 feet, and the foot subdivided into 12 inches. The multiples of the yard are the pole or perch, the furlong, 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. Furlongs. Miles. 1 0.083 0.028 0.00505 0.00012626 0.0000157828 12 1. 0.333 0.06060 0.00151515 0.00018939 36 3. 1. 0.1818 0.004545 0.00056818 198 16.5 5.5 1. 0.025 0.003125 7920 660. 220. 40. 1. 0.125 63360 5280. 1760. 320. 8. 1. MJfeasures of Superficies. 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. Acres. Sq. mile. 1. 0.1111 0.00367309 0.000091827 0.000022957 9. 1. 0.0330579 0.000826448 0.000206612 272.25 30.25 1. 0.025 0.00625 10890. 1210. 40. 1. 0.25 43560. 4840. J 160. 4. 1. 292800. 3097600. 102400. 2560. 640. 1, Log. 3097600 - 6.4910253. 2 10 WEIGHTS AND MEASURES. III. Jleasures of Volume. Solids are measured by cubic yards, feet, and inches; 17'28 cubic inches making a cubic foot, and 27 cubic feet a cubic yard. For all sorts of liquids, grain, and other dry goods, the standard measure is declared, by the act of 1824, to be the imperial gallon, the capacity of which is determined immediately by weight, and remotely by the'standard of length, in the following manner: According to the act, the imperial standard gallon contains 10 pounds avoirdupois weight of distilled water, weighed in air at the temperature of 62~ Fahrenheit's Thermometer, the Barometer being at 30 inches. The pound avoirdupois contains 7,000 troy grains; and it is declared that a cubic inch of distilled water (temperature 62~, barometer 30 inches) weighs 252.458 grains. Hence the contents of the imperial standard gallon are 277.274 cubic inches. The parts of the gallon are quarts and pints. Its multiples are the peck, the bushel, and the quarter. The following are the relations: Pints. Quarts. Gallons. Pecks. Bushels. Quarters. 1 0.5 0.125 0.0625 0.015625 0.001953125 2 1. 0.25 0.125 0.03125 0.00390625 8 4. 1. 0.5 0.125 0.015625 16 8. 2. 1. 0.25 0.03125 64 32. 8. 4. 1. 0.125 512 256. 64. 32. 8. 1. WEIGHTS AND MEASURES. 1l IV. Tables of British Weights. 1.-Imperial Troy Weight. Standard: One cubic inch of distilled water, at 620 Fahrenheit's Thermometer, the Barometer being 30 inches, weighs 252.458 Troy grains. grs. dwt. 24- 1 oz. 480- 20= 1 lb. 5760 - 240 - 12 = 1 Troy weight is used in weighing gold, silver, jewels, &c., and in philosophical experiments. 2.-Imperial./voirdupois Weight. Standard: The same as in Troy weight, and one avoirdupois pound = 7000 Troy grains. drs. oz. 16 1 lb. 256= 16- I qr. 7168=- 448- 28 1 cwt. 28672=- 1792= 112= 4= 1 ton. 573440 - 35840 2240=- 80 =20= 1 This weight is used for the general purposes of commerce. 12 WEIGHTS AND MEASURES. V. JIliscellaneous. Length. —Gunter's chain -66 feet =-4 poles = 100 links of 7.92 inches. 1 fathom =-6 feet; 1 cable lengoth - 120 fathoms. 1 hand - 4 inches; 1 palm - 3 inches; 1 span - 9 inches. Solid. —1 cubic yard = 27 cubic feet (B. M.) = 1728 cubic inches. 1 reduced foot (B. M.) = 1 square foot X 1 inch thick - 144 cubic inches. 1 perch of masonry = 1 perch (162 feet) long X 1 foot high X 1~ foot thick - 24.75 cub. feet; 25 cubic feet has generally been adopted for convenience. 1 cord fire wood = 8 feet long X 4 feet high X 4 feet deep = 128 cubic feet. 1 chaldron coal = 36 bushels = 57.25 cub. feet. Paper. —24 sheets 1 quire. 20 quires = 1 ream = 480 sheets. Dimensions of Drawing Paper. Cap - - 13 X 16 in. Elephant - 27*X 22~in. Demy - - 19aX 15~ Columbia - 33X 23 Medium - - 22 X 18 Atlas- - - 33 X 26 Royal - - 24 X 19 Theorem - 34 X 28 Super Royal 27 X 19 Double eleph't 40 X 26 Imperial- - 29 X 211 Antiquarian - 52 X 31 Capacities. A box 16 X 16.8 X 8. in. contains I bushel 12 X 11.2 X 8. Ad ~ bushel dry measure 8 X 8.4 X 8. "' 1 peck ) 6 X 6 X 6.4 "9 1 gallon I liquid meas. 4 X 4 X 3.6 "' 1 quart WEIGHTS AND MEASURES. 13 VI. French System of Measures. The unit of measures of length is the metre. The unit of superficial measure is the are, a surface of 10 metres each way, or 100 square metres. The unit of measures of capacity is the litre, a vessel containing the cube of the tenth part of the metre. The standard temperature is that of melting ice. The measures of length are: Myriametre _ 10000 metres. Kilometre - 1000 Hectometre -- 100 Metre - 1 Decimetre = 0.1 Centimetre - 0.01 Millimetre -- 0.001 The measures of surface are: Hectare = 10000 sq. metres. Are - 100 Centiare -- 1 The measures of capacity are: Kilolitre = 1000 litres. Hectolitre - 100 Decalitre - 10 Litre -- 1 Decilitre = 0.1 Centilitre - 0.01 The unit of solid measure is the stere or cube of the metre, equal to 35.31658 English cubic feet. 14 WEIGHTS AND MEASURES. Table for converting JMletres into Toises and French and English feet and inches. French. English. Met. Toises. Feet. In. Lines. Feet. I Inches. 1 0.51307 3 0 11.296 3 3.3708 2 1.02615 6 1 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 5 2.56537 15 4 8.480 16 4.8539 6 3.07844 18 5 7.776 19 8.2247 7 3.59152 21 6 7.072 22 11.5955 8 4.10459 24 7 6.368 26 2.9663 9 4.61767 27 8 5.664 29 6.3371 10 5.13074 30 9 4.960 32 9.7079 20 10.26148 61 6 9.920 65 7.4158 30 15.39222 92 4 2.880 98 5.1237 40 20.52296 123 1 7.840 131 2.8316 50 25.65370 153 11 0.800 164 0.5395 60 30.78444 184 8 5.760 196 10.2474 70 35.91519 215 6 10.720 229 7.9553 80 41.04593 246 3 3.680 262 5.6632 90 46.17667 277 0 8.640 295 3.3711 100 51.30741 307 10 1.600 328 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 1312 4.3160 500 256.53704 1539 12 8.000 1640 5.3950 600 307.84444 1847 0 9.600 1968 6.4740 700 359.15185 2154 10 11.200 2296 7.5530 800 410.45926 2462 9 0.800 2624 8.6320 900 461.76667 2770 7 2.400 2952 9.7110 1000 513.07407 3078 5 4.000 3280 10.7900 2000 1026.14815 6156 10 8.000 6561 9.5800 3000 1539.22222 9235 4 0.000 9842 8.3700 4000 2052.29630 12313 9 4.000 13123 7.1600 5000 2565.37037 15392 2 8.000 16404 5.9500 6000 3078.44444 18470 8 0.000 19685 4.7400 7000 3591.51852 21549 1 4.000 22966 3.5300 8000 4104.59259 24627 6 8.000 26247 2.3200 9000 4617.66667 27706 0 0.000 29528 1.1100 10000 5130.74074 30784 5 4.000 32808 11.9000 Log. to reduce metres to Eng. feet - 0.5159929. WEIGHTS AND MEASURES. 15 Table for converting English Feet into French Toises, Metres, and Feet. French. English Toises. Metres. Feet. In. Lines. 1 0.15638 0.30479 0 11 3.114 2 0.31276 0.60959 1 10 6.228 3 0.46915 0.91438 2 9 9.343 4 0.62553 1.21918 3 9 0.457 5 0.78191 1.52397 4 8 3.571 6 0.93829 1.82877 5 7 6.685 7 1.09468 2.13356 6 6 9.799 8 1.25106 2.43836 7 6 0.913 9 1.40744 2.74315 8 5 4.028 10 1.56382 3.04794 9 4 7.142 20 3.12764 6.09589 18 9 2.284 30 4.69146 9.14383 28 1 9.425 40 6.25529 12.19178 37 6 4.567 50 7.81911 15.23972 46 10 11.709 60 9.38293 18.28767 56 3 6.851 70 10.94675 21.33561 65 8 1.993 80 12.51057 24.38536 75 0 9.134 90 14.07439 27.43150 84 5 4.276 100 15.63822 30.47945 93 9 11.418 200 31.27643 60.95850 187 7 10.836 300 46.91465 91.43835 281 5 10.254 400 62.55286 121.91780 375 3 9.672 500 78.19108 152.39725 469 1 9.090 600 93.82929 182.87670 562 11 8.508 700 109.46751 213.35615 656 9 7.926 800 125.10572 243.83559 750 7 7.344 900 140.74394 274.31504 844 5 6.762 1000 156.38215 304.79449 938 3 6.180 2000 312.76431 609.58899 1876 7 0.360 3000 469.14646 914.38348 2814 10 6.539 4000 625.52861 1219.17797 3753 2 0.719 5000 781.91076 1523.97246 4691 5 6.899 6000 938.29292 1828.76696 5629 9 1.079 7000 1094.67507 2133.56145 6568 0 7.259 8000 1251.05722 2438.35594 7506 4 1.438 9000 1407.43937 2743.15044 8444 7 7.618 10000 1563.82153 3047.94493 9382 11 1.798 Log. to reduce English feet to metres = 9.4840071. Table of relations between the Linear lfeasures of several countries, with corresponding logarithms. France. England. Prussia. Bavaria. Saxony. Baden. Austria. Spain. Metre. Russia. Denmark. Switzerland. Mexico. Paris foot. foot. foot. foot. foot. foot. Vienna foot. foot. 3.078444 3.280899 3.186199 3.426310 3.531197 3.333333 3.163446 3.537877 # I 0. 4883313 0. 5159929 0. 5032730 0. 5348266 0.5479220 0.5228787 0.5001605 0.5487427 0.3248394 1.065765 1.035003 1.113000 1.147072 1.082798 1.027612 1.149242 9. 5116687 1 0. 0276616 0.0149417 0. 0464954 0.0595907 0.0345475 0. 0118292 0.0604114 0 0.3047945 0.938293 0.971136 1.044320 1.076290 1.015982 0.964201 1.078325 9,. 4840071 9.9723384 1 9. 9872801 0. 0188337 0. 0319291 0. 0068859 9.9841676 0. 0327498 0.3138535 0.966181 1.029722 1.075359 1.108279 1.046178 0.922859 1.110375 9. 4967270 9.9850583 0.0127199 1 0.0315536 0. 0446490 0. 0196058 9.9968875 0. 0454697 C 0.2918592 0.898472 0.957561 0.929922 1.030612 0.972864 0.923281 1.032562 M 9. 4651734 9.9535047 9.9811663 9.9684464 0. 0130954 9.9880521 9.9653339 0.0139161 C 0.2831901 0.871785 0.929118 0.902300 0.970297 0.943967 0.895856 1.001892 9.4520780 9. 9404093 9. 9680709 9. 9553510 9.9869046 1 9.9749567 9. 9522385 0.0008207 0.3000000 0.923533 0.984270 0.955860 1.027893 1.059359 0.949034 1.061361 9.4771213 9.9654525 9. 9931141 9. 9803942 0.0119479. 0250433 9.9772817 0.0258630 0.3161109 0.973130 1.037128 1.007193 1.083094 1.116250 1.053703 1.118361 9.4998395 9. 9881708 0. 0158324 0. 0031125 0.0346661 0. 0477615 0.0227183 1 0.0485822 0.2826553 0.870139 0.927364 0.900597 0.968465 0.998112 0.942184 0.894165 1 9.4512573 9. 9395886 9. 9672502 9. 9545303 9.9860839 9. 9991793 9.9741360 9. 9514178 o0 > France. England. Prussia. Austria Russia Spain. Germany. England. (1 1 Denmark Mexico France. C o = g::: Myriametre Stat. mile mile mile verst Jud. league Geo. mile Naut. l'gue -10000M. ==- 5280'. = 24000'. 24000'. 3500'. - 15000'. 15 = 1 deg. 20- leg. e;. CD c' CD 6.213824 1.327583 1.318103 9.373997 2.358584 1.347680 1.796907! 30. o.7933590 0. 1230617 0. 1199492 0. 9719248 0. 3726514 0.o 1295869 0. 2545256 C 0.1609315 0.213650 0.212124 1.508571 0.379570 0.216884 0.289179 Ct c 9.2066410 9.3297028 9.3265903 0.1785659 9.5792924 9.3362279 9.4611666 II Il II I! 0 0.7532485 4.680554 0.992859 7.060950 1.776600 1.015138 1.353518 o 9. 8769383 0.6702972 9.9968875 0. 848863 O 0.2495897 0. 0065251 0. 1314639 Z-, -.a ~,1 c 0.7586663 4.714219 1.007193 7.111736 1.789379 1.022440 1.363253 9.8800508 0.6734097 0. 0031125 0.8519756 0. 2527022 0. 0096376 0. 1345764; = = ~ - 0.1066781 0.662879 0.141624 0.140613 1 2.516092 0.143768 0.191691 0 P' UQ =, s= 1 0,'-~ a< > cD c 9.0280752 9.8214341 9. 1511369 9. 1480244 0. 4007266 9. 1576620 9. 2826008 c b a-n. a- a I t, 8= o 5 5 0.4239831 2.634556 0.562873 0.558853 0.397442 0.571394 0.761868 -: B I ~ 9.6273486 0.4207076 9.7504103 9. 7472978 9.5992734 9.7569355 9. 8818742. a 0 0.7420158 4.610755 0.985088 0.978053 6.955654 1.750107 1.333333 ~a a CD 9.8704131 0.6637721 9. 9934749 9.9903624 0. 8423380 0. 2430645 0.1249387 = 0.5565118 3.458067 0.738816 0.733540 5.216740 1.312580 0.750000 9. 7454744 0. 5388334 9. 8685361 9. 8654236 0. 7173992 0.1181258 9. 8750613 o *,o 1 English or French geographical mile = 1-60 of a degree of longitude at the Equator = CD "O4 - = 2028.7 English yards. 18 FOREIGN MEASURES. Comparison of French and English JMeasures. Metre..... 39.37079 inches. 3.28089 feet. 1.09363 yards. Kilometre.... 0.62138 miles. Myriametre. 6.2138 miles. Square metre.. 1.196033 square yards. Are...119.6033 square yards. Hectare... 2.471143 acres. Litre... 1.760773 pints. cc e e e e 0. 220096 gallons. Decalitre.... 2.200967 gallons. Hectolitre..... 2.009668 gallons. Gramme.... 15.438 grains, troy. 0.032 ounce, troy. Kilogramme... 2.680 pounds, troy. (... 9 2.205 pounds, avoirdupois. IX. Spanish and MJlexican.Measures of length. 1 Castilian foot = 11.1284 English inches. 3 Castilian feet —..i vara = 33.3852 English inches. - 0.927365 English yards. 5000 varas =- judicial league = 4637. English yards. SPECIFIC GRAVITIES. 19 X. Specific Gravities. Specific Weight Specific Weight Substance. gravity. of 1 Substance. gravity. of 1 cub.inch. cub. in. Lbs. Lbs. Brass, (cast).,...... 8.396 0.3037 Sand............ 1.800 0.0652 Bronze, (gun metal) 8.700 0.3147 Stone, (common) 2.520 0.0911 Copper, (cast)..... 8.788 0.3179 Wood, ash...... 0.722 0.0261 Iron, (bar)......788 0.2817 " cypress.. 0.441 0.0160 Iron, (cast)...... 7.207 0.2607 " hickory... 0.838 0.0303 Lead, (cast)........ 11.352 0.4106 " oak...... 0.687 0.0248 Tin, (cast)......... 7.291 0.2637."' ie..... 0.541 0.0196 Bricks............ 1.900 0.0690 Coai, (bitumin's) 1.270 0.0460 Earth, (common)... 1.500 0.0543 Water, (distilled) 1.000 0.0361 The weight of dry atmospheric air at the temperature of 32~, the barometer being at 30 in., is,rA-d of that of distilled water. The weight of a cubic foot of distilled water at the maximum density being nearly 1000 ounces avoirdupois, the specific gravity of a solid or liquid body expresses the weight of a cubic foot, in ounces; therefore the weight of such a body in ounces will be found by multiplying its contents in cubic feet by its specific gravity. According to Mr. Hassler's comparisons, the weight of a cubic foot of water at its maximum density, the barometei being at 30 in., is 998.068 oz. According to the British imperial standards, the weight of a cubic foot of water, at 620, the barometer being at 30 in., is 997.136 oz.; this would give for the cubic foot of water, at the maximum density, 998.224 oz. By Ihe investigations of Prof. R. S. McCulloch, the maximum density of water is at the temperature of 39~.6 Fahr.; this agrees very nearly with Mr. Hassler's determination of the maximum density, 39~.83. 20 MENSURATION. XI. Analytical Expressions for different Lines, Surfaces, and Solids. 1. Lines. Circle. Ratio of circumference to diameter - 3.1415926 - s-%5 nearly. Length of an arc - 10; r being the radius of the circle, and a the number of degrees in the arc; or, nearly 8e c, C; c being the chord of the arc, and c/ (the chord of half the arc) = 0/ cS + versine2. Ellipse. Circumference _='O o A, / ( a2+ b2 ) nearly; a and b being the axes. Parabola. Length of an arc, commencing at vertex j4 a + V b nearly; a being the abcissa, and b the ordinate. 2. Surfaces. 1. Triangle in terms ofits base and its altitude.. A two sides and the included angle.a b sin C its three sides.. = [(s - a) (s - b) (s- c) ] X where A = the altitude; a, b, c = the three sides, and C the angle included between a and b; s a + b + c MENSURATION. 21 2. Parallelogram in terms ofits base and its altitude.. = b A two sides and the included angle.... a b gin C two sides and their corresponding diagonal = 2 [s (s - a) (s - b) (s - c) 2 where C = the angle included between two adjacent sides a, b; c = the diagonal opposite, and s a + + 3. Trapezium in terms ofits two parallel bases and its altitude B + 6A its two parallel bases, one of its oblique) sides and the angle between one of - B 1 sin C these bases and this side...... where A = the distance between the two parallel bases B, b; I= 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. 5. Regular Polygon 1800 tang where n — the number of sides; a the length of one of them. 6. Circle..- 7. R2 7. Ellipse..... ab a and b being the semi-axes. 8. Right cylinder, exclusive of its bases. = 2 A R A 9. Sphere.. = 4 n R2 10. Zone. -4a R2 sin (L - L) cos (L L + L) 22 MENSURATION. 11. Spherical Quadrilateral, formed by two parallels of Latitude and two meridians 9 (Ml — M) R2 sin a (L- ) cos + ( + L) where R = the radius of the sphere; L, LL' = the latitudes of the bases of the zone, + when North, - South; M', M = the longitudes of the extreme meridians of the quadrilateral. (MI — M ) being expressed in degrees and decimals. In the place of R, the normal N, of the mean Latitude (L-+-L), can be used. 12. Right cone...e. R L 13. Frustrum of cone with parallel bases =- I (R + r) When R and 2r - the radii of the bases of these solids, L and 1= the lengths of their generating elements. 3. Solids. 14. Prism...... B A where B the area of the base, A = the altitude. 15. Rectangular parallelopiped... = p X q X r Cube.... =p3 where p, q, r, = the lengths of the three contiguous edges. BA 16. Pyramid....... 3 The area B being found from No. 5. 17. Right cylinder.... ~ R2 A 18. Right cone.= eX R2 A 19. Sphere..... e.t R. 3 MENSURATION. 23 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 - (b+ h') h'q-(b — rh hh) b + r Zi 2 It' h h where b the breadth at the bottom of the cutting h the perpendicular depth of cutting at higher end h-= the perpendicular depth of cutting at lower end I = the length of the solid r the ratio of the perpendicular height of the slope to its horizontal base. Lengths of Circular drcs, Taking the base of Segments as unity. __.. _ _. 0......01 1.000.11 1.032.21 1.114.31 1.239.41 1.401.02,".12 1.038.22 1.124.32 1.254.42 1.418.03 ".13 1.044.23 1.135.33 1.269.43 1.437.04'".14 1.051.24 1.147.34 1.284.44 1.455.05.15 1.059.25 1.159.35 1.300.45 1.474.06 1.006.16 1.067.26 1.171.36 1.316.46 1.493.07 1.014.17 1.075.27 1.184.37 1.332.47 1.512.08 1.018.18 1.084.28 1.197.38 1.349.48 1.531.09 1.020.19 1.093.29 1.212.39 1.366.49 1.551.10 1.026.20 1.103.30 1.225.40 1.383.50 1.571 24 HYDROMETRY. XII. Hydrometry. 1. To determine the mean velocity of a stream from observations of the velocity at its surface. Let,- the observed surface velocity, in inches, p the bottom velocity, o =- the mean velocity, = M V y = a +a 2 Prony has given the very simple formula y = 0.816458 a which is, perhaps, more correct than the above of Dubuat. 2. In open streams which are flowing with an uniform motion, calling o, the area of the section of the stream, x, that portion of the perimeter of the section of the bed, which is in contact with the water, I, the fall divided by the length, v the mean velocity per second Q, Q, the discharge per second, R, the hydraulic depth, or -, (the unit being English feet), the relations, according to Eytelwein, between these several quantities, may be expressed by the following, 0.0000242651 v + 0.0001114155 = R I. Whence v- 0. 1088941604+-V/0.0118580490+8975.414285R I. And 0.00002426,51 Q - 0.0001114155 Q2 I. HYDROMETRY. 25 Whence (8975414285 - I+ 0.01213425 2) — 0.1088942 co Log 0.01213425 - 8.0840130 Log 0.1088942 - 9.0370065 Log 8975.414285 = 3.9530545 Log 0.0000242651 5.3849821 Log 0.0001114155 = 6.0469456 To realize in practice what would be called uniform motion, the canal or stream should be straight, and with the same section and inclination from one end to the other. In proportion as it varies from these conditions, we may expect to find the formula in fault. Table of Surface, Bottom and Miean Velocities. VELOCITY IN INCHES. Surface. Bottom. Mean. Surface. Bottom. Mean. 5 1.527 3.263 55 41.167 48.083 10 4.675 7.337 60 45.508 52.754 15 8.254 11.627 65 49.875 57.436 20 12.055 16,027 70 54.266 62.133 25 16.000 20.500 75 58.679 66.839 30 20.045 25.022 80 63.111 71.555 35 24.167 29.583 85 67.561 76.280 40 28.350 34.175 90 72.026 81.006 45 32.583 38.791 95 76.506 85.753 50 36.857 43.428 100 81.000 90.500 4 26 RAILROAD SURVEYING. XIII. 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 centre 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 centre, subtended by either chord. 3. A line bisecting the angle of deflection formed by any two equal chords, is a tangent to the are at the point where the two chords meet. 4. If an are 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. TABLE 1. Table of deflections for chords and tangents with radii and versed sines corresponding. _*-' _-' -O.. D. D. M. D. M. feet. 1; D. M. D. M. feet. o Io 0 o 0.15 0.30 11460.106 3.45 7.30 764. 1.635 0.30 1.00 5730..217 4.00 8.00 716.2 1.744 0.45 1.30 3820..328 4.15 8.30 674.1 1.853 1.00 2.00 2865..435 4.30 9.00 636.6 1.960 1.15 2.30 2292..545 4.45 9.30 603.1 2.070 1.30 3.00 1910..655 5.00 10.00 573. 2.180 1.45 3.30 1637.1.762 5.15 10.30 545.7 2.286 2.00 4.00 1432.5.872 5.30 11.00 520.9 2.394 2.15 4.30 1273.3.981 5.45 11.30 498.2 2.505 2.30 5.00 1146. 1.090 6.00 12.00 477.5 2.613 2.45 5.30 1041.8 1.199 6.15 12.30 458.4 2.722 3.00 6.00 955. 1.309 6.30 13.00 440.7 2.828 3.15 6.30 881.5 1.416 6.45 13.30 424.4 2.940 3.30 7.00 818.5 1.525 7.00 14.00 409.2 3.048 3.45 7.30 764. 1.635 7.15 14.30 395.2 3.157 RAILROAD CURVES. 27 TABLE 2. Table showing,, for arcs of different riadii, the lengths of lines of deflection from a tangential point to points on the arc 100 feet apart, with the angles of deflection and versed sines corresponding. Angle of de- Length of Versed sine Angle of de- Length of Versed sine flection from line of for line of flection from line of for line of Tangent. deflection, deflection. Tangent. deflection. deflection. deg. min. feet. feet. deg. min. feet. feet. 2~ Radius 2865 ft. 2 Rad. 2292 ft. 10.00' 100.00.43 10~.15'1 100.00.54 2.00 199.97 1.74 2.30 199 95 2.18 3.00 299.88 3.93 3.45 299.81 4.90 4.00 399.70 6.98 5.00 399.53 8.72 5.00 499.39 10.90 6.15 499.05 13.62 3~ Rad. 1910 ft. 31~ Rad. 1637.1 ft, 10o.30t 1C0.00.65 1~.45' 100.00.76 3.00 199.93 2.62 3.30 199.91 3.05 4.30 299.73 5.89 5.15 299.63 6.87 6.00 399.32 10.46 7.00 399.07 12.20 7.30 498.63 16.34 8.45 498.14 19.05 4~ Rad. 1432.5 ft. 41~ Rad. 1273.3 ft. 2~.00'1 100.00.87 20.15' 100.00.98 4.00 199.88 3.49 4.30 199.85 3.92 6.00 299.51 7.84 6.45 299.38 8.90 8.00 398.78 13.93 9.00 398.46 15.67 10.00 497.57 21.75 11.15 496.92 24.46 5~ Rad. 1146 ft. 510 Rad. 1041.8 ft. 20. 30' 100.00 1.09 20.45' 100.00 1.20 5.00 199.81 4.36 5.30 199.77 4.79 7.30 299.24 9.80 8.15 299.08 10.77 10.00 398.10 17.41 11.00 397.70 19.1.2 12.30 496.20 27.16 13.45 495.41 29.83 ~28 RAILROAD CURVES. Table 2-Continued. Angle of de- Length of Versed sine Angle of de- Length of Versed sine flection from line of for line of fleetion from line of for line of Tangent. deflection. deflection. Tangent. deflection. deflection. deg. min. feet. feet. deg. min. feet. feet. 60 Rad. 955 ft. 6~0 Rad. 881 ft. 30.001 100.00 1.31 30.151 100.00 1.41 6.00 199.73 5.23 6.30 199.68 5.66 9.00 298.90 11.'15 9.45 298.71 12.72 12.00 397.26 20.86 13.00 396.79 22.58 15.00 494.53 32.54 16.15 493.59 35.19 7~0 Rad. 818.5 ft. 710~ Rad. 764 ft. 30.30' 100.00 1.52 30.451 100.00 1.63 7.00 199.63 6.09 7.30 199.57 6.53 10.30 298.51 13.69 11.15 298.29 14.68 14.00 396.28 24.30 15.00 395.73 26.03 17.30 492.57 37.86 18.45 491.47 40.54 8~ Rad. 716.2 ft. 8~~ Rad. 674.1 ft. 40.001 100.00 1.74 40.15' 100.00 1.85 8.00 199.51 6.97 8.30 199.48 7.40 12.00 298.05 15.64 12,.45 297.84 16.62 16.00 395.14 27.73 17.00 394.57 29.45 20.00 490.28 43.18 21.15 489.13 45.83 90 Rad. 636.6 ft. 910 Rad. 603.1 ft. 40.301 100.00 1.96 40.451 100.00 2.07 9.00 199.39 7.83 9.30 199.31 8.27 13.30 297.54 17.57 14.15 297.26 18.55 18.00 393.86 31.13 19.00 393.16 32.85 22.30 487.75 48.41 23.45 486.36 51.07 10~ Rad. 573 ft. 10O~ Rad. 545.7 ft. 5~.001 100.00 2.18 5~.15f 100.00 2.28 10.00 199.24 8.70 10.30 199.16 9.12 15.00 296.96 19.52 15.45 296.65 20.46 20.00 392.42 34.55 21.00 391.65 36.19 25.00 484.90 53.68 26.15 483.37 56.20 RAILROAD CURVES. 29 Table 2 —Continued. Angle of de- Length of Versed sine Angle of de- Length of Versed sine flection from line of for line of flection from line of for line of Tangent. deflection. deflection. Tangent. deflection. deflection. deg. min. feet. feet. deg. min. feet. feet. 110 Rad. 520.9 ft 11~~ Rod. 498.2 ft. 5~.30' 100.00 2.39 50~.45' 100.00 2.50 11.00 199.08 9.56 11.30 198.99 9.99 16.30 296.33 21.41 17.15 295.99 22.40 22.00 390.84 37.86 23.00 390.00 39.58 27.30 481.76 58.75 28.45 480.10 61.39 120 Rad. 477.5 ft. 120-~ Rad. 458.4 ft. 60.00' 100.00 2.61 60.15' 100.00 2.72 12.00 198.90 10.42 12.30 198.81 10.85 18.00 295.63 23.34 18.45 295.26 24.30 24.00 389.12 41.24 25.00 388.20 42.91 30.00 478.34 63.90 31.15 476.52 66.45 13~ Rad. 440.7 ft. 132~ 60.301 100.00 2.82 60.45' 100.00 2.94 13.00 198.71 11.27 13.30 198.61 11.71 19.30 294.87 25.23 20.15 294.47 26.20 26.00 387.24 44.53 27.00 386.25 46.21 32.30 474.63 68.90 33.45 472.68 71.45 140 1410 7~0. 001 100.00 3.04 7~0.15' 100.00 3.15 14.00 198.51 12. 14 14.30 198.40 12.58 21.00 294.06 27.16 21.45 293.63 28.12 28.00 385.23 47.87 29.00 384.16 49.52 35.00 470.65 73.96 36.15 468.55 76.45 42.00 549.06 105.05 43.30 545.45 108.47 49.00 619.28 140.67 50.45 613.63 145.08 56.00 680.27 180.29 58.00 671.99 185.65 63.00 731.12 223.32 65.15 719.61 229.63 70.00 771.07 269.11 72.30 755.73 276.22 30 RAILROAD CURVES. TABLE 3. Table of ordinates to circular arcs on a chord of 100feet. rwl, ABSCISSA IN FEET. ~{ 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 95. 90. 85. 80. 75. 70. 65. 60. 55. 1~.00.04.07.11.14.16.18.20.21 1.22 1.30.06.10.16.21.25.28.30.31.32.33 2.00.08.14.22.28.33.37.40.41.43.44 2.30.11.20.28.35.41.46.50.52.54.55 3.00.13.24.33.42.49.55.59.63.65.66 3.30.15.28.39.49.57.64.69.74.76.77 4.00.17.32.44.56.66.73.79.84.86.87 4.30.19.36.50.63.74.83.89.95.97.98 5.00.21.40.56.70.82.92.99 1.05 1.08 1.09 5.30.23.43.61.77.91 1.01 1.09 1.16 1.19 1.20 6.00.25.47.67.84.99 1.10 1.19 1.26 1.30 1.31 6.30.27.51.72.91 1.07 1.19 1.29 1.37 1.41 1.42 7.00.29.55.78.98 1.15 1.28 1.39 1.47 1.52 1.53 7.30.31.59.83 1.05 1.23 1.38 1.49 1.57 1.63 1.64 8.00.33.63.89 1.13 1.31 1.47 1.59 1.68 1.74 1.75 8.30.35.67.94 1.20 1.39 1.56 1.69 1.78 1.84 1.86 9.00.37.71 1.00 1.26 1.47 1.65 1.78 1.89 1.95 1.97 9.30.40.75 1.06 1.33 1.56 1.74 1.88 1.99 2.05 2.08 10.00.42.79 1.11 1.40 1.64 1.83 1.98 2.10 2.15 2.19 10.30.44.82 1.17 1.47 1.72 1.93 2.08 2.20 2.26 2.29 11.00.46.86 1.22 1.54 1.80 2.02 2.18 2.31 2.36 2.40 11.30.48.90 1.28 1.61 1.88 2.11 2.28 2.41 2.47 2.51 12.00.50.94 1.34 1.68 1.97 2.20 2.38 2.52 2.57 2.62 12.30.52.98 1.39 1.75 2.05 2.29 2.48 2.62 2.68 2.72 13.00.54 1.02 1.45 1.82 2.13 2.38 2.58 2.73 2.78 2.84 13.30.56 1.06 1.50 1.89 2.21 2.48 2.68 2.83 2.89 2.94 14.00.58 1.10 1.56 1.96 2.29 2.57 2.78 2.93 3.00 3.05 14.30.60 1.12 1.61 2.02 2.37 2.65 2.87 3.03 3.13 3.16 LAND SURVEYING. 31 XIV. Land Surveying with Compass and Chain. To calculate the J.rea or Content of Land. If the sum of each adjacent pair of distances perpendicular to a meridian (departures) assumed without the survey, be multiplied 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 area1. Multiply the double meridian distance of each course by its northing or southing. 2. Place all the plus products in one column, and all the minus products 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 proportionAs the sum of the courses, is to the error of latitude, (or departure,) 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. DIFF. LAT. DEPARTURE. BALANCED. d m B r Dist. __ D.M.D. Area Area O Courses. E. - o g clhains N. S. E. W. + + O 1Ic. 121. I I I t.2 4. P X OCM! N4. 46Q W. 20.00 13.77 -. 14.51 13.88 -14.56 14.56 20209 28 ~- 2 N. 517- E. 13.80 8.54 2 0.84 - + 8.41 - 10.81 10.81 93.0741 P. c. cQ tt, 3 Eaus..2.120 3. - 21.25 - - + 21.20 42.82 -3 9 5. Error in Northing. 6 E.. 7.60 - 15.44 22.88 - - 159Westig 22.82 86.84 27.07836 Answer 046 P............ 0 S. 3-0. 18-80 8 1.7 - 10.1 1.63 10.36 99.30 9 1552 0590 * B 6 N. 74 W. 30.95 8.27 - - 2983 + 8.43 29.91 59.03 497.6229 Sums. 132.40 30.581. 4 97 54.65 792.7898 29 6 0I o30.58 5465 792.7898 TRAVERSE TABLE. 33 XIII. Traverse Table, Showing differences of Latitude and the Departures. 00 10 Po El C~ Lat. Dep. Lat. Dep. Lat. Dep. q S 1 1.00000 0.00000 0.99984 0.01745 0.99939 0.03490 i 1 2 2.00000 0.00000 1.99969 0.03490 1.99878 0.06980 1 2 3 3.00000 0.00000 2.9995410.05235 2.99817 0.10470 3 4 4.00000 0.00000 3.999390.06980 3.9975610.13960 4 0' 5 5.00000 0.00000 4.99923 0.087'26 4.99695 0.17450 1 5 60 6 6.00000 0.00000 5.9990810.10471 5.99634 0.20940 6 7 7.00000 0.00000 6.99893 0.12216 6.99573 0.24430 7 8 8.00000 0.00000 7.99878 0.13961 7.99512 0.27920 8 9 9.00000 0.00000 8.9986210.15707 8.99451 0.31410 9 1 0.99999 0.00436 0.99976 0.02181 0.99922 0.03925 1 2 1.9999810.00872 1.99952 0.04363 1.99845 0.07851 2 3 2.99997 0.01308 2.99928 0.06544 2.9976810.11777 3 15' 4 3.999360.01745 3.9990410.08725 3.99691 0.15703 4 5 4.99995.02181 4.99881 0.10907 4.99614 0.19629 5 45' 6 5.9999410.02617 5.99857 0.13089 5.99537 0.23555 6 7 6.99993 0.03054 65.99833 0.15270 6.99460 0.27481 7 8 7.99992 0.03490 7.99809 0.17452 7.99383 0.31407 8 9 8.99991 0.03926 8.99785 0.19633 8.99306 0.35333 9 1 0.99996 0.00872 0.99965 0.02617 0.99904 0.04361 1 2 1.99992 0.01745 1.99931 0.05235 1.99809 0.08723 2 3 2.99988 0.02617 2.99897 0.07853 2.99714 0.13085 3 4 3.99984 0.03490 3.99862 0.10470 3.99619 0.17447 4 30' 5 4.99981 0.04363 4.99828 0.13088 4.99524 0.21809 5 30t 6 5.9997710.05235 5.99794 0.15706 5.99428 0.26171 6 7 I 6.99973 0.06108 6.997600.18323 6.9933310.30533 7 8 7.99969 0.06981 7.9972510.20941 7.99238 0.34895 8 9 8.99965 0.07853 8.99691 0.23559 8.9914310.39257 9 1 0.99991 0.01308 0.99953 0.03053 0.99884 0.04797 1 2 1.99982 0.02617 1.99906 0.06107 1.99769 0.09595 2 3 2. 99974 0.03926 2.99860. 091 61 2.99654 0.14393 3 4 3.39965 0.05235 3.99813 0.12215 3.99539 0.19191 4 45' 5 4.99957 0.06544 4.99766 0.15269 4.99424 0.23989 5 15' 6 5.99948 0.07853 5.99720 0.18323 5.99309 0.28786 6 7 6.99940 0.09162 6.99673 0.21376 6.99193 0.33584 7 8 7.99931 0.10471 7.99626 0.24430 7.99078 0.38382 8 9 8.99922 0.11780 8.99580 0.27484 8.98963 0.43180 9 Dep. Lat. Dep. Lat. Dep. Lal. 890 880 870 5 _:: 34 TRAVERSE TABLE. Differences of Latitude and Departures-Continued. lO 30 40 50 Lat. Dep. Lat. Dep. Lat. Dep. 1 0.99863 0.05233 0.9975610.06975 0.9961910.08715 1 2 1.99726 0.10467 1.9951210.13951 1.99238 0.17431 2 3 2.99589 0.15700 2.9926910.20926 2.9885810.26146 3 4 3.9945210.20934 3.99025 0.27902 3.98477 0.34862 4 0' 5 4.99315 0.26168 4.9878210.34878 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 7.98904 0.41868 7.98051 0.55805 7.96955 0.69724 8 9 8.9876710.47102 8.97807 0.62780 8.96575 0.78440 9 1 0.99839 0.05669 0.99725 0.07410 0.99580 0.09150 1 2 1.99678 0.11338 1.99450 0.14821 1.9916010.18300 2 3 2.99517 0.17007 2.99175 0.22232 2.98741 0.27450 3 4 3.99356 0.22677 3.9890010.29643 3.98321 0.36600 4 15' 5 4.9919510.28346 4.98625 0.37054 4.97902 0.45750 5 45' 6 5.9903510.34015 5.98350 0.44465 -5.97482 0.54900 6 7 6.9887410.39684 6.9807510.51875 6.9706310.64051 7 8 7.98713 0.45354 7.97800 0.59286 7.96643 0.73201 8 9 8.98552 0.51023 8.97525 0.66697 8.96224 0.82351 9 1 0.99813 0.06104 0.99691 0.07845 0.99539 0.09584 1 2 1.9962610.12209 1.99383 0.15691 1.-99079 0.19169 2 3 2.99440 0.18314 2.9907510.23537 2.98618 0.28753 3 4 3.99253 0.24419 3.9876610.31383 3.98158 0.38338 4 30' 5 4.9906710.30524 4.98458 0.39229 4.97698 0.47922 5 30' 6 5.9888010.36629 5.98150 0.47075 5.9723710.57507 6 7 6.9869410.42733 6.97842 0.54921 6.96777 0.67092 7 8 7.9850710.48838 7.97533 0.62767 7.9631610.76676 8 9 8.98321 0.54943 8.9722510.70613 8.95856 0.86261 9 1 9.99785 0.06540 0.99656 0.08280 0.99496 0.10018 1 2 1.99571 0.13080 1.99313 0.16561 1.9899310.20037 2 3 2.99357 0.19620 2.98969 0.24842 2.9849010.30056 3 4 3.99143 0.26161 3.9862610.33123 3.9798710.40075 4 45' 5 4.98929 0.32701 4.9828210.41404 4.9748410.50094 5 15' 6 5.98715 0.39241 5.9793910.49684 5.96981 0.60112 6 7 6.98501 0.45782 6.9759510.57965 6.96477 0.70131 7 8 7.98287 0.52322 7.97252 0.66246 7.95974i0.80150 8 9 8.98073 0.58862 8.969080.74527 8.95471 0.90169 9 5r Dep. Lat. Dep. Lat. Dep. Lat. 860 850 840 m O36~6 ~ 8 L~ p.~~~~~~~~~~~~~~~~/4 TRAVERSE TABLE. 35 Differences of Latitude and Departures-Continued. 60 70 80 0;'4~~~8 Lat. Dep. Lat. Dep. Lat. Dep. o 1 0.99452 0.10452 0.99254 0.12186 0.99026 0.13917 1 2 1.98904 0.20905 1.98509 0.24373 1.98053 0.27834 2 3 2.98356 0.31358 2.97763 0.36560 2.97080 0.41751 3 4 3.97808 0.41811 3.97018 0.48747 3.96107 0.55669 4 0' 5 4.97261 0.52264 4.96273 0.60934 4.95134 0.69586 5 60' 6 5.96713 0.62717 5.95519 0.73121 5.94160 0.83503 6 7 6.96165 0.73169 6.94782 0.85308 6.93187 0.97421 7 8 7.95617 0.83622 7.94038 0.97495 7.92214 0.11338 8 9 8.95069 0.94075 8.93291 0.09682 8.91241 0.25255 9 1 0.99405 0.10886 0.99200 0.12619 0.98965 0.14349 1 2 1.98811 0.21773 1.98400 0.25239 1.97930 0.28698 2 3 2.98216 0.32660 2.97601 0.37859 2.96895 0.43047 3 4 3.97622 0.43546 3.96801 0.50479 3.95860 0.57397 4 15'1 5 4.97028 0.54433 4.96002 0.63099 4.94825 0.71746 5 45' 6 5.96433 0.65320 5.95202 0.75719 5.93790 0.86095 6 7 6.95839 0.76206 6.94403 0.88339 6.92755 1.00444 7 8 7.'95245 0.87093 7.93603 1.00959 7.91721 1.14794 8 9 8.94650 0.97980 8.92804 1.13579 8.906861.29143 9 1 0.99357 0.11320 0.99144 0.13052 0.98901 0.14780 1 2 1.98714 0.22640 1.98288 0.26105 1.97803 0.29561 2 3 2.98071 0.33960 2.97433 0.39157 2.96704 0.44342 3 4 3.97428 0.45281 3.96577 0.52210 3.95606 0.59123 4 30' 5 4.96786 0.56601 4.95722 0.65263 4.94508 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 6.923111.03466 7 8 7.94857 0.90562.93155 1.04420 7.91212 1.18247 8 9 8.94214 1.01882 8.92300 1.17473 8.90114 1.33028 9 1 0.99306 0.11753 0.99086 0.13485 0.98836 0.15212 1 2 1.98613 0.23507 1.98173 0O.26970 1.97672 0.30424 2 3 2.97920 0.35261 2.97259 0.40455 2.96508 0.45637 3 4 3.97227 0.47014 3.96346 0.53940 3.95344 0.60849 4 45' 5 4.96534 0.58768 4.95432 0.67425 4.94180 0.76061 5 15' 6 5.95841 0.70522 5.94519 0.80910 5.93016 0.91274 6 7 6.95147 0.82276 6.93606 0.94395 6.91853 1.06486 7 8 7.94454 0.94029 7.92692 1.07880 7.90689 1.21698 8 9 8.93761 1.05783 8.91779 1.21365 8.89525 1.36911 9 Dep. Lat. Dep. Lat. Dep. Lat.. 21 a 830 82 81ID 36 TRAVERSE TABLE. Differences of Latitude and Departures —Continued. 9 0 _ _ 100 110 1 Lat. Dep. Lat. Dep. Lat. Dep. } 1 0.98768 0.15643 0.98480 0.17364 0.9816'20.19081 1 2 1.97537 0.31286 1.96961 0.34729 1.9632510.38162 2 3 2.96306 0.46930 2.95442 0.52094 2.944880().57243 3 4 3.95075 0.62573 3.93923 0.69459 3.92650 0.76324 4 0' 5 4.93844 0.78217 4.924010.86824 4.90813 0.95405 5 60' 6 5.92612 0.93860 5.90884 1.04188 5.889761.14486 6 7 6.91381 1.09504 6.89365 1.21553 6.87139 1.33566 7 8 7.901501.25147 7.87846 1.38918 7.85301 1.52648 8 9 8.88919 1.40791 8.86327 1.56283 8.83464 1.71729 9 1 0.98699 0.16074 0.98404 0.17794 0.98078 0.19509 1 2 1.97399 0.32148 1.968080.35588 1.961570.39018 2 3 2.960980.48222 2.95212 0.53383 2.94235 0.58527 3 4 3.94798 0.64297 3.93616 0.71177 3.92314 0.78036 4 15' 5 4.93493 0.80371 4.92020 0.88971 4.90392 0.97545 5 45' 6 5.92197 0.96445 5.90424 1.06766 5.884711.17054 6 7 6.90897 1.12519 6.88828 1.24560 6.86549 1.36563 7 8 7.89597 1.28594 7.872321.42354 7.84628 1.56072 8 9 8.88296 1.44668 8.85636 1.60149 8.827061.75581 9 1 0.98628 0.16504 0.983250.18223 0.97992 0.19936 1 2 1.97257 0.33009 1.96650 0.36447 1.95984 0.39873 2 3 2.95885 0.49514 2.94976 0.54670 2.93977 0.59810 3 4 3.945140.66019 3.93301 0.72894 3.91969 0.79747 14 30' 5 4.93142 0.82523 4.91627 0.91117 4.89962 0.99683 5 30' 6 5.91771 0.99028 5.89952 1.09341 5.87954 1.19620 6 7 6.90399 1.15533 6.88278 1.27564 6.85947 1.39557 7 8 7.89028 1.32038 7.86603 1.45788 7.83939 1.59494 8 9 8.87657 1.48542 8.84929 1.64011 8.81932 1.79431 9 1 0.98555 0.16935 0.98245 0.18652 0.97904 0.20364 1 2 1.97111 0.33870 1.96490 0.37304 1.95809 0.40728 2 3 2.95666 0.50805 2.94735 0.55957 2.9371330.61092 3 4 3.942220.67740 3.92980 0.74609 3.91618 0.81456 4 45' 5 4.92778 0.84675 4.912'250.93262 4.89522 1.01820 5 15' 6 5.91333 1.01610 5.89470 1.11914 5.87427 1.22185 6 7 6.89889 1.18545 6.87715 1.30566 6.85331 1.42549 7 8 7.88444 1.35480 7.85960 1.49219 7.83236 1.62913 8 9 8.87000 1.52415 8.84205 1.67871 8.81140 1.83277 9 li MDep. Lat. Dep. Lat. Dep. bat. 80~ 790 780. TRAVERSE TABLE. 37 Differences of Latitude and Departures-Continued. = 3 120 130 14 Lat. Dep. Lat. Dep. Lat. Dep. 1' 0.97814 0.20791 0.974370.22495 0.97029 0.24192 1 2 1.95629 0.41582 1.94874 0.44990 1.94059 0.48384 2 3 2.93444 0.62373 2.92311 0.67485 2.91088 0.72576 3 4 3.91259 0.83164 3.89748 0.89980 3.88118 0.96768 4 0' 5 1 4.89073 1.03955 4.87185 1.12475 4.85147 1.20961 5 60' 6 5.86888 1.24747 5.84622 1.34970 5.82177 1.45153 6 7 6.8470311.45538 6.8205911.57465 6.79206 1.69345 7 8 7.82518 1.66329 7.79496 1.79960 7.76236 1.93537 8 9 8.80332 1.87120 8.7693312.02455 8.73'266 2.17729 9 1 0.97723 0.21217 0.9733710.22920 0.96923 0.24615 1 2 1.954460,.42435 1.94675 0.45840 1.93846 0.49230 2 3 2.93169 0.63653 2.92013 0.68760 2.90769 0.73845 3 4 3.90892 0.84871 3.89351 0.91680) 3.87692 0.98461 4 15' 51 4.88615 1.06088 4.86689 1.14600 4.84615 1.23076 5 45' 6 5.86338 1.27306 5.84027 1.37520 5.81538 1.47691 6 71 6.84061 1.48524 6.813651.60440 6.78461 1.72307 7 81 7.81784 1.69742 7.78703 1.83360 7.75384 1.96922 8 9; 8.79507 1.90959 8.76041 2.06280 8.72307 2.21537 9 1I 0.97629 0.21644 0.97237 0.23344 0.96814 0.25038 1 2 1.95259 0.43288 1.94474 0.46689 1.93629 0.50076 2 3 2.92888 0.64932 2.91711 0.70033 2.90444 0.75114 3 4 3.9051810.86576 3.88948 0.93378 3.87259 1.00152 4 30' E 4.8814811.08220 4.86185 1.16722 4.84073 1.25190 5 30' 6 5.85777 1.29864 5.83422 1.40067 5.80888 1.50228 6 7 6.83407 1.51508 6.80659 1.63411 6.77703 1.75266 7 8 7.8103611.73152 7.77896 1.86756 7.74518 2.00304 8 9: 8.786661.94796 8.75133 2.10100 8.71332 2.25342 9 1 0.9753410.22069 0.97134 0.23768 0.96704 0.25460 1 2 1.9506810.44139 1.94268 0.47537 1.93409 0.50920 2 3 2.926nC2 0.66209 2.91402 0.71305 2.90113 0.76380 3 4 3.9013610.88278 3.88536 0.95074 3.8681811.01840 4 45' 5 4.87671 1.10348 4.85671 1.18843 4.8352311.27301 5 15' 6 5.85205 1.32418 5.82805 1.42611 5.80227 1.52761 6 7 6.82739 1.54488 6.79939 1.66380 6.76932 1.78221 7 8 7.8027311.76557 7.77073 1.90148 7.73636 2.03681 8 9 8.7780811.98627 8.74207 2.13917 8.703412.29141 9 Dep. Lat. Dep. Lat. Dep. Lat. a 770 760 750 D CD ~ ~ ~ ~ ~ ~ ~~8731.94 9t 38 TRAVERSE TABLE. Differences of Latitude and Departures-Continued. 15~ 160 17~ 5 Lat. Dep. Lat. Dep. Lat. Dep. 1 0.96592 0.25881 0.96126 0.27563 0.95630 0.29237 1 2 1.931850.51763 1.92252 0.55127 1.91260 0.58474 2 3 2.89777 0.77645 2.88378 0.82691 2.86891 0.87711 3 4 3.86370 1.03527 3.84504 1.10254 3.82521 1.16948 4 0'5 4.82962 1.29409 4.80630 1.37818 4.78152 1.46185 5 60' 6 5.79555 1.55291 5.76757 1.65382 5.73782 1.75423 6 7 6.76148 1.81173 6.72883 1.92946 6.69413 2.04660 7 8 7.7274012.07055 7.69009 2.20509 7.65043 2.33897 8 9 8.69333 2.32937 8.65135 2.48073 8.60674 2.63134 9 1 0.964780.26303 0.96005 0.27982 0.95502 0.29654 1 2 1.92957 0.52606 1.92010 0.55965 1.91004 0.59308 2 3 2.89436 0.78909 2.88015 0.83948 2.86506 0.88962 3 4 3.85914 1.05212 3.84020 1.11931 3.82008 1.18616 4 15/ 5 4.82393 1.31515 4.80025 1.39914 4.77510 1.48270 5 45' 6 5.788721.57818 5.7603011.67897 5.73012 1.77924 6 7 6.75351 1.84121 6.72035 1.95880 6.68514 2.07579 7 8 7.71829 2.10424 7.68040 2.23863 7.64016 2.37233 8 9 8.68308 2.36728 8.64045 2.51846 8.59518 2.66887 9 1 0.96363 0.26723 0.95882 0.28401 0.95371 0.30070 1 2 1.92726 0.53447 1.91764 0.56803 1.90743 0.60141 2 3 2.8908910.80171 2.87646 0.85204 2.86115 0.90211 3 4 3.85452 1.06895 3.83528 1.13606 3.81486 1.20282 4 30, 5 4.81815 1.33619 4.79410 1.42007 4.76858 1.50352 5 30' 6 5.78178 1.60343 5.752921.70409 5.72230 1.80423 6 7 6.74541 1.87066 6.71174 1.98810 6.676012.10494 7 8 7.70904 2.13790 7.67056 2.27212 7.62973 2.40564 8 9 8.67267 2.40514 8.62938 2.55613 8.58345 2.70635 9 1 0.96245 0.27144 0.95757 0.28819 0.95239 0.30486 1 2 1.92491 0.54288 1.91514 0.57639 1.90479 0.60972 2 3 2.88736 0.81432 2.87271 0.86458 2.8571810.91459 3 4 3.84982 1.08576 3.83028 1.15278 3.80958 1.21945 4 45' 5 4.81227 1.35720 4.78785 1.44098 4.76197 1.52432 5 15' 6 5.7741331.62864 5.74542 1.72917 5.71437 1.82918 6 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 8.662092.44296 8.618142.59376 8.57156 2.74377 9 Dep. Lat. Dep. Lat. Dep. Lat. a 740 730 720 _...... TRAVERSE TABLE. 39 Differences of Latitude and Departures-Continued. 180 190 200I Lat. La Dep. at. Lat. Dep. 1 0.95105 0.30901 1 0.94551 0.32556 0.93969 0.34202 1 2 1.90211 0.61803 1.89103 0.65113 1.87938 0.68404 2 3 2.85316 0.92705 2.83655 0.97670 t 2.81907 1.02606 3 4 3.8042211.23606 3.78207 1.30227 3.7587711.36808 4 0' 5 4.75528 1.54508 4.72759 1.62784 4.69846 1.71010 5 60' 6 5.70633 1.85410 5.6731111.95340 1 5.63815 2.05212 6 7 6.6573912.16311 6.61863 2.27897 ] 6.57784 2.39414 i7 8 7.60845 2.47213 7.56414 2.60454 7.51754 2.73616 8 9 8.55950 2.78115 8.5096612.93011 J 8.45723 3.07818 19 1 0.94969 0.31316 0.94408 0.32969 0.93819 0.34611 1 2 1.89939 0.62632 1.88817 0.65938 1.8763810.69223.2 3 2.84909 0.93949 2.8322610.98907 2.8145711.03835 13 4 3.7987911.25265 3.7763511.31876 3.75276 1.38446 4 151 5 4.74849 1.56581 4.72044 1.64845 4.69095 1.73058 5 45' 6 5.69819 1.87898 5.66453 1.97814 5.62914 2.07670 6 7 6.64789 2.19214 6.6086212.30783 6.56733 2.44281 17 8 7.59759 2.50531 7.552712.63752 7.50553 2.76893 i8 9 8.54729 2.81847 8.49680 2.96721' 8.44372 3.11505 9 1 0.94832 0.31730 0.94264 0.33380 0.93667 0.35020 1 2 1.8966410.63460 1.88528 0.66761 1.87334 0.70041 *2 3 2.84497 0.95191 2.8279211.00142 2.81001 1.05062 J3 4 3.79329 1.26921 3.77056 1.33522! 3.74668 1.40082 4 30' 5 4.741611.58652 4.713201.66903 4.68336 1.75103 5 30' 6 5.68994 1.90382 5.65584J2.00284 1 5.62003 2.10124 6 7 6.6382612.22113 6.5984912.33664 1 6.556701.45145 7 8 7.58658 2.53843 7.54113 2.67045 i 7.49337 2.80165 8 9 8.5349112.85574 8.48377 3.00426 1 8.4300413.15186 9 1 0.9469310.32143 0.94117 0.33791 i 0.93513 0.35429 1 2 1.8938610.64287 1.8823510.67583 1 1.87027 0.70858 12 3 2.84079 0.96431 2.82352 1.01375 2.80540 1.06287 3 4 3.78772 1.28575 3.7647011.35166 3.74054 1.41716 4 45' 5 4.73465 1.60719 4.705881.68958 4.67567 1.77145 5 15t 6 5.6815811.92863 5.64705 2.02750 5.61081 2.12574 6 7 6.62851 2.25007 6.58823 2.36541 1 6.54594 2.48003 17 8 7.57544 2.57151 7.5294012.70333 7.48108 2.83432 18 9 8.52237 2.89295 8.4705813.04125 8.41621 3.18861'9 t Dep. Lat. Dep. Lat. Dep. Dat. 1 710 70~ 690 __________________'a) 40 TRAVERSE TABLE. Differences of Latitude and Departures-Continued. 2212 | 220 230 1.4 Lat. Dep. Lat. Dep. Lat. Dep. d 1 0.9335810.35836 0.9271810.37460 0.9205010.39073 1 2 1.86716 0.71673 1.8543610.74921 1.84100 0.78146 1 3 2.8007411.07510 2.78155 1.12381 2,76151 1.17219 3 4 3.7343211.43347 3.7087311.49842 3.68201 1.56292 4 0O 5 4.66790 1.79183 4.63591 1.87303 4.60252 1.95365 5 60' 6 5.60148 2.15020 5.56310 2.24763 5.52302 2.34438 6 7 6.53506 2.50857 6.49028 2.62224 6.44353 2.73511 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 1 0.93200 0.36243 0.92554 0.37864 0.9187 9 0.39474 1 2 1.86401 0.72487 1.8510810.75729 1.83758 0.78948 2 3 2.796021.08731'2.776621.13594 2.75637 1.18423 3 4 3.72803 1.44975 3.7021611.51459 3.67516 1.57897 4 15' 5 4.66004 1.81219 4.627701.89324 4.5939511.97372 5 45' 6 5.5920412.17462 5.55324 2.27189 5.51274 2.36846 6 7 6.52405 2.53706 6.478782.65054 6.4315312.76320 7 8 7.45606J2.89950 7.40432 3.02918 7.35032 3.15795 8 9 8.38807 3.26194 8.32986 3.40783 8.26912 3.55269 9 1 0.93041 0.36650 0.92388 0.38268 0.91706 0.39874 1 2 1.86083 0.73300 1.8477610.76536 1.8341210.79749 2 3 2.79125 1.09950 2.77164 1.14805 2.7511811.19624 3 4 3.72167 1.46600 3.6955211.53073 3.66824 1.59499 4 30' 5 4.6520811.83250 4.619401]..91341 4.5853011.99374 5 30' 6 5.5825012.19900 5.54328 2.29610 5.50236 2.39249 6 7 6.51292/2.56550 6.4671612.67878 6.41942 2.79124 7 8 7.44334 2.93200 7.39104 3.06146 7.33648 3.18999 8 9 8.37375 3.29851 8.3149213.44415 8.25354 3.58874 9 1 0.92881 0.37055 0.92220 0.38671 0.91531 0.40274 1 2 1.85762 0.74111 1.84440 0.77342 1.8306210.80549 2 3 2.78643 1.11167 2.76660 1.16013 2.74593 1.20824 3 4 3.7152411.48222 3.6888011.54684 3.66124 1.61098 4 45' 5 4.6440511.85278 4.61100 1.93355 4.5765512.01373 5 15' 6 5.57286 2.22334 i 5.5332012.32026 5.49186 2.41648 6 7 6.50167 2.59390 1 6.45540 2.70697 6.40718 2.81922 7 8 7.43048 2.96445 7.37760 3.09368 7.32249 3.22197 8 9 8.35929 3.33501 8.29980 3.48039 8.23780 3.612472 9 Dep. Lat. Dep. Lat. Dep. Lat. xX680 670 660 TRAVERSE TABLE. 41 Differences of Latitude and Departures-Continued. 240 25 260 1; Lat. Dep. Lat. Dep. Lat. Dep. C 1 0.91.354 0.40673 0.90630 0.42261 0.89879 0.43837 1 2 1.82709 0.81347 1.81261 0.84523 1.79758 0.87674 1 2 3 2.7406311.22020 2.7189211.26785 2.6963811.31511 3 41 3.6541811.62694 3.62523 1.69047 3.59517 1.75348 /4 0'1 5 4.56772 2.03368 4.5315312.11309 4.4939712.19185'5 60, 6 5.48127 2.44041 5.43784-2.53570 5.39276 2.63022 6 7 6.39481 2.84715 6.3441512.95832 6.291553:.06859 7 8 7.30836 3.25389 7.25046 3.38094 7.1903513.50696 /8 9 8.2219013.66062 8.1567713.80356 8.08914 3.94533 9 1 0.91176 0.41071 0.90445 0.42656 0.89687 0.44228 i 1 2 1.82352 0.82143 1.8089110.85313 1.7937410.88457 li2 3 2.7352811.23215 2.7133611.27970 2.69061 1.32686 3 4 3.64704 1.64287 3.61782 1.70627 3.587491.76915 4 15' 5 4.55881 2.05359 4.52227 2.13284 4.4843612.21144 5 45' 6 5.4705712.46431 5.42673 2.55941 5.38123 2.65373 1,6 7 6.3823312.87503 6.33118 2.98598 6.2781013.09602 7 8 7.29409 3.28575 7.23564 3.41254 7.1749813.53830!8 9 8.2058513.69647 i 8.14009 3.83911 8.07185 3.98059 19 1 0.90996 0.41469 0.90258 0.43051 0.89493 0.44619 /1 2 1.81992 0.82938 1.80517 0.86102 1.78986 0.89239 12 3 2.7298811.24407 2.70775 1.29'153 2.68480 1.33859 1/3 4 3.63984 1.65877 3.61034 1.72204 3.5797311.78479 i 4 30' 5 4.5498012.07346 4.51292 2.15255 4.47467 2.23098 5 30' 6 5.415976 2.48815 5.41551 2.58306 5.36960 I.67718!6 7 6.36972 2.90285 6.31809J3.01357 6.26454 3.12338 7 81 7.2796913.31754 7.22068 3.44408 7.15947 3.56958 8 91 8.18965 3.73223 8.12326!3.87459 8.05440 4.01578 9 1 0.90814 0.41866 0.90069 0.43444 0.89297 0.45009 1 2 1.81628 0.83732 1.80139 0.86889 1.78595 0.90019 2 3 2.72442 1.25598 2.7020911.30333 2.6789'31.35029 3 41 3.63257 1.67464 3.602791.73778 3.5719111.80039 4 451 5 4.5407112.09330 4.503492. 17222 4. 46489 2.25049 5 15' 6 5.4488512.51196 5.40418K2.60667 5.3578712.70059 / 6 7 i 6.3570012.93062 6.3048813.04111 6.2508513.15068 7 8 7.2651413.34928 7.20558]3.47556 7.14383 3.60078 18 9 8.17328 3.76794 8.1062813.91000 8.036814.05088 9 Dep. Lat. Dep. Lat. Dep. Lat. CD r 650 640 630 CD 42 TRAVERSE TABLE. Differences of Latitude and Departures —Continued. 270 280 290 S, Lat. Dep. Lat. Dep. Lat. Dep. = 1 0.89100 0.45399 0.882940.46947 0.87462 0.48481 1 2 1.7820110.90798 1.76589 0.93894 1.74924 0.96962 2 3 2.67301 1.36197 2.6488411.40841 2.62386 1.45443 3 4 3.564021.81596 3.53179 1.87788 3.4984811.93924 4 O' 5 4.4550312.26995 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 6.1806313.28630 6.12234 3.39367 7 8 7.1280513.63193 7.0635813.75577 6.9969613.87848 8 9 8.01905 4.08591 7.94652 4.22524 7.87156 4.36329 9 1 0.889010.45787 0.88089 0.47332 0.87249 0.48862 1 2 1.77803 0.91574 1.7617810.94664 1.74499 0.97724 2 3 2.66705 1.37362 2.6426711.41996 2.61748 1.46566 3 4 3.5560611.83149 3.5235611.89328 3.48998 1.95448 4 151 5 4.44508 2.28937 4.4044512.36660 4.36248 2.44310 5 45' 6 5.33410 2.74724 5.285341 2.83992 5.234971 293172 6 7 6.22311 3.20511 6.1662313.31324 6.1074713.42034 7 8 7.1121313.66299 -7.04712 3.78656 6.9799613.90896 18 9 8.00115 4.12086 7.92801 4.25988 7.85246 4.39759 9 1 0.88701 0.46174 0.87881 0.47715 0.87035 0.49242 1 2 1.77402 0.92349 1.7576310.95431 1.7407110.98484 2 3 2.6610311.38524 2.63645 1.43147 2.61106 1.47727 3 4 3.5480411.84699 3.51526 1.90863 3.48142 1.96969 4 30' 5 4.43505 2.30874 4.39408 2.38579 4.35177 2.46211 5 30 6 5.32206[2.77049 5.2729012.86295 5.22213 2.95454 6 7 6.20907 3.23224 6.15171 3.34011 6.09248 3.44696 7 8 7.09608[3.69398 7.03053 3.81727 6.9628413.93938 8 9 7.9830914.15573 7.90935 4.29442 7.83320 4.43181 9 1 0.88498 0.46561 0.87672 0.48098 0.8681 9 0.49621 1 2 1.76997 0.93122 1.7534510.96197 1.73639 0.99243 2 3 2.6549611.39684 2.63018 1.44296 2.60459 1.48864 3 4 3.53)995 1.86245 3.50690 1.92395 3.47279 1.98486 - 4 45'1 5 4.42493 2.32807 4.38363 2.40494 4.34099 2.48108 5 15'.6 5.30992 2.79368 5.26036 2.88593 5.2919 2.97729 t 6 7 6.19491 3.25930 6.13708 3.36692 6.077393.47351 7 8 7.0799013.72491 7.01381 3.84791 6.94559 3.96973 8 9 7.96488 4.19053 7.89054 4.32889 7.81378 4.46594 9.-1 -. H -.H -.- a'o620 610 600~ TRAVERSE TABLE. 43 Diferences of Latitude and Departures-Continued. ~; 30~0 31~ 320 0 -_.. c0 Y b Lat. Dep. Lat. Dep. Lat. Dep. _ 1 0.86602 0.50000 0.85716 0.51503 0.848040.52991 1 2 1.73205 1.00000 1.71433 1.03007 1.69609 1.05983 2 3 2.598071.50000 2.57150 1.54511 2.54414 1.58975 3 4 3.46410 2.00000 3.42866 2.06015 3.39219 2.11967 4 0' 5 4.33012 2.50000 4.28583 2.57519 4.2402412.64959 5 60' 6 5.19615 3.00000 5.14300 3.09022 5.08828 3.17951 6 7 6.06217 3.50000 6.00017 3.60526 5.93633 3.70943 7 8 6.92820 4.00000 6.85733 4.12030 6.78438 4.23935 8 9 7.79422 4.50000 7.714504.63534 7.63243 4.76927 9 1 0.86383 0.50377 0.85491 0.51877 0.845720.53361 1 2 1.727671.00754 1.70982 1.03754 1.69145 1.06722 2 3 2.59150 1.51132 2.564731.55631 2.53718 1.60084 3 4 3.45534 2.01509 3.41964 2.07509 3.3829.12.13445 4 15' 5 4.31917 2.51887 4.27456 2.59386 4.22863 2.66807 5 45' 6 5.18301 3.02264 5.12947 3.11263 5.07436 3.20168 6 7 6.04684 3.52641 5.98438 3.63141 5.92009 3.73530 7 8 6.91068 4.03019 6.83929 4.15018 6.76582 4.26891 8 9 7.77451 4.53396 7.694204.66895 7.611554.80253 9 1 0.86162 0.50753 0.85264 0.52249 0.843390.53730 1 2 1.72325 1.01507 1.70528 1.04 499 1.686781.07460 2 3 2.58488 1.52261 2.557921.56749 2.53017 1.61190 3 4 3.44651 2.03015 3.41056 2.08999 3.37356 2.14920 4 30' 5 4.30814 1. 53769 4.26320 2.61249 4.21695 2.68650 5 30' 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 4.70218 7.59052 4.83570 9 1 0.85940 0.51129 0.85035 0.52621 0.84103 0.54097 1 2 1.718811.02258 1.70)70 1.05242 1.68207 1.08194 2 3 2.57821 1.53387 2.55105 1.57864 2.52311 1.62292 3 4 3.437622.04517 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 8 1 6.87525 4.09034 6.80281 4.20971 6.7283114.32779 8 9 7.73465 4.60163 7.65316 4.73592 7.56935 4.86877 9 4 t Dep. at ep. at. Dep. Lat. CD 590 580 570 44 TRAVERSE TABLE. Differences of Latitude and Departures-Continued. m 330 340 350 4 % Lat. Dep. Lat. Dep. Lat. Dep. 1 0.83867 0.54463 0.82903 0.55919 0.81915 0.57357 1 2 1.67734 1.08927 1.65807 1.11838 1.63830 1.14715 2 3 2.51601 1.63391 2.487111.67757 2.457451.72072 3 4 3.35468 2.17855 3.31615 2.23677 3.27660 2.29430 4 01 5 4.19335 2.72319 4.14518 2.79596 4.09576 2.86788 5 60' 6 5.03202 3.26783 4.9742213.35515 4.914913.44145 6 7 5.87069 3.81247 5.803263.91435 5.734064.01503 7 8 6.70936 4.35711 6.632304.47354 6.553214.58861 8 9 7.54803 4.90175 7.46133 5.03273 7.37236 5.16218 9 1 0.83628 0.54829 0.82659 0.56280 0.81664 0.57714 1 2 1.67257 1.09658 1.65318 1.12560 1.633281.15429 2 3 2.50885 1.64487 2.47977 1.68841 2.4499211.73143 3 4 3.345142.19317 3.306362.25121 3.266562.30858 4 15' 5 4.18143 2.74146 4.13295 2.81402 4.0832012.88572 5 45' 6 5.01771 3.28975 4.9595413.37682 4.89984 3.46287 6 7 5.85400 3.83805 5.7861313.93963 5.71649 4.04001 7 8 6.69028 4.38634 6.61272 4.50243 6.53313 4.61716 8 9 7.52657 4.93463 7.4393115.06524 7.34977 5.19430 9 1 0.83388 0.55193 0.824120.56640 0.81411 0.58070 1 2 1.66777 1.10387 1.648251.13281 1.62823 1.16140 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 30' 5 4.16942 2.75968 4.120632.83203 4.07057 2.90351 5 30' 6 5.00331 3.31162 4.9447513.39843 4.88469 3.48421 6 7 5. 83720 3.86355 5.768883.96484 5.69880 4.06492 7 8 6.67108 4.41549 6.59300 4.53124 6.51292 4.64562 8 9 7.50497 4.96743 7.41713 5.09765 7.32703 5.22632 9 1 0.83147 0.55557 0.82164 0.56999 0.81157 0.58425 1 2 1.66294 1.11114 1.64329 1.13999 1.62314 1.16850 2 3 2.49441 1.66671 2.46494 1.70999 2.43472 1.75275 3 4 3.32588 2.22228 3.28658 2.27998 3.24629 2.33700 4 45 5 4.15735 2.77785 4.10823 2.84998 4.05787 2.92125 5 15' 6 4.98882 3.33342 4.92988 3.41998 4.869443.50550 6 7 5.82029 3.88899 5.75152 3.98997 5.68101J4.08975 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 c Dep. Lat. Dep. Lat. Dep. Lat. 2 560 550 540 o a,.., IIr TRAVERSE TABLE. 45 Differences of Latitude and Departures-Continued. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.80901 0.58778 0.79863 0.60181 0.78801 0.61566 1 2 1.61803J1.17557 1.59727 1.20363 1.5760211.23132 2 3 2.4270511.76335 2.395901.80544 2.36403[1.84698 3 4 3.23606 2.35114 3.19454 2.40726,3.15204[2.46264 4 0' 5 4.04508J2.93892 3.9931713.00907 3.94005]3.07830 5 60' 6 4.8541013.52671 4.79181 3,61089 4.7280613.69396 6 7 5.66311 4.11449 5.59044 4.21270 5.51607J4.30963 7 8 6.47213 4.70228 6.38908 4.81452 6.30408 4.92529 8 9 7.28115 5.29006 7.1 87715.41633 7. 09209 5.54095 9 1 0.8064410.59130 0.796000.60529 1.7853110.61909 11 2 1.61288 1.18261 1.5920011.21058 1.570631.23818 2 3 2.4193311.77392 2.388001.81588 2.355951.85728 13 4 3.2257712.36523 3.18400 2.42117 3.1412612.47637 4 15' 5 4.0322221.95654 J[3.980013.02647 3.9265813.09547 5 45' 6 4.838663.54785 4.776013.63176 4.7119013.71456 6 7 5.64511[4.13916 5.57201,4.23705 5.4972114.33365 7 8 6.45155J4.73047 6.36801 4.84235 6.2825314.95275 8 9 7.25800 5.32178 7.1 6401 5.44764 7.06785-5.57184 9 1 0.80385 0.59482 0.79335 0.60876 0.78260 0.62251 1 2 1.60771 1.18964 1.58670 1.21752 1.5652111.24502 2 3 2.41157[1.78446 2.3800511.82628 2.347821.86754 3 4 3.21542 2.37929 3.1734112.43504 3.13043]2.49005 4 30' 5 4.0192812.97411 3.96676J3.04380 3.913043.11257 530' 6 4.82314J3.56893 4.760113.65256 4.69564[3.73508 6 7 5.62699 4.16375 5.55347 4.26132 5.478254.35760 7 8 6.4308514.75858 6.3468324.87009 6.260864.98011 8 9 7.23471 15.35340 7.14017 5.47885 7.04347 5.60263 9 1 0.80125 0.59832 0.79068 0.61221 0.77988 0.62592 1 2 1.602501.19664 1.58137/1.22443 1.5594611.25184 2 3 2.40376 1.79497 2.37206 1.83665 2.3396511.87777 3 4 3.20501J2.39329 3.1627512.44886 3.1195312.50369 4 45'15 4.0062612.99162 3.95344J3.06108 3.89942 3.12961 5/15' 6I 4.8075213.58994 4.74413j3.67330 4.67930[3.75554 6 7 5.60877[4.18827 5.5348214.28552 5.45919J4.38146 7 8 6.41003J4.78659 6.32551[4.89773 6.-2390715.00738 8 9 7.21128 5.38492 7.1162015.50995 7.01896 5.63331 9 ~Dep. Lat. Dep. Lat. Dep. Lat. 530 520 510 46 TRAVERSE TABLE. Differences of Latitude and Departures-Continued. 390 400 410 Lat. Dep. Lat. Dep. Lat. Dep., 1 0.77714 0.62932 0.76604 0.64278 0.75470 0.65605 1 2 1.55429 1.25864 1.532081.28557 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 2.57115 3.01883 2.62423 4 0' 5 3.88573 3.14660 3.830223.21393 3.77354 3.28029 5 60' 6 4. i62873.77592 4.59626 3.85672 4.52825 3.93635 6 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 6.79235 5.90453 9 1 0.77439 0.63270 0.76323 0.64612 0.75184 0.65934 1 2 1.54878 1.26541 1.52646 1.29224 1.50368 1.31869 2 3 2.32317 1.89811 2.28969 1.93837 2.25552 1.97803 3 4 3.09757 2.53082 3.05293 2.58449 3.00736 2.63738 4 15' 5 3.87196 3.1635'2 3.81616 3.23062 3.759203.29672 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 6.195145.06164 6.10586 5.16899 6.01472 25.27476 8 9 6.96953 5.69434 6.86909 5.81511 6.76656 5.93411 9 1 0.77162 0.63607 0.76040 0.64944 0.74895 0.66262 1 2 1.54324 1.27215 1.52081 1.29889 1.49791 1.32524 2 3 2.31487 1.90823 2.28121 1.94834 2.24686 1.98786 3 4 3.08649 2.54431 3.04162 2.59779 2.99582 2.65048 4 30' 5 3.85812'3.18039 3.80203 3.24724 3.7447713.31310 5 30' 6 4.62974j 3.81646 4.56243 3.89668 4.49373 3.97572 6 7 5.4013714.45254 5.32284 4.54613 5.24268 4.63834 7 8 6.1729915.08862 6,08324 5.19558 5.99164 5.30096 8 9 6.94462 5.72470 6.84365 5.84503 6.74060 5.96358 9 1 0.76884 0.63943 0.'I 5756 0.65276 0.74605 0.66588 1 2 1.53768 1.27887 1.51513 1.30552 1.49211 1.33176 2 3 2.30652 1.91831 2.27269 1.95828 2.23817 1.99764 3 4 3.07536 2.55775 3.03026 2.61104 2.98422 2.66352 4 45' 5 3.84420 3.19719 3.78782 2.26380 3.73028 3.32940 5 15' 6 4.61305 3.83663 4.54539 3.91656 4.47634 3.99529 6 7 5.3818914.47607 5.30295 4.56932 5.2224'04.66117 7 8 6.1507315.11551 6.06052 5.22208 5.96845 5.32705 8 9 6.91957 5.75495 6.81808 5.87484 6.71451 5.99293 9 ep. Lat. Dep. Lat. Dep. Lat. D = 500 490 480 a TRAVERSE TABLE. 47 Differences of Latitude and Departures-Continued. Q; 42o 430 440 _ Lat. Dep. Lat. Dep. Lat. Dep. 1 0. 74314 0.C6913 0.73135 0.68199 0.71933 0.69465 1 2 1.4862811.33826 1.46270 1.36399 1.43867 1.38931 2 3 2.22943 2.00739 2.19406 2.04599 2.15801 2.08397 3 4 2.97257 2.67652 2.92541 2.72799 2.87735 2.77863 4 0' 5 3.71572 3.34565 3.65676 3.40999 3.59669 3.47329 5 60' 6 4.4588614.01478 4.38812 4.09199 4.31603 4.16795 6 7 5.20201 4.'68391 5.11947 4.77398 5.03537 4.86260 7 8 5.94515 5.35304 5.8508215.45598 5.75471 5.55726 8 9 6.68830 6.02217 6.58218 6.13798 6.47405 6.251'2 9 1 0.74021o0.67236 0.72837 1.68518 0.71630 0.69779 1 2 1.48043 1.34473 1.45674 0.37036 1.43260 1.39558 2 3 2.2206512.01710 2.18511 2.05554 2.14890 2.09337 3 4 2.96087'2.68946 2.91348 2.74073 2.865202.79116 4 15' 5 3.70109 3.36183 3.6418513.42591 3.58151] 3.48895 5 45' 6 4.44130 4.03420 4.37022 4.11109 4.2978] 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 5.7304] 5.58232 8 9 6.66196 6.05130 6.55533 6.16664 6.44671 6.28C11 9 1 0.73727 0.67559 0.72537 0.68835 0.71325 0.70090 1 2 1.47455 1.35118 1.45074 1.37670 1.426501.40181 2 3 2.211832.02677 2.17612 2.06506 2.13975 2.10272 3 4 2.94910 2.70236 2.90149 2.75341 2.853002.80363 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 4.27950 4.20545 6 7 5.16094 4.72913 5.07762 4.81848 5.99275 4.90636 7 8 5.89821 5.40472 5.80299 5.50683 5.70600 5.60727 8 9 6.63549 6.08031 6.52836 6.19519 6.41925 6.30818 9 1 0.73432 0.67880 0.72236 0.69151 0.71018 0.70401 1 2 1.46864 1.35760 1.44472 1.38302 1.42037 1.40802 2 3 2.20296 2.03640 2.16709 2.07453 2.13055 2.11204 3 4 2.93729 2.71520 2.88945 2.76605 2.84074 2.81605 4 451 5 3.67161 3.39400 3.61182 3.45756 3.55092 3.52007 5 15' 6 4.40593 4.07280 4.33418 4.14907 4.26111 4.22408 6 7K 5.14025 4.75160 5.05654 4.84059 4.97129 4.92810 7 81 5.87458 5.43040 5.778915.53210 5.68148 5.63211 8 91i 6.60890 6.10920 6.50127 6.22361 6.39166 6.33613 9 I | Dep. Lat. I Dep. Lat. Dep. Lat 470 460 450~ cD c '48 TRAVERSE TABLE. Differences of Latitude and Departure-Continued. 450 Lat. Dep. 1 0.70710 0.70710 1 2 1.41421 1.41421 2 3 2.12132 2.12132 3 4 2.82842 2.82842 4 5 3.53553 3.53553 5 6 4.24264 4.24264 6 7 4.94974 4.94974 7 8 5.65685 5.65685 8 9 6.36396 6.36396 9 Dep. Lat. 450 Chlains, Yards, and Feet, WITH THEIR RECIPROCAL EQUIVALENTS. Link = 7.92 inches. Chain = 66 feet = 792 inches. CHAINS INTO FEET. FEET INTO CHAINS. = Yards. Feet. Feet. Yards. Links. U. 0. 1 0.22 0.66 0.10.033 0.15 0.2 0.44 1.32 0.20.066 0.30 0.3 0.66 1.98 0.25.082 0.38 0.4 0.88 2.64 0.30.010 0.45 0.5 1.10 3.30 0.40.133 0.60 0. 6 1.32 3.96 0.50.166 0.76 0.7 1.54 4.62 0.60.200 0.91 0.8 1.76 5.28 0.70.233 1.06 0. 9 1.98 5.94 0.75.250 1.13 0. 10 2.20 6.60 0.80.266 1.21 MISCELLANEOUS. 49 Chains, Yards, and Feet-Continued. CHAINS INTO FEET. FEET INTO CHAINS. 8Yadrds. Feet. Feet. Yards. Links. 0.20 4.40 13.20 0.90.300 1.36 0. 30 6.60 19.80 1.00.330 1.51 0. 40.80 26.40 2.0.660 3.0 0.50 11.00 33.00 3.0 1.000 4.5 0. 60 13.20 39.60 4.0 1.330 6.0 0.70 15.40 46.20 5.0 1.66 7.5 0.80 17.60 52.80 6.0 2.00 9.L 0.90 19.80 59.40 7.0 2.33 10.6 I. 00 22.00 66.00 8.0 2.66 12.1 2. 00 44.00 132 9.0 3.00 13.6 3 66.00 198 10.0 3.33 15.1 4 88.00 264 15.0 5.00 22.7 5 110 330 20 6.66 30.3 6 132 396 24 8.00 36.3 7 154 462 27 9.00 40.9 8 176 528 30 10.00 45.4 9 198 594 33 11.00 50.0 10 220 660 36 12.00 54.5 20 440 1320 39 13.00 59.1 30 660 1980 40 13.33 60.6 35 770 2310 42 14.00 63.3 40 880 2640 45 15.00 68.2 45 990 2970 48 16.00 72.7 50 1100 3300 50 16.66 75.7 55 1210 3630 51 17.00 77.3 60 1320 3960 54 18.00 81.8 65 1430 4290 57 19.00 86.3 70 1540 4620 60 20.00 90.9 75 1650 4950 63 21.00 95.4 80 1760 5280 66 22.00 100 7 50 ARMY RATION. The army Ration. Table showing the weight and bulk of 1000 rations. One thousand Nett weight Gross weight Bulk in 100 rations consist of rations of in pounds. in pounds. barrels. Pork - - 750. 1218.75 3.75 75 lbs. or Bacon - - 750. 903.19 4.90 75 lbs. Flour - - 1125. 1234.06 5.74 112.5 lbs. or Pilot bread- 750. 921.69 9.03 75 lbs. or Do. - - - 1000. 1228.91 12.05 100 lbs. in the field j Beans - - 155. 177.32 0.71 8 quarts, or Rice- - - 100. 114.50 0.46 10 lbs. Coffee - - 60. 70.90 0.35 6 lbs. Sugar - - 120. 135.62 0.50 12 lbs. Vinegar - 92.5 107.50 0.33 4 quarts. Candles — 15. 17.50 0.09 11 lb. Soap - - 40. 46.89 0.19 4 lbs. Salt - - - 33.75 38.63 0.16 2 quarts. Forage. 14 lbs. hay or fodder) h when pressed 11 lbs. to cub. foot. 12 quarts oats, or per hose 40 lbs. to bus., 33.14 lbs. cub. foot 8 quarts corn 55 lbs. to bus., 45.65 lbs. cub. foot Daily allowance of water for a horse, 4 gallons. Average mule pack, New Mexico, 175 lbs. Average load to mule team across the Prairies, 2000 lbs. TABLES AND FORMULjE. PART II. GEODE S Y. GEODESY. I. Reduction to centre of station. Call P the place of the instrument, C the centre of the station, O the angle at P, between two objects A and B, y the angle at P, between C and the left hand object B, r the distance C P, C the unknown angle at C, D the distance A C, G the distance B C, C o r sin (O + y)) r sin y D sin ItI G sin 1II In the use of this formula proper attention should be paid to the signs of sin ( O + y) and sin y; for the first term will bepositive when ( O + y) is less than 180~, (the reverse with sin y); D being the distance of the right 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. II Reduction to centre of signal observed, or correction for phase in tin cones used as signals. r cos2 I Z Correction _ D 2i Where r = radius of the signal Z = angle at the point of observation between the Sun and the signal, D = the distance. 54 GEODESY. III. Spherical Excess. S a b sin C rs sin 1I,, 2? 2 sin 1! S, being the area of the triangle. r = the radius of the Earth a b sin c S - 2 -s (s-a) (s-b) (s —c), sbeing = 2 Between latitudes 45~ and 250 the spherical excess amounts to about 1"I for an area of 75.5 square miles. Hence, if the area in square miles be known, a close approximation to the spherical excess will be had by dividing the area by 75.5. Log. mean radius of the Earth in yards = 6.8427917. If the three angles of a triangle are assumed to have been equally well determined, the previous determination of the spherical excess is not necessary for the calculation of the sides, though it will be required for estimating the relative accuracy of the observations. For the sides of a spherical triangle may be computed as if they were rectilineal, when ~ 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 - E) sin ( A — E). GEODESY. 55 IV. 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, o the inclination, b - B cos 0 But as 0 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 o to be given in minutes. 0 sin~ll B —b B (1-cos 0)-2 —B 2 B s 2sin211 2 B in I B-6= B o)~ 22 2 0 B, or B - b - 0.00000004231 02 B or by logarithms, Log (B-b)- const. log 2.626422 - 2 log 0 qt log B V. 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 1= 80~ - o, o being small, and cos o -- 1- a 0 sin21I a b 02 then, sine c - a + b - 2 - 2 a+b a b 02 _ a +qb - 0.00000004231 X o being expressed in minutes. log. 0.00000004231= 2.6264222 56 GEODESY. VI. 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, d, 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 portion of a base which cannot be directly measured. 4 a b sin p sin (y —a) tangle — (a-b)2 sin X sin (y —) x a+b a- b 2 2 cos E VII. 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 + a the radius referred to the level of the measured base B, thenr+ a: r::B b: B X a r -+- a and B-b -B —B r =BX(- + etc. But the radius of the Earth being very great in comparison to the difference of level a, we have the correction 8 sufficiently accurate by retaining only the first term. Hence, Ba r GEODESY. 57 VIII. Correction for temperature in metallic rods. Let e = the linear expansion for 1~ of Fahrenheit, I = the length of the rod before expansion, 1' - the length of the rod after expansion, 1 - the number of degrees, Fahrenheit, Total expansion = e t and lf l(1+et) The following expansions were adopted by Mr. Hassler in his comparisons of weights and measures, (Report of 1832.) Expansion for 1~ Fahr. = e For 1~ in a yard's length. Platinum - 0.0000051344; 0.0001848384 Eng. In. Brass Bar- 0.00001050903; = 0.00037832508 cc Iron Bar = 0.000006963535; = 0.000250687260 cc Other authorities: Expansion for 1.~ Fahr. - e For 1~ in a yard's length. Brass bar 0.000010480 0.0003772800 Eng. In. Bailey. Brass rod 0.0000105155 0.0003785580 " Roy. 106666 0.0003839976 " Troughton. Brass wire 107407 0.0003866652 " Smeaton. Iron bar 0.0000069907 0.0002516652 " Smeaton. Steel rod 63596 0.0002289456 " Roy. Glass, Barom. tubes 43119 0.0001552284 " Roy. White NDorway pine 22685 0.0000816660 " Kater. 58 GEODESY. IX..Measurement of distances by sound. The velocity of Sound, in one second of time at 320 Fahrenheit in dry air, is about 1090 English feet. For any higher temperature, add 1 foot for every degree of the Thermometer above 32~. The measurement of distances by sound should always be made, if possible, in calm, dry weather. In cases of wind, the velocity per second must be corrected by the quantity, f cos d; f being the force of the wind in feet per second, and d the angle which its direction makes with that of the sound. Or, in general, in dry air, v - 1090 feet + ( t~ - 32) fi: f cos d. Velocity and force of winds. Velocity in A wind, when it does not exceed the velocity op- Veloci- Force on miles per posite to it, may be denominated ty per a square hour. sec'nd. foot. feet. lbs. 6.8 a gentle, pleasant wind............... 10 0.129 13.6 a brisk gale...............2.......... 20 0.915 19.5 a very brisk gale................. 30 2.059 34.1 a high wind.................. 50 5.718 47.7 a very high wind.............. 70 11.207 54.5 a storm or tempest..................... 80 14.638 68.2 a great storm......................... 100 22.872 81.8 a hurricane........................... 120 32.926. 102.3 a violent hurricane, that tears up trees, etc. 150 51.426 GEODESY. 59 X. For Reconnoissances. " Three point problem." At a point P, from whence are to be seen three points A, C, B, forming a triangle, the elements (i. e. 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 -- a; B C=-b; B C A=-C; A P C=-P and C P B- P; also, make R = 360~- P - Pt - C; x- CAP; y P B C. Then will Cotx-cot R (b sin PcosR +y- R- -x The use of these formule need not be embarrassing if care is taken in properly applying the sign of cos. and cot. R. When R is less than 90~ both cos. and cot. are plus; between 90 and 180~ both are minus; between 180~ and 270~ the cos. is minus and the cot. plus; between 270 and 3600, cos. is plus and the cot. minus. This problem is indeterminate when P falls upon the circumference of the circle passing through A, B, C. A case of this nature is of rare occurrence, however, in practice. 60 GEODESY. XI. For computing the principal Geodetic quantities dew pending on the spheroidal figure of the earth, at any given latitude. Eccentricity of the Earth e (ab b Ellipticity = E ab 1- a a or, very nearly en = 2 E; E -2 Normal ending at minor axis (or radius of curvature of a section perpendicular to the meridian) = N=- a ( 1- e2 sin2 L )2 Normal ending at major axis = N=- N ( 1- e2) a (1 — e2) (1 - e sin- L ) Tangent ending at minor axis- t N cot L Tangent ending at major axis T N tang L ( 1 - e) Radius of the parallel - p N cos L No Radius of curvature of the merid. R - (1 - e2) a (1 - es) ( 1 -- es sin2 L)3 Radius of curvature of a section making an angle Z with the meridian, = Rz =N N2 cos2 Z + R' sin' Z Radius of the earth- r a (1 e- 1 _ e2 ) sin'2L a = Equatorial Radius, b = Polar Radius, L - the given Latitude. GEODESY. 61 XII. Numerical values of some of the preceding 9nantities, from a discussion by BESSEL in the "1/lstronomische JVachrichten, JV/o. 438." a = Eq. Rad. 3272077.14 toises; log - 6.5148235337 b - Polar Rad. = 3261139.33 toises; log - 6.5133693539 Ratio of the Toise to the Metre-law of France, Dec. 10, 1799. M T - 1.9490363 Log = 0.2898199300 whence in metres M a 6377397.15; Log = 6.8046434637 M b 6356078.96; Log - 6.8031892839 Ratio of the axes, a: b:: 299.1528: 298.1528; mean uncertainty ~k 4.667 units. T M Length of the Earth's quad. - 5131179.81 — 10000855.76; mean uncertainty - 498.23 metres. e Eccentricity _( -- ) 0. 0816967; Log 8.9122052271 t,; = Ellipticity -- e0 Log-u 7.5233789824 Length, in toises, of a meridional degree whose middle latitude is p. T T T Dmn = 57013.109 - 286.337 cos 2 T + 0.611 cos 4 t ) T - 0.001 cos 6 ) Length of a degree of the parallel, in toises, D 2 - 57156.285 cos -- 47.825 cos 3 p + 0.060 cos 5 p or making sin -- e sin p Log Dp = 4.7567009.0 + log cos --- log cos q 62 GEODESY. XIII. Ratio of the Jletre to the English Yard. The value of the French metre in English imperial inches, in general use in this country and in Europe, is that derived from Kater's Experiments in 1818, viz: 39.37079 inches of Sir G. Shuckburg's scale at 62~ Faht., the metre being at 32~ Faht. From the more recent and accurate comparisons of Mr. Baily in 1835, when engaged in constructing a new standard scale for the Royal Astronomical Society (Mem. R. A. S., vol. ix); 39.369678 inches is the value of the standard metre, in mean inches of the centre yard of the Astronomical Society's scale, each being reduced to its standard temperature, namely, the platina metre to 320 and the brass scale to 62~0 of Fahrenheit's Thermometer. This very change of temperature, however, involves the result in some degree of uncertainty. The centre yard of the Astronomical Society's scale exceeds the imperial standard yard by 0.000377 inches. Whence, according to these experiments 39.370092 inches is the value of the standard metre in imperial standard inches, both being at their respective standard temperatures. The value of the metre, as reported to Congress by Mr. Hassler in his report on Weights and Measures in 1832, is 39.38091714 inches of the English imperial standard at 320 Fht., the comparisons having been made at that temperature upon an 82 inches scale by Troughton, said to be identical with the English standard; or, correcting for expansion - 39.36850154 imperial standard inches at 62~ Fht., a value materially smaller than the two preceding. According to Baily this discordance has probably GEODESY. 63 arisen from inaccuracy in the length of the copy of Troughton's scale employed by Mr. Hassler. This 82 inches scale is the standard of the United States, but in the absence of a direct comparison between it and the English standard, and not to add to a confusion already too great, it is as well to adhere for the present to the old value of Kater, as being that which is still most in use. To recapitulate: 1 metre 39.3707900 English imperial inches, according to Kater, (1818,) Log — 1.5951741293 39.3700920 English imperial inches, according to Baily, (1835,) Log -1.5951664297 39.36850154, American std'd inches, according to Hassler, (1832,) being the ratio, for the present, in use upon the Survey of the Coast,. - Log = 1.5951489169 The metre being at 32~, and the inches at 62~ Fht. XIV. NJ __ _ tang I I-L — L tang — = 2 cos Z /3 sin I K --- ut N sin 1I' See the note to the preceding formulae. The algebraic sign of the azimuth Z will determine the sign of,, and consequently whether the quantity u11 is to be added to or subtracted from q. GEODESY. 69 XIX. 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. tang, -= sin L tang Z tang Lt sin ( - m) sin. 3 = e2 (L — Lt) cos2 (L -+ ); L't); L= Ltt l —L- L 2 /,11iL + m cos 1! utl - I = u1" N sin 1,/ sin Z m = 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 q, and consequently whether it is to be increased or diminished by m. The formulhe on page 67 can be presented in a different form, thus: From the formulae on page 65, (MI - M) cos L- u1l sin Z and, (L- L') — B u"2 sin2 Z cos2 L' tan L sin 1f (1 4- e' cos2 L) u"l cos Z- - 1 - e2 coa2 L Substituting, in this last, the value of u"' sin Z, and dividing one by the other; tang Z ( Ml M) cos LI ( 1 + e2 cos2 L) (L-L')-2 (M'-M)2 cos2 L' tang L sin 1 ( 1 + e2 cos5 L) Then knowing Z; u (MI -M) cos L sinZ and, K = ull N sin 11t N, being the normal for the mean latitude. 70 GEODESY. XX. Forms for record Survey of E Names of. Observed M 4 = Final plane o Statins. o Angles. Angles. o 0 I " I~ Io 0 Sought Cedar Point 18 66 34 04.80 -0.36104.44 1.58 66 34 02.86 XIII Right. Buck Hill 18 64 08 37.78- 0.3637.42 1.58 64 08 35.84 Left. Fort Flats 18 49 17 23.24 -0.3622.88 1.58 47 17 21.30 180 00 00.00 Example of Survey of LATITUDES. NAMES OF LI = L L u".(1 +- e2 Cos.2 L) Cos. Z STATIONS. 2 Sin. ~2 Sin.2 Z U'12 (1 + e2 CoS.? L) tang. L. Fort Flats Latitude L — 4539113".89 Log. K (yards) = 4.7295212 Sin. 1 -=4.38454 Leg. N Sin. 1, = 8.4701676 2 Log. Sin. Z = 9.09522 Log. u" = 3.1996888 2 Log. u" = 6.39936 Log.(l+e2 Cos. L)= 0.0014140......... = 0.00141 Log. Cos. Z (-) = 9.9711210 Log. tang. L = 0.00991 Log. 1st term = 3.1722268 Log. 2d term = 9.89034 1st term () =-+1486".71 2d term = 0".77 2d term (-) = - 0.77 L = 0~ 24' 45".94 L =45 39 13.89L + L' =91~43 13".7-2 L - =-45 51 36.86 Cedar Point Latitude Lt = 460031 59t. 83 2 GEODESY. 71 and computation. Calculations of Triangles of the first order. c Logarithms Sides in their Calculation of the Sides. Designation. Sines. Yards. S 9.9626198 Log. RL =4.7379524 =54695.61 BFort Flat. Comp. Log. Sin. S =0.0373802 Log. Sin. R =9.9541886 R 9.9541886 Log. LS =4.7295212 =53644.00 i Fort FlatLog. RL ~ 47753326 Comp.Log.Sin.S } = Log. Sin. L =9.8796760 L 19.8796760 Log. RS =4.6550086 =45186.49 IBuck Hill-Po.Method 1, (page 65.) Geodetic Determination of Positions. (Secondary.) LONGITUDES AZIMUTHS u" Sin. Z L-L' REMARKS. M'-=M+- Cos. L' Z'=180o+Z-(j M)Sin 2 Lon. M= 84~421 22'. 19 Azim. Z - 159020 1311. 62 Log.Sin.Z=( +- )9.5476117 1800 Log u" = 3.1996888 1800~ Z = 339 20 13.62 20 39 46.38 2.7473005 L L' Log. Cos. L' = 9.8412474 Log.Sin -9.8559089 Log. J M = 2.9060529..........(+) - 2.9060529 Log. J'Z = 2.7619618 - 578".05 SM = 0013'25".48'Z - 0 o091 38'/.05 M =84 42 22.19 1800~ 4- Z =339 20 1362 Lon. M-= 840551471!.67 Azim. Z — 339010! 351-.57 72 GEODESY. 1 Ellipticity = 300, Equat. Rad. = 6974532 yds. Log 6.8435151 Normal or Radius of Curvature of the perpendicular to the Meridian. Latitude. N ( _- e2sinL Log. (1 + e2 cos2 L) Corn. Differ. Log. N. differ'ce Log Differ. N- differlee N sin I" for 10o for 10f 20 0 6.8436847 27 3 8.4707404 0.0025521 15 6888 27.6 7363 5439 56 30 699 27.8 7322 5356 45 6971 28.1 7280 524 5 21 0 7013 28.4 7238 5191 15 7056 28 7 196 5106 30 7099 2 90 7153 5021 57 45 7142 29.3 7109 4934 58 22 0 7186 29.5 7066 4847 58 15 7230 29.7 7021 4760 30 7274 300 6977 4671 45 7319 30.2 6932 4582 60 23 0 7365 30.5 6887 4492 61 15 7410 30.7 6841 4401 62 30 7457310 6795 4309 62 45 7503 6748 4217 31.2 62 24 0 7550 31 5 6701 4124 63 15 7597 31.7 6654 4030 63 30 7645 320 6607 3935 45 7693 32:2 6559 3840 64 32.2 64 25 0 7741 32 6510 3744 64 15 7790 32.7 6462 3648 65 30 7839 32.9 6413 3550 65 45 7888 33.2 6363 3452 66 26 0 7938 33 6313 3353 66 15 7988 336 6263 3254 67 30 8038 338 6213 3154 67 45 8089 34.0 6162 3053 68 27 0 8140 34.3 6111 2951 68 15 8192 34.5 6060 2849 69 30 8243 6008 2746 69 45 8295 34:9 5956 2643 69 28 0 8348 351 5904 2539 70 15 8400 35.3 5851 2434 70 30 8453 5.5 5798 2329 71 45 6.8438507 8.4705745 0.0022223 35.7 71 GEODESY. 73 Corn. 1 Lo Differ. Latitude. Log N. diff Nsin ( + e cos2 L) for 10' for 10' 0 \ 29 0 6.8438560 359 8.4705691 0.0022117 15 8614 1 5637 2010 30 8668 36.3 5583 190 72 45 8723 36.5 5529 1794 72 30 0 8777 36.7 5474 1686 36.9 73 15 8832 366.97 5419 1576 30 8888 37.1 5364 1466 73 45 8943 37.2 5308 1356 31 0 8999 5252 1245 15 9055 376 5196 1134 30 9111 37.7 5140 102 7 45 9168 37.9 5084 0910 32 0 9225 38.1 5027 0797 15 9282 381 4970 0684 75 30 9339 38.2 4912 0570 45 9397 385 4855 0455 77 38.5 /77 33 0 9454 7 4797 0340 15 9512 38.9 4737 0225 30 9571 390 4681.0020109 77 45 9629 39:1 4622.0019993 34 0 9688 2 4564 9877 15 9747 45(5 960 8 39.4 78 30 9806 4446 9643 7 45 9865 4387 9525 35 0 9924 8 4327 9407 80 15.8439984 400 4267 9288 80 30.8440044 400 4208 9169 80 45 0104 40.1 4148 9050 80 36 0 0164 40.3 4087 8931 80 15 0224 403 4027 8811 81 30 0285 40 5 3966 8690 80 45 0346 40.6 3906 8570 81 37 0 0406 3845 8449 15 0467 407 3784 8328 81 30 0529 40.7 3723 8206 81 45 0590 41:9 3661 8084 81 41.0 81 38 0 0651 41.1 3600 7963 82 15 0713 411 3538 7840 82 30 0775 412 3477 7717 82 45 0837 41.4 3415 7594 82 39 0 0898 1 4 3353 7471 82 15 0961 3291 7348 41.4 83 30 1023 415 3229 7224 82 45 6.8441085 41:6 8.4703166 0.0017101 83 10 74 GEODESY., Log Latitude. Log N. Cor Log ( L)Differ. for 10' 40 0 6.8441147 8.4703104 0.0016977 3J 1210 41 3(41 6853 1273 41.8 2979 6728 45 1335 41.9 2916 6604 84 0 1398 2853 6479 30 1524 41.9 2728 6229 84 45 1587 42. 2665 6104 84 42 0 1650 42.1 2602 5979 84 15 1713 42.1 2539 5853 84 30 1776 42.1 245 5728 45 1839 42.2 2412 5602 42.1 3 0 1903 42.1 2349 5477 15 1967 42.2 2286 5351 30 2(29 42.2 2222 5299 45 2093 42159 5099 42.3 4 0 2156 2095 4973 15 2219 42.3 2032 4847 30 2283 42.3 1969 4721 45 2346 42.3 1905 4595 42.3 45 0 2410 42.3 1812 4469 84 15 2473 3 1778 4343 84 30 2537 42.3 1715 4217 84 45 2600 423 16 4091 42.3 84 46 0 2663 42.3 1588 3965 84 15 2727 42.3 1525 3839 84 30 2790 42.3 1461 3713 84 45 2854 42.2 1398 3587 84 42.2 47 0 2917 420 1334 3461 84 15 2980 421 1271 3336 84 30 3043 421 1208 3210 84 45 3107 421 1145 3084 48 0 3170 42 1 1082 2959 8 15 3233 420 1018 2833 84 30 3296 420 0955 2708 84 45 3359 42 0892 2583 84 49 0 3422 419 0830 2458 84 15 3485 41.9 0767 2333 30 3547 41.9 0704 22209 83 45 3610 418 0641 2084 84 50 0 6.8443673 41.8 8.4700579 0.0011960 GEODESY. 75 Ellipticity = -00, Equatorial Radius 6974532 Yards. Radius of curvature of Meridian. R — f a(1- e2) Latitude. (1 - e2 sin2 L) - Corm.'Log 1 Log R ddiffer. R sin 1"e for 10'. 20 0 6.8411155 81.9 8.4733096 15 1278 82.7 2973 30 1402 83.5 2849 45 1527 84.3 2724 14.3 21 0 1654 85.1 2598 15 1781 86.0 2470 86.8 30 1910 86.8 2341 45 2040 87.5 2211 22 0 2172 2080 88.4 2 5 2304 8 1947 Jo 6 2438 80. 1813 45 2573.0 1679 91.0 23 0 2709 1543 15 -2846 9. 1405 30'2984 93.0 1267 45: 3124 1128 93.6 2:~r 3264 0987 3406 9 0845 30 3549 9. 0702 145 3693 96. 0559 96.7 25 - 0, 3838 04"A 15 3984 026, 30 4131 98.1 4730o20 45 4279.4729972 99.4 26 0 4428 100.1 9823 15 4578 9673 100.9 30 4730 101.5 9522 45 4882 9370 27 0 5035 102.8 96 15 5189 10. 9062 30 5344 104 8907 45 5500 104. 8751 104.7 28 0 5657 105.3 8594 15 5815 106.0 8436 30 5974 106. 8277 45 6.8416134 8.4728117 107.1 78 GEODESY. XXI. Trigonometrical Levelling. In the following formule let: A At represent the observed zenith distance of which A is the smaller. d A, d Al the height of each signal above the telescope of the instrument. K the distance in linear units between the two stations. a the known altitude of the station from which the zenith distance A was measured. N the normal for the mean latitude of the two stations. M the modulus of common logarithms, having for its log 9.6377843. 1st. To compute the difference of level of two points by reciprocal zenith distances. If possible, the zenith distances should be simultaneously taken, that the results may be independent of refraction. dA sin A d Al sin A1 S=a+-K sin III K sin 11i Log diff. of level = log K tang a (8'- ) + a 4- 2N K tang w (8- ) s- Ki The third term of this formula will be positive, if a is the altitude of the point from which the smallest zenith distance, always represented by a, has been observed; otherwise negative. TRIGONOMETRICAL ALTITUDES. 79 2d. To compute the difference of level of two points by a single zenith distance: d A sin A +K sin 111 Log diff. of level = log 2 N sin 11, tan 8tang ( 8 2 - N s 1s KJ N 2rNsi \ The third term will be positive when A is less than 900, which will be the case when A is observed from the lowest point. In this formula r represents the coefficient of terrestrial refraction, a variable quantity; its mean value is generally stated to be 0.08 with variation of 0.02 less in summer and more in winter. 1- 2r If we assume r=0.08, the factor 2 N 011004133; 2 N sin 1/1 assuming also N to be constant and = to the normal at latitude 450, log N (in English feet) = 7.3213623; M - M log 2.3164220; log 2N = 2.0153920; log 12 N.9158785. In ordinary cases, as an approximation, we may take: difference of level = K cot (a — 0.004133 K.) K being in English feet and ( 0.004133 K ) seconds of arc. Log 0.004133 = 7.6163121. 80 GEODESY. 3d. The method of reciprocal zenith distances gives the means of obtaining the coefficient of refraction r, which, using the same notation as before, is: K 180~ + N sin 1- (8 + 8) N sin 1-1 In the trigonometrical survey of Massachusetts, Mr. Borden used 0.0784 as a mean coefficient for the sea coast, and 0.0697 for the interior of the State. 4th. To compute the altitude of a station from the observed zenith distance of the sea horizon; using the same notation as before: N/sin I'_ 2 log Alt. = log (sn 1)+ log ( -_ 900 ) M sin If' + M i ( ) -- 900)2 It would be as well, to ensure greater accuracy, to observe the zenith distance of points of the horizon on several days, taking a mean of the whole; and also to note the state of the tide at the time of observation. Should r be assumed - 0.08 /sin 1',\ 2 log ~ ( -)= 9.1425441 lM /sin 11"\2 log ~ s- r) 8.4792985 The last term can generally be neglected, and N may be assumed as the Normal of latitude 45~. TRIGONGMETRICAL ALTITUDES. 81 Corrections for Curvature and Refraction, showing the difference of the apparent and true level, in feet and decimals of afoot, for distances in feet and miles. C; CORRECTION IN FEET. CORRECTION IN FEET. For curva- For curvaFor cur- For re- ore a For curva- For refrac- tre and vature. fracction. tore. tion, refraction. 100.00024.00004.00020 4.0417.0060.0357 150.00054.00008.00046 2.1668.0238.1430 200.00094.00013.00083 4.3752.0536.3216 250.00149.00021.00128 1.6670.0953.5717 300.00215.00031.00184 14 1.5008.2144 1.2864 350.00293.00042.00251 2 2.6680.3811 2.2869 400.00383.00055.00328 2z 4.1688.5955 3.5733 450.00484.00069.00415 3 6.0030.8561 5.1469 500.00598.0008.5.00513 34 8.1708 1.1673 7.0035 550.00724.00103.00621 4 10.6720 1.5246 9.1474 600.00861.00123.00738 44 13.5468 1.9295 11.5773 650.01010.00144.00866 5 16.6750 2.3821 14.2929 700.01172.00167.01005 51 20.1769 2.8824 17.945 750.01345.00192.01153 6 24.0120 3.4303 20.5817 800.01531.00219.01312 62 28.1809 4.0258 24.1551 850.01728.00247.01481 7 32.6830 4.6690 28.0143 900.01938.00277.01661 74 37.5190 5.3599 32.1591 950.02159.00308.01851 8 42.6880 6.0997 36.5883 1000.02392.00333.02059 81 48.1910 6.8844 41.3066 1050.02638.00377.02261 9 54.0270 7.7181 46.3089 1100.02895.00414.02481 94 60.1971 8.5996 51.5975 1150.03164.00452.02712 10 66.7000 9.5286 57.1714 1200.03445.00492.02953 11 80.7070 11.5296 69.1774 1250.03738.00534.03204 12 96.0480 13.7211 82.3269 1300.04043.00578.03465 13 112.7230 16.1033 96.6197 1350.04361.00623.03738 14 130.7320 18.6760 112.0560 1400.04689.00670.04019 15 150.0750 21.4393 128.6357 1450.05030.00719.04311 16 170.7520 24.3931 146.3589 1500.05383.00769.04614 17 192.7630 27.5376 165.2254 1550.05748.00821.04927 18 216.1086 30.8727 185.2359 1600.06125.00875.05250 19 240.7870 34.3981 206.3889 1650.06514.00931.05583 20 266.8000 38.1143 228.6857 1700.06914.00988.05926 1750.07327.01047.06280 1800.07792.01107.06645 For a very close approximation, 180.07792~.01107.06642 D2 1850.08188.01170.07018 correc'n for curvature, in ft., 2 D2 — 1900.08637.01234.07403 1950.09098.01300.07798 D being the distance in miles. 2000.09570.01367.08203 11 82 GEODESY. Reduction, in feet and decimals, upon 100 feet, for the Jollowing vertical angles. Angle. Reduct. Angle. Reduct. Angle. Reduct. Angle. Reduct. Ot 0 0 O 0 t 3 0.137 7 30.856 12 0 2.185 16 30 4.118 3 15.161 7 45.913 12 15 2.277 16 45 4.243 3 30.187 8 0.973 12 30 2.370 17 0 4.370 3 45.214 8 15 1.035 12 45 2.466 17 15 4.498 4 0.244 8 30 1.098 13 0 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 0 1.231 13 30 2.763 18 0 4.894 4 45.343 9 15 1.300 13 45 2.866 18 15 5.030 5 0.381 9 30 1.371 14 0 2.9170 18 30 5.168 5 15.420 9 45 1.444 14 15 2.077 18 45 5.307 5 30.460 10 0 1.519 14 30 3.185 19 0 5.448 5 45.503 10 15 1.596 14 45 3.295 19 15 5.591 6 0.548 10 30 1.675 15 0 3.407 19 30 5.736 6 15.594 10 45 1.755 15 15 3.521 19 45 5.882 6 30.643 11 0 1.837 15 30 3.637 20 0 6.031 6 45.663 11 15 1.921 15 45 3.754 7 0.745 11 30 2.008 16 0 3.874 7 15.800 11 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. perpendicular. dicular. dicular. dicular. 0 0, 0' 0 I 0 15 229 3 35 16 8 8 7 18 26 3 0 30 115 3 49 15 8 45 64 19 59 29 0 45 76 4 6 14 9 27 6 2148 22 1 0 57 4 24 13 9 52 53 23 58 24 1 15 46 4 45 12 10 18 54 26 34 2 130 39 5 0 114 10 47 54 29 44 13 1 45 33 5 12 11 11 19 5 33 42 1~ 2 0 28 5 27 101 11 53 43 38 40 14 2 15 25 5 42 10 12 32 4~ 45 0 1 2 30 23 6 0 9 13 15 44 53 8 2 45 21 6 21 9 14 2 4 63 28 2 3 0 19 6 43 84 14 55 33 75 58 3 15 18 7 7 8 15 56 34 78 41 - 328 17 736 74 17 6 34 _..~~~~~3 MEASUREMENT OF HEIGHTS. 83 XXII. Bairometrical JMeasurement of Heights. For computing the difference in the heights of two places, by means of the Barometer. X = 60345.51 ( 1 +.. 001111 ( t + it - 64~) I X logof X -.0001(-d) XX 1 +. 002695 cos 2 Where q = the latitude of the place. /3= the height of the barometer,: the temperature (Faht.) of the t th l mercury,. station. t = the temperature (Faht.) of the air. pt = the height of the barometer, ft the temperature (Faht.) of the t rcurat the upper mercury.(. * - r it station. tt - the temperature (Faht.) of the air,.. Make A = the log of the first term, in English feet. B = the log of 1 +-. 0001 ( -d') C - the log of the last term. D = log - (log /' + B) Then, by the tables which follow, the logarithm of the difference of altitude in English feet, =A + C + log D 84 GEODESY. TABLE I. —Thermometers in the open air. tbt' A. t+t' A. t+t' A. t- t' A. 1 4.74914 46 4.77187 91 4.79348 136 4.81407 2.74966 47.77236 92.79395 137.81452 3.75017 48.77285 93.79442 138.81496 4.75069 49.77334 94.79488 139.81541 5.75120 50,77383 95.79535 140.81585 6.75172 51.77432 96.79582 141.81630 7.75223 52.77481 97.79629 142.81675 8.75274 53.77530 98.79675 143.81719 9.75326 54.77579 99.79722 144.81763 10.75371 55.77628 100.79768 145.81807 11.75428 56.77677 101.79814 146.81851 12.75479 57.77726 102.79860 147.81895 13.75531 58.77774 103.79907 148.81939 14.75582 59.77823 104.79953 149.81983 15.75633 60.77871 105.79999 150.82027 16.75684 61.77919 106.80045 151.82071 17.75735 62.77968 107.80091 152.82115 18.75786 63.78016 108.80137 153.82159 19.75837 64.78065 109.80183 154.82203 20.75888 65.78113 110.80229 155.82247 21.75938 66.78161 111.80275 156.82291 22.75989 67.78209 112.80321 157.82335 23.76039 68.78257 113.80367 158.82379 24.76090 69.78305 114.80412 159.82423 25.76140 70.78352 115.80458 160.82466 26.76190 71.78400 116.80504 161.82510 27.76241 72.748449 117.80550 162.82553 28.76291 73-.78497 118.80595 163. 82596 29.76342 74.78544 119.80641 164.82640 30.76392 75.78592 120.80687 165.82683 31.76442 76.78640 121.80732 166.82727 32.76492 77.78688 122.80777 167.82770 33.76542 78.78735 123.80822 168.82813 34.76592 79.78783 124.80867 169.82857 35.76642 80.78830 125.80912 170.82900 36.76692 81.78878 126.80957 171.82943 37.76742 82.78925 127.81002 172.82986 38.76792 83.78972 128.8,1047 173.83030 39.76842 84.79019 129.81092 174.83073 40.76891 85.79066 130.81137 175.83116 41.76941 86.79113 131.81182 176.83159 42.76990 87.79160 132'.81227 177.83201 43.77039 88.79207 133.81272 178.83244 44.77089 89.79254 134.81317 179.83287 45 4.77138 90 4.79301 135 4.81362 180 4.83329.; —-----------------. BAROMETRICAL MEASUREMENT OF HEIGHTS. 85 TABLE 11I. TABLE II. —./ttached Thermometer. Latitude of the place. o I0 0 0 0.00o00 20 0.00087 40 0.00174 0 0.00117 1.00004 i 21.00091 41.00178 5 0.00115 2.00009 22 r.00096 42.00182 10 0.00110 3.00013 23.00100 43.00187 15 0.00100 4.00017 24.00104 44.00191 20 0.00090 5.00022 25.00109 45.00195 25 0.00075 6.00026 26.00113 46.00200 30 0.00058 7.00030 27.00117 47.00204 35 0.00040 8.00035 28.00122 48.00208 40 0.00020 9.00039 29.00126 49.00213 45 0.00000 10.00043 30.00130 50.00217 50 9.99980 11.00048 31.00135 51.00221 55 9.99960 12.00052 32.00139 52.00226 60 9.99942 13.00056 33.00143 53.00230 65 9.99925 14.00061 34.00148 54.00234 70 0.99910 15.00065 35.00152 55.00239 75 9.99900 16.00069 36.00156 56.00243 80 9.99890 17.00074 37.00161 57.00247 85 9.99885 18.00078 38.00165 58.00252 90 9.99883 19 0.00083 39 0.00169 59 0.00256 Example, latitude 21~. Upper Station. Lower Station. Thermometer in open air t' 70. 4 t 77. 6 Attached Thermometer..r' 70. 4 r 77. 6 Barometer...........,' = 23.66 P 130.05 B =0.00031 Log D = 9.01502 Log R'-1.37401 C =0.00087 - A 4.81939 1.37432 _ Log = 1.47784 3.83528 D -0.10352'6843.7 feet. 86 GEODESY. Table of comparison of Fahrenheit's Thermometer with Reaumur's and the Centesimal. Fah. Reaum. Centes. Fvh. Reaum. Centes. Fah. Reaum. Centes. o 0 O 33 + 0.4 + 0.6 67 +15.6 +19.4 0 -14.2 -17.8 34 0.9 1.1 68 16.0 20.0 1 13.8 17.2 35 1.3 1.7 60 16.4 20.6 2 13.3 16.7 36 1.8 2.2 70 16.9 21.1 3 12.9 16.1 37 2.2 2.8 71 17.3 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 11.6 14.4 40 3.6 4.4 74 18.7 23.3 7 11.1 13.9 41 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 10.2 12.8 43 4.9 6.1 77 20.0 25.0 10 9.8 12.2 44 5.3 6.7 78 20.4 25.6 11 9.3 11.7 45 5.8 7.2 79 20.9 26.1 12 8.9 11.1 46 6.2 7.8 80 21.3 26.7 13 8.4 10.6 47 6.7 8.3 81 21.8 27.2 14 8.0 10.0 48 7.1 8.9 82 22.2 27.8 15 7.6 9.4 49 7.6 9.4 83 22.7 28.3 16 7.1 8.9 50 8.0 10.0 84 23.1 28.9 17 6.7 8.3 51 8.4 10.6 85 23.6 29.4 18 6.2 7.8 52 8.9 11.1 86 24.0 30.0 19 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 31.1 21 4.9 6.1 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 23 4.0 5.0 57 11.1 13.9 91 26.2 32.8 24 3.6 4.4 58 11.6 14.4 92 26.7 33.3 25 3.1 3.9 59 12.0 15.0 93 27.1 33.9 26 2.7 3.3 60 12.4 15.6 94 27.6 34.4 27 2.2 2.8 61 12.9 16.1 95 28.0 35.0 28 1.8 2.8 62 13.3 16.7 96 28.4 35.6 29 1.3 1.7 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 - 0.4 - 0.6 65 14.7 18.3 99 29.8 37.2 32 0.0 0.0 66 +15.1 +18.9 100 +30.2 +37.8 x~ Reaumur = (320 + I x~) Fah. = - x~ Centes. x~ Centes. = (320 + - x~) Fah. = - x~ Reaum. x~ Fah. (x - 32~0) Reau.- (x~ - 32~) ~ Cen. BAROMETRICAL MEASUREMENT OF HEIGHTS. 87 Table for the comparison of French and English Barometers. Milli- English inches. Millime- English inches. Millime- English inches. metres. tres. tres. 501 19.725 531 20.906 561 22.087 502.764 532.945 562.126 503.803 533 20.985 563.166 504.843 534 21.024 564.205 505.882 535.063 565.244 506.921 536.103 - 566.284 507 19.961 537.142 567.323 508 20.000 538.181 568.363 509.040 539.221 569.402 510.079 540.266 570.441 511.118 541.300 571.481 512.158 542.339 572.520 513.197 543.378 573.559 514.236 544.417 574.599 515.276 545.457 575.638 516.315 546.496 576.678 517.354 547.536 577.717 518.394 548.575 578.756 519.433 549.614 579.796 520.473 550.654 580.835 521.512 551.693 581.875 522.551 552.733 582.914 523,591 553.772 583.953 524.630 554.811 584 22.993 525.670 555.851 585 23.032 526.709 556.890 586.071 527.748 557.930 587.111 528.788 558 21.969 588.150 529 827 559 22.009 589.189 530 20.867 560 22.048 590 23.229 88 GEODESY. Table for the comparison of French and English Barometers, Millime- English inches. Millime- English inches. Millime- English inches. tres. tres. tres. 591 23.268 621 24.449 651 25.630 592.308 622.489 652.670 593.347 623 [.528 653.709 594.386 624.567 654.748 595.426 625.607 655.788 596.465 626.646 656.827 597.504 627.685 657.867 598.544 628.725 658.906 599.583 629.764 659.945 600.622 630.804 660 25.985 601.662 631.843 661 26.024 602.701 632.882 662.063 603.741 633.922 663.103 604.780 634.961 664.142 605.819 635 25.000 665.181 606.859 636.040 666.221 607.898 637.079 667.260 608.937 638.118 668.300 609 23.977 639.158 669.339 610 24.016 640.197 670.378 611.056 641.237 671.418 612.095 642.276 672.457 613.134 643.315 673.496 614.174 644.355 674.536 615.213 645.394 675.575 616.252 646.433 676.615 617.292 647.473 677.654 618.331 648.512 678.693 619.371 649.552 679.733 620 24.410 650 25.591 680 26.772 BAROMETRICAL MEASUREMENT OF HEIGHTS. 89 Table for the comparison of French and English Barometers. Millime- English inches. Millime- English inches. Millime- English inches. tres. tres. tres. 681 26.811 711 27.992 741 29.173 682.851 712 28.032 742.213 683.890 713.071 743.252 684.930 714.110 744.292 685 26.969 715.150 745.331 686 27.008 716.189 746.370 687.048 717.229 747.410 688.087 718.268 748.449 689.]26 719.307 749.488 690.166 720.347 750.528 691.205 721.386 751.567 692.245 722.425 752.606 693.284 723.465 753.646 694.323 724.504 754.685 695.363 725.543 755.725 696.402 726.583 756.764 697.441 727.622 757.803 698.481 728.662 758.843 699.520 729.701 759.882 700.559 730.740 760.921 701.599 731.780 761 29.961 702.638 732.819 762 30.000 703.677 733.858 763.040 704.717 734.898 764.079 705.756 735.937 765.118 706.795 736 28.977 766.158 707.835 737 29.016 767.197 708.874 738.055 768.236 709.914 739.095 769.276 710 27.953 740 29.134 770 30.315 90 GEODESY. -Table forthe comparison of French and English Barometers. Millime- English inches. Millime- English inches. PROPORTIONAL PARTS. tres. tres. Millim. English inches. 771 30.355 781 30.748 0.1 0.0039 772.394 782.788.2.0079 773.433 783.827.3.0118 774.473 784.866.4.0157 775.512 785.906.5.0197 776.551 786.945.6.0236 777.591 787 30.984.7.0276 778.630 788 31.024.8.0315 779.670 789.063 0.9.0354 780 30.709 790 31.103 1.0 0.0394 I Metre =- 39.3707:English inches =443.296 Paris lines. I English foot= 0.304794 metre =135.114 Paris lines. I French foot-:1.0658 -English feet 0.32484 metre. French inches English inches. French lines. English inches. 1 1.0658 1 0.0888 2:2.1315 2.1776 3 3.1973 3.2664 4 4.2631 4.3553 5 5.3288 5.4441 6 6.3946 6.5329 7 7.4604 7.6217 8 8.5261 8.7105 9 9.-5919 9.7993 10 10.6577 10.8881 11 11.7234 11.9770 12 12.7899 12 1.0658 BAROMETRICAL MEASUREMENT OF HEIGHTS. 91 Table for the comparison of French and English Barometers. Hundredths of an inch. English inches O 4 6 8 and tenths. - Millimetres. 21.0 533.39 533.90 534.41 534.91 535.42.1 535.93 536.44 536.95 537.45 537.96.2 538.47 538.98 539.49- 539.99 540.50.3 541.01 541.52 542.03 542.53 5-43.04.4 543.55 544.061 544.57 545.07 545.58.5 546.09 546.60 547.11 547.61 548.12.6 548.63 549.14 549.65 550.15 550.66.7 551.17 551.68 552.19 552.69 553.20.8 553.71 554.22 554.73 555.23 555.74.9 556.25 556.76 557.27 557.77 558.28 22.0 558.79 559.30 559.81 560.31 560.82.1 561.33 561.84 562.35 562.85 563.36.2 563.87 564.38 564.89 565.39 565.90.3 566.41 566.92 567.43 567.93 568.44.4 568.95 569.46 569.97 570.47 570.98.5 571.49 572.00 572.51 573.01 573.55.6 574.03 574.54 575.05 575.55 576.06.7 576.57 577.08 577.59 578.09 578.60.8 579.11 579.62 580.13 580.63 581.14.9 581.65 582.16 582.67 583.17 583.68 23.0 584.19 584.70 585.21 585.71 586.22.1 586.73 587.24 587.75 588.25 588.76.2 589.27 589.78 590.29 590.79 591.30.3 591.81 592.32 592.83 593.33 593.84.4 594.35 594.86 595.37 595.87 596.38.5 596.89 597.40 597.91 598.41 598.92.6 599.43 599.94 600.45 600.95 601.46.7 601.97 602.22 602.99 603.49 604.00.8 604.51 605.02 605.53 606.03 606.54.9 607.05 607.56 608.07 608.57 609.08 92 GEODESY. Table for the comparison of French and English Barometers. Hundredths of an inch. English inches 0 2 4 6 8 and tenths. Millimetres, 24.0 609.59 610.10 610.61 611.11 611.62.1 612.13 612.64 613.15 613.65 614.16.2 614.67 615.18 615.69 616.19 616.70.3 617.21 617.72 618.23 618.73 619.24.4 619.75 620.26 620.77 621.27 621.78.5 6222.29 6-22.80 623.31 623.81 624.32.6 624.83 625.34 625.85 626.34 626.86.7 627.37 627.88 628.39 628.89 629.40.8 629.91 630.42 630.93 631.43 631.94.9 632.45 632.96 633.47 633.97 634.48 25.0 634.99 635.50 636.01 636.51 637.02.1 637.53 638.04 638.55 639.05 639.56.2 640.07 640.58 641.09 641.59 642.10.3 642.61 643.12 643.63 644.13 644.64.4 645.15 645.66 646.17 646.67 647.18.5 647.69 648.20 648.71 649.21 649.72.6 650.23 650.74 651.25 65.1.75 652.26.7 652.77 653.28 653.79 654.29 654.80.8 655.3L 655.82 656.33 656.83 657.34.9 657.85 658.36 658.87 659.37 659.88 26.0 660.39 660.90 661.41 661.91 662.42.1 662.93 663.44 663.95 664.45 664.96.2 665.47 665.98 666.49 666.99 667.50.3 668.01 668.52 669.03 669.53 670.04.4 670.55 671.06 671.57 672.07 672.58.5 673.09.673.60 674.11 674.41 675.1 2.6 675.63 676.14 676.65 677.15 677.66.7 678.17 678.68 679.19 679.69 680.20.8 680.71 681.22 681.73 682.23 682.74.9 683.25 683.76 684.27 684.77 685.28 BAROMETRICAL MEASUREMENT OF HEIGHTS. 93 Table for the comparison of French and English Barometers. Hundredths of an inch. English inches and 0 2 4 6 8 tenths., a. Millimetres. 27.0 685.79 686.30 686.81 687.31 687.82.1 688.33 688.84 689.35 689.85 690.36.2 690.87 691.38 691.89 692.39 692.90.3 693.41 693.92 694.43 694.93 695.44.4 695.95 696.46 696.97 697.47 697.98.5 698.49 699.00 699.51 700.01 700.52.6 701.03 701.54 702.05 702.55 703.06.7 703.57 704.08 704.59 705.09 705.60.8 706.11 706.62 707.13 707.63 708.14.9 708.65 709.16 709.67 710.17 710.68 28.0 711.19 711.70 712.21 712.71 713.22.1 713.73 714.24 714.75 715.25 715.77.2 716.27 716.78 717.29 717.79 718.30.3 718.81 719.32 719.83 720.33 720.84.4 721.35 721.86 722.37 722.87 723.38.5 723.89 724.4U 724.91 725.41 725.92.6 726.43 726.94 727.45 727.95 728.46.7 728.97 729.48 729.99 730.49 731.00.8 731.51 732.02 732.53 733.03 733.54.9 734.05 734.56 735.07 735.57 736.08 29.0 736.59 737.10 737.61 738.11 738.62.1 739.13 739.64 740.15 740.65 740.16.2 741.67 742.18 742.69 743.19 743.70.3 744.21 744.72 745.23 745.73 746.24.4 746.75 747.26 747.77 748.27 748.78.5 749.29 749.80 750.31 750.81 751.32.6 751.83 752.34 752.85 753.35 753.86.7 754.37 754.88 755.39 755.89 756.40.8 756.91 757.42 757.93 758.43 758.94.9 759.45 759.96 760.47 760.97 761.48 94 GEODESY. Table for the comparison of French and English Barometers. Hundredths of an inch. English inchesnd O 2 4 6 8 tenths. Millimetres. 30.0 761.99 762.50 763.01 763.51 764.02.1 764.53 765.04 765.55 766.05 766.56.2 767.07 767.58 768.09 768.59 769.10.3 769.61 770.12 770.63 771.13 771.64.4 772.15 772.66 773.17 773.67 774.18.5 774.69 775.20 775.71 776.21 776.72.6 777.23 777.74 778.25 778.75 779.26.7 779.77 780.28 780.79 781.29 781.80.8 782.31 782.82 783.33 783.83 784.34.9 784.85 785.36 385.87 786.37 786.88 Table of corrections for capillary action to be added to English Barometers. Com. of Physics, &c. Royal Soc., 1840. Diameter of Ivory. Young. Laplace. tube. Unboiled Boiled tubes. tubes. Inches. Inches. Inches, Inches. Inches. Inches. 0.05 0.2949 0.2964 0..10.1404.1424.1394 0.142 0.070.15.0865.0880.0854.088.044.20.0583.0589.0580.060.029.25.0409.0404.0412.040.020.30.0293.0280.0296.028.014.35.0212.0196.0216.020.010.40.0154.0139.0159.014.007.45.0112.0100.0117.010.005.50.0082.0074.0087.007.003.60.0043.0645.0046 0.004 0.002.70.0023...........0024 0.80 0.0012.......... 0.0013 PROJECTION OF MAPS. 95 XXIII. Thermometrical.Measurement of Heights. Table of Barometric pressures corresponding to temperatures of boiling water. TENTHS OF A DEGREE OF FAHRENHEIT. Degrees of Fahrenheit. 0. 2. 4. 6. 8. 185 17.048 17.123 17.199 17.274 17.350 186.425.502.578.655.731 187.808.886.964 18.042 18.120 188 18.198 18.277 18.357.436.516 189.595.676.756.837.918 190.999 19.081 19.163 19.245 19.328 191 19.410 -493.577.661.744 192.828.913.998 20.083 20.169 193 20.254 20.341 20.427.514.601 194.688.776.864.952 21.041 195 21.129 21i.219 21.309 21.398.488 196.578.669.761.853.944 197 22.036 22.129 22.222 22.315 22.409 198.502.597.692.786.881 199.976 23.072 23.169 23.265 23.362 200 23.458.556.654.752.850 201.948 24.047 24.147 24.247 24.346 202 24.446.547.648,750 851 203.952 25.055 25.158 25.261 25.364 204 25.467.572.677.781.886 205.991 26.097 26.204 26.311 26.417 206 26.524.632.741.849.957 207 27.066 27.176 27.286 27.397 27.507 208.617.729.841.954 28.066 209 28.178 28.292 28.406 28.521.635 210.749.865.981 29.098 29.214 211 29.330 29.448 29.566.685.803 212.921 30.041 30.161 30.281 30.402 96 GEODESY. XXIV. Formulce for computing the elements for the projection of.Maps. 1. For surfaces of not more than four degrees of latitude and longitude, the formulce being approximate. 1. Normal -N - (1 - e2 sin' L)' 2. Tangent = T - N cot L 3. Radius of the parallel _ (Rp) = N cos L 4. Degree of the parallel -(Dp) - (p180) 180~ 5. Number of minutes of the parallel Dp =(n')p = (n)p 60 6. Angle between the tangent and the chord *sinZ (n')p = sin ~ - 2 T Then, difference of parallels = y p = (nl )p sin ~ z [(n' )p]2 2T difference of meridians x 8= m =( n' ) cos ~ z The values 6 m and 8 p will be found in the following tables. Example of their use. -Let it be required to make a projection containing 401 of longitude between the parallels of 41~ 301 and 42~ 10', to be subdivided to 51. Assume the centre 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'. PROJECTION OF MAPS. 97 Through A draw the central meridian and a line at right angles to it. Beginning at A, lay off above and below, on the central meridian, the values of D m from 41~ 501 to 41~ 55'; 41~ 551 to 42~; 420 to 42~ 5/, etc.; and from 41~ 50' to 410 45'; 41~ 45' to 410 40', etc. These values to be taken from the table of JMeridional arcs-values of Dm in yards, by interpolation from the values there given for the middie latitudes of 41~ and 42~. Through each of the points....A. At.A.A A\... thus found, lay off perpendiculars to the central meridian. Now turn to the table of Co-ordinates a m and a p in yards and lay off, from each of the points.... At.A.A.A\A\\... to the right and left of the central meridian, the values of 6 m for successively 5', 10/, 15! and 20', corresponding (by interpolation from the columns of 41~ 301 and 42~) to each parallel of latitude required; and, from the points thus found, the corresponding values of 8p at right angles to the lines already drawn. Lines passing through the extremities of ap will be the required meridians and parallels. The projection being made, any point whose latitude and longitude is known will be projected on the map from elements taken from the table of values of Dm and Dp, which are measured from the meridians and arallels, and not from the axes of co-ordinates used in making the projection. 2. When the map extends over several degrees of latitude and longitude, the preceding approximate formulae will not answer; a middle latitude is assumed where the developing cone is tangent, and the projection made as follows: Through the centre of the map, A, two lines are drawn, A y representing the central meridian, and the other, A x, 13 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 98 GEODESY. a line perpendicular to it; from the point A, along the line Ay (above and below A) the lengths s. sL s2 s3. s r (in miles or yards) of degrees or minutes, as the case may be, of the meridian are laid down, and the remaining intersections of meridians and parallels are projected by means of co-ordinates y and x from the central point A, as follows: - d = difference of meridians = p sin 0 y - dp = difference of parallels = s + x tang ~ 0 Rp 0 (n )2 TR p;- T =S where s= the length on the meridian from minute to minute or degree to degree as desired. T = N cot L - the tangent at the central point of the map, L being the latitude and N the normal at that point. (Rp) = N' cos 1 = the radius of the parallel at any point of latitude (above or below the central point) of which the ordinate is required. (nf )p = the number of minutes of the parallel of the new point of which the ordinate is required. 3. In maps of large portions of the Larth's surface deviations from real magnitudes may be lessened by making the developing cone cut two parallels equidistant from the middle parallel; say through one-third of the length of the middle meridian of the map. It will then be necessary to substitute for T in the preceding equation, the distance from the vertex of the cone to either intersection of the Earth's surface Rpn S being the angle at the vertex between the sill S elements of the cone and its axis; equal, in a spheroid, to one-half the sum of the geocentric latitudes of the two points of intersection. PROJECTION OF MAPS. 99 Co-ordinates, mn, op, in Yards. Long. Lat. 22~ 0' Lat. 9220 30' Lat. 230 0' value of Z. m Jp Sm cp p m 1/ 1882.0 0.1 1875.3 0.1 1868.5 0.1 2 3763.9 0.4 3750.6 0.4 3737.0 0.4 3 5645.9 0.9 5625.9 1.0 5605.4 1.0 4 7527.8 1.6 1 7501.2 1.7 7473.9 1.7 5 9409.9 2.6 1 9376.4 2.6 9342.4 2.7 6 11291.8 3.7 } 11251.7 3.7 11210.9 3.8 7 13173.7 5.0 13127.0 5.1 13079.4 5.2 8 15055.7 6.6 1 15002.3 6.7 14947.8 6.8 9 16937.6 8.3 16877.6 8.5 16816.3 8-.6 10 18819.6 10.3 l 18752.9 10.5 18684.8 10.6 11 20701.6 12.4 20628.2 12.6 20553.3 1-.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 26254.1 20.5 26158.7 20.8 15 28229.4 23.1 28129.3 23.5 28027.2 23.9 16 30111.4 26.2 30004.6 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 37.0 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 75011.5 167.0 74739.0 169.9 50 94097.7 256.3 93764.2 260.9 93423.7 265.5 10 00 112917.0 369.1 112516.9 375.8 112108.2 382.3 1 20 150555.4 656.2 150021.9 668.0 149476.9 679.6 1 30 169374.4 830.5 168774.2 845.4 168161.1 860.1 i 40 188193.3 1025.3 187526.3 1043.8 186845.1 1061.8 2 00 225830.5 1476.5 1225030.0 1503.0 224212.5 1529.1 2 30 282284.7 2307. 1 281284.0 2348.5 280261.9 2389.2 3 00 338736.6 3322.2 337535.6 3381.8 336309.0 3440.4 3 30 395186.0 4521.9 393784.5 4603.0 392353.1 4682.8 4 00 451632.0 5906.2 [l 450029.9 6012.1 448393.7 6116.2 100 GEODESY. Co-ordinates, mn, S p, in Yards. Long. Lat. 230 30' Lat. 240 0, Lat. 240 30' value of Z. Jm Sp Sm in p nm Sp 1' 1861.5 0.1 1854.4 0.1 1847.2 0.1 2 3723.1 0.4 3708.9 0.4 3694.4 0.4 3 5584.6. 1.0 5563.3 1.0 5541.6 1.0 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 8 14892.2 6.9 14835.5 7.0 14777.6 7.1 9 16753.7 8.7 16689.9 8.9 16624.8 9.0 10 18615.3 10.8 18544.3 11.0 18472.0 11.1 11 20476.8 13.1 20398.8 13.3 20319.2 13.5 12 22338.3 15.6 2-2253.-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 31.2 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 74177.1 175.5 73887.7 178.3 50 93076.0 269.9 92721.3 274.3 92359.5 278.5 10 00 111691.0 388.7 111265.3 394.9 110831.4 401.1 1 20 148920.6 690.9 148353.0 702.1 147774.1 713.0 1 30 167535.2 874.5 1668)6.6 888.6 166245.4 902.4 1 40 186149.7 1079.6 185440.1 1097.0 184716.5 1114.1 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 00 335056.8 3497.9 333779.1 3554.4 332476.1 3609.8 3 30 390892.0 4761.1 389401.1 4837.9 387880.6 4913.3 4 00 446723.4 6218.5 445019.2 6318.9 443281.1 6417.4 PROJECTION,OF MAPS. 101 Co-ordinates, 8 m, & p, in Yards. Long. Lat. 250 0' Lat. 25~ 30' Lat. 260 0' value of Z. mn P p dJnm Jp Jm cp 1' 1839.8 0.1 1832.3 0.1 1824.7 0.1 2 3679.6 0.5 3664.6 0.5 3649.3 0.5 3 5519.5 1.0 5496.9 1.0 5474.0 1.0 4 7359.3 1.8 7329.2 1.8 7298.6 1.9 5 9199.1 2.8 9161.5 2.9 9123.3 2.9 6 11038.9 4.1 10993.8 4.1 10947.9 4.2 7 12878.8 5.5 12826.1 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 11.3 18323.1 11.5 18246.5 11.6 11 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 22.8 15 27597.3 25.4 27484.6 25.8 27369.8 26.2 16 29437.1 29.0 29316.9 29.4 29194.4 29.8 17 31277.0 32.7 31149.2 33.2 31019.1 33.6 18 33116.8 36.6 32981.5 37.2 32843.7 37.7 19 34956.6 40.8 34813.8 41.4 34668.4 42.0 20 36796.4 45.2 36646.1 45.9 36493.0 46.5 25 45995.5 70.7 45807.6 71.7 45616.2 72.7 30 55194.6 101.8 54969.1 103.2 54739.5 104.7 40 73592.7 180.9 73292.0 183.6 72985.8 186.1 50 91990.7 282.7 91614.9 286.8 91232.1 290.8 10 00 110388.6 407.1 109937.6 413.0 109478.3 418.8 1 20 147184.0 723.8 146582.7 734.3 145970.3 744.6 1 30 165581.5 916.0 164905.0 929.3 164216.0 942.3 1 40 183978.8 1130.9 183227.1 1147.3 182461.5 1163.4 2 00 220772.7 1628.5 219870.6 1652.1 218951.9 1675.2 2 30 275961.6 2544.5 274833.9 2581.4 273685.3 2617.6 3 00 331147.8 3664.1 329794.3 3717.3 328415.8 3769.3 3 30 386330.6 4987.2 384751.3 5059.6 383142.7 5130.5 4 00 441509.4 6513.9 439704.0 6608.5 437865.3 6701.0 102 GEODESY. Co-ordinates, a m, 6 p, in Yards. Long. Lat. 260 30' Lat. 270 O Lat. 270 30' value ofZ j'm o p osn o2 p pm p 1' 1816.9 0.1 1808.9 0.1 1800.9 0.1 2 3633.7 0.5 3617.9 0.5 3601.7 0.5 3 5450.6 1.1 5426.8 1.1 5402.6 1.1 4 7267.4 1.9 7235.7 1.9,7203.4 1.9 5 9084.3 2.9 9044.6 3.0 9004.3 3.0 6 10901.2 4.2 10853.6 4.3 10805.1 4.4 7 12, 18.0 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 11.8 18089.3 11.9 18008.6 12.1 11 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 27133.9 26.9 27012.8 97.2 16 29069.8 30.2 28942.8 30.6 28813.7 30.9 17 30886.6 34.1 30751.7 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 72356.8 191.1 72034.0 193.5 50 90842.4 294.8 90445.8 298.6 90042.4 302.4 1~ 00 109010.7 424.5 108534.8 430.0 108050.6 435.4 1 20 145346.7 754.6 144712.1 764.4 144066.5 774.0 1 30 163514.5 955.1 162800.5 967.5 162074.2 979.6 1 40 181682.0 1179.1 180888.7 1194.4 180081.7 1209.4 2 00 218016.4 1697.9 217064.4 1720.0 216095.9 1741.6 2 30 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 3/9838.2 5267.5 378142.5 5333.6 4 00 435993.2 6791.5 434088.0 6880.0 432149.7 6966.3 PROJECTION OF MAPS. 103 Co-ordinates, 8 m, S p, in Yards. Long. Lat. 280 0 Lat. 280 30' Lat. 290~. value value Z. m _ _J_ m __ _...dp 1' 1792.7 0.1 1784.3 0.1 1775.8 0.1 2 3585.3 0.5 3568.6 0.5 3551.7 0.5 3 5378.0 1.1 5352.9 1.1 5327.5 1.1 4 7170.6- 2.0 7137.2 2.0 7103.3 2.0 5 8963.3 3.1 8921.5 3.1 8879.1 3.1 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 8.0 9 16133.9 9.9 16058.8 10.0 15982.5 10.1 10 17926.5 12.2 17843.1 12.4. 17758.3 12.5 11 19719.2 14.8 19627.4 15.0 19534.1 15.2 12 1 21511.8 17.6 21411.7 17.8 21309.9 18.0 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 32117.2 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 35516.6 50.1 25{ 44816.S2 76.5 44607.6 78.4 44395.7 78.3 30 i 53779.4 110.1 53529.1 111.4 53274.8 112.7 40 71705.8 195.8 71372.1 198.1 71032.9 200.3 501 89632.0 306.0 89214.9 309.6 88790.9 313.0 10 00 107558.2 440.7 107057.6 445.8 106548.8 450.8 1 20 1 143410.0 783.4 142742.5 792.5 142064.1 801.4 1 30 i 161335.6 991.5 160584.6 1003.0 159821.5 1014.3 1 40 179260.9 1224.2 178426.5 1238.3 177578.6 1252.2 2 00!215110.9 1762.6 214109.6 1783.2 213092.0 1803.1 2 30 268883.6 2754.1 267631.8 2786.2 266359.6 2817.4 3 00 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 00 ] 430178.5 7050.6 428174.6 7132.7 426138.2 7212.6 104 GEODESY. Co-ordinates, in, m p, in Yards. Long. Lat. 290 30' Lat. 30~ 0' Lat. 300 30' value ofZ..Jm.p Jm cp Jn.p 1' 1767.2 0.1 1758.5 0.1 1749.6 0.1 2 3534.4 0.5 3516.9 0.5 3499.2 0.5 3 5301.6 1.1 5275.4 1.2 5248.8 1.2 4 7068.9 2.0 7033.9 2.0 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 12247.1 6.3 8 14137.7 8.1 14067.8 8.2 13996.7 8.2 9 15904.9 10.3 15826.2 10.4 15746.3 10.5 10 17672.7 12.7 17584.7 12.8 17495.9 12.9 11 19439.4 15.3 19343.1 15.5 19245.4 15.6 12 21206.6 18.2 21101.6 18.4 20995.0 18.6 13 22973.8 21.4 22860.1 21.6 22744.6 21.8 14 24741.0 24.8 24618.5 25.1 23494.2 25.3 15 26508.2 28.5 26377.0 28.8 25243.8 29.1 16 28275.4 32.4 28135.5 32.7 26993.3 33.1 17 30042.6 36.6 29893.9 37.0 28742.9 37.4 18 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 51.6 35169.3 51.2 33991.7 51.7 25 44180.3 79.1 43961.6 79.9 43239.6 80.7 30 53016.4 113.9 52753.9 115.1 52487.4 116.3 40 70688.4 202.5 70338.4 204.6 69983.1 206.6 50 88360.2 316.4 87922.8 319.7 87478.7 322.9 1~ 00 106032.0 455.6 105507.1 460.4 104974.1 464.9 1 20 141375.0 810.0 140675.1 818.4 139964.4 826.6 1 30 159046.2 1025.2 158258.7 1035.8 157459.2 1046.1 1 40 176717.1 1265.7 175842.2 1278.8 174953.8 1291.5 2 00 212058.1 1822.6 211008.1 1841.5 209942.1 1859.8 2 30 265067.1 2847.8 263754.5 2877.3 262421.8 2905.9 3 00 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 00 424069.3 7290.3 421968.0 7365.9 419834.7 7439.1 PROJECTION OF MAPS. 105 Co-ordinates, 8 m, 8 p, in Yards. Long. Lat. 310 0'. Lat. 310 30' Lat. 320 0' value _ of Z. jmn' p p m p in p 1' 1740.6 0.1 1731.4 0.1 1 122.1 0.1 2 3481.1 0.5 3462.8 0.5 3444.3 0.5 3 5221.7 1.2 5194.3 1.2 5166.4 1.2 4 6962.3 2.1 6925.7 2.1 6888.6 2.1 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 13777.1 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 11 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 22.0 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 25971.3 29.6 25832. 1 29.9 16 27849.1 33.4 27702.7 33.7 27554.3 34.0 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 47.1 32897.0 47.5 32720.7 47.9 20 34811.4 52.2 34628.4 52.6 34442.8 53.1 25 43514.2 81.5 43286.0 82.3 43053.5 83.0 30 52217.0 117.3 51942.5 118.4 51664.1 119.5 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 1~ 00 104433.2 469.4 103884.3 473.7 103327.4 477.8 1 20 139243.1 834.5 138511.2 842.1 137768.8 849.5 1 30 156647.8 1056.1 155824.4 1065.8 154989.1 1075.1 1 40 174052.2 1303.8 173137.2 1315.8 172209.1 1327.3 2 00 208860.0 1877.5 207762.0 1894.7 206648.2 1911.3 2 30 261069.1 2933.7 2,59696.5 2960.5 258304.1'2986.5 3 00 313274.2 4224.5 311626.9 4263.1 309955.8 4300.5 3 30 365474.6 5750.0 363552.4 5802.6 361602.5 5853.5 4 00 417669.4 7510.2 415472.3 7578.9 413243.4 7645.3 14 106 GEODESY. Co-ordinates, m, 8 p, in Yards. Lnng. Lat 320 30' Lat. 330 0' Lat. 333 30' value of Z. cm' m Sp m p p J' p 1' 1712.7. 0.1 1703.2 0.1 1693.5 0.1 2 3425.5 0.5 3406.4 0.5.3387.0 0.5 3 5138.2 1.2 5109.6 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 10276.4 4.8 10219.1 4.9 10161.1 4.9 7 11989.1 6.6 11922.3 6.6 18154.6 6.7 8 13701.8 8.6 13625.5 8.6 13548.1 8.7 9 15414.6 10.8 15328.? 10.9 15241.7 11.0 10 17127.3 13.4 17031.9 13.5 16935.2 13.6 11 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 22.6 221 41.4 2.8 22015.7 23.0 14 23978.2 26.2 23844.6 26.4 23709.2 26.6 15 25690.9 30.1 25547.8 30.4 25402.7 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 48.5 32360.6 49.0 32176.8 49.1 20 31254.6 53.5 34063.8 54.0 33870.3 54.4 25 42818.2 83.6 42579.6 84. 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 10 00 102762.7 481.8 102190(.2 485.7 101609.9 489.4 1 20 137015.8 856.6 136252.4 86:.5 135478.6 870.1 1 30 154142.0 1084.1 153283.1 1092.8 152412.6 1101.2 1 40 171267.0 1338.4 170313.6 1349.2 169346.3 1359.5 2 00 205518.7 1927.4 204373.5 1942.8 203212.7 1957.7 2 30 2568932.0 3011.5 255460.4 303S.6 254009.2 3058.8 3 00 308261.1 4336.6 306542.9 4371.3 304801.4 4404.7 3 30 359625.1 5902.6 357620.3 5949.9 355588.2 5995.3 4 00 410983.2 7709.5 408691.6 7771.2 406368.8 7830.6 PROJECTION OF BIAPS. 107 Co-ordinates, 8 m 6 p, in Yards. Lon-g. Lat. 340 0' Lat. 34~0 30' Lat. 35~ 0' ot'Z. n m I p Jm p j it p 1' 1683.7 0.1 1673.8 0.1 1663.7 0.1 2 3367.4 0.5 3347.6 0.6 3327.5 0.6 3 5051.1 1.2 5021.4 1.2 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 10102.3 4.9 10042.7 5.0 9982.4 5.0 7 11786.0 i 6.7 11716.5 6.8 11616.1 6.8 8 13469.7 8.8 13390.3 8.8 13309.8 8.9 9 15153.4 11.1 15064.1 11.2 14973.6 11.2 10 16837.2 13.7 16737.9 13.8 16637.3 13.9 11 18520.9 16.6 18411.6 16.7 18301.0 16.8 12 20204.6 19.7 20085 4 19.9 19964.8 20.0 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 31.9 16 2693!.4 35.1 26780.6 35.3 26619.7 35.5 17 28623.1 39.6 28454.4 39.8 28283.4 49.1 18 30306.9 44.4 30128.1 44.7 29947.1 45.0 19 31990.6 49.4 31801.9 49.8 31610.9 50.1. 20 33674.3 54.8 33475.7 55.2 33274.6 55.5 2i5 42092.8 85.6 41844.6 86.2 41593.2 86.7 30 50511.4 123.2 50213.5 125.1 49911.8 124.9 40 67348.3 219.1 66951.1 220.6 65548.9 222.1 50 84185.1 342.3 83688.7 344.7 83185.8 347.0 10 00 101021.8 493.0 1 100426.0 496.4 99822.6 499.7 1 20 134694.5 876.4 1 133900.1 882.5 133095.5 883.3 1 30 151530.5 1109.2 150636.8 1116.9 149731.5 1124.2 1 40 168366.1 1369.4 167373.1 1378.9 166367.3 1387.9 2 00 202036.4 1971.9 200844.7 1985.6 199637.8 1998.6 2 30 252538.7 3081.1 251049.0 3102.5 249540.1 3122.8 3 00 303036.6 4436.8 301248.7 4467.5 29943'M.8 4496.9 3 30 3535929.0 6039.0 351442.8 6080.8 349329.8 6120.8 4 00 404015.1 7887.7 401630.5 7942.3 399215.4 7994.5 108 GEODESY. Co-ordinates, a m, 3p, in Yards. Long. Lat. 350 30' Lat. 36~ 0' Lat. 360 30' value of Z. Jm $p ln Ju p' in J1p 1' 1653.5 0.1 1643.2 0.1 1632.8 0.1 2 3307.1 0.6 3286.5 0.6 3265.6 0.6 3 4960.6 1.3 4929.7 1.3 4898.4 1.3 4 6614.2 2.2 6572.9 2.2 6531.2 2.3 5 8267.7 3.5 8216.2 3.5 8164.0 3.5 6 9921.3 5.0 9859.4 5.1 9796.8 5.1 7 11574.8 6.8 11502.7 6.9 11429.6 6.9 8 13228.4 8.9 13145.9 9.0 13062.4 9.0 9 14881.9 11.3 14789.1 11.4 14695.2 11.4 10 16535.5 14.0 16432.4 14.0 ]6328.0 14.1 11 18189.0 16.9 18075.6 17.0 17960.8 17.0 12 19842.5 20.1 19718.8 20.2 19593.6 20.3 13 21496.1 23.6 21362.0 23.7 21226.4 23.9 14 23149.6 27.4 23005.3 27.5 22859.2 27.7 15 24803.2 31.4 24648.5 31.6 24492.0 31.8 16 26456.7 35.7 26291.8 36.0 26124.8 36.2 17 28110.3 40.4 27935.0 40.6 27757.6 40.8 18 29763.8 45.2 29578.2 45.5 29390.4 45.8 19 31417.3 50.4 31221.2 50.7 31023.2 51.0 20 33070.9 55.9 32864.7 56.2 32656.0 56.5 25 41338.6 87.2 41080.8 87.8 40819.9 88.3 30 49606.2 125.7 49296.9 126.4 48983.9 127.1 40 66141.5 223.4 65729.1 224.8 65311.7 226.0 50 82676.6 349.1 82161.1 351.2 81639.3 353.1 1~ 00 99211.5 502.7 98592.9 507.7 97966.7 508.5 1 20 132280.7 893.8 131455.9 899.1 130621.0 904.1 1 30 148814.9 1131.2 147886.9 1137.9 146947.7 1144.2 1 40 165348.8 1396.6 164317.7 1404.8 163274.0 1412.6 2 00 198415.4 2011.1 197178.0 2022.9 195925.6 2034.1 2 30- 248012.1 3142.3 246465.3 3160.8 244899.6 3178.3 3 00 297604.0 4524.9 295747.6 4551.6 293868.6 4576.8 3 30 347190.2 6158.9 345024.1 6195.2 342831.7 6229.5 4 O0 396769.7 8044.3 394293.8 8091.6 391787.8 8136.5 PROJECTION OF MAPS. 109 Co-ordinates, 8 m, 8 p, in Yards. Long. Lat. 37~ 01 Lat. 370 301 Lat. 380~ 0' value of Z. Sm J' p n m Jp J'mn J p 1' 1622.2 -0.1 1611.6 0.1 1600.7 0.1 2 3244.5 0.6 3223.1 0.6 3201.5 0.6 3 4866.7 1.3 4834.7 1.3 4802.2 1.3 4 6489.0 2.3 6446.2 2.3 6403.0 2.3 5 8111.2 3.5 8C57.8 3.6 8003.7 3.6 6 9733.4 5.1 9669.3 5.1 9604.5 5.2 7 11355.7 7.0 11280.9 7.0 11205.2 7.0 8 12977.9 9.1 12892.4 9.1 12806.0 9.2 9 14600.2 11.5 14504.0 11.6 14406.7 11.6 10 16222.4 14.2 16115.6 14.3 16007.5 14.3 11 17844.6 17.2 17727.1 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 29008.0 46.2 28813.4 46.4 19 30822.6 51.3 30619.5 51.5 30414.2 51.7 20 32444.8 56.8 32231.1 57.1 32015.0 57.3 25 40555.9 88.7 40288.8 89.1 40018.6 89.6 30 48667.1 1'27.8 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 10 00 97333.1 511.2 96692.0 513.7 96043.6 516.0 1 20 129776.1 908.8 128921.3 913.2 128056.7 917.4 1 30 145997.2 1150.2 145035.5 1155./8 144062.6 1161.0 1 40 162217.9 1419.9 161149.4 1426.9 160068.5 1433.4 2 00 194658.2 2044.7 193375.9 2054.7 192078.9 2064.1 2 30 243315.2 3194.9 241712.2 3210.5 240090.8 3225.1 3 00 291967.1 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 00 389251.9 8178..9 386686.3 821-8.8 384091.9 8256.3 110 GEODESY. Co-ordinates, a m, a p, in Yards. Long. Lat. 38~ 30' Lat. 390 0' Lat. 390 30I value of Z. Jn p jn Jp mm m p 1' 1589.8 0.1 1578.8 0.1 1567.6 0.1 2 3179.6 0.6 3157.5 0.6 3135.2 0.6 3 4769.5 1.3 4736.3 1.3 4702.8 1.3 4 6359.3 2.3 6315.1 2.3 6270.4 2.3 5'949.1 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 11051.4 7.1 10973.2 7.1 8 12718.6 9.2 12630.1 9.2 12540.8 9.3 9 14308.4 11.7 14208.9 11.7 14108.4 11.7 10 15898.2 14.4 15787.7 14.5 15676.0 14.5 11 17488.0 17.4 17366.4 17.5 17243.6 17.5 12 19077.8 20.7 18945.2 20.8 18811.1 20.9 13 W0667.6 24.3 20524.0 24.4 20378.7 24.5 14 22257.4 28.2 22102.7 28.3 219416.3 28.4 15 23847.3 32.4 23681.5 32.5 23513.9 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 31351.9 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 10 00 95387.8 518.2 94724.7 520.2 94054.4 522.1 1 20 127182.3 921.2 126298.1 924.8 125404.3 928.2 1 30 143079.0 1165.9 142084.4 1170.5 141078.8 1174.7 1 40 158975.5 1439.4 157870.3 1445.1 156753.0 1450.2 2 00 190767.1 2072.8 189440.8 2080.9 188100.1 *2088.3 2 30 238451.0 3238.7 236793.0 3251.4 235116.9 3263.0 3 00 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 00 381466.7 8291.2 378813.1 8323.6 376130.5 8353.4 PROJECTION OF MAPS. 111 Co-ordinates, a m, 8 p, in Yards. Long. Lat. 40~0 O Lat. 400 30' Lat. 410 0' value of Z. j'm cp jm n p dnm 1' 1556.3 0.1 1544.9 0.1 1533.4 0.1 2 3112.6 0.6 3089.8 0.6 3066.7 0.6 3 4668.9 1.3 4634.7 1.3 4600.1 1.3 4 6225.2 2.3 i 6179.6 2.3 6133.5 2.3 5 7781.5 3.6 7724.5 3.6 7666.8 3.7 6 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 11.8 13904.0 11.8 13800.3 11.9 10 15563.0 14.6 15448.9 14.6 15333.7 14.6 11 17119.3 17.6 16993.8 17.7 16867.0 17.7 12 18675.6 21.0 18538.7 21.0 18400.4 21.4 13 20231.9 24.6 i 0083.6 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 46346.7 131.3 46000.S 131.7 40 62251.8 232.8 61795.3 233.5 61334.2 234.1 50 77814.4 363.7 77243.9 364.8 76667.4 3C5.8 10 00 93376.9 523.8 92692.2 525.3 92000.4 526.7 1 20 124500.9 931.2 123588.0 933.9 122665.7 936.4 1 30 140062.5 1178.5 139035.5 1182.0 137997.8 1185.1 1 40 155623.7 1455.0 154482.6 1459.3 153329.6 1463.1 2 00 186744.9 2095.2 185375.4 2)101.4 183991.8!2106.9 2 30 233422.9 3273.7 231710.9 3283.4 229998.3 3292.0 3 00 280095.3 4714.1 278040.9 4728.1 275965.1 4740.6 3 30 326761.1 6416.5 324364.1 6435.4 321942.2 16452.4 4 00 373419.3 8380.7 370679.4 8405.5 367911.3 8427.7 112 GEODESY. Co-ordinates, 6 m, 8 P, in Yards. Long. Lat. 410 301 Lat. 420 01 Lat. 420 30' value of Z. JAn Jp Jrn dp A'm p 1 1521.7 0.1 1510.0 0.1 1498.1 0.1 2 3043.4 0.6 3019.9 0.6 2996.2 0.6 3 4565.2 1.3 4529.9 1.3 4494.2 1.3 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 9130.3 5.3 9059.7 5.3 8988.6 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 11.9 13482.7 11.9 10 15217.2 14.7 15099.6 14.7 14980.8 14.7 11 16738.9 17.7 16609.5 17.8 16478.8 17.8 12 18260.6 21.1 18119.5 21.2 17976.9 21.2 13 19782.3 24.8 19629.4 24.8 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 2891'2.6 52.9 28689.1 53.0 28463.4 53.1 20 30434.3 58.6 30199.1 58.8 29961.5 58.9 25 38042.9 91.6 37748.8 91.8 37451.3 92.0 30 45651.4 132.0 45298.5 132.3 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 10 00 91301.6 527.9 90595.9 529.0 89883.2 529.9 1 20 1 121733.9 938.6 120792.9 940.5 119842.6 942.1 1 30 136949.6 1187.9 135890.9 1190.3 134821.8 1192.3 1 40 152164.9 1466.5 15C988.5 1469.5 149800.6 1472.0 2 00 182594.1 2111.8 181182.5 2116.1 179756.9 2119.7 2 30 228234.1 3299.7 226469.4 3306.4 224687.4 3312.0 3 00 273868.3 4751.6 271750.5 4761.2 269612.0 4769.3 3 30 319495.6 6467.5 317024.7 6480.5 314529.5 6491.6 4 00 365115.0 8447.3 362290.8 8464.4 359438.9 8478.8 PROJECTION OF MAPS. 113 Co-ordinates, a m, a p, in Yards. Long. Lat. 430 0' Lat. 430 30' Lat. 440 0' value ofZ. i p di mp fp Sm p 1' 1486.1 0.1 1474.0 0.1 1461.8 0.1 2 2972.2 0.6 2948.0 0.6 2923.5 0.6 3 4458.3 1.3 4421.9 1.3 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 13155.8 12.0 10 14860.8 14.7 14739.8 14.8 14617.6 14.8 111 16346.9 17.8 16213.7[ 17.8 16079.3 17.9 12 17833.0 21.2 17687.7 21.2 17541.1 21.3 13 19319.1 24.9 19161.7 24.9 19002.8 25.0 14 20805.2 28.9 20635.7 28.9 2C464.6 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 59.0 29479.5 59.0 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 59443.0 235.8 58958.7 236.1 58469.9 236.3 50 74303.4 368.5 73698.0 368.9 73087.0 369. 2 1~ 00 89163.6 530.7 88437.1 531.2 87703.9 531.7 1 20 118883.1 943.4 117914.5 944.4 116936.9 945.2 1 30 133742.4 1194.0 132652.7 1195.3 131552.9 1196.3 1 40 148601.2 1474.1 147390.5 1475.7 146168.4 1476.9 2 00 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 00 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 00 356559.5 8490.7 353652.8 8500.1 350719.0 8506.8 15 114 GEODESY. Co-ordinates, 8 m, ap, in Yards. Long. Lat. 440 30' Lat. 450 0' Lat. 450 30' value of Z. J m a p J m r p Jm Sp 1' 1449.4 0.1 1437.0 0.1 1424.4 0.1 2 2898.9 0.6 2874.0 0.6 2818.9 0.6 3 4348.3 1.3 4311.0 1.3 4273.3 1.3 4 5797.7 2.4 5747.9 2.4 5697.7 2.4 5 7247.1 3.7 7184.9 3.7 7122.2 3.7 6 8696.6 5.3 8621.9 5.3 8546.6 5.3 7 10146.0 7.2 10058.9 7.2 9971.0 7.2 8 11595.4 9.5 11495.9 9.5 11395.4 9.5 9 13044.8 12.0 12932.9 12.0 12819.9 12.0 10 14494.3 14.8 14369.8 14.8 14244.3 14.8 11 15943.7 17.9 15806.8 17.9 15668.7 17.9 12 17393.1 21..3 17243.8 21.3 17093.1 21.3 13 18812.5 25.0 18680.8 25.0 I 18517.6 25.0 14 20292.0 29.0 20117.7 29.0 1 9942.0 29.0 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 43109.3 133.0 42732.7 133.0 4011 57976.6 236.4 57478.9 236.5 56976.8 236.4 50 72470.4 369.4 71848.3 369.5 71220.6 369.4 10 00 86964.0 531.9 86217.4 5320 85464.2 532.0 1 20 115950.3 945.7 114954.9 945.8 113950.6 945.7 1 30 130442.9 1196.8 129323.0 1197.1 128193.2 1196.9 1 40 144935.2 1477.6 143690.8 1477.9 142435.5 1477.7 2 00 173918.3 2127.7 172425.0 2128.1 170918.5 2127.8 2 30 217388.7 3324.6 215522.0 3325.2 213638.8 3324.8 3 00 260853.0 4787.4 258612.9 4788.3 256352.9 4787.7 3 30 304310.0 6516.2 301696.3 6517.3 299059.6 6516,.5 4 00 347758.4 8510.9 344771.2 8512.5 341757.6 8511.4 PROJECTION OF MfAPS. 115 Co-ordinates, a m, p, in Yards. Long. | Lat. 460 O't Lat. 46~ 30' |l Lat. 470 0' value of Z. J pm m Jp inm,p 1' 1411.8 0.1 1399.0 0.1 1386.1 0.1 2 2823.5 0.6 2798.0 0.6 1 2772.2 0.6 3 4235.3 1.3 4197.0 1.3 1 4158.4 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 98S2.4 7.2 9793.0 7.2 9702.8 7.2 8 11294.1 9.5 11192.0 9.4 1 1089.0 9.4 9 12705.9 12.0 12591.0 12.0! 12475.1 11.9 10 14117.7 14.8 13990.0 14.8 13861.2 14.7 11 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 1 18353.0 25.0 18186.9 24.9 i 18019.5 24.9 14 19764.7 28.9 19585.9 28.9 19405.7 28.9 15 21176.5 33.2 20984.9 33.2 20791.8 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 25411.8 47.9 25181.9 47.8 24950.1 47.8 19 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 10 00 84704.5 531.7 83938.2 531.3 83165.6 530.8 1 20 112937.6 945.3 111915.9 944.6 110885.7 943.6 1 30 127053.6 1196.4 125904.2 1195.5 124747.2 1194.3 1 40 141169.2 1477.0 139892.1 1476.0 138604.3 1474.4 2 00 169399.0 2126.9 167866.4 2125.4 166321.0 2123.2 2 30 211739.3 3323.3 209823.5 3320.9 207891.7 3317.5 3 00 254073.4 4785.6 251774.4 4782.1 249456.0 4777.2 3 30 296400.0 6513.8 293717.6 6509.0 291012.8 6502.2 4 00 338717.8 8507.8 335652.1 8501.5 332560.6 8492.7 116 GEODESY. Co-ordinates, a m, 8 p, in Yards, Long. Lat. 47~0 30' Lat. 48~ 01 Lat. 48~0 30' value of Z. Jm tp J m'p cm J' p 1' 1373.1 0.1 1360.0 0.1 1346.9 0.1 2 2746.3 0.6 2720.1 0.6 2693.7 0.6 3 4119.4 1.3 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 8081.1 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 12358.2 11.9 12240.4 11.9 12121.7 11.9 10 13731.4 14.7 13600.5 14.7 13468.5 14.7 11 15104.5 17.8 14960.5 17.8 14815.4 17.8 12 16477.6 21.2 16320.5 21.2 16162.2 21.1 13 17850.8 24.9 17680.6 24.8 17509.1 24.8 14 19223.9 28.9 19040.6 28.8 18855.9 28.8 15 20597.0 33.1 20400.7 33.1 20202.8 33.0 16 21970.1 37.7 21760.7 37.6 21549.6 37.6 17 23343.3 42.6 23120.7 42.5 22896.5 42.4 18 21716.4 47.7 24480.8 47.6 24243.3 47.5 19 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 68001.4 367.5 67341.7 366.8 10 00 82386.5 530.1 81601.1 529.2 80809.5 528.2 1 20 109846.9 942.4 108799.7 940.8 107744.2 939.0 1 30 123576.6 1192.7 122398.5 1190.7 121211.0 1188.4 1 40 137305.6 1472.4 135996.8 1470.0 134677.3 1467.1 2 00 164762.8 2120.3 163191.9 2116.8 161608.6 [2112.7 2 30 205943.9 3313.0 203980.3 3307.5 202001.0 3301.1 3 00 247118.6 4770.7 244762.2 4762.9 242387.0 4753.5 3 30 288285.6 6493.5 285536.3 6482.8 282765.2 6470.1 4 00 329443.7 8481.3 326301.5 8467.3 1 323134.3 8450.7 PROJECTION OF MAPS. 117 Co-ordinates, a&m, 6p, in Yards. Long. Lat. 490 0' Lat. 490 301 Lat. 500 Of value of Z. Jn m p m Up Jm $p 1' 1333.6 0.1 1320.2 0.1 1306.7 0.1 2 2667.1 0.6 2640.3 0.6 2213.3 0.6 3 4000.7 1.3 2960.5 1.3 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 12002.0 11.9 11881.4 11.8 11759.9 11.8 10 13335.5 14.6 13201.6 14.6 13066.5 14.6 11 14669.1 17.7 14521.7 17.7 14373.2 17.6 12 16002.7 21.1 15841.8 21.0 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 15 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 23519.7 47.2 19 25337.5 52.8 25082.9 52.7 24826.4 52.6 20 26671.1 58.6 26403.1 58.4 26133.0 58.2 25 33338.8 91.5 33003.8 91.2- 32666.2 91.0 30 40006.5 131.7 39604.5 131.4 39199.4 131.0 40 53341.7 234.2 52805.7 233.6 52265.7 232.9 50 66676.8 365.9 66006.8 365.0 65331.7 364.0 10 00 80011.6 527.0 79207.6 525.6 78397.5 524.1 1 20 106680.4 936.8 105608.3 934.4 104528.2 931.7 1 30 120014.2 1185.7 118808.2 1182.6 117593.0 1179.2 1 40 133347.5 1463.8 132007.5 1460.0 130657.4 1455.8 2 00 160012.8 2107.9 158404.8 2102.5 156784.7 2096.4 2 30 200006.3 3293.6 197996.2 3285.1 195970.8 3275.6 3 00 239993.2 4742.8 237581.0 4730.5 235150.6 4716.9 3 30 279972.3 6455.4 277158.0 6438.8 274322.4 6420.2 4 00 319942.3 4831.,6 316725.8 8409.9 313485.0 8385.6 118 GEODESY..frcs of the Parallel — Values of Dp, in Yards. L. 200 30' L. 210 0' L. 210 30' L. 22~ 0' L. 220 30' L. 230 0O 7 221.8 221.1 220.3 219.6 218.8 218.0 8 253.5 252.6 251.8 250.9 250.0 249.1 9 28j.2 28 4.2 283.3 282.3 281.3 280.3 10 316.9 315.8 314.7 313.7 312.5 311.4 20 633.7 631.6 629.5 627.3 625.1 622.8 30 95().6 947.4 944.2 941.0 937.6 934.2 40 1267.4 1263.2 1259.0 1254. 6 1250.2 1245.7 50 1584.3 1579. 1 1573.7 156.3 1562.7 1557.1 60 1901.1 1894.9 1888.5 1881.9 1875.3 1668.5 7 - 13307.8 13264.1 113219.4 13173.7 13127. 0 13079.4 8 - 15208.9 15159.0 15107.9 15055. 7 15002.3 14947.9 9 - 17 10.0 17053.8 16996.4 16937.6 16877.6 16816.3 10 _- I 19011.1 189487 18884. 9 18819.6 18752.9 18684 8 20 - 380222.1 35897.4 37769.7 37639.2 37505.8 37369.6 30 - 57033.2 56846.1 56654.6 56458. 8 36258.8 56054.4 40 - 76044.3 75794.8 75539.5 75278.4 75011.7 74739.3 50 - 95055.4 94743.4 94424.3 94098.0 93764.6 93424.1 60 - 114066.4 113692.1 113309.2 112917.7 112517.6 112108.9 llieridional./rcs- Values of D m, in Yards. L. 210 0. 22~ 0' L. 23~ 0' 7 235.4 235.4 235.5 8 269.0 269.1 269.1 9 302.7 302.7 302.8 10 336.3 336.4 336.4 20 6 72. 6 672.7 672.8 30 1008.9 1009.1 1009.2 40 1345.2 1345.4 1345.6 50 1681.6 1681.8 1682.0 60 2017.9 2018.1 2018.4 7 - 14125.0 14126.7 14128.5 8 - 16142.9 16144.8 16146.8 9 - 18160.8 18162.9 18165.2 10 - 201'8.6 20181.0 20183.5 20 - 40357.2 40362.1 40367.1 30 - 60535.9 60543.1 60550.6 40 - 80714.5 80724.1 80734.1 50 - 100893.1 100905.2 100917.6 60 - 121071.7 121086.2 121101.2 Intermediate minutes and seconds will be found'by moving the decimal point. PROJECTION OF MAPS. 119 Jdrcs of the Parallel- Values of D p, in Yards. L. 230 30' L. 240 0' L. 240 30' L. 250 0' L. 250 30' L. 260 0' 7 217.2 21f6.4 215.4 214.6 213.8 212.9 8 248.2 247.3 246.3 245.3 244.3 243.3 9 279.2 278.2 277.1 276.0 274.8 273.7 10 310.3 309.1 307.9 306.6 305.4 304.1 20 620.5 618.1 615.7 613.3 610.8 608.2 30 930.8 927.6 923.6 919.9 916.2 912.3 40 1241.0 1236.3 1231.5 1226.5 1221.5 1216.4 50 1551.3 1545.4 1539.3 1533.2 1526.9 1520.5 60 1861.5 1854.4 1847.2 1839.8 1832.3 1824.7 7 - 13030.7 12981.0 12930.4 12878.8 12826.1 12772.6 8 - 14892.2 14835.5 14777.6 14718.6 14658.5 14597.2 9 - 16753.8 16689.9 16624.8 16558.4 16490.8 16421.9 10 - 18615.3 18544.3 18472.0 1839!8.2 18323. 1 18246.5 20 - 37230.6 37G88.7 36944.0 36796.5 36646.1 36493.0 30 - 55845.8 55633.0 55416.0 55194.7 54969.2 54739.6 40 - 74461.1 74177.4 73887.9 73592.9 73292.3 72986.1 a0 - 93076.4 92721.7 9?359.9 91991.1 91615.3 91232.6 60 - 111691.7 111266.0 110831.9 110389.4 109938.4 1109479.1.Meridional.frcs- Values of D m, in Yards. L. 240 0' L. 250 0' L. 260 0' 7 235.5 235.5 235.6 8 269.1 269.2 269.2 9 302.8 30'2.8 3(12.9 10 336.4 336.5 336.5 o0 672.9 673.0 673.1 30 1009.3 1009.4 1009.6 40 1345.7 1345.9 1346.1 50- 1682.2 1682.4 1682.6 60 2018.6 2018.9 2019.2 7 - 14130.3 14132.1 14134.1 8 - 16148.9 16151.0 16153.2 9 - 18167.5 18169.9 18172.4 10 - 20186.1 20188.8 20191.5 20 - 40372.2 40377.5 4(0382.0 30 - 60558.3 60566.3 60574.6 40 - 80744.4 80755.1 80766.1 5() - 1]00930.5 100943.9 100957.6 60 - 121116.6 121132.6 121149.1 Intermediate minutes and seconds will be found by moving the decimal point. 120 GEODESY..qrcs of the Parallel —Values of D p, in Yards. L. 260 30' L. 270 0' L. 270 30' L. 28 0 L. 280 30' L. 290 0' 7 212.0 211.0 210.1 209.1 208.2 207.2 8 242.2 241.2 240.1 239.0 237.9 236.8 9 272.5 271.3 270.1 268.9 267.6 266.4 10 302.8 301.5 300.1 298.8 297.4 296.0 20 605.6 603.0 600.3 597.6 594.8 591.9 30 908.4 904.5 900.4 896.3 892.2 887.9 40 1211.2 1206.0 1200.6 1195.1 1189.5 1183.9 50 1514.0 1507.4 1500.7 1493.9 1486.9 1479.9 60 1816.9 1808.9 1800.9 1792.7 1784.3 1775.8 7 - 12718.0 12662.5 12606.0 12548.6 12490.1 12430.8 8 - 14534.9 14471.4 14406.9 14341.2 14274.5 14206.6 9 - 16351.7 16280.3 16207.7 16133.9 16058.8 15982.5 10 - 18168.6 18089. 3 18008.6 17926.5 17843.1 17758.3 20 - 36337.2 36178.5 36017.1 35853.0 35686.2 35516.6 30 - 54505.8 54267.8 54025.7 53779.5 53529.3 53274.9 40 - 72674.4 72357.1 72034.3 71706.1 71372.4 71033.2 50 - 9)843.0 90446.3 90042.9 89632.6 89215.4 88791.5 60 - 109011.5 108535.6 108051.4 107559.1 107058.5 106549.8 Jlleridional d.rcs- Values of D m, in Yards. L. 27~ 0' L. 28~ 0' L. 29~ 0' 7 235.6 235.6 235.7 8 269.3 269.3 269.3 9 302.9 303.0 303.0 10 336.6 336.6 336.7 20 673.1 673.2 673.3 30 1009.7 1009.9 1010.0 40 1346.3 1346.5 1346.7 50 1662.9 1683.1 1683.3 60 2019.4 2019.7 2020.0 7 - 14136.0 14138.1 14140.1 8 - 16155.5 16157.8 16160.2 9 - 18174.9 18177.5 18180.2 10 - 20194.3 20197.2 20200.2 20 - 40388.7 40394.5 40400.4 30 60583.0 60591.7 60600.6 40 _ 80777.4 80788.9 80800.8 50 100971.7 100986.2 101001.0 60 121166.0 121183.4 121201.2 Intermediate minutes and seconds will be found by moving the decimal point. PROJECTION OF MAPS. 121,drcs of the Parallel- Values of D p, in Yards. L. 290 30r L. 300 0' L. 30~ 30' L. 310 0' L. 310 30' L. 320 0' 7 206.2 205.2 204.1 203.1 202.0 200.9 8 235.6 234.5 233.3 232.1 230.9 229.6 9 265.1 263.8 262.4 261.1 259.7 258.3 10 294.5 293.1 291.6 290.1 288.6 287.0 20 589.1 586.2 583.2 580. 577.1 574.0 30 883.6 879.2 874.8 870.3 865.7 861.1 40 1178.1 1172.3 1166.4 1160.4 1154.3 1148.1.50 1472.7 1465.4 1458.0 1450.5 1442.9 1435.1 60 1767.2 1758.5 1749.6 1740.6 1731.4 1722.1 7 - 12370.5 12309.3 12247.1 12184.0 12120.0 12055.0 8 - 14137.7 14067.7 13996.7 13924.6 13851.4 13777.1 9 - 15904.9 15826.2 15746.3 15665.1 15582.8 15499.3 10 - 17672.2 17584.7 17495.9 17405.7 17314.2 17221.4 20 _ 35344.3 35169.4 34991.7 34811.4 34628.4 34442.9 30 - 53016.5 52754.0 52487.6 52217.1 51942.7 51664.3 40 _ 70688.7 70338.7 69983.4 69622.8 69256.9 68885.7 50 _ 88360.8 87923.4 87479.3 87028.5 86571.1 86107.1 60 _ 106033.0 105508.1 104975.2 104434.2 103885.3 103328.6.Meridional drcs- Values of D m, in Yards. L. 300 0' L. 310 0' L. 323 0' 7 235.7 235.7 235.8 8 269.4 269.4 269.5 9 303.0 303.1 303.1 10 336.7 336.8 336.8 20 673.4 673.5 673.6 30 1010.2 1010.3 1010.5 40 1346.9 1347.1 1347.3 50 1683.6 1683.9 1684.1 60 2020.3 2020.6 2020.9 7 - 14142.3 14144.4 14146.6 8 - 16162.6 16165.1 16167.6 9 - 18182.9 18185.7 18188.5 10 - 20203.2 20206.3 20209.5 20 - 40406.5 40412.6 40418.9 30 - 60609.7 60619.0 60628.4 40 - 80812.9 80825.3 80837.9 50 - 101016.1 101031.6 101047.4 60 - 121219.4 121237.9 121256.8 Intermediate minutes and seconds will be found by moving the decimal point. 16 122 GEODESY. JTrcs of the parallel- Values of D p, in Yards. L. 320 301 L. 330 0" L. 330 30'1 L 340 0' L. 340 30'1 L.350 0' 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 20 570.9 567.7 564.5 561.2 557.9 554.6 30 856.4 851.6 846.8 841.9 836.9 831.9 40 1141.8 1135.5 1129.0 1122.5 1115.9 1109.2 50 1427.3 1419.3 1411.3 1403.1 1394.8 1386.4 60 1712.7 1703.2 1693.5 1683.7 1673.8 1663.7 7 - 11989.1 11922.3 11854.6 11786.0 11716.5 11646.1 8 - 13701.9 13625.5 13548.1 13469.7 13390.3 13309.8 9 - 15414.6 15328.7 15241.7 15153.5 15064.1 14973.6 10 - 17127.3 17031.9 16935.2 16837.2 16737.9 16637.3 20 - 34254.6 34063.8 33870.4 33674.3 33475.8 33274.6 30 - 51381.9 51095.7 50805.5 50511.5 50213.6 49911.9 40 - 68509.3 68127.6 67740.7 67348.7 66951.5 66549.2 50 - 85636.6 85159.5 84675.9 84185.8 83689.4 83186.5 60 - 102763.9 102191.4 101611.1 101023.0 100427.3 99823.8 Jferidional.drcs- Values of D m, in Yards. L. 330 0' L. 340 01 L. 350 0' 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 20 673.8 673.9 674.0 30 1010.6 1010.8 1011.0 40 1347.5 1347.7 1347.9 50 1684.4 1684.7 1684.9 60 2021.3 2021.6 2021.9 7 - 14148.9 14151.2 14153.5 8 - 16170.1 16172.7 16175.4 9 - 18191.4 18194.3 18197.3 10 - 20212.7 20215.9 20219.2 20 - 40425.4 40431.9 40438.5 30 - 60638.0 60647.8 60657.7 40 - 80850.7 80863.7 80877.0 50 - 101063.4 101079.7 101096.2 60 - 121276.1 121295.6 121315.4 Intermediate minntes and seconds will be found by moving the decimal point. PROJECT1GN OF MAPS. 123.arcs of the Parallel- Values of D p, in Yards. L. 350 30'1 L. 360 0 L.360 30' L. 370 01. 370 30' L. 38 0 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 30 826.8 821.6 816.4 811.1 805.8 800.4 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 1632.9 1622.2 1611.6 1600.7 7 - 11574.8 11502.7 11429.6 11355.7 11280.9 11205.2 8 - 13228.4 13145.9 13062.4 12977.9 12892.5 12806.0 9 - 14881.9 14789.1 14695.2 14600.2 14504.0 14406.7 10 - 16535.5 16432.4 16328.0 16222.4 16115.6 16007.5 20 - 33070.9 32864.7 32656.0 32444.8 32231.1 32015.0 30 - 49606.4 49297.1 48984.0 48667.2 48346.7 48022.5 40 - 66141.9 65729.5 65312.1 64889.6 64462.3 64030.0 50 - 82677.3 82161.8 81640.1 81112.0 80577.8 80037.5 60 - 99212.8 98594.2 97968.1 97334.4 96693.4 96045. JMeridional Arcs- Values of D m, in Yards. Log 36~ 0' L. 370 Ot L. 380 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 1011.1 1011.3 1011.5 40 1348.2 1348.4 1348.6 50 1685.2 1685.5 1685.8 60 2022.3 2022.6 2022.9 7 - 14155.8 14158.2 14160.6 8 - 16178.1 16180.8 16183.5 9 - 18200.3 18203.4 18206.5 10 - 20222.6 20226.0 20229.4 20 - 40445.2 40451.9 40458.8 30 - 60667.5 60677.9 60688.2 40 - 80890.3 80903.9 80917.6 50 - 101112.9 101129.9 101147.0 60 - 121335.5 121355.8 121376.4 Intermediate minutes and seconds will be found by moving the decimal point. 124 GEODESY. ~.rcs of the Parallel- Values of D p, in Yards. L. 380 30' L. 390~ 0 L. 390~ 30' L. 40~01 L. 40~ 30 L. 41~ 00 7 185.5 184.2 182.9 181.6 180.2 178.9 8 212.0 210.5 209.0 207.5 206.0 204.4 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 30 794.9 789.4 783.8 778.2 772.4 766.7 40 1059.9 1052.5 1045.1 1037.5 1029.9 1022.2 50 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 - 11128.7 11051.4 10973.2 10894.1 10814.3 10733.6 8 - 12718.6 12630.2 12540.8 12450.4 12359.2 12266.9 9. 14308.4 14208.9 14108.4 14006.8 13904.1 13800.3 10 - 15898.2 15787.7 15676.0 15563.1 15448.9 15333.4 20 - 31796.4 31575.4 31351.9 31126.1 30897.9 30667.3 30 - 47694.6 47363.1 47027.9 46689.2 46346.8 46001.0 40 - 63592.6 63150.8 62703.9 62252.2 61795.8 61334.6 50 - 79491.0 78938.4 78379.9 77815.3 77244.7 76668.3 60 - 95389.2 94726.1 94055.8 93378.3 92693.7 92001.9 MJeridional./rcs- Values of D m, in Yards. L. 39~ 0' L. 40~ 0' L. 400 0' 7 236.0 236.1 236.1 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 1011.6 1011.8 1012.0 40 1348.9 1349.1 1349.3 50 1686.1 1686.4 1686.7 60 2023.3 2023.6 2024.0 7 - 14163.0 14165.4 14167.9 8 - 16186.3 16189.1 16191.9 9 - 18209.6 18212.7 18215.8 10 - 20232.8 20236.3 20239.8 20 - 40465.7 40472.7 40479.7 30 - 60698.5 60709.0 60719.5 40 - 80931.4 80945.3 80959.3 50 - 101164.2 101181.6 101199.2 60 - 121397.1 121418.0 121439.0 Intermediate minutes and seconds will be found by moving the decimal point. PROJECTION OF MAPS. 125.lrcs of the Parallel- Values of D p, in Yards. L. 410 30' L. 420 0' L. 420 30' L. 430 0' L. 436 30' L. 440 0' 7 177.5 176.2 174.8 173.4 172.0 170.5 8 202.9 201.3 199.7 198.1 196.5 194.9 9 228.,3 226.5 224.7 222.9 221.1 219.3 10 253.6 251.7 249.7 247.7 245.7 243.6 20 507.2 503.3 449.4 495.4 491.3 487.3 30 760.9 755.0 749.0 743.0 737.0 730.9 40 1014.5 1006.6 998.7 990.7 982.7 974.5 50 1268. 1 1258.3 1248.4 1238.4 1228.3 1218.1 60 1521.7 1510.0 1498.1 1486.1 1474.0 1461.8 7 - 10652.0 10569.7 10486.6 10402.6 10317.9,10232.3 8 - 12173.8 12079.7 11984.6 11888.7 11791.8 11'694.1 9 - 13695.5 13589.6 13482.7 13374.8 13265.8 13155.8 10 - 15.217.2 15099.6 14980.8 14860.9 14739.8 14617.6 20 - 30434.4 30199.1 29961.6 29721.7 j 29479.6 29235.2 30 - 45651.6 45298.7 44942.4 44582.6 44219.4 43852.8 40 - 60868.8 60398.3] 59923.2 59443.4 i 58959.2 58470.4 50 - 76086.0 75497.9 74903.9 74304.3 1 73698.9 73088.0 60 _ 91303.2 90597.4 89884.7 89165.1 8 &8438.7 87705.6.Meridional s/rcs- Values of D m, in Yards. L. 420 0' L.4300' L.44~ 0' 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 1012.2 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 - 14170.3 14172.8 14175.3 8 - 16194.7 16197.5 16200.3 9 - 18219.0 18222.2 18225.4 1-0 - 20243.4 20246.9 20250.4 20 - 40486.7 40493.8 40500.9 30 - 60730.1 60740.7 60751.3 40 - 80973.4 80987.5 81001.7 50] -_. 101216.8 101234.4 101252.2 60 - 121460.1 121481.3 121502.6 Intermediate minutes and seconds will be found by moving the decimal point. 126 GEODESY. Jdrcs of the Parallel- Values of D p, in Yards. L. 44030 L.45' L. 45030 L. 460 0' L. 4630' L. 470~ 0 7 169.1 167.6 166.2 164.7 163.2 161.7 8 193.3 191.6 189.9 188.2 186.5 184.8 9 217.4 215.5 213.7 211.8 209.8 207.9 10 241.6 239.5 237.4 235.3 233.2 331.0 20 483.1 479.0 474.8 470.6 466.3 462.0 30 724.7 718.5 712.2 705.9 699.5 693.1 40 966.3 958.0 949.6 941.2 932.7 924.1 50 1207.9 1197.5 1187.0 1176.5 1165.8 1155.1 60 1449.4 1437.0 1424.4 1411.8 1399.0 1386.1 7 - 10146.0 10058.9 9971.0 9882.4 9793.0 9702.8 8 - 11595.4 11495.9 11395.5 11294.2 11192.0 11089.0 9 - 13044.8 12932.9 12819.9 12705.9 12591.0 12475.1 10 - 14494.3 14369.8 14244.3 14117.7 13990.0 13861.2 20 - 28988.6 28739.7 28488.6 28235.4 27980.0 27722.4 30 - 43482.8 43109.5 42733.0 42353.1 41970.0 41583.6 40 - 57977.1 57479.4 56977.3 56470.8 55960.0 55444.8 50 - 72471.4 71849.2 71221.6 70588.5 69950.0 69306.0 60 - 86965.7 86219.1 85465.9 84706.2 83940.0 83167.3 Meridional d.rcs- Values of D m, in Yards. L. 450 0' L. 460 0' L. 470~ O0' 7 236.3 236.3 236.4 8 270.1 270.1 270.1 9 303.8 303.9 303.9 10 337.6 337.6 337.7 20 675.1 675.3 675.4 30 1012.7 1012.9 1013.1 40 1350.3 1350.5 1350.7 50 1687.8 1688.1 1688.4 60 2025.4 2025.8 2026.1 7 - 14177.8 14180.3 14182.8 8 - 16203.2 16206.0 16208.9 9 - 18228.6 18231.8 18235.0 10 - 20254.0 20257.5 20261.1 20 - 40508.0 40515.1 40522.2 30 - 60761.9 60772.6 60783.2 40 - 81015.9 81030.1 81044.3 50 - 101269.9 101287.7 101305.4 60 - 121523.9 121545.2 121566.5 Intermediate minutes and seconds will be found by moving the decimal point. PROJECTION OF MAPS. 127 J.rcs of the Parallel- Values of D p, in Yards. L. 470 30' L. 480 0' L. 480 30'1 L. 490 0' L. 490 30' L. 50~ 0' 7 160.2 158.7 157.1 155.6 154.0 152.4 8 183.1 181.3 179.6 177.8 176.0 174.2 9 206.0 204.0 202.0 200.0 198.0 196.0 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 30 686.6 680.0 673.4 666.8 660.1 653.3 40 915.4 906.7 897.9 889.0 880.1 871.1 50 1144.3 1133.4 1122.4 1111.3 1100.1 1088.9 60 1373.1 1360.0 1346.9 1333.6 1320.2 1306.7 7 - 9612.0 9520.3 9428.0 9334.9 9241.1 9146.6 8 - 10985.1 10880.4 10774.8 10668.5 10561.2 10453.2 9 - 12358.2 12240.4 12121.7 12002.0 11881.4 11759.9 10 - 13731.4 13600.5 ]3468.5 13335.6 13201.6 13066.5 20 - 27462.7 27200.9 26937.1 26671.1 26403.1 26133.1 30 - 41194.1 40801.4 40405.6 40006.7 39604.7 39199.6 40 - 54925.5 54401.9 53874.1 53342.3 52806.2 52266.2 50 - 68656.8 68002.4 67342.7 66677.8 66007.8 65332.7 60 - 82388.2 81602.8 80811.2 80013.4 79209.4 78399.3 JMIeridional./rcs- Values of D m, in Yards. L. 480 0' L. 49 0 L. 50~ 0' 7 236.4 236.5 236.5 8 270.2 270.2 270.3 9 304.0 304.0 304.1 10 337.7 337.8 337.9 20 675.5 675.6 675.7 30 1013.2 1013.4 1013.6 40 1351.0 1351.2 1351.4 50 1688.7 1689.0 1689.3 60.2026.5 2026.8 2027.2 7 - 14185.2 14187.7 14190.2 8 - 16211.7 16214.5 16217.3 9 - 18238.2 18241.3 18244.5 10 - 20264.6 20268.1 20271.7 20 - 40529.2 40536.3 40543.3 30 - 60793.9 60804.4 60815.0 40 - 81058.5 81072.6 81086.6 50 - 101323.1 101340.7 101358.9 60 - 121587.7 121608.9 121629.9 Intermediate minutes and seconds will be found by moving the decimal point. 128 GEODESY. Lengths in JVautical miles and Statute miles of degrees of Latitude and Longitude in different Latitudes. DEGREE OF THE PARALLEL. DEGREE OF THE MERIDIAN. Latitude of Nautical Statute Latitude of Nautical Statute Parallel. miles. miles. middle point. miles. miles. 200 56.404 65.018 200 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 28 53.016 61.113 29 52.518 60.540 30 52.005 59.948 30 59.749 68.875 31 51.476 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 - 38 47.341 54.571 39 46.960 53.822 40 46.026 53.056 40 59.847 68.987 41 45.348 52.274 42 44.654 51.476 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. APPENDIX. MAGNETIC OBSERVATIONS. On the use of the Portable Declinometer in the determination of the magnetic Declination (variation) and Horizontal Intensity. ABSOLUTE DECLINATION. The adjustment of the Declinometer consists in bringing the line of detorsion (or the direction in which the force of torsion of the thread acts,) first approximately, then accurately, into the magnetic meridian; in determining the zero division of the scale corresponding to the magnetic axis of the collimator magnet, and in bringing the line of collimation of the Theodolite telescope into the magnetic meridian, its vertical wire coinciding with this division. Having determined the readings of the verniers of the Azimuth circle of the Theodolite corresponding to the magnetic axis of the collimator magnet, turn the telescope into the direction of some object whose azimuth from true north is known, or can be determined, and read off again. The difference of these readings added to or subtracted from the true azimuth of the obk)t~ referred to, will give the absolute declination. As the direction of the magnetic meridian is continually changing, the instrument should be left in adjustment, and observed half-hourly, or hourly, for as long a period as possible, in order that the mean declination may be obtained, 17 130 MAGNETIC OBSERVATIONS. instead of the declination at one period only. By this means, also, variations of declination will be obtained. The angular value of the magnet's scale is determined by measuring with the theodolite the horizontal angle subtended by a certain number of its divisions, the magnet being temporarily fixed. As the interval of the lens and scale is adjusted by the maker, so that the divisions shall be most clearly seen at infini-distant focus, if the adjustment is very accurately made the angular value of the scale will be the same at whatever distance the telescope of the Theodolite may be placed. This, however, should always be tested. If a, denote the angular value of one division of the scale, and F the ratio of the torsion and magnetic forces, the true declination changes are deduced from the observed readings, by multiplying their differences by the constant co-efficient a X ( + The value of H is determined by turning the torsion circle through two or more large angles (for example 900) and noting the corresponding differences of reading; if w equals the mean of the former, and u that of the latter, reH w duced to angular value, F-w The value of the co-efficient a (1 + — must be given with every abstract of observations; and also a statement of whether -increasing numbers denote an easterly or westerly movement of the north end of the magnet. M1 AGNETIC OBSERVATIONS. 131 ABSOLUTE HORIZONTAL INTENSITY. The determination of the horizontal intensity requires two distinct series of observations or experiments; those of deflection and those of vibration. Experiments of Deflection give the ratio of the magnetic moment of the deflecting magnet to the Horizontal Intensity. Experiments of Vibration, the product of the same quantities. The absolute value of either is obtained by comparing the two. The experiments of deflection consist in observing the angles through which a freely suspended magnet is deflected, by the action of a second magnet, placed at different distances from it in the direction of a line passing through its centre, perpendicular either to the axis of the suspended magnet, or to the magnetic meridian. The experiments of deflection with the portable declinometer should be made, if possible, at three distances at each side of the suspended magnet, reversing the deflecting magnet each time, and repeating the reversals so as to obtain four readings at each distance on each side. The first distance should be as near as the length of the collimator scale will allow. The second distance, if possible, one-third greater than the first, and the third intermediate to the two first. When the deflecting magnet is shifted from the east to the west side of the suspended magnet, reverse the order of distances, so that the mean results may correspond nearly to the same time of observation. The corrections for changes of temperature and intensity (during the time employed in the observations) will, in the absence of pro 13'2 MAGNETIC OBSERVATIONS. per instruments for observing them, in most cases, be compensated by this system of observation. Calculation. —The ratio of the magnetic moment of the deflecting magnet to the horizontal intensity is to be calculated in any case from the observations at each distance taken separately; if m = the magnetic moment of the deflecting magnet, X = the horizontal intensity, r = the distance between the centres of the deflecting and suspended magnets expressed in feet and decimals of a foot, u = the corresponding angle of deflection, obtained by multiplying half the mean of each partial result (at each distance) by the angular value of one scale division corrected for torsion, then: X = r3 tang u. The experiments of vibration consist in suspending the magnet, used as a deflecting bar, in a wooden box of suitable dimensions, and noting the times at which some central division of the scale passes across the vertical wire of the telescope (of the Theodolite) during at least 300 vibrations; the magnet having been first made to vibrate steadily, and the arc of vibration reduced to as small a size as the observations will allow. As the time of vibration depends on the form and weight of the suspended mass, as well as on the product of the magnetic moment of the magnet into the horizontal intensity, its moment of inertia must be experiment MAGNETIC OBSERVATIONS. 133 ally determined before the required value of this product can be obtained. The experiment consists in observing a second series of vibrations with two cylindrical weights of equal dimensions, whose moment of inertia is known, suspended at opposite ends of the magnet; if, K = the moment of the suspended magnet and stirrup, (the value required,) Kt = the moment of inertia of the weights, Tt and T = the times of vibration with and without the weights, K K= T 12 - T.) the moment of inertia of the weights is calculated by the formula, KI=- (I Z1 +, 2p) where: 1 — interval of the points of suspension, or length of the magnet diminished by the depth of the grooves in which the threads rest, r — the radius of the cylinders, and 2 p their mass, expressed respectively in feet and grains. The value of K must be ascertained with very great accuracy from the mean of several series of observations with and without the weights; once satisfactorily determined, it may be employed as a constant quantity, and the observations with weights need not be repeated in after determinations of the intensity. A small correction must be applied for the changes in the dimensions or form of the suspended mass, produced by changes of temperature; the correction consists in multiplying the 134 MAGNETIC OBSERVATIONS. value of K by 1 + 2 e (it — t), where It denotes the actual temperature of the magnet, t the temperature corresponding to. the time of the original observations, and e the coefficient of dilatation of steel for 1~ Fahrenheit - 0.0000068. Calculation.-The product of the horizontal intensity into the magnetic moment of the suspended magnet is obtained by the formula:,z K mX- T= T8 where -- circumference of circle to diameter 1. K=- the moment of inertia of suspended magnet and stirrup. T - the time of vibration, corrected. The observed times of vibration must be corrected for the force of torsion of the suspension thread, the rate of chronometer, the arc of vibration, the change of horizontal force between the observations of deflection-and vibration, and the differences in the magnetic moment of the deflecting magnet, produced by an increase or decrease' of temperature, and by the earth's inducing action. The corrections are to be applied according to the formula: T2 T —'(1-86400) (1'(1+-) (+ x) -(it t) (1+ m) - t) AX the change of horizontal force between the experiA m ments of deflection and vibration, and m the difference MAGNETIC'OBSERVATIONS. 135 in the magnetic moment of the deflecting magnet, produced by the earth's inducing action, are determined from experiments with the Bifilar magnetometer-when this instrument is not used, as is the case in our observations and upon magnetic surveys generally, the formula becomes: T2 T' (186;400 6 X 1+ F X (1-( = -- I) g) where: T and T' - the true and observed times of vibration, r the rate of the chronometer - when lgaing a and a' - the initial and terminal semi-arcs of vibration in parts of radius, H the ratio of the torsion and magnetic forces, t -the temperature of the deflecting magnet during the experiments of deflection, t= the temperature of the deflecting magnet during the experiments of vibration, _= the change.of magnetic moment of the deflecting magnet for 1~ of temperature. ag al The value of - is found by the formula: a at a= a a2 d d' X 0.000072722 16 136 MAGNETIC OBSERVATIONS. where d, d' denote the semi-arcs of vibration in divisions of scale, and a, the angular value of 1 division.* = — 1 a. n. cot U. where a the are value of 1 division of scale in terms of radius, h = the difference of scale readings corrected for changes of declination, t- to the corresponding difference of temperature, u = the angle of deflection at the lowest mean temperature. Final calculation of the results. By the observations of deflection, we have X A By those of vibration,.. m X = B whence, X and m = A B. The correction for are, in an extreme case when the initial semiare was 36 scale divisions (10 40' 30"), amounted to only 0.000037. The correction for rate, when the chronometer loses 3 seconds per day, is 0.000084. From examples in Riddell's Instructions, the value of the horizontal intensity is not carried beyond four decimal figures, and as these two corrections only change the fifth decimal, it will be, in most instances, needless to compute them, and the final formula would then be further reduced to'2 =T 2 v I 1 H t1-_ d to ) - ----- -- MAGNETIC OBSERVATIONS. 137' The horizontal intensity being thus determined, the Total intensily will be found by dividing the horizontal intensity by the cosine of the Dip, deduced from observations with the dipping needle. OBSERVATIONS FOR THE DIP, OR ABSOLUTE INCLINATION. In addition to the usual method of conducting observations of the Dip Circle, it is desirable that a few series should be made in different azimuths, for the purpose of testing the axles of the needles and the limb of the instrument (which is often magnetic;) if r denote the observed inclination of the needle, 0 the inclination sought, a the azimuth of the vertical circle. tang 0 = tang v co-sec a The true inclination may be inferred also from the observed inclinations of the needle in any two planes at right angles to one another, without the knowledge of the angle a, according to the formula cot2 0 = cot's 7 + cot' a7 The difference between the mean of the results obtained by observations in different azimuths, and the result obtained by observations in the magnetic meridian, should be applied as a constant correction for the errors of axl' and limb to all preceding or subsequent observations made in the meridian. 18 138 MAGNETIC OBSERVATIONS. To compute, theoretically, the variations in the magnetic declination, due to changes in the Latitude and Longitude. In a system of rectangular co-ordinates of which zo is the origin, letz - declination at P. M = increase of declination in the direction, x. N = increase of declination in the direction, y, x-= difference of longitude. y = difference of latitude. At the origin, x —o; ~y-o; I z n L - z; L- = M x +- N I x y- X x z. M x y +N X y' = y z. Let z - L = K; then, Equation of the line passing through all the points of equal declination is, M x + N y- I K and the angle which it makes with the meridian. -arc (tang = ) TABLES AND FORMUL}. PART III. ASTRONOMY..,,......,................ >..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ASTRONO MY. I. 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. JIean 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 always 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 ~- equation 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 * Sidereal time at mean moon = AR mean 0 at mean moon; and, generally, Sidereal time = sidereal time at mean moon i solar time from mean moon (expressed in sidereal intervals.) Solar time = sidereal time - sidereal time at mean moon, (the difference being reduced to a solar interval.) 142 ASTRONOMY. ExampleTo find the mean solar time of the passage of Altair over the meridian of Washington, on the 10th July, 1849. h m s AR: Altair, July 10th, 1849.... 19 43 27 39 Sidereal time at mean moon at Washington.. 7 14 00.96 Sidereal interval past Washington mean moon.. = 12 29 26.43 Retardation of mean on sidereal time.. 2 02.27 Corresponding mean time interval past mean moon or mean time of culmination. -12 27 23.66 The nautical almanacs give the sidereal time at mean moon 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 24h 03m 56s.55 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 3m 56s.5554 of sidereal time; hence the sidereal time at mean moon under any meridian other than that of the nautical almanac used, will be found by allowing the proportion of this quantity 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 23h 56m 04s.09 of those solar hours, or the daily retardation of mean solar on sidereal time is 3m 55s.9093 of solar time. The astronomical day begins at noon. In the civil or common method of reckoning, the day is supposed'o commence at the preceding midnight. The civil reckoning is therefore 12 hours in advance of the astronomical reckoning, and in the above example, July 10th, 12h 27m 23S.66 astronomical time, corresponds to July 11th, Oh 27m 23s.66 A. M. civil time. TIME. 143 II. To find the time by an altitude of the Sun or a star. Sidereal time -- AR * i *'s hour angle, Solar time - 24hrs i.-'s hour angle, 2 m- L + - A, cos ra. sin (mn-A) Sin2 ~P 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. = obs. alt. -(refrac'n-paral'x) 1- semi-diam. p — the hour angle of the sun or star. The formula gives the are 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 centre, the hour angle, p, 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 means 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 sideral time is at once deduced from the hour angle p. 4. And if, in this last instance, solar time should be required, convert this sidereal time into solar time by means of the equation, Solar time = AR *-AR 0 -4- p. In which the sign + 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 (sid. time of obs'n-sid'l time of preced'g mean moon at place.) 144 ASTRONOMY. For converting intervals of SIDEREAL into corresponding intervals of MEAN SOLAR time. HOURS. MINUTES. SECONDS. h. rn. s. m. S. m. s. S. S. S. S. 1 0 9.830 1 0.164 31 5.079 1 0.003 31 0.085 2 0 19.659 2 0.328 32 5.242 2 0.005 32 0.087 3 0 29.489 3 0.491 33 5.406 3 0.008 33 0.090 4 0 39.318 4 0.655 34 5.570 4 0.011 34 0.093 5 0 49.148 5 0.819 35 5.734 5 0.014 35 0.096 6 0 58.977 6 0.983 36 5.898 6 0.016 36 0.098 7 1 8.807 7 1.147 37 6.062 7 0.019 37 0.101 8 1 18.636 8 1.311 38 6.225 8 0.022 38 0.104 9 1 28.466 9 1.474 39 6.389 9 0.025 39 0.106 10 1 38.296 10 1.638 40 6.553 10 0.027 40 0.109 11 148.125 11 1.802 41 6.717 11 0.030 41 0.112 12 1 57.955 12 1.966 42 6.881 12 0.033 42 0.115 13 2 7.784 13 2.130 43 7.044 13 0.036 43 0.118 14 2 17.614 14 2.294 44 7.208 14 0.038 44 0.120 15 2 27.443 15 2.457 45 7.372 15 0.041 45 0.123 16 2 37.273 16 2.621 46 7.536 16 0.044 46 0. 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 19 3.113 49 8.027 19 0.052 49 0.134 20 3 16.591 20 3.277 50 8.191 20 0.055 50 0.137 21 3 26.421 21 3.440 51 8.355 21 0.057 51 0.140 22 3 36.250 22 3.604 52 8.519 22 0.060 52 0.142 23 3 46.080 23 3.768 53 8.683 23 0.063 53 0.145 24 3 55.909 24 3.932 54 8.847 24 0.066 54 0.148 25 4.096 55 9.010 25 0.08 55 0.150 26 4.259 56 9.174 26 0.071 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 29 0.079 59 0.161 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 sdlar timne. TIME. 145 For converting intervals of MEAN SOLAR Time into corresponding intervals of SIDEREAL Time. HOURS. MINUTES. SECONDS. h. m. s. Mm. SS. S.. S. S. 1 0 9.856 1 0.164 31 5.092 1 0.003 31 0.085 2 019.713 2 0.329 32 5.257 2 0.005 32 0.088 3 029.569 3 0.493 33 5.421 3 0.008 33 0.090 4 039.426 4 0.657 34 5.585 4 0.011 34 0.093 50 49.282 5 0.821 35 5.750 5 0.014 35 0.096 6 059.139 6 0.986 36 5.914 6 0.016 36 0.098 7 1 8.995 7 1.150 37 6.078 7 0.019 37 0.101 8 1 18.852 8 1.314 38 6.242 8 0.022 38 0.104 9 128.708 9 1.478 39 6.407 9 0.025 39 0.106 10 1 38.565 10 1.643 40 6.571 10 0.027 40 0.109 11 1 48.421 11 1.807 41 6.735 11 0.030 41 0.112 12 1 58.278 12 1.971 42 6.900 12 0.033 42 0.115 13 2 8.134 13 2.136 43 7.064 13 0.036 43 0.118 14 2 17.991 14 2.300 44 7.228 14 0.038 44 0.120 15 2 27.847 15 2.464 45 7.392 15 0.041 45 0.123 162 37.704 16 2.628 46 7.557 16 0.044 46 0.126 172 47.560 17 2.793 47 7.721 17 0.047 47 0.128 182 57.416 18 2.957 48 7.885 18 0.049 48 0.131 19 3 7.273 19 3.121 49 8.050 19 0.052 49 0.134 20 317.129 20 3.285 50 8.214 20 0.055 50 0.137 21 326.986 21 3.450 51 8.378 21 0.057 51 0.140 22 336.842 22 3.614 52 8.542 22 0.060 52 0.142 23 346.699 23 3.778 53 8.707 23 0.063 53 0.145 24 456.555 24 3.943 54 8.871 24 0.066 54 0.148 25 4.107 55 9.035 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 0.156 28 4.600 58 9.528 28 0.077 58 0.159 29 4.764 59 9.692 29 0.079 59 0.161 30 4.928 60 9.856 30 0.082 60 0.164 The quantities taken from this table must be added to a mean interval, to obtain the corresponding interval in sidereal time. 19 146 ASTRONOMY. To convert parts of the Equator in.Jrc into Sidereal Time, or to convert Terrestrial Longitude in.drc into Time. DEGREES. Arc. Time. Arc. Time. |Arc. Time. Arc. Time. O hm h mn o hm o hm 1 0 4 31 2 4 61 4 4 91 6 4 2 0 8 32 2 8 62 4 8 92 6 8 3 0 12 33 2 12 63 4 12 93 6 12 4 0 16 34 2 16 64 4 16 91 6 16 5 020 35 2 20 65 4 20 95 6 20 6 024 36 2 24 66 4 24 96 624 7 028 37 2 28 67 4 28 97 628 8 032 38 2 32 68 4 32 98 632 9 036 39 2 36 69 4 36 99 636 10o 40 40' 2 40 70 4 40 100 640 11 0 44 41 2 44 71 4 44 J01 6 44 12 048 42 2 48 72 448 102 648 13 0 52 43 2 52 73 4 52 103 6 52 14 0 56 44 2 56 74 4 56 104 656 15 1 0 45 3 0 75 5 0 105 7 0 16 1 4 46 3 4 76 5 4 106 7 4 17 1 8 47 3 8 77 5 8 107 7 8 18 112 48 3 12 78 5 12 108 712 19 116 49 3 16 79 5 16 109 716 20 120 50 3 20 80 5 20 110 720 21 1 24 51 3 24 81 5 24 III 7 24 22 128 52 3 28 82 5 28 112 728 23 1 32 53 3 32 83 5 32 113 7 32 24 1 36 54 3 36 84 5 36 114 7 36 25 1 40 55 3 40 85 5 40 115 7 40 26 1 44 56 3 44 86 5 44 116 7 44 27 148 57 3 48 87 5 48 117 748 28 152 58 3 52 88 5 52 118 752 29 1 56 59 3 56 89 5 56 119 756 30 20 60 4 0 90 6 0 120 8 0 SPACE INTO TIME. 147 To convert parts of the Equator in Jdrc into Sidereal Time, or to convert Terrestril Longitlude in d.rc into Time. DEGREES. Arc. Time. Are. Time. Arc. Time. Arc. Time. 0 h m o 1I mi 0 h m h m 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 16 214 14 16 125 8 20 155 10 20 185 12 20 215 14 20 126 8 24 156 10 24 186 12 24 216 14 24 127 8 28 157 10 28 i187 12 28 217 14 28 128 8 32 158 10 32 188 12 32 218 14 32 129 8 36 159 10 36 189 12 36 219 14 36 130 8 40 160 1040 190 1240 220 14 40 131 8 44 161 10 44 191 12 44 221 14 44 132 8 48. 162 10 48 192 12 48 222 14 48 133 8 52 163 10 52 193 12 52 223 14 52 131 8 56 164 10 56 194 12 56 224 14 56 135 9 0 165 11 0 195 13 0 225 15 0 136 9 4 166 11 4 196 13 4 226 15 4 137 9 8 167 11 8 197 13 8 227 15 8 138 9 12 168 11 12 198 13 12 228 15 12 139 9 16 169( 111 6 199 13 16 229 15 16 140 9 20 170 11 20 200 13 20 230 15 20 141 9 24 171 11 24 201t 13 24 231 15 21 142 9 28 172 11 28 202 13 28 232 15 28 113 9 32 173 11 32 203 13 32 233 15 32. 144 9 36 174 11 36 204 13 36 234 15 36 145 9 40 175 11 40 205 13 40 235 15 40 146 9 44 176 11 44 206 13 44 236 15 44 147 9 48 177 11 48 207 13 48 237 15 48 148 9 52 178 11 52 208 13 52 238 15 52 149 9 56 179 11 56 209 13 56 239 15 56 150 10 0 180 12 0 210 14 0 240 16 0 148 ASTRONOMY. To convert parts of the Equator in drc into Sidereal Time, or to convert Terrestrial Longitude in.Jrc into Time. DEGREES. Are. Time. Arc. Time. Arc. Time. Arc. Time. o h m o h m o h m o 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 18 12 303 20 12 333 22 12 244 16 16 274 18 16 304 20 16 334 22 16 245 16 20 275 18 20 305 20 20 335 22 20 216 16 24 276 18 24 306 20 24 336 22 24 247 16 28 277 18 28 307 20 28 337 22 28 248 16 32 278 18 32 308 20 32 338 22 32 249 16 36 279 18 36 309 20 36 3:9 22 36 250 16 40 280 18 40 310 20 40 340 22 40 251 16 44 281 18 44 311 20 44 341 22 44 252 16 48 282 18 48 312 20 48 342 22 48 253 16 52 283 18 52 313 20 52 343 22 52 254 16 56 284 18 56 314 20 56 344 22 56 255 17 0 285 19 0 315 21 01 345 23 0 256 17 4 286 19 4 316 21 4 346 23 4 257 17 8 287 19 8 317 21 8 347 23 8 258 17 12 288 19 12 318 21 12 348 23 12 259 17 16 289 19 16 319 21 16 349 23 16 260 17 20 290 19 20 320 21 20 350 23 20 261 17 24 291 19 24 321 21 24 351 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 0 300 20 0 330 22 0 360 24 0 SPACE INTO TIME. 149 To convert parts of the Equator in dArc into Sidereal Time or to convert Terrestrial Longitude in.drc 2nto Time. MINUTES. SECONDS. Arc. Time. A rc. Time. Arc. Time. A rc. Time. m s' m s a A 1 0 4 31 2 4 1 0.067 31 2.067 2 0 8 32 2 8 2 0.133 32 2.133 3 0 12 33 2 12 3 0.200 33 2.200 4 0 16 31 2 16 4 0.267 34 2.267 5 020 35 2 20 5 0.333 35 2.333 6 0 24 36 2 24 6 0.400 36 2.400 7 0 28 37 2 28 7 0.467 37 2.467 8 0 32 38 2 32 8 0.533 38 2.533 9 0 36 39 2 36 9 0.600 39 2.600 10 0 40 40 2 40 10 0.667 40 2.667 11 0 44 41 2 44 11 0.733 41 2.733 12 0 48 42 2 48 12 0.800 42 1.800 13 0 52 43 2 52 13 0.867 43 2.867 14 0 56 44 2 56 14 0.933 44 2.933 15 1 0 45 3 0 15 1.000 45 3.000 16 1 4 46 3 4 16 1.067 46 3.067 17 1 8 47 3 8 17 1.133 47 3.133 18 1 12 48 3 12 18 1.200 48 3.200 19 1 16 49 3 16 19 1.267 49 3.267 20 1 20 50 3 20 20 1.333 50 3.333 21 1 24 51 3 24 21 1.400 51 3.400 22 1 28 52 3 28 22 1.467 52 3.467 23 1 32 53 3 32 23 1.533 53 3.533 24 1 36 54 3 36 24 1.600 54 3.600 25 1 40 55 3 40 25 1.667 55 3.667 26 1 44 56 3 44 26 1.733 56 3.733 27 1 48 57 3 48 27 1.800 57 3.800 28 1 52 58 3 52 28 1.867 58 3.867 29 1 56 59 3 56 29 1.933 59 3.933 30 2 0 60 4 0 30 2.000 60 4.000 150 ASTRONOMY. To convert Sidereal Time into parts of the Equator in d.rc, or to convert Tzme into Terrestrial Longitude in.Trc. HOURS. INUTES. SECONDS. Time Are. Tim'lnle Arc.'ilea Arc. Tiime Are. Tillle Arc. h 0 m o m' s I',,, 1 15 1 0 15 31 7 45 1 0 15 31 7 45 2 30 2 0 30 32 8 C 2 0 30 32 8 0 3 4.5 3 0 45 33 8 15 3 0 45 33 8 15 4 60 4 1 0 34 830 4 1 0 34 8 30 5 75 5 1 15 35 8 45 5 1 15 35 8 45 G 90 6 1 30 36 9 0 6 1 30 36 9 0 7 105' 7 1 45' 37 9 15 7 1 45 37 9 15 8 120 8 2 0 38 9 30 8 2 0 38 9 30 9 135 9 2 15 39 9 45 9 2 15 39 9 45 10 150 10 2 30 40 10 0 10 2 30 40 - 10 0 11 165 11 2 45 41 1015 I11 2 45 41 10 15 12 180 12 3 0 42 10 30 12 3 0 42 10 30 13 1.95 13 3 15 43 10 45 13 3 15 43 10 45 14 210 14 3 30 44 11 0 14 3 30 44 11 0 15 225 15 3 45 45 1 115 15 3 45 45 11 15 16 240 16 4 O0 46 11 30 16 4 0 49 11 30 17 255 17 4 15 47 11 45 17 41 15 47 11 45 18 2710 18 4 30 48 12 0 18 430 48 12 0 19 285 19 4 45 49 12 15 19 4 45 49 12 15 20 300 20 5 O 30 12 30 20 5 0 50 13 30 21 315 21 5 15 51 12 45 21 5 15 51 12 45 22 330 22 530 52 13 0 22 5 30 52 13 0 23 345 23 5 45 53 13 15 23 5 45 53 13 15 24 360 24 6 0 54 13 30 2i G 0 54 13 30 25 6 15 55 13 45 25 6 15 55 13 45 26 6 30 56 14 0 26 6 30 56 14 0 27 6 45 57 14 15 27 6 45 57 14 15 28 7 0 58 14 30 28 7 0 58 14 33 29 7 15 59 14 45 29 7 15 59 14 45 30 7 30 60 15 0 30 7 30 60 15 0 TIME INTO SPACE. 151 To convert Sidereal Time into parts of the Equator in.dlrc, or to convert Time into Terrestrial Longitude in.rc. TENTHS OF SECONDS. Time. Arc. Time. Arc. Time. Arc. Time. Arc. s S' 1 fs 0.01 0.15 0.31 4.65 0.61 9.15 0.91 13.65 0.02 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 0.64 9.60 0.94 14.10 0.05 0.75 0.35 5.25 0.65 9.75 0.95 14.25 0.06 0.90 0.36 5.40 0.66 9.90 0.96 14.40 0.07 1.05 0.37 5.55 0.67 10.05 0.97 14.55 0.08 1.20~/ 0.38 5.70 0.68 10.20 0.98 14.70 0.0(9 1.35 01 o.39 5.85 0.69 10.35 0.99 14.85 0.10 1.50 0.40 6. 0 0.70 1050 1.00 15.00 0.11 1.65 0.41 6.15 0.71 10.65 0.12 1.80 0.42 6.30 0.72 10.80, 0.13 1.95 0.43 6.45 0.73 10.95 R 0.14 2.10 0.44 6.60 0.74 11.10 0.15 2.25 0.45 6.75 0.75 11.25: Arc. 0.16 2.40 0.46 6.90 0.76!1.40 o 0.17 2.55 0.47 7.05 0.77 11.55 0.18 2.70 0.48 7.20 0.78 11.70 a 0.19 2.85 0.49 7.35 0.79 11.85 5 0.20 3.00 0.50 7.50 0c80 12.00 0.21 3.15 0.51 7.65 0.81 [2.15.001 0.015 0.22 3.30 0. 52 7.80 0.82 12.30.0 ()2 0.030 0.23 3.45 0.53 7.95 0.83 12.45.003 0.045 0.'24 3.60[ 0.54 8.10 0.84 12.0i(.004 ]0.060 0.25 3.75 0.55 8.25 0.85 12.75.005 0.075 0.26 3.90 0.56 8.40 0.86 12.90.06 0, 90 0.27 4.05 0.57 8.55 0.87 13.05.007 0.105 0.28 4.20 0.58 8.70 0.88 13.20.008 0.120 0.29 4.35 0.59 8.85 0.89 13.35.009 0.135 0.30 4.50 0.60 9.00 0.90 13.50.010 0.150 15-2 ASTRONOMY. To convert Right dscension in dirc into JMean Time. DEGREES. R. A. R. A. R. Mean Time. RAr Mean Time. Ar. Mean Time. Mean Tie. in Arc. in Arc. in Arc. / h m s o hm s 0 h m s 1 0 3 59.345 31 2 3 39.686 61 4 3 20.027 2 0 7 58.689 32 2 7 39.030 62 4 7 19.371 3 0 11 58.034 33 2 11 38.375 63 5 11 18.716 4 0 15 57.379 34 2 15 37.720 64 4 15 18.061 5 0 19 56.724 35 2 19 37.064 65 4 19 17.405 6 0 23 56.068 36 2 23 36.409 66 4 23 16.750 7 0 27 55.413 37 2 27 35.754 67 4 27 16.095 8 0 31 54.758 38 2 31 35.099 68 4 31 15.639 9 0 35 54.102 39 2 35 34.443 69 4 35 14.784 10 0 39 53.447 40 2 39 33.788 70 4 39 14.129 11 0 43 52.792 41 2 43 33.133 71 4 43 13.474 12 0 47 52.136 42 2 47 32.477 72 4 47 12.818 13 0 51 51.481 43 2 51 31.822 73 4 51 12.163 14 0 55 50.826 44 2 55 31.167 74 4 55 11.508 15 0 59 50.170 45 2 59 30.511 75 4 59 10.852 16 1 3 49.515 46 3 3 29.856 76 5 3 10.197 17 1 7 48.860 47 3 7 29.201 77 5 7 9.542 18 1 11 48.205 48 3 11 28.545 78 5 11 8.886 19 1 15 47.549 49 3 15 27.890 79 5 15 8.231 20 1 19 46.894 50 3 19 27.235 80 5 19 7.576 21 1 23 46.239 51 3 23 26.580 81 5 23 6.920 22 1 27 45.583 52 3 27 25.924 82 5 27 6.265 23 1 31 44.928 53 3 31 25.269 83 5 31 5.610 24 1 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 1 43 42.962 56 3 43 23.303 86 5 43 3.644 27 1 47 42.307 57 3 47 22.648 87 5 47 2.989 28 1 51 41.652 58 3 51 21.992 88 5 51 2.333 29 1 55 40.996 59 3 55 21.337 89 5 55 1.678 30 1 59 40.341 60 3 59 20.682 90 5 59 1.023 AR. IN ARC INTO TiME. 153 To convert Right JIscension in /lrc into.Mean Time. DEGREES. R.. Mean ime. ean.. Mean Time. in Are. in Ar. in Arc. o h m s o h m s 0 h m s 91 6 3 0.367 121 8 2 40.708 151 10 2 21.049 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 8 14 38.742 154 10 14 19.083 95 6 18 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 36.776 157 10 26 17.117 98 6 30 55.780 128 8 30 36.121 158 10 30 16.462 99 6 34 55,125 129 8 34 35.466 159 10 34 15.807 100 6 38 54.470 130 8 38 34.810 160 10 38 15.151 101 6 42 53.814 131 8 42 34.155 161 10 42 14.496 102 6 46 53.159 132 8 46 33;500 162 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 51.193 135 8 58 31.534 165 10 58 11.875 106 7 2 50.538 136 9 2 30.879 166 11 2 11.220 107 7 6 49.883 137 9 6 30.223 167 11 6 10.564 108 7 10 49.227 138 9 10 29.568 168 11 10 9.909 109 7 14 48.572 139 9 14 28.913 169 11 14 9.254 110 7 18 47.917 140 9 18 28.257 170 11 18 8.598 111 7 22 47.261 141 9 22 27.602 171 11 22 7.943 112 7 26 46.606 142 9 26 26.947 172 11 26 7.288 113 7 30 45.951 143 9 30 26.292 173 11 30 6.632 114 7 34 45.295 144 9 34 25.636 174 11 34 5.977 115 7 38 44.640 145 9 38 24.981 175 11 38 5.322 116 7 42 43.985 146 9 42 24.326 176 11 42 4.666 117 7 46 43.329 147 9 46 23.670 177 11 46 4.011 118 7 50 42.674 148 9 50 23.015 178 11 50 3.356 119 7 54 42.019 149 9 54 22.360 179 11 54 2.701 120 7 58 41.364 150 9 58 21.704 180 11 58 2.045 20 154 ASTRONOMY. To convert Right ascension in arc into Jiean Time. MLNUTES. SECONDS. R. A. R.A. R.A. R.A. in Arc. Mean Time. in Are. Mean Time. in Are. Mean Time. Arc. Mean Time. m, s. i l. s.'I s. i s, 1 0 3.989 31 2 3.661 1 0.066 31P 2.061 2 0 7.978 32 2 7.650 2 0.133 32 2.128 3 0 11.969 33 2 11.640 3 0.199 33 2.194 4 0 15.956 34 2 15.629 4 0.266 34 2.261 5 0 19.945 35 2 19.618 5 0.332 35 2.327 6 0 23.935 36 2 23.607 6 0.399 36 2.393 7 0 27.924 37 2 27.596' 0.465 37 2.460 8 0 31.913 38 2 31.585 8 0.532 38 2.526 9 0 35.902 39 2 35.574 9 0.598 39 2.593 10 0 39.891 40 2 39.563 10 0.665 40 2.659 11 0 43.880 41 2 43.552 11 0.731 41 2.726 12 0 47.869 42 2 47.541 12 0.798 42 2.792 13 0 51.858 43 2 51.530 13 0.864 43 2.859 14 0 55.847 44 2 55.519 14 0.931 44 2.925 15 0 59.836 45 2 59.509 15 0.997 45 2.992 16 1 3.825 46 3 3.498 16 1.064 46 3.058 17 1 7.814 47 3 7.487 17 1.130 47 3.125 18 1 11.803 48 3 11.476 18 1.197 48 3.191 19 1 15.793 49 3 15.465 19 1.263 49 3.258 20 1 19.782 50 3 19.454 20 1.330 50 3.324 21 1 23.771 51 3 23.443 21 1.396 51 3.391 22 1 27.760 52 3 27.432 22 1.463 52 3.457 22 1 31.749 53 3 31.421 23 1.529 53 3.524 24 1 35.738 54 3 35.410 24 1.596 54 3.590 25 1 93.72? 55 3 39.399 25 1.662 55 3.657 26 1 43.716 56 3 43.388 26 1.729 56 3.723 27 1 47.705 57 3 47.377 27 1.795 57 3.790 28 1 51.694 58 3 51.367 28 1.862 58 3.856 29 1 55.683 59 3 55.356 29 1.928 59 3.923 30 1 59.672 60 3 59.345 30 1.995 60 3.989 TIME INTO AR. IN ARC. 155 To convert.Mean Time into Right,dscension in.drc. HOURS. MINUTES. Mean R. in Ar. M R.. in Are. Mean R AinAr. Mean R. A. in Arc. Time. Time. Time. h 0O I 0,t t in 1 15 2 27.85 1 0 15 2.46 31 7 46 16.39 2 30 4 52.69 2 0 30 4.93 32 8 1 18.85 3 45 7 23.54 3 0 45 30.39 33 8 16 21.31 4 60 9 51.39 4 1 0 9.86 34 8 31 23.78 5 75 12 19.24 5 1 15 12.32 35 8 46 26.24 6 90 14 47.08 6 1 30 14.79 36 9 1 28.71 7 105 17 14.93 7 1 45 17.25 37 9 16 31.17 8 120 19 42.78 8 2 0 19.71 38 9 31 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 1 38.57 11 165 27 6.32 1l 2 45 27.11 41 10 16 41.03 12 180 29 34.16 12 3 0 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 11 1 48.42 15 225 36 57.70 15 3 45 36.96 45 11 16 50.89 16 240 39 25.55 16 4 0 39.43 46 11 31 53.35 17 255 41 53.40 17 4 15 41.89 47 11 46 55.81 18 270 44 21.24 18 4 30 44.35 48 12 1 58.38 19 285 46 49.09 19 4 45 46.82 49 12 17 0.74 20 300 49 16.94 20 5 0 49.28 50 12 32 3.21 21 315 51 44.78 21 5 15 51.75 51 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 0 59.14 54 13 32 13.06 25 6 16 1.60 55 13 47 15.53 26 6 31 4.07 56 14 2 17.99 27 6 46 6.53 57 14 17 20.45 28 7 1 9.00 58 14 32 22.92 29 7 16 11.46 59 14 47 25.38 30 7 31 13.92 60 15 2 27.85 156 ASTRONOMY. To convert.Mean Time into Right.dscension in dIrc. SECONDS AND TENTHS. Mean R. A. in Mean R. A in Mean R. A. Mean R. A. Time. Arc. Time. Arc. Time. in Arc. Timre. in Arc. 1 0 15.04 31 7 46.27 0.01 0.15 0.31 4.66 2 0 30.08 32 8 1.31 0.02 0.30 0.32 4.81 3 0 45.12 33 8 16.36 0.03 0.45 0.33 4.96 4 1 0.16 34 8 31.40 0.04 0.60 0.34 5.12 5 1 15.21 35 8 46.44 0.05 0.75 0.35 5.27 6 1 30.25 36 9 1.483 0.06 0.90 0.36 5.42 7 1 45.29 37 9 16.52 0.07 1.05 0.37 5.57 8 2 0.33 38 9 31.56 0.08 1.20 0.38 5.72 9 2 15.37 39 9 46.60 0.09 1.35 0.39 5.87 10 2 30.41 40 10 1.64 0.10 1.50 0.40 6.02 11 2 45.45 41 10 16.68 0.11 1.65 0.41 6.17 12 3 0.49 42 10 31.73 0.12 1.81 0.42 6.32 13 3 15.53 43 10 46.77 0.13 1.96 0.43 6.47 14 3 30.58 44 11 1.81 0.14 2o11 0.44 6.62 15 3 45.62 45 11 16.85 0.15 2.26 0.45 6.77 16 4 0.66 46 11 31.89 0.16 2.41 0.46 6.92 17 4 15.70 47 11 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 45.78 49 12 17.01 0.19 2.86 0.49 7.37 20 5 0.82 50 12 32.05 0.20 3.01 0.50 7.52 21 5 15.86 51 12 47.09 0.21 3.16 0.51 7.67 22 5 30.90 52 13 2.14 0.22 3.31 0.52 7.82 23 5 45.94 53 13 17.18 0.23 3.46 0.53 7.97 24 6 1.00 54 13 32.22 0.24 3.61 0.54 8.12 25 6 16.03 55 13 47.26 0.25 3.7[6 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 28 7 1.15 58 14 32.38 0.28 4.21 0.58 8.73 29 7 16.19 59 14 47.42 0.29 4.36 0.59 8.88 30 7 31.23 60 15 2.46 0.30 4.51 0.60 9.03 TIME INTO AR. IN ARC. 157 To convert.Jlean Time into Right,1scension in./rc. SECONDS AND TENTHS. Mean R. A. in Mean R. A. in Mean R. A. in Time. Arc. Time. Arc. T ime. Arc. s i 5,,, s s. ~ 0.61 9.18 0.76 11.43 0.91 13.69 0.62 9.33 0.77 11.58 0.92 13.84 a 0.63 9.48 0.78 11.74 0.93 13.99 J 0.64 9.63 0.79 11.89 0.94 14.14 [5 0.65 9.78 0.80 12.04 0.95 14.29 0.66 9.93 0.81 12.19 0.96 14.44.001 0.02 0.67 10.08 0.82 12.34 0.97 14.59.002 0.03 0.68 10.23 0.83 12.49 0.98 14.74.003 0.05 0.69 10.38 0.84 12.64 0.99 14.89.004 0.06 0.70 10.53 0.85 12.79 1.00 15.05.005 0.08 0.71 10.68 0.86 12.94.006 0.09 0.72 10.83 0.87 13.09.007 0.11 0.73 10.98 0.88 13.24.008 0.12 0.74 11.13 0.89 13.39.009 0.14 0.75 11.28 0.90 13.54. 010 0.15 Logarithms. 12 hours, expressed in seconds = 43200. 4.6354837 Complement to the same =.00002315 5.3645163 24 hours, expressed in seconds- 86400. 4.9365137 Complement to the same =.00001157 5.0634863 360 degrees, expressed in seconds = 1296000 6.1126050 To convert Sidereal time to M. solar time - 9.9988126 158 ASTRONOMY. FORM FOR SURVEY OF DETERMINATION OF'TIME DATE AND STATION.-1843, October 13 —Jouth of the Big Black river, I Sextant No. 2197, by Troughton ~j Simnms, and INSTRUMENTS. JMlean Solar Chronometer No. 76, by Charles O I lh. $~h. h.Sm.. 9I2 18 00 46 10 09.3 7 07 a 8.67 6 58 43.2 NAME93 4 05 46 33 1.6 7 09 44.37 7 00 59.6 o I 0I h. m. s. h. is..91 43 40 45 52 58.8 7 05 47.69 6 57 02.4 92 18 00 46 10 09.3 7 07 28.67 6 58 43.2 A'-.Indromeened 92 41 15 46 21 47.3 7 08 37.15 6 59 52.8 93 04 05 46 33 12.6 7 09 44.37 7 00 59.6 (East.) 93 45 20 46 53 50.8 7 11 45.92 7 03 01.2 94 13 45 47 08 03.7 7 13 09.73 7 04 25.6 94 40 50 47 21 36.6 7 14 29.64 7 05 45 95 07 25 47 34 54.5 7 15 48.14 7 07 03.6 Mean result of 8 observations on Mt.dndromnedae, in the East, O 1 I hm. s. h.m. s. 95 20 05 47 41 14.7 8 55 32.36 8 46 49.2 95 00 00 47 31 11.6 8 56 32.06 8 47 50.4 eLyre 94 30 40 47 16 31.2 8 57 59.42 8 49 16 94 12 20 47 07 21 8 58 54 8 50 10.8 (West.) 93 53 45 46 58 03.1 8 59 49.4 8 51 06.9 93 29 20 46 45 50.2 9 01 02.1 8 52 19.4 93 07 35 46 34 57.S 9 02 07 8 53 24.8 92 46 50 46 24 34.5 9 03 09 8 54 26 92 28 45 46 15 31.7 9 04 02.96 8 55 21.2 Mean result of nine observations on the Star a Lyrce in the West Mean result of eight observations on the Star cc d.ndromedze in the East as above CHRONOMETER ERROR.-Slow of Mean Solar time at 8 h. p. in., by a mean of these results from E. and W. Stars. TIME BY OBSERVED ALTITUDES. 159 RECORD AND COMPUTATION. by observed double altitudes of East and West Stars. a tributary to the river St. John, JDMaine. artificial horizon of Mercury. Young. E~~~~ o;;REMARKS. h. m. s. index error of Sextant = + 2' 40" O 08 45.29 Error of eccentricity of Sextant = + 1' 32" 8 45.47 Thermometer 31~.5 Fahr. 8 44.35 Barometer 29.14 inches. 8 44.77 Apparent AR. of Star, -=Oh OOm 21S.72 8 44.72 Apparent declination of Star, = 28~ 13' 59".5 N. 8 44.13 Appt. N. Polar distance of Star = 61 46 00.5 A 8 44.64 Approximatelat. of this Station = 46 57 0ON. L 8 44.54 Approximate longitude of do. = 411 37m 47s Sider'l time of mean noon at Station = 13 26 20.83 Oh 081n 44s.74 h. m. s. 0 08 43.16 Thermometer 290 Fahr. 8 41.66 Barometer 29.14 inches. 8 43.42 Apparent A. R. of Star, -=18h 31m 39s.16 8 43.20 Apparent declination of Star N. 38~ 38' 46".5 8 42.50 App't N. Polar distance of Star = 51 21 13.5 8 42.70 Index error of Sextant - + 2' 40"r 8 42.20 Error of eccentricity of Sextant - + 1' 32'" 8 43.00 8 41.76 Oh 08m 42S.6 0 08 44.7 Oh 08m 43s.6 Observer, Mlajor J. D. Graham. Computer, Private F. Herbst. 160 ASTRONOMY. Computation of the 5th of the preceding altitudes of a jndromeda. Formula page 143, Observed double altitude = 930 45' 201" Index error, sextant = + 2 40 Excentricity, sextant = + 1 32 Double altitude, corrected = 93 49 32 Altitude = 46 54 46 Refraction (Therm. 310.5-Bar. 29.1) = - 55.2 True altitude of * = A = 460 53' 50".8 2 m-L + A + A L = 46057' Cos... = 9.8341894 A - 61 46 00.5..... Sin.... 9.9449899 A 46 53 50.8 Cos L. Sin A = 9.7791793 2 m =1550 36' 51".3 m = 77 48 25.6.... Cos. =.... 9.3247069 (m —A)- 30 54 34.8..... Sin..... =.. 9.7106984 Cos mn Sin (m - A) = 9.0354053 Sin2 Cos. Sin (m - A) = 1902562259 Cos L. Sin A Sinl p = 9.6281129 P p = 250 08' 00''.7 p in arc = 50 16 01.4 (page 146) p in time - 3h 21in 04s.09 AR. * = 24 00 21.72 Sidereal time of observation = AR ip = 20 39 17.63 Sidereal time, mean noon, at place, (Naut. Aim.) = 13 26 20.83 Sidereal interval past mean noon 7 12 56.8 Retardation of mean on sidereal interval, (page 144) = - 1 10.9 Mean solar interval past mean noon, or mean 7h 111 45F.9 time, P. M., of observation................ Time of observation by Chronometer 7 03 01.2 Chronometer slow 8m 44s.7 OBSERVATIONS FOR THE TIME. 161 III. To find the time by equal.iltitudes of the Sun. 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. 8 T [ tangD tangL 48h X 30( tang7o T sin 7 T/ T T = ~t. tang D 1440 tang 7~ T - S tang L 1440 sin 71 T T make A 1440 sin 7~ T T - -B 1440 tang 7 T x- = A. a. tang L + B. 8. tang D apparent noon = a (t + it) + c t, f -= the times of observation. T = (t - t) -= the interval of time between the observations, expressed in hours and decimals. L = the latitude of the place of observation: (minus when south.) D the declination, at the time of noon, on the given day: (minus when south.) 6- the double daily variation in the declination, deduced from the noon of the preceding day to noon of the following day: (minus when the Sun is proceeding towards the south.) x = required correction in seconds: where A is to be minus when time of noon is required, andplus when time of midnight is required, i. e. when the first observation is made in the afternoon, and the corresponding one the morning following. Log. values of A and B are given in the tables. 21 162 ASTRONOMY. Equations to equal dltitudes. Interval. Log A. Log B. Interval. Log A. Log B. Ii. M, l. M. 2 0 7.7297 7.7146 3 0 7.7359 7.7015 2.7298.7143 2.7362.7010 4.7300.7139 4.7364.7005 6.7302.7136 6.7367.6999 8.7304.7132 8.7369.6993 10.7305.7128 10.7372.6988 12.7307.7125 12.7374.6982 14.7309.7121 14.7377.6976 16.7311.7117 16.7380.6970 18.7313.7113 18.7383.6964 20.7315.7109 20.7386.6958 22.7317.7105 22.7388.6952 24.7319.7101 24.7391.6946 26.7321.7097 26-.7394.6940 28.7323.7092 28.7397.6934 30.7325.7088 30.7400.6927 32.7327.7083 32.7403.6921 34.7329.7079 34.7406.6914 36.7331.7075 36.7409.6908 38.7333.7070 38.7412.6901 40.7336.7065 40.7415.6894 42.7338.7061 42.7418.6888 44.7340'.7056 44.7421.6881 46.7342.7051 46.7424.6874 48.7345.7046 48.7428.6867 50.7347.7041 50.7431.6859 52.7319.7036 52.7434.6852 54.7352.7031 54.7437.6845 56.7354.7026 56.7441.6838 58 7.7357 7.7021 58 7.7444 7.6830 x =:F A a tang L + B tang D. OBSERVATIONS FOR THE TIME. 163 Equations to equal dltitudes. Interval. Log A. Log B. Interval. Log A. Log B. II. M. H. M. 4 0 7.7447 7.6823 5 0 7.7562 7.6556 2.7451.6815 2.7566.6546 4.7454.6807 4.7570.6536 6.7458.6800 6.7575.6525 8.7461.6792 8.7579.6514 10.7464.6784 10.7583.6504 12.7468.6776 12.7588.6493 14.7472.6768 14.7592.6482 16.7475.6759 16.7597.6471 18.7479.6751 18.7601.6460 20.7482.6743 20.7606.6448 22.7486.6734 22.7610.6437 24.7490.6726 24.7615.6425 26.7494.6717 26.7620.6414 28.7497.6708 28.7624.6402 30.7501.6700 30.7629.6390 32.7505.6691 32.7634.6378 34.7509.6682 34.7638.6366 36.7513.6673 36.7643.6354 38.7517.6663 38.7648.6342 40.7521.6654 40.7653.6329 42.7525.6645 42.7658.6317 44.7529.6635 44.7663.6304 46.7533.6626 46.7668.6291 48.7537.6616 48.7673.6278 50.7541.6606 50.7678.6265 52.7545.6597 52.7683.6252 54.7549.6587 54.7688.6239 56.7553.6577 56.7693.6225 58 7.7557 7.6567 58 7.7698 7.6212 x = Z A 8 tang L + B,& tang D. 164 ASTRONOMY. Equations to Equal.1ltiludes. Interval. Log A. Log B. Interval. Log A. Log B. H.M. H. M. 6 0 7.7703 7.6198 7 0 7.7873 7.5717 2.7708.6184 2.7879.5699 4.7713.6170 4.7885.5680 6.7719.6156 6.7891.5661 8.7724.6142 8.7898.5641 10.7729.6127 10.7904.5622 12.7735.6113 12.7910.5602 14.7740.6098 14.7916.5582 16.7745.6083 16.7923.5562 18.7751.6068 18.7929.5542 20.7756.6053 20.7936.5522 22.7762.6038 22.7942.5501 24.7767.6023 24.7949.5480 26.7773.6007 26.7955.5459 28.7779.5991 28.7962.5437 30.7784.5975 30.7969.5416 32.7790.5959 32.7975.5394 34.7796.5943 34.7982.5372 36.7801.5927 36.7989.5350 38.7807.5910 38.7995.5327 40.7813.5894 40.8002.5304.42.7819.5877 42.8009.5281 44.7825.5860 44.8016.5258 46.7831.5843 46.8023.5234 48.7836.5825 48.8030.5211 50.7842.5808 50.8037.5186 52.7848.5790 52.8044.5162 54.7854.5772 54.8051.5137 56.7860.5754 56.8058.5112 58 7.7867 7.5736 58 17.8065 7.5087 x-= F A a tang L + B a tang D. OBSERVATIONS FOR THE TIME. 165 Equations to Equal.dltitudes. Interval. Log A. Log B. Interval. Log A. Log B. H. M. I. M. 8 0 7.8072 7.5062 15 0 8.0521 -7.6350 2.8079.5036 2.0539.6413 4.8086.5010 4.0556.6475 6.8094.4983 6.0574.6537 8.8101.4957 8.0592.6599 10.8108.4930 10.0610.6660 12.8116.4902 12.0628.6721 14.8123.4874 14.0646.6781 16.8130.4816 16.0664.6841 18.8138.4818 18.0682 *6900 20.8145.4789 20.0700.6959 22.8153.4760 22.0718.7018 24.8160.4731 24.0737.7077 26.8168.4701 26.0755.7135 28.8176.4671 28.0774.7192 30.8183.4640 30.0792.7249 32.8191.4609 32.0811.7306 34.8199.4578 34.0830.7363 36.8206.4546 36.0849.7419 38.8214.4514 38.0868.7475 40.8222.4482 40.0887.7531 42.8230.4449 42.0906.7586 44.6238.4415 44.0925.7641 46.8246.4381 46.0945.7696 48.8254.4347 48.0964.7751 50.8262.4312 50.0983.7805 52.8270.4277 52.i003.7859 54.8278.4241 54.1023.7912 56.8286.4205 56.1042.7966 58 7.8294 7.4168 58 8.1062 -7.8019 x =: A a tang L + B a tang D. 166 ASTRONOMY. Equations to Efual d/ltitudes. Interval. Log A. Log B. Interval. Log A. Log B. H. M. H. M. 16 0 8.1082 -7.8072 17 0 8.1726 -7.9571 2.1102.8125 2.1749.9618 4.1122.8177 4.1773.9666 6.1143.8229 6.1796.9713 8.1163.8281 8.1819.9761 10.1183.8333 10.1843.9808 12.1204.8385 12.1867.9855 14.1224.8436 14.1890.9902 16.1245.8487 16.1914.9949 18.1266.8538 18.1938 — 7.9996 20.1287.8589 20.1963 -8.0043 22.1308.8640 22.1987.0090 24.1329.8690 24.2011.0137 26.1350.8740 26.2036.0184 28.1371.8790 28.2061.0230 30.1393.8840 30.2086.0277 32.1414.8890 32.2111.0323 34.1436.8939 34.2136.0370 36.1458.8989 36.2161.0416 38.1479.9038 38.2186.0462 40.1501.9087 40.2212.0508 42.1523.9136 42.2237.0555 44.1545.9185 44.2263.0601 46.1568.9234 46.2289.0647 48.1590.9282 48.2315.0693 50.1612.9330 50.2341.0739 52.1635.9379 52.2367.0785 54.1658.9427 54.2394.0831 56.1680.9475 56.2420.0877 58 8.1703 — 7.9523 58 8.2447 -8.0923 x = -- A tang L + B 8 tang D. OBSERVATIONS FOR THE TIME. 167 Equations to Equal.dltitudes. Interval. Log A. Log B. Interval. Log A. Log B. H. M. H. M. 18 0 8.2474 -8.0969 19 0 8.3359 -8.2354 2.2501.1015 2.3392.2401 4.2529.1061 4.3424.2448 6.2556.1107 6.3457.2495 8.2583.1153 8.3490.2542 10.2611.1199 10.3524.2589 12.2639.1245 12.3557.2637 14.2667.1291 14.3591.2684 16.2695.1336 16.3625.2732 18.2723.1382 18.3659.2779 20.2752.1428 20.3694.2827 22.2781.1474 22.3728.2875 21.2809.1520 24.3763.2923 26.2838.1566 26.3798.2971 28.2868.1612 28.3834.3019 30.2897.1658 30.3869.3068 32.2926.1704 32.3905.3116 34.2956.1750 34.3941.3165 36.2986.1797 36.3978.3214 38.3016.1842 38.4015.3263 40.3046.1889 40.4052.3312 42.3077.1935 42.4089.3361 44.3107.1981 44.4126.3410 46.3138.2028 46.4164.3460 48.3169.2074 48.4202.3510 50.3200.2121 50.4241.3560 52.3232.2167 52.4279.3610 54.3263.2214 54.4318.3660 56.3295.2261 56.4357.3711 58 8.3327 -8.2307 58 8.4397 -8.3761 X = F A 8 tang L + B 8 tang D. 168 ASTRONOMY. Equations to Equal d.ltitudes. Interval. Log A. Log B. Interval. Log A. Log B. H. M. H. M. 20 0 8.4437 -8.3812 21 0 8.5810 -8.5466 2.4477.3863 2.5863.5527 4.4518.3915 4.5917.5588 6.4559.3966 6.5971.5650 8.4600.4018 8.6025.5712 10.4641.4070 10.6081.5775 12.4683.4122 12.6136.5838 14.4726.4175 14.6193.5902 16.4768.4227 16.6250.5966 18.4811.4280 18.6308.6031 20.4854.4334 20.6366.6096 22.4898.4387 22.6426.6162 24.4942.4441 24.6486.6229 26.4987.4495 26.6546.6296 28 e5032.4549 28.6608.6364 30.5077.4604 30.6670.6433 32.5W13.4659 32.6733.6502 34.5169.4714 34.6796.6572 36.5215.4770 36.6861.6643 38.5262.4826 38.6927.6715 40.5310.4882 40.6993.6788 42.5357.4939 42.7060.6860 44.5406.4996 44.7128.6934 46.5455.5053 46.7197.7009 48.5504.5111 48.7268.7085 50.5554.5169 50.7339.7162 52.5604.5228 52.7411.7239 54.5655.5287 54.7484.7318 56.5706.5346 56.7558.7398 58 8.5758 -8.5406 58 8.7634 -8.7478 x = T A 8 tang L + B 8 tang D. OBSERVATIONS FOR THE TIMEo 169 Equations to Equal./ltiutdes. Interval. Log A. Log B. Interval. Log A. Log B. H. M H. HM. 22 0 8.7711 -8.7560 23 0 9.0877 -9.0839 2.7789.7643 1 2.1029.0995 4.7868.7727 1 4.1187.1155 6.7948.7813 1 6.1351.1321 8.8030.7899 8.1520.1492 10.8113.7987 i 10.1696.1670 12.8198.8076 1 12.1879.1855 14.8284.8167' 14.2069.2047 16.8372.8259 16.2268.2248 18.8461.8353 18.2476.2456 20.8553.8448 20.2693.2677 22.8645.8545, 22.2922.2907 24.8740.8644 24.3162.3149 26.8837.8745 26.3416.3404 28.8935.8847 28.3685.3674 30.9036.8952 30.3971.3962 32.9139.9058 32.4276.4268 34.9244.9167 34.4604.4597 36.9351.9278 36,4957.4952 38.9461.9391 38.5341.5336 40.9574.9507 40.5761.5757 42.9689.9626 42.6224.6221 44.9807.9747 44.6742.6739 46 8.9928.9871 46.7328.7326 48 9.0052 -8.9999 48.8003.8001 50.0180 -9.0129 50.8801.8800 52.0311.0263 52 9.9776 -9.9775 54.0446.0401 54 0.1031 -0.1031 56.0585.0543 56.2798.2798 58 9.0729 -9.0689 58 0.5814 -0.5814 x =F A 8 tang L + B, tang D. 22 170 ASTRONOMY. SURVEY OF DETERMINATION OF THE TIME, Chronometer DATE AND STATION. —1844, Jlutgust 9 —r/merican Camp, near Tasche( Sextant No. 2197, by T'ougthlon 8 Simms, and JMean Solar Chronometer No. 2440, by Parkinson Times by Chronometer, of Observed double observed equal altitudes. t' -t = the Equation altitudes of thegust 9h elapsed time, of equ.al Sun'supperand = T. altitudes lower limbs. x. A. M. = t. P. M.= t'. Upper Limb. h. m. s. h. m. s. h. m. s. 780 50' 00" 1 28 23 8 03 16.5 6 33 -10.63 79 19 30 1 28 52.8 8 01 46.5 Lower Limb. 830 10' 00" 1 45 01 7 46 40.5' 83 40 00 1 46 34.5 7 45 06.2 I 5 592 +10.24 84 00 00 1 47 38 7 44 04 J Upper Limb. 850~ 36 00" 1 49 23 7 42 18 87 02 10 1 53 55.5 737 46.2 5 48 +10.1 CHRONOMETER ERROR.-Fast of mean solar time at apparent noon of.August 9, 1844, by a mean of 7 pairs of equal altitudes of the.Sun, TIME BY OBSERVED EQUAL ALTS. 171' by observed equal altitudes of the Sun's limbs, to correct the at noon. reau's house, on the highland boundary between MJIaine and Canada. artificial horizon of Mercury. A' Frodsham. a. a.= E as REMARKS. h. m. s. Index error of Sextant.. 4 40 51.29 Error of excentricity of Sextant 4 40 51.17 Thermr. (A. M.) 700 Fahr. Barom. Thermr. (P. M.) 690 Fahr. Barom. 4 40 51.9 Sun's appt. declination at appt. noon (D) -15043' 12" N. 4 40 51.5 Double daily variation of Sun's declination (,) = 34' 54" 4 40 52.15 = 2094!' in arc. Equation of time at apparent noon + 5m 09'.09 4 40 51.51 Latitude of station (approximate) + 450 48' - (L.) 4 40 51.86 Observer, JMajor J. D. Graham, 4h 40m 51s.6 Computer, Do. 172 ASTRONOMY. Computation of the equation of equal altitudes to correct the chronometer for noon of.dugust 9, 1844, by the first of the preceding equal altitudes of the Sun's upper and lower limbs. -(-A.. tang L) + (B. S. tang D.) Ist Set. T 6h 33m, log A (page 164) =-7.77930 log B = 7.59510 = 2094, log — 3.32097 log -=-3.32097 L -- 45~ 48', log tang =+ 0.01213 logtang D= 9.44933 1st term = + 12S.95 -+ 1.11240 — 2s.32 — 0.36540 2d term -- 2.32 x = + 1Cs.63 Equation of equal altitudes. Computatton of the first two of the foregoing pairs of equal altitudes of the Sun's limbs. 1st pair. 2d pair. A. M. t = 1h 28m 23s.0 1h 29m 52s. 8 P.M. - t'M =8 03 16.5 8 01 46.5 t -- t'- 9 31 39.5 9 31 39.3 t+- t' 4 45 49.75 4 45 39.65 Equat'n of equal altitudes x += 10.63 10.63 Time by chron. of appt. noon = 4 46 00.38 4 46 00.28 Correct mean time at apparent noon (Naut. Alm.) = 0 05 09.09 0 05 09.09 Chron. fast of mean time at appt. noon, August 9, 1844 = 411 40m 51s.29 4h 40m 51s.17 ASTRONOMY. 173 Sun's Parallax in.Jltitude. Sun's Sun's Horizontal Sun's Sun's Horizontal Altit. Parallax. Altit. Parallax. 8".4 8".5 8".6 8"'.7 8".8 8".4 8'.5 8".6 8".7 8".8 o -0 - - 0 8.40 8.50 8.60 8.70 8.80 45 5.94 6.01 6.0816.156.22 5 8.37 8.4718.57 8.67 8.77 50 5.4015.46 5.53 5.59 5.66 10 8.27 8.37 8.47 8.5718.67 55 4.8214.88 4.9314.99 5.05 15 8.11 8.21 8.31 8.40 8.50 60 4.204.25 4.30 4.35 4.40 20 7.89 7.99 8.08 8.188.27 65 3.55 3.59 3.63 3. 683.72 25 7.61 7.70 7.79 7.8817.98 70 2.87 2.91 2.94 2.98 3.01 30 7.28 7.3617.4517.5317.62 75 2.172.202.232.252.28 35 6.886.9617.04 7.13 7.21 80 1.46 1.4811.4911.51 1.53 40 6.44 6.516.5916.6616.74 85 0.73 0.74 0.750.7650.77 45 5.946'.016.08 6.15 6.22 90 0.000.00ooo.oo000.0 0.00 Parallax in Altitude =- Hor. Par X Cos. Altitude. Decimals of an fHour. MINUTES. SECONDS. M. Deem M. Decem. M. Deem. s. Deem. s. Decm. s. Deem. 1.01667 21.35000 41.68333 1.00028 21.00583 41.01139 2.03333 22.36667 42.70000 2.0005622.00611 42.01167 3.05000 23.38333 43.71667 3.00083 231.0063943.01194 4.06667 24.40000 44.73333 4.00111124.00667 44.01222 5.08333 25.41667 45.75000 5.06139 25.00694 45.01250 6.10000 26.43333 46.76667 6.00167;26.00722 46.01278 7. 11667 27.45000 47.78333 7.00194 27.00750 47.01306 8.13333 28.46667 48.80000 8.00222 28.00778 48.01333 9.15000 29.48333 49.81667 9.00250 29.00806 49.01361 10.16667 30.50000 50.83333 10.00278130.00833 50.01389 11.18333 31.51667 51.85000 11.00306 31.00861 51.01417 12.:20000 32.53333 52.86667 1 2.00333 32.00889 52.01444 13.21667 33.55000 53.88333 13.00361 33.00917 53.01472 14.23333 34.56667 54.90000 14.00389 34.00944 54.01500 15.25000 35.58333 55.91667 15.00417 35.00972 55.01528 16.26667 36.60000 56.93333 1 6.00444 36.01000 56.01556 17.28333 37.61667 57.95000 17.00472137.01028 57.01583 18.30000 38.63333 58.96667 11 8.00500 38.0105658.0161.1 19.31667 39.65000 59.98333 19.00528 39.01083 59.01639 20.33333 40.66667 601.00000 20.00556140.01111160.01667 174 ASTRONOMY. Fahrenheit's Thermometer 50~. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 0 I 0 I I I 90 00 0 0.00 0.0000 83 00 0 7.17 0.8557 102 89 50 0.17 9.2304 3011 82 50 7.34 0.8659 101 40 0.34 9.5315 1761 40 7.52 0.8760 30 0.51 9.7076 1249 30 7.69 0.8859 20 0.68 9.8325 1249 20 7.86 0.8956 969 95 10 0.85 9.9294 791 10 8.04 0.9051 93 89 00 0 1.02 0.0085 82 00 0 8.21 0.9144 670 90 88 50 1.19 0.0755 81 50 8.38 0.9234 580. 89 40 1.36 0.1335 51 40 8.56 0.9323 87 30 1.53 0.1847 30 8.73 0.9410 457 85 20 1.70 0.2304 414 20 8.90 0.9495 84 10 1.87 0.2718 10 9.08 0.9579 84 88 00 0 2.04 0.3097 81 00 0 9.25 0.9663 347 80 87 50 2.21 0.3444 80 50 9.42 (0.9743 80 40 2.38 0.3766 301 40 9.60 0.9823 78 30 2.55 0.4067 280 30 9.77 0.9901 20 2. 72 0.4347 263 20 9.95 0.998 76 10 2.89 0.4610 20 130 10.12 1.0054 87 00 0 3.06 0.4860 235 80 00 10.30 1.0129 72 86 50 3.23 0.5095 224 79 50 0 10.47 1.0201 72 40 3.40 0.5319 40 10.65 1.0273 211 71 30 3.57 0.55s0 203 30 10.82 1.0344 70 20 3.74 0.5733 193 20 11.00 1.0414 69 10 3.91 0(.5926 1 10 11.17 1.0483 69 86 00 0 4.08 0.6112 178 79 00 0 11.35 1.0552 66 85 50 4.26 0.6290 171 78 50 11.53 1.0618 66 40 4.43 0.6461 165 40 11.71 1.0684 66 30 4.60 0.6626 30 11.89 1.0750 478 65 20 4.77 0.6784 153 20 12.06 1.0815 64 10 4.94 0.6937 10 12.21 1.0879 62 85 00 0 5.11 0.7086 142 78 00 0 12.42 1.0941 62 84 50 5.28 0.7'228 77 50 12.60 1.1003 61 40 5.45 0.7367 3 40 ]12.78 1. 1061 60 30 5.63 0.7502 131 30 12 95 1.1121 60 20 5.80 0.7633 7 20 13.13 1.1184 58 10 5.97 0.7760 10 13.31 1.1242 84 00 0 6.14 0.7882 12 77 00 0 13.49 1.1300 83 50 6.31 0.8002 116 76 50 13.67 1.1357 57 40 6.48 0.8118 114 40 13.85 1.1414 55 30 6.66 0.8232 30 14.02 1 1469 55 20 6.83 0.8343 20 14.20 1.1524 10 7.00 0.84551 10 14.38 1.1578 106 54 83.00 0 7.17 0.8557 1 76.00 0 14.56 1.1632' 102 54 MEAN REFRACTIONS. 175 Fahrenheit's Thermometer 500. Barometer 30 Inches. Alt. r. Log. r. Diff. At. r. Log. r. Diff. 0, I' o 0 I 1 76 00 0 14.56 1.1632 54 69 00 1022.42 1.3507 37 75 50 14.74 1.1686 68 50 22.62 1.3544 8 40 14.93 1.1740 5) 40 22.81 1.3582 3 30 15.11 1.1793 52 30 23.01 1.3619 37 20 15.29 1.1845 52 20 23.21 1.36356 3 10 15.48 1.1897 50 10 23.40 1.3693 36 75 00 0 15.66 1.1'947 51 68 00 0 23.60 1.3729 37 74 50 15.84 1.1998 50 67 50 23.80 1.3766 36 40 16.03 1'.048 40 24.00 1.3302 30 16.'21 1.2U98 50 30 24.20 1.3838 36 20 16.39 1.2147 20 24.40 1.3874 10 16.58 1.2195 46 10 24.60 1.3909 36 74 00 0 16.75 1.2241 46 67 00 0 24.t0O 1.3945 36 73 50 16.93 1.2287 66 50 /25.00 1.3981 40 17.12 1.2334 46 40 25.20 1.4015 30 1*7.30 1.2380 46 30 25.41 1.41)49 20 17.48 1.2426 46 20 25.61 1.4084 34 10 17.67 1.2472, 660 25.81 1.4118 73 00 0 17.86 1.2519 45 66 00 0 26.01 1.4151 3 72 50 18.05 1.2564 45 65 50 26.21 1.4185 3 40 18.23 1.26091 40 26.42 1.4219 30 ]8.42 1.2653 30 26.62 1.4253 20 18.61 1.2'697 43 20 26.83 1.4286 33 10 18.79 1.2740 10 27.03 1.4319 72 00 0 18.98 1.2784 42 65 00 0 27.24 1.4352 33 71 50 19.17 1.2826 42 64 50 27.45 1.4385 40 19.36 1.2868 40 2l7.66 1.4418 30 19.55 1.2910 42 30 27.86 1.4451 32 20 19.73 1.2952 20 28.07 1.4483 32 10 19.92 1.2994 42 10 28.28 1.4515 32 71 00 0 20.11 1.30)36 39 64 00 0 28.49 1.4547 32 70 50 20.30 1.3075 39 63 50 28.70 1.4579 32 40 20.49 1.3116 40 28.91 1.4611 32 30 20.69 1.3157 41 30 29.13 1.4643 31 20 20.88 1.3197 41 20 29.34 1.4674 3 10 21.07 1.3237 40 10 29.55 1.4706 32 40 300 70 00 0 21.26 1.3277 38 63 00 0 29.76 1.4736 32 69 50 21.45 1.3'315 39 62 50 29.97 1.4768 31 40 21.65 1.3354 40 30.19 1.4799 30 30 22.84 1.3393 30 30.40 1.4829 20 22.03 1.3431 38 20 30.62 1.4860 30 10.22.23 1.3469 38 10 30.83 1.4890 31 69 00 022.42 1.3507 38 62 00 0 31.05 1.4921 3 37 31 176 ASTRONOMY. Fahrenheit's Thermometer 50~. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 0. 11 o 0 1 r It 62 00 0 31.05 1.4921 31 55 00 0 40.89 1.6116 27 61 50 31.27 1.4952 30 54 50 41.14 1.6143 27 40 31.49 1.4982 31 40 41.40 1.6170 27 30 31.72 1.5013 30 30 41.65 1.6197 26 20 31.94] 1.504 30 20 41.91 1.6223 27 10 32.16 1.5073 29 10 42.16 1.6250 26 61 00 0 32.38 1.5102 31 54 00 0 42.42 1.6276 27 60 50 32.60 1.5133 29 53 50 42.68 1.6303 27 40 32.83 1.5162 30 40 42.95 1.6330 2 30 33105 1.5192 29 30 43.21 1.6356 26 20 33.27 1.5221 29 20 43.47 1.6382 26 10 33.50 1.5250 29 10 43.74 1.6408 27 60 00 0 33.72 1.5279 29 53 00 0 44.00 1.6435 26 59 50 33.95 1.5308 29 52 50 44.27 1.6461 26 40 34.18 1.5337 29 40 44.54 1.6487 26 30 34.40 1.5366 29 30 44.80 1.6513 26 20 34.63 1.5395 29 20 45.07 1.6539 26 59 00 0 35 09 1.5452 29 52 00 0 45 61 1.6591 26 58 50 35.32 1.5481 29 51 50 45.89 1.6617 26 40 35.56 1.5510 40 46.16 1.6643 26 30 35.79 1.5538 28 30 46.44 1.6669 26 20 36.02 1.5566 28 20 46.72 1.6695 25 10. 36.26 1.5594 28 10 46.99 1.6720 26 58 00 0 36.49 1.5622 28 51 00 0 47.27 1.6746 26 40 36.97 1.5678 29 40 47.84 1.6798 30 37.21 1.5707 28 30 48.13 1.6824 26 20 37.45 1.5735 27 20 48.42 1.6850 26 10 37.69 1.5762 28 10 48.70 1.6876 26 57 00 0 37.93 1.5790 28 50 00 0 48.99 1.6901 257 56 50 38.17 1.5818 27 49 50 49.28 1.69267 256 40 38.42 1.5845 28 40 49.58 1.69523 30 38.66 1.5873 27 30 49.87 1.69780 257 20 281 20 38.90.5900 50.16 1.570037 25 10 39.15 1.5927 27 10 50.46 1.70293 257 56 00 0 39.39 1.5954 27 49 00 0 50 75 1.70550 254 55 50 39.64 1.5981 28 48 50 51.06 1.70804 254 40 39.89 1.6009 27 40 51.36 1.71058 253 30 40.14 1.6036 7 30 51.66 1.71311 253 20 40.39 1].6063 2 20 51.96 1.71564 10 40.64 1.6090 26 10 52.27 1.71818 252 55 00 0 40.89 1.6116 48 00 0 52.57 1.72070 252 27 252 MEAN REFRACTIONS. 177 Fahrenheit's Thermometer 500. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 0c' I II o, 1 i 48 0 0 52.57 1.72070 252 41 00 1 7.11 1.82678 47 50 52.88 1.72322 252 40 50 7.51 1.82933 255 40 53.19 1.72574 252 40 7.91 1.83188 255 30 53.50 1.72826 252 30 8.32 1.83443 255 20 53.81 1.73078 251 20 8.72 1.83698 255 10 54.12 1.73329 251 10 9.12 1.83953 255 47 00 0 54.43 1.73580 253 40 00 1 9.52 1.84208 256 253 256 46 50 54.75 1.73833 24 39 50 9.94 1.84464 257 40 55.07 1.74087 253 40 10.35 1.84721 256 30.55.40 1.74340 253 30 10.77 1.84977 257 20 55.72 1.74593 254 20 11.19 1.85234 256 10 56.04 1.74847 253 10 11.60 1.85490 257 46 00 0 56.35 1.75100 252 39 00 1 12.02 1.85747 258 45 50 56.68 1.75352 252 38 50 12.46 1.86005 259 40 57.02 1.75604 252 40 12.89 1.86264 258 30 57.35 1.75856 252 30 13.33 1.86522 259 20 57.69 1.76108 252 20 13.77 1.86781 258 10 58.02 1.76360 251 10 14.20 1.87039 259 45 00 0 58.36 1.76611 252 38 00 1 14.64 1.87298 260 44 50 58.70 1.76863 252 37 50 15.10 1.87558 261 40 59.05 1.77115 252 40 15.55 1.87819 261 30 59.39 1.77367 25 30 16.01 1.88080 261 20 59.74 1.77619 252 20 16.47 1.88341 260 10 1 0.08 1.77871 252 10 16.92 1.88601 262 44 00 1 0.43 1.78123 252 37 00 1 17.38 1.88863 262 43 50 0.79 1.78375 253 36 50 17.86 1.89125 262 40 1.15 1.78628 252 40 18.33 1.89387 263 30 1.50 1.78880 252 30 18.81 1.89650 263 20 1.86 1.79132 253 20 19.29 1.89913 263 10 2.21 1.79385 252 10 19.76 1.90176 264 43 00 1 2.57 1.79637 253 36 00 1 20.24 1.90440 265 42 50 2.94 1.79890 253 35 50 20.74 1.90705 265 40 3.31 1.80143 253 40 21.24 1.90970 266 30 3.69 1.80396 253 30 21.75 1.91236 266 20 4.06 1.80649 253 20 22.25 1.91502 267 10 4.43 1.80902 253 10 22.75 1.91769 267 42 00 1 4.80 1.81155 35 00 1 23.25 1.92036 254 268 41 50 5.18 1.81409 254 34 50 23.78 1.92304 269 40 5.57 1.81663 253 40 24.30 1.92573 2 30 5.95 1.81916 254 30 24.83 1.92841 271 20 6.34 1.82170 254 20 25.36 1.93112 27 10 6.72 1.82424 254 10 25.88 1.93382 270 41 00 1 7.11 1.82678 255 34 00 1 26.41 1.93653 271 23 178 ASTRONOMY. Fahrenheit's Thermometer 50~. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 34 00 1 26.41 1.93653 271 27 00 1 54.17 2.05754 310 33 50 26.96 1.93924 27 26 50 54.99 2.06064 40 27.52 1.94196 27 40 55.81 3.06376 30 28.07 1.94469 273 30 56.66 2.06688 312 20 28.62 1.94742 274 20 57.50 2.07003 315 10 29.18 1.95016 275 10 58.36 2.07318 315 33 00 1 29.73 1.95291 275 26 00 159.22 2.07635 317 r~~~~275 ~318 32 50 30.31 1.95566 277 25 50 2 0.09 2.07953 40 30.90 1.95843 277 40 0.99 2.08273 32 30 31.48 1.96120 278 30 1.88 2.08594 321 20 32.06 1.96397 279 20 2.80 2.08917 323 10 32.65 1.96676 279 10 3.71 2.09241 324 32 00 1 33.23 1.96955 280 25 00 2 4.65 2.09567 326 280 327 31 50 33.85 1.97235 281 24 50 5.59 2.09894 40 34.46 1.97516 281 40 6.54 2.10224 330 30 35.08 1.97797 283 30 7.51 2.10554 33 20 35.70 1.98080 282 20 8.49 2.10886 332 10 36.31 1.98362 284 10 9.48 2.11220 31 00 1 36.93 1.98646 24 00 2 10.48 2.11555 2~~~~~85 ~337 30 50 37.58 1.98931 285 23 50 11.50 2.11892 40 38.24 1.99216 287 40 12.52 2.12231 339 30 38.89 1.99503 287 30 13.57 2.12571 340 20 39.54 1.99790 289 20 14.62 2.12913 34 10 40.20 2.00079 289 10 15.70 2.13258 345 30 00 1 40.85 2.00368 290 23 00 2 16.78 2.13603 348 29.50 41.52 2.00658 291 22 50 17.88 2.13951 40 42.21 2.00949 292 40 19.00 2.14300 49 30 42.90 2.01241 293 30 20.13 2.14652 354 20 43.59 2.01535 20 21.28 2.15006 10 44.30 2.01829 294 10 22.43 2.15361 35 29 00 1 45.01 2.02124 295 22 00 2 23.61 2.15719 358 296 359 28 50 45.73 2.02420 298 21 50 24.81 2.16078 40 46.46 2.02718 299 40 26.02 2.16440 362 30 47.18 2.03016 300 30 27.25 2:16804 364 20 47.93 2.03316 301 20 28.50 2.17171 366 10 48.68 2.03617 301 10 29.76 2.17539 368 28 00 1 49.44 2.03918 303 21 00 2 31.04 2.17910 303 373 27 50 50.21 2.04221 304 20 50 32.34 2.18283 35 40 50.99 3.04525 305 40 33.67 2.18658 37 30 51.77 2.04830 307 30 35.01 2.19036 381 20 52.57 2.05137 308 20 36.37 2.19417 383 10 53.36 2.05445 309 10 37.76 2.19800 385 27 00 1 54.17 2.05754 310 20 00 2 39.16 2.20185 388 MEAN REFRACTIONS. 179 Fahrenheit's Thermometer 50~. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 0 l,f If1 o I 1 20 00 2 39.16 2.201!5 388 13 00 4 7.91 2.39430 19 50 40.59 2.20573 39 12 50 11.11 2.39987 3 40 42.04 2.20963 40 14.39 2.40550 30 43.52 2.21356 396 30 17.74 2.41119 576 20 45.02 2.21752 398 20 21.19 2.41695 573 10 46.53 2.22150 10 24.72 2.42278 589 19 00 2 48.08 2.22552 404 12 00 4 28.33 2.42867 596 404 596 18 50 49.65 2.22956 407 11 50 32.04 2.43463 603 40 51.25 2.23363 40 35.84 2.44066 611 30 52.87 2.23773 41 30 39.75 2.44677 618 20 54.53 2.24186 41 - 20 43.76 2.45295 626 10 56.21 2.24603 419 10 47.88 2.45921 635 18 00 2 57.92 2.25022 423 11 00 4 52.12 2.46556 642 17 50 59.66 2.25445 425 10 50 56.47 2.47198 650 40 3 1.43 2.25870 A29 40 5 0.94 2.47848 659 30 3.23 2.26299 3() 5.54 2.48507 669 20 5.06 2.26732 43 20 10.28 2.49176 677 10 6.93 2.27168 44 10 15.16 2.49853 678 17 00 3 8.83 2.27608 44 10 00 5 2019 2.50541 696 16 50 10.77 2.28051 447 9 50 25.86 2.51237 707 40 12.74 2.28498 450 40 30.70 2.51944 716 30 14.75 2.28948 30 36.20 2.52660 727 20 16.80 2.29402 458 20 41.88 2.53387 738 10 18.88- 2 2.29860 462 10 47.74 2.54125 749 16 00 3 21.01 2.30322 467 9 00 5 53.79 2.54874 759 15 50 23.18 2.30789 470 8 50 6 0.04 2.55635 772 40 25.39 2.31259 4 40 6.50 2.56407 785 30 27.66 2.31734 30 13.18 2.57192 20 29.95 2.32213 483 20 20.G9 2.57989 811 10 32.30 2.32696 488 10 27.26 2.58800 824 15 00 3 34.70 2.33184 4 8 00 6 3468 2.59624 838 14 50 37.16 2.33677 7 50 42.37 2.60462 851 40 39.65 2.34174 502 40 50.33 2.61313 866 30 42.21 2.34676 507 30 58.59 2.62179 883 20 44.82 2.35183 12 20 7.19 2.63062 899 10 47.48 2.35695 517 10 16.13 2.63961 914 14 00 3 50.21 2.36212 523 7 00 7 25.40 2.64875 931 13 50 53.00 2.36735 528 6 50 35.05 2.65806 949 40 55.85 2.37263 40 45.10 2.66755 967 30 58.76 2.37796 538 30 55.58 2.67722 986 20 4 1.74 2.38334 45 20 8 6.50 2.68708 1006 10 4.79 2.38879 55 1 1 17.90 2.69714 1026 13 00 4 7.91 2.39430 557 6 00 8 29.80 2.70740 1047 180 ASTRONOMY. Fahrenheit's Thermometer 50~. Barometer 30 Inches. Alt. r. Log. r. Diff. Alt. r. Log. r. Diff. 6 00 8 29.80 2.70740 1047 3 00 14 26.04 2.93754 1608 5 50 42.24 2 71787 1069 2 50 58.71 2.95362 15 40 55.25 2.72856 1092 40 15 33.60 2.97016 1701 30 9 8.88 2.73948 30 16 10.89 2.98717 1749 20 23.16 2.75063 20 50.8 3.00466 1801 10 38.12 2.762021 10 17 33.6 3.02267 1855 5 00 9 53.84 2.773671 2 00 18 19.6 3.04122 1909 4 50 10 10.35 2.78558 1 50 19 9.0 3.06031 40 27.73 2.79777 40 20 2.2 3.07998 67 30 46.03 2.81025 1248 30 59.6 3.10024 202 20 11 5.30 2.82302 127 20 22 1.7 3.12113 2089 10 25.66 2.83611 1340 10 23 8.9 3.14268 222 4 00 11 47.151 2.84951 1374 1 00 24 21.8 3.16489 2221 1.374 I 2290 3 50 12 9.68 2.86325 0 50 25 40.9 3.18779 40 33.97 2.87735 1410 40 27 7.1 3.21140 2434 30 59.51 2.89182 1 30 28 40.8 3.23574 2509 20 13 26.61 2.90666 1523 20 30 23.2 3.26083 2584 10 55.40 2.921891 10 32 15.0 3.28667 2667 3 00 14 26.04 2.93754 1608 0 00 34 17.5 3.31334 In ordinary cases it will be sufficient to apply to the observed altitude the Mean Refraction standing against it in the adjoining column. Where greater accuracy is required, the corresponding log. r must be taken. In the Table of Corrections on the following page, for the Barometer and Thermometers, the proportional parts of log. t, for tenths of a degree of Fahrenheit, will be found in the column adjoining that of log. t, standing against the corresponding units of the argument. In the same manner the proportional parts of log. 2, for hundredths of an inch, will be found standing against the corresponding tenths. These must be added or subtracted according to the sign at the top of the column. The proportional parts of log. xr, for tenths of a degree, will be found at the bottom. The sum of logs. r, t, 2, and xr, will be the log. of the refraction, which must be subtracted from the observed altitude, or added to the observed zenith distance. The column Barometer contains the Logarithms of 0, p being the height of the Barometer in English Inches. MEAN REFRACTIONS. 181 Corrections, depending on the state of the Thermometer and Barometer, to be applied to the foregoing JMean Refractions. External Thermometer. Barometer. Internal Thermometer. Th. Log. t. t. P.Th. Log. t. P. P. Bar. Log. P. P. Th Log. r. Th. Log. qr 10 0.03779 - 50 0.00000 - 26.6 9.94776 10 0,00173 50 0.00000 1 0.03680 10 1 9.99910 9 7 9.94939 11 0.00169 51 9.99996 2 0.03582 20 2 9.99820 18 8 9.95101 12 0.00164 52 9.99991 3 0.03484 29 3 9.99730 27 9 9.95263 13 0.00160 53 9.99987 4 0.03386 39 4 9.99640 36 14 0.00156 54 9.99983 5 0.03288 49 5 9.99550 45 27.0 9.95464 15 0.00151 55 9.99978 6 0.03191 59 6 9.99460 54 1 9.95584 16 0.00147 56 9.99974 7 0.03094 69 7 9.99371 63 2 9.96745 17 0.00143 57 9.99970 8 0.02997 78 8 9.99282 72 3 9.95904 18 0 00138 58 9.99965 9 0.02900 88 9 9.99193 81 4 9.96063 19 0.00134 59 9.99961 5 9.96221 20 0.02803 60 0.99104 6 9.96379 20 0.00130 60 9.99957 1 0.02706 10 1 9.99016 9 7 9.96536 21 0.00126 61 9.99953 2 0.02609 19 2 9,98927 18 8 9.96692 22 0.00121 62 9.99948 3 0.02514 29 3 9,98839 26 9 9.96848 23 000117 63 999944 4 0.02418 38 4 9,98751 35 24 0.00113 64 9.99940 5 0.02323 48 5 9.98663 44 28.0 9.97004 + — 25 0.00108 65 9.99935 6 0.02227 58 6 9.98575 53 1 9.97158 15 26 0.00104 66 9.99931 7 0.02132 67 7 9.98488 62 2 9 97313 30 27 0.00100 67 9.99927 8 0.02037 77 8 9.98401 70 3 9.97466 46 28'10.00095 68 9.99922 9 0.01942 86 9 9.98314 79 4 9.97620 61 29 0.00091 69 9.99918 5 9.97772 76 30 0.01848 70 9.98227 6 997924 91 30 0.00087 70 9.99913 1 0.01754 9 1 9.98140 9 7 9.98076 106 31 0.00083 71 9.99909 2 0.01660 19 2 9.98054 17 8 998227 122 32 0.00078 72 9.99904 3 0.01566 28 3 9.97967 26 9 9.98378 137 33 0.00074 73 9.99900 4 0.01472 38 4 9.97881 34 34 0.00070 74 9.99896 5 0.01379 47 5 9.97795 43 29.0 9.98528 35 0.00065 75 9.99891 6 0.01285 56 6 9.97709 52 1 9.98677 15 36 0.00061 76 9.99887 7 0.01192 66 7 9.97623 60 2 9.98826 29 37 0.00057 77 9.99883 8 0.01099 75 8 9.97537 69 3 9.98975 44 38 0.00052 78 9.99878 9 0.01006 85 9 9,97452 77 4 9.99123 59 39 0.00048 79 9.99874 5 9.99270 73 40 0.00914 80 9.97367 6 9.99417 88 40 0.00043 80 9.99870 1 0.00822 9 1 9.97282 8 7 9.99563 103 41 0.00039 81 9.99866 2 0.00730 18 2 9.97197 1 7 8 9.99709 118 42 0.00034 82 9.99861 3 0.00638 28 3 9.97112 25 9 9,99855 132 43 0.00030 83 9.99857 4 0.00546 37 4 9,97027 34 44 0.00026 84 9.99853 5 0.00455 46 5 9 96943 42 30.0 0.00000 45 0.00021 85 9.99848 6 0.00363 55 6 9.96859 50 1 0.00145 14 46 0.00017 86 9.99844 7 0.00272 64 7 9.96775 59 2 0.00289 29 47 0.00013 87 9.99840 8 0.00181 74 8 9.96691 67 3 0.00432 43 48 0.00008 88 9 99835 9 0.00090 83 9 9.96607 76 4 0.00575 57 49 0.00004 89 9.99831 50 0.00000 90 9.96524 5 0.00718 71 50 0.00000 90 9,99827 6 0.00860 86 7 0.01002 100 8 0.01143 114 P. P. to tenths of a Degree. 9 0.01284 129.1.2.3.4.5.6.7.8.9 31.0 0.01424 - 0 1 1 2 3 3 3 3 4 18~2 ASTRONOMY. IV. 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 instrnment-in Azimuth, in the Inclination of the axis, and in Collimation-in order to obtain the correct clock error. sin (L — D) cos (L — D) c cos D cos D cos D E - 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 of the telescope is 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 optical axis of the telescope (pointing to the South) falls to the East; and minus, when it falls to the West; and vice versa when pointing to the North. 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. TIME BY TRANSITS. 183 1. To determine the value (in time) of the co-efficients a, b, c, in the preceding formula. For inclination of the axis of the telescope: b 6 (w + wl') ( —e + e) Where wt and el 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. e = the inclination of the level to the East For collimation: c = - (t' - t) cos D + I (bl - b) cos (L - D.) Where t' and bt denote respectively the values of t and b, after reversing the instrument, D - the declination of a circumpolar star. t the time of the transit of the circumpolar star, (leduced from an observation at a given side wire of the instrument. For the deviation in azimuth: By observations of a circumpolar star: 12h — (T' —T) b cos (L - D) - b' cos (L + D) + 2 c a 212 cos L tangD + 2 cos L sin D Where T' and bl denote respectively the values of T and b, at the lower culmination. 184 ASTRONOMY. Deviation in azimuth by transits of a high and low star. cos D.' cos D a = (ARt - AR) - (TI- T) X cosL sin (D' - D) Where T', AR', and DI, denote respectively the values of T, AR, and D of the second star observed, or make in(L D) for the first star = n cos D and s ) for the second star n' cos D' (AR - AR) — (T' - T) thena= - - 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 declination 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 Reduction to middle wire = (I + ) - (IV + ) 5 cos D TRANSIT INSTRUMENT. 185 Numerical values of sin ( L-D ) cos (L - D ) 1 cos D' cosD c os DD for facilitating the method of determining the deviation of the Transit Instrument in.dzimuth, by means of " high and low stars." Deviton. Star's Declination D For Level. Star's Z D 10 10~ 200 300 400 500 600 Star's Z D =(L -D) (L - D) 10 02.02.02.02.02.03.03 890 5.08.08.09.10.11 013.17 85 10 I.17.17.18.20.23.27.35 80 15 26.26.27.30.34.40.52 75 0.34.34.36.39.45.53.68 70 25.42.43.45.48 -.55.66.84 65 30.50.51.53.57.65.77 1.00 60 35 |.57.58.61.66.75.89 1.15 55 40.64.65.68.74.84 1.00 1.28 50 45.71.72.75.81.92 1.10 1.41 45 50.76.78.81.88 1.00 1.19 1.53 40 55 1.82.83.87.94 1.07 1.27 1.64 35 60.86.88.92 1.00 1.13 1.35 1.73 30 65 I.90.92.96 1.05 1.18 1.41 1.81 25 70.94.95 1.00 1.08 1.23 1.46 1.88 20 75 96.98 1.03 1.11 1.26 1.50 1.93 15 80.98 1.00 1.05 1.14 1.28 1.53 1.97 10 85.99 1.01 1.06 1.15 1.30 1.55 1.99 5 89 1.00 1.01 1.06 1.15 1.30 1.55 1.99 1 For Colli- i maFtion l.oo0 1.015 1.064 1.154 1.305 1.555 2.000 Cos mation.CosD 24 186 ASTRONOMY. FORM FOR RECORD AND COMPUTATION. SURVEY OF STATION Transits of Stars with inch transit No. sidereal Chronometer, Hlardy, tro. 50. Illuminated end of axis, west. Date (1847) - October 6th. October 6th. October 6th. Observer - - T. J. L. T. J. L. T. J. L. Object - cr Capricorni 14 Capricorni ca Cygni. Level ~ E. 32.2 W. 330 1E. 32.7 W. 32.5 E. 32.7 W. 32.5 alue of 1 division 1E. 32 2 W. 33.0 E. 32.5 W. 33.3 E. 33.0 W. 32.5 of scale 711.5 h1 m s h m s h l m s Wires i 20. 17. 33.0 200 29. 43.7 20. 35. 00.0 Is 17. 53.5 30. 02.7 35. 26.0 HII 18. 12.7 30. 22.0 35. 52.0 IV 18. 32.7 30. 41.7 36. 18.7 v 20. 18. 52.5 20. 31. 00.7 20. 36. 45.5 Sum - - 184.4 110.8 142.2 Mean. - 20. 18. 12.88 20. 30. 22.16 20. 35. 52.44 Reduc'n to mid. wire -.07 -.07 -.10 Transit on instrument 12.81 22.09 52.39 ) for collim'n for level - - ~.10.04 -.1 = for dev'ninaz'h -+ 17.1.. 01 TransitbyChronom'r 20. 18. 13.08 20. 30. 22.31 20. 35. 52.21 AR. of star - 20. 18. 36.66 20. 30. 45.8920. 36. 15.80 Error of Chronometer 23s.58 23s.58 233.59 Chronometer slow of time at p. m., October 6th, 1847. TRANSIT INSTRUENT.o 187 Computation of the corrections a and b, in the preceding Transits. Declination of xr Capri. 180 42 S. Latitude of Station = L == 430 13' 14 Capri. 150 29 S. 4 Cygni 440 44 N. Level correction of - Capricorni. L- 43~13' E. 32.2 W. 33 D - 180 421 32.2 33 (L- D) = 61~ 55' 64.4 66 from table pape 185. 66 - 64.4 = 1.6 Cos (L- D)050 b X 1.6 -0.20 CosD 60 Cos (L - D) Level correction b (L ) OS.20 X 0.50 0.10 Cos D Deviation in azimuth. (AR' - AR) - (TI - T) a=' - 5% T' and T being the times of transit corrected for level and collimation. Combining qr Capri. and a Cygni. H.. S B..I. S. AR = 20 36i 5.80 T' = 20 35 52.22 AR = 20 18 36.66 T = 20 18 12.91 17 39.14 17 39.31 17 39.31 (sin L - D) sin (- 1~ 31t) ___ _ - _ =...._ 0,03 (AR/ - AR) - (T' - T) = - 0.17 Cos D/ Cos 440 44' sin (L- D) sin 610 551 - - — ___- = - + 0.93 Cos D Cos 180 42t os.17.17 O -0.03 - 0.93.96 0 Combining 14 Capri. and a Cygni, a - + 05.19. Correction for deviation in Azimuth of vr Capricorni = a sin (L D) Coo18 s 0.93 =0,1.8 > 0.93 = 0%.17 188 ASTRONOMY. Transit Instrument-Continued. Rules for the direction of the deviation in azimuth, in the method of fixing a'Transit Instrument in the meridian by "h igh and low stars." Position of Stars. lCulminatiol. Precedence. Relative mag- Deviation. nitude of Intervals. Both south, or both Both upper Highest or near- Obs'd greater. W. of S. north, or one south'" est to the Pole. Obs'd less. E. of S. and theothernorth " Furthest from Obs'dgreater. E. of S. of the zenith. " the Pole. Obs'd less. WV. of S. The north- Nearest to the ern being Obs'd less. E. of N. One north, and the the lower Pole. culm'n. Obs'd greater. W. of N. other south of theI The south- Obs'd less. W. of N. zenith. ern being Farthest from upper cul e. Obs'dgreater. E. of N. mination. the Pole. Upper. Obs'd greater. E. of N. One upper Both north of the Upper. Obs'd less. W. of N. and one zenith. Lower. Obs'd greater. W. of N. lower. Lower. Obs'd less. E. of N. LATITUDE. 189 LATITUDE. V. To determine the Latitude from the meridional altitude of an object whose declination is known. 1. When the object observed is south of the zenith: L=900 -- D-A=Z + D =900+Z- A= 180~- (A+-a) 2. When the star is between the zenith and the pole: L= A-a - Z = 90~ - (Z -+ ) A+-D —90~ 3. When the star is between the pole and the horizon to the north: L -A + =900+ a-Z =90~+A —D = 180 —(Z +D) where L = the latitude sought. D the declination of the object, minus when south, A- its north polar distance, A= its meridional altitude, 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) t= semi-diam. 190 ASTRONOMY. VI. Determination of the Latitude of a place by the method of circum-meridian altitudes. Reduction to meridian k cos Icos D tang i cos I cos D cos a cos a 2 sin2 ~ p 2 sin4 p sin 1l sin 1fl a =90~- +D — A a + x -- the meridional altitude of the object, a - its observed altitude-(refractioln-parallax) = semidiameter. p = its correct hour angle, D its declination, I the assumed latitude of the place, x - the required correction in seconds. When a star is the object observed and the chronometer marks mean time, i = 1.005473, log i - 0.0023708 When the sun is observed and the chronometer marks sidereal time, i — 0.99455418, log i- 9.9976285; and generally, when the chronometer has a large losing rate, x must be multiplied by 1 + 0.00002315 r; when it has a gaining rate it must be divided by 1 + 00002315 r; r being the rate in 24 hours, which must be assumed minus when gaining, and plus when losing. LATITUDE. 191 The values of k and m for each value of p, are given in the following tables. The meridian altitude A = a + x for each observaat + al +. tion; for any number of observations n, + x + X+I- the mean, a, of all the observed n altitudes + 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', pll, etc.. in time, for which the corresponding values of k', k"l, etc., and mn, in1, etc., must be taken from the tables. 3. The means k and mn 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 computation should be repeated with the new value for 1. 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. G. The second correction, nz, is seldom necessary, unless great accuracy is desired, and the object is observed more than ten minutes of time from the meridian. 192 ASTRONOMY. 2 sin2 ~p Reduction to the JMeridian; values of k in sin 1I Sec. Om' 1Tl 2m 3m 4m 5m 6I 7rm I f It t t II It It It I1 0 0.0 2.0 7.8 17.7 31.4 49.1 70.7 96.2 1 0.0 2.0 8.0 17.9 31.7 49.4 71.1 96.7 2 0.0 2.1 8.1 18.1 31.9 49.7- 71.5 97.1 3 0.0 2.2 8.2 18.3 32.2 50.1 71.9 97.6 4 0.0 2.2 8.4 18.5 32.5 50.4 72.3 98.0 5 0.0 2.3 8.5 18.7 32.7 50.7 72.7 98.5 6 0.0 2.4 8.7 18.9 33.0 51.1 73.1 99.0 7 0.0 2.4 8.8 19.1 33.3 51.4 73.5 99.4 8 0.0 2.5 8.9 19.3 33.5 51.7 73.9 99.9 9 0.0 2.6 9.1 19.5 33.8 52.1 74.3 100.4 10 0.1 2.7 9.2 19.7 34.1 52.4 74.7 100.8 11 0.1 2.7 9.4 19.9 34.4 52.7 75.1 101.3 12 0.1 2.8 9.5 20.1 34.6 53.1 75.5 101.8 13 0.1 2.9 9.6 20.3 34.9 53.4 75.9 102.3 14 0.1 3.0 9.8 20.5 35.2 53.8 76.3 102.7 15 0.1 3.1 9.9 20.7 35.5 54.1 76.7 103.2 16 0.1 3.1 10.1 20.9 35.7 54.5 77.1 103.7 17 0.2 3.2 10.2 21.2 36.0 54.8 77.5 104.2 18 0.2 3.3 10.4 21.4 36.3 55.1 77.9 104.6 19 0.2 3D4 10.5 21.6 36.6 55.5 78.3 105.1 20 0.2 3.5 10.7 21.8 36.9 55.8 78.8 105.6 21 0.2 3.6 10.8 22.0 37.2 56.2 79.2 106.1 -22 0.3 3.7 11.0 22.3 37.4 56.5 79.6 106.6 23 0.3 3.8 11.2 22.5 37.7 56.9 80.0 107.0 24 0.3 3.8 11.3 22.7 38.0 57.3 80.4 107.5 25 0.3 3.9 11.5 22.9 38.3 57.6 80.8 108.0 26 0.4 4.0 11.6 23.1 38.6 58.0 81.3 108.5 27 0.4 4.1 11.8 23.4 38.9 58.3 81.7 109.0 28 0.4 4.2 11.9 23.6 39.2 58.7 82.1 109.5 29 0.5 4.3 12.1 23.8 39.5 59.0 82.5 110.0 LATITUDE. 193 2 sin2 p Reduction to the Mderidian; values of k si 1 Sec. Om Im 2m 3m 4m 5m 6m 7m 30 0.5 4.4 12.3 24.0 39.8 59.4 83.0 110.4 31 0.5 4.5 12.4 24.3 40.1 59.8 83.4 110.9 32 0.6 4.6 12.6 24.5 40.3 60.1 83.8 111.4 33 06 4.7 12.8 24.7 40.6 60.5 84.2 111.9 34 0.6 4.8 12.9 25.0 40.9 60.8 84.7 112.4 35 0.7 4.9 13.1 25.2 41.2 61.2 85.1 112.9 36 0.7 5.0 13.3 25.4 41.5 61.6 85.5 113.4 37 0.7 5.1 13.4 25.7 41.8 61.9 86.0 113.9 38 0.8 5.2 13.6 25.9 42.1 62.3 86.4 114.4 39 0.8 5.3 13.8 26.2 42.5 62.7 86.8 114.9 40 0.9 5.4 14.0 26.4 42.8 63.0 87.3 115.4 41 0.9 5.6 14.1 26.6 43.1 63.4 87.7 115.9 42 1.0 5.7 14.3 26.9 43.4 63.8 88.1 116.4 43 1.0 5.8 14.5 27.1 43.7 64.2 88.6 116.9 44 1.1 5.9 14.7 27.4 44.0 64.5 89.0 117.4 45 1.1 6.0 14.8 27.6 44.3 64.9 89.5 117.9 46 1.2 6.1 15.0 27.9 44.6 65.3 89.9 118.4 47 1.2 6.2 15.2 28.1 44.9 65.7 90.3 118.9 48 1.3 6.4 15.4 28.3 45.2 66.0 90.8 119.5 49 1.3 6.5 15.6 28.6 45.5 66.4 91.2 120.0 50 1.4 6.6 15.8 28.8 45.9 66.8 91.7 120.5 51 1.4 6.7 15.9 29.1 46.2 67.2 92.1 121.0 52 1.5 6.8 16,1 29.4 46.5 67.6 92.6 121.5 53 1.5 7.0 16.3 29.6 46.8 68.0 93.0 122.0 54 1.6 7.1 16.5 29.9 47.1 68.3 93.5 122.5 55 1.6 7.2 16.7 30.1 47.5 68.7 93.9 123.1 56 1.7 7.3 16.9 30.4 47,8 69.1 94.4 123.6 57 1.8 7.5 17.1 30.6 48.1 69.5 94.8 124.1 58 1.8 7.6 17.3 30.9 48.4 69.9 95.3 124.6 59 1 9 7.7 17.5 31.1 48.8 70.3 95.7 125.1 25 194 ASTRONOMY. 2 sin'2 p Reduction to the JIleridian; values of k sin 1"1 Sec. 8m 9m 10m rlm 12m 13m 14m _.., I I. II 0 125.7 159.0 196.3 237.5 282.7 331.8 384.7 1 126.2 159.6 197.0 238.3 283.5 332.6 385.6 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 5 128.3 162.0 199.6 241.2 286.6 336.0 389.3 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..4 164.4 202.2 244.1 289.8 339.4 393.0 10 131.0 165.0 202.9 244.8 290.6 340.3 393.9 11 131.5 165.6 203.6 245.5 291.4 341.2 394.8 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 14 133.1 167.4 205.6 247.7 293.8 343.7 397.6 15 133.6 168.0 206.3 248.5 294.6 344.6 398.6 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 20 136.3 171.0 209.6 252.2 298.6 349.0 403.3 21 136.9 171.6 210.3 253.0 299.4 349.8 404.2 22 137.4 172.2 211.0 253.6 300.2a 350.7 405.1 23 138.0 172.9 211.7 254.4 301.0 351.6 406.0 24 138.5 173.5 212.3 255.11 301.8 352.5 407.0 25 139.1 174.1 213.0 255.9 302.6 353.3 408.0 26 139.6 174.7 213.7 256.6 303.5 354.2 408.9 27 140.2 175.3 214.4 257.4 304.3 355.1 409.9 28 140.7 175.9 215.1 258.1 305.1 356.0 410.8 29 141.3 176.6 215.8 258.9 305.9 356.9 411.7 LATITUDE. 195 2 sin' 1 Reduction to the JMeridian; values of k 2 sin III Sec. 8m 9m i9m 1m 12m 13m 14m myI I IP I r; t t I 30 141.8 177.2 216.4 259.6 306.7 357.7 412.7 31 142.4 177.8 217.1 260.4 307.5 358.6 413.6 32 143.0 178.4 217.8 261.1 308.4 359.5 414.6 33 143.5 179.0 218.5 261.9 309.2 360.4 415.5 34 144.1 179.7 219.2 262.6 310.0 361.3 416.5 35 144.6 180.3 219.9 263.4 310.8 362.2 417.5 36 145.2 180.9 220.6 264.1 311.6 363.1 418.4 37 145.8 181.6 221.3 264.9 312.5 364.0 419.4 38 146.3 182.2 222.0 265.7 313.3 364.8 ) 420.3 39 146.9 182.8 222.7 266.4 314.1 365.7 421.3 40 147.5 183.5 223.4 267.2 315.0 366.6 422.2 41 148.0 184.1 224.1 267.9 315.8 367.5 423.2 42 148.6 184.7 224.8 268.7 316.6 368.4 424.2 43 149.2 185.4 225.5 269.5 317.4 369.3 425.1 44 149.7 186.0 226.2 270.3 318.3 370.2 426.1 45 150.3 186.6 226.9 271.0 319.1 371.1 427.0 46 150.9 187.3 227.6 271.8 319.9 372.0 428.0 47 151.5 187.9 228.3 272.6 320.8 372.9 429.0 48 152.0 188.5 229.0 273.3 321.6 373.8 429.9 49 152.6 189.2 229.7 274.1 322.4 374.7 430.9 50 153.2 189.8 230.4 274.9 323.3 375.6 431.9 51 153.8 190.5 231.1 275.6 324.1 376.5 432.8 52 154.4 191.1 231.8 276.4 325.0 377.4 433.8 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 55 156.1 193.1 234.0 278.8 327.5 380.2 436.7 56 156.7 193.7 234.7 279.5 328.4 381.1 437.7 57 157.3 194.4 235.4 280.3 329.2 382.0 438.7 58 157.8 195.0 236.1 281.1 330.0 382.9 439.7 59 158.4 195.7 236.8 281.9 330.9 383.8 1 440.6 196 ASTRONOMY. 2 sin2. Reduction to the JlMeridian; values of k 2 in sin Iu Sec. 15m 16m 17m 18m 19m1 20m 21m I1 i I it 1,. fr 0 441.6 502.5 567.2 635.9 708.4 784.9 865.3 1 442.6 503.5 568.3 637.0 709.7 786.2 866.6 2 443.6 504.6 569.4 638.2 710.9 787.5 868.0 3 444.6 505.6 570.5 639.4 712.1 783.8 869.4 4 445.6 506.7 571.6 640.6 713.4 790.1. 870.8 5 446.5 507.7 572.8 641.7 714.6 791.4 872.1 6 447.5 508.8 573.9 642.9 715.9 792.7 873.5 7 448.5 539.8 575.0 644.1 717.1 794.0 874.9 8 449.5 510.9 576.1 645.3 718.4 795.4 876.3 9 450.5 511.9 577.2 646.5 719.6 796.7 877.6 10 451.5 513.0 578.4 647.7 720.9 798.0 879.0 11 452.5 514.0 579.5 648.9 722.1 799.3 880.4 12 453.5 515.1 580.6 650.0 723.4 800.7 881.8 13 454.5 516.1 581.7 651.2 724.6 802.0 883.2 14 455.5 517.2 582.9 652.4 725.9 803.3 884.6 15 456.5 518.3 584.0 653.6 727.2 804.6 886.0 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 22 463.5 525.7 591.9 662.0 736.0 813.9 895.8 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 25 466.5 528.9 595.3 665.6 739.8 817.9 900.0 26 467.5 530.0 596.5 666.8 741.1 819.2 901.4 27 468.5 531.1 597.6 668.0 742.3 820.5 902.8 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. 197 2 sin2 1 p Reduction to the.Meridian; values of k = sin 1.2 Sec. 15m 16m 1m7m 18 19m 20m 21ml 30 471.5 534.3 601.0 671.6 746.2 824.6 907.0 31 472.6 535.4 602.2 672.8 747.4 825.9 908.4 32 473.6 536.5 603.3 674.1 748.7 827.3 909.8 33 474.6 537.6 604.5 675.3 750.0 828.6 911.2 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 37 478.7 541.9 609.1 680.1 755.1 833.9 916.9 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 6J2.5 683.8 759.0 838.0 921.1 41 482.8 546.3 613.7 685.0 760.2 839.3 922.5 42 483.8 547.4 614.8 686.2 761.5 840.7 923.9 43 484.8 548.4 616.0 687.4 762.8 842.0 925 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 46 487.9 551.7 619.5 691.1 766.7 846.1 929.6 47 488.9 552.8 620.6 692.4 768.0 847.5 931.0 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 50 492.0 556.1 624.1 696.0 771.9 851.6 935.2 51 493.1 557.2 625.3 697.3 773.1 852.9 936.6 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 54 496.2 560.5 628.8 701.0 777.1 857.1 940.9 55 497.2 561.6 630.0 702.2 778.4 858.4 942.3 56 498.3 562.7 631.2 703.5 779.7 859.8 943.8 57 499.3 563.9 632.3 704.7 781.0 861.1 945.2 58 500.3 565.0 633.5 705.9 782.3 862.5 946.6 59 501.4 566.1 634.7 707.1 783.6 863.9 948.1 198 ASTRONOMY. 2 sin2 p Reduction to the.Meridian; values of k sin 1p sin 111 Sec. 22m 23m 24m Sec. 22m 23m 24m _t l! I, I. If II 0 949.6 1037.8 1129.9 30 993.2 1083.3 1177.5 1 951.0 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 1137.8 35 1000.6 1091.0 1185.5 6 958.2 1046.8 1139.3 36 1002.1 1092.6 1187.1 7 959.6 1048.3 1140.9 37 1003.5 1094.1 1188.7 8 961.1 1049.8 1142.5 38 1005.0 1095.7 1190.3 9 962.5 1051.3 1144.0 39 1006.5 1097.2 1191.9 10 963.9 1052.8 1145.6 40 1008.0 1098.8 1193.5 11 965.4 1054.3 1147.2 41 1009.4 1100.3 1195.1 12 966.9 1055.9 1148.8 42 1010.9 1101.9 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 1105.0 1199.9 15 971.2 1060.4 1153.6 45 1015.4 1106.5 1201.5 16 972.7 1062.0 1155.2 46 1016.9 1108.1 1203.1 17 974.1 1063.5 1156.8 47 1018.4 1109.6 1204.7 18 975.5 1065.0 1158.3 48 1019.9 1111.2 1206.4 19 977.0 1066.5 1159.9 49 1021.4 1112.7 1208.0 20 978.5 1068.1 1161.5 50 1022.8 1114.3 1209.6 21 979.9 1069.6 1163.1 51 1024.3 1115.8 1211.2 22 981.4 1071.1 1164.7 52 1025.8 1117.4 1212.9 23 982.9 1072.6 1166.3 53 1027.3 1118.9 1214.5 24 984.4 1074.2 1167.9 54 1028.8 1120.5 1216.1 25 985.8 1075.7 1169.5 55 1030.3 1122.0 1217.7 26 987.3 1077.2 1171.1 56 1031.8 1123.6 1219.4 27 988.8 1078.7 1172.7 57 1033.3 1125.1 1221.0 28 990.3 1080.3 1174.3 58 1034.8 1126.7 1222.6 29 991.8 1081.8 1175.9 59 1036.3 1128.3 1224.2 LATITUDE. 199 Second part of the Reduction to the Mferidian; 2 sin4 ap values of m - si 1p sin It/ Minutes. Os 10s 20s 30s 40s 50s 5 0.01 0.01 0.01 0.01 0.01 0.01 6 0.01 o.o0 0.01 0.02 0.02 0.02 7 0.02 0.02 0.03 0.03 0.03 0.04 8 0.04 0.04 0.05 0.05 0.05 0.06 9 0.06 0.06 0.08 0.08 0.08 0.09 10 0.09 0.10 0.11 0.11 0.12 0.13 11 0.14 0.15 0.15 0.16 0.17 0.18 12 0.19 0.20 0.22 0.23 0.24 0.25 13 0.27 0.28 0.30 0.31 0.33 0.34 14 0.36 0.38 0.39 0.41 0.43 0.45 15 0.47 0.49 0.52 0.54 0.56 0.59 16 0.61 0.64 0.67 0.69 0.72 0.75 17 0.78 0.81 0.84 0.88 0.91 0.95 18 0.98 1.02 1.06 1.09 1.13 1.18 19 1.22 1.26 1.30 1.35 1.40 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 3.64 3.74 3.84 3.94 4.05 4.15 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 30 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 11.04 11.27 11.50 11.73 11.96 12.20 34 12.44 12.69 12.94 13.19 13.45 13.71 35 13.97 14.24 14.51 14.78 15.06 15.35 200 ASTRONOMY. FORM FOR RECORD SURVEY OF DETERMINATION OF THE LATITUDE, V'orth and South DATE AND STATION.-1843, October 13-M-3outh of the Big Black river, NAME OF STAR, / Pegasi, South of the Zenith. INSTRUMENTS. Sextant No. 2197, by Troughton 8r Simms, and INSTRUMENTS. ean Solar Chronometer No. 76, by Charles o.. Times of ob- MERIDIAN DISTANCES, 2 Sin" 2p $ servationby = p. Sin 1" Chronome- ->__' 5 = ter.ne In mean In Sidereal -k o, iz Solar time. time. V h m s m s m s 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 51.9 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 1 47.6 5 22 44.4 5 40.2 5 41.1 63.4 C 117.8 6 23 54 4 30.5 4 31.2 40.1. 0 49.2 7 25 12 3 12.6 3 13.1 20.3. 0 24.9 8 26 46 1 38.6 1 38.8 5.2 0 06.3 9 28 16.4 0 08.2 0 08.2 0.0 E 0 00.0 10 29 42 1 17.4 1 17.6 3.2 1 0 03.9 11 31 42 3 17.4 3 17.9 21.4 ~ 0 26.2 12 32 54.4 4 29.8 4 30.5 40.0 0 0 49 13 34 18 5 53.4 5 54.3 68.5 1 24 14 36 14.2 7 49.6 7 50.9 123.5 2 31.5 15 38 32.2 10 07.6 10 09.2 202.3 4 08.2 16 40 06 11 41.4 11 43.3 269.9 5 31.1 Observer, 3Major J. D. Graham. Computer, do. do. LATITUDE. 201 AND COMPUTATION. from observed double circum-meridian altitudes of Stars, of the Zenith. a tributary to the river St. John, MJaine. artificial horizon of Mercury. Young. Obser'd double True circum-meridi- True meridian al- Latitude, circum - meri- an altitudeof tar, titudes deduced, deduced from as corrected for redian altitudes fraction and errors - (a -+- x) -A each observaof Star. of instrument, tion = L - = a. (900+D-A) 0 o II 0 o ft 0 o II 0 i f! 114 34 15 57 18 38.5 57 22 28.3 46 56 42.55 36 15 57 19 38.5 57 22 30.4 56 40.45 37 10 57 2()0 06 57 22 33.3 56 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 41 50 57 22 26 57 22 32.3 56 34.55 41 50 57 22 26 57 22 26 56 40.85 41 50 57 22 26 57 22 29.9 56 36.95 41 00 57 22 01 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 11 57 22 19.2 56 51.85 30 50 57 16 56 57 22 27.1 46 56 39.75 LATITUDnE-Deduced from a mean of 16 altitudes of Star -, Pegasi.... 46~ 561 43".4 Deduced from a mean of 10 altitudes of Star?v Cephei, observed this night with same Sextant... 46 57 10.7 Mean; or Latitude adopted..... 46 56 57 26 202 ASTRONOMY. D = apparent declination of Star 140 19' 10".85 N. Log cos 9.9869,9 I = approximate Lat. of place 460 57.. Log cos 9.83418 Sum - 19.82048 a = approximate merid. alt. of Star 57~ 221 10"' Log cos 9.73176 Cos 1 cos D Cos cs D = constant multiple = 1.227. Log 0.08872 Refraction (Ther. 280, Bar. 29.14 in.) for mean obs'd alts. - 39" Index error of Sextant....... - 2' 40"1 *Error of excentricity, &c., of Sextant.. + 1' 401" hIm s Apparent AR. of of the Star;v Pegasi. 0 05 14.09 Sidereal time at mean noon at this station.... 13 26 20.83 Sidereal interval from mean noon, of Star's culmination 10 38 53.16 Retardation of mean on Sidereal time 1.. - 1 44.96 Mean time of culmination of Star? 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 10 28 24.6 On this night, Oct. 13, 1843, Major Graham obtained for the Latitude of this station, from 75 observations on 5 stars South of the zenith, combined with 21 observations on? Cephei and Polaris, to the North... 460 56' 56."3 On the night of Oct. 24, by 43 observations on 4 southern stars, combined with 2 observations on? Cephei, the Latitude deduced was... 46 56 57.2 On Sept. 17, 1844, 66 observations on N. and S. stars gave for the Latitude of this station.... 46 56 60.4 *NOTE.-The error of excentricity is approximately ascertained by comparing Latitudes, well determined, by observations on N. and S. stars, with that which will result from N. or S. stars individually of various meridional altitudes. It varies with the altitudes observed. That is to say, it is different for different parts of the limb of the instrument. LATITUDE. 203 VIJ. To determine the Latitude by an altitude of a star near the pole, at any hour. L= A- (a cos p) -t - (A sinp)2 tang A-p (A sinp)2 (A cosp) where A - the observed altitude, corrected for refraction, etc. - the polar distance of the star, in seconds of arc. a ~ sin 11t log a = 4.3845449, -p =- sin" lII log p 8.89403, p the hour angle of the star. p = sidereal time — AR = solar time + AR — AR * p is plus when the star is west, and minus when it is east of the meridian. The sign of cos p should also be attended to, for when p is greater than 6hr, 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 011.55. This formula is only applicable to stars within a very few degrees of the pole. For other circumpolar stars, tang x tang A cos p cos x sin A sin y = cos A L=yTx In which the upper sign is used when the star is above the pole, the under when below the pole. 204~ ASTRONOMY. FORM FOR SURVEY OF DETERMINATION OF THE DATE AND STATION.-1843, September 6 —Woodstock, YNew Brunswick, NAME OF STAR.-Polaris (z UrsCm JVMinoris,) observed on, between four and INST. Sextant No. 2197, by Troughton 8r Simms, and artiJVMean Solar Chronometer No. 2440, by Parkinson &, Times of ob- True Sidereal MrERIDIAN DISTANCES. servation by times of obJMeean Solar servation. _ i cos p. Chronometer In Sid'l time In arc Y Jo. 2440. RP. -o _op. h.m. s. h. mn. s. h.ms. s. o t, 1 1 33 02.5 20 05 34.1 4 58 23.2 74 35 48 — 24 18.1 2 1 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 1 36 38.2 20 09 10.4 4 54 46.9 73 41 43.5 -25 41.4 5 1 39 07.5 20 11 40.1 4 52 17.2 73 04 18 -26 34.7 6 1 41 11.2 20 13 44.1 4 50 13.2 72 33 22.5 -27 27.1 7 1 44 28.2 20 17 01.7 446 55.6 7143 54 -28 40.8 Observer, Major J. D. Graham. Computer, Do. LATITUDE. 205 RECORD AND COMPUTATION. LATITUDE, from observed double altitudes of Polaris. (Grover's Inn.) five hours before its upper meridian passage. ficial horizon of Mercury. Frodsham. Obser'd dou- True altitudes Latitude de( ble alts. of of Star, as coy- duced from + a (A sin~ p)2. __ (A sin p)t Polaris out reeled for retang Ad.. (Acosp.) of the Meri- fraction and each obserdian. errors of in- vation strument,. j=A. I L. 0 I 0 0 r I 0 /I o + 1 11.63 - 0.32 93 01.30 46 31 58.6 46 08 51.8 + 111.41 — 0.33 93 02.45 46 32 36 46 08 52.6 - 1 11.20 -0.33 93 03.50 46 33 08.6 46 08 59.7 + 1 1 1.04 — 0.33 93 04.40 46 33 33.6 46 09 02.9 + 1 10.63 - 0.34 93 06.15 46 34 21 46 08 56.6 + 1 10.28 — 0.35 93 08.20 46 35 23.5 46 09 06.3 + 1 09.68 - 0.37 93 10.50 46 36 38.5 46 09 07 LATITUDE-deduced from a mean of 7 altitudes of Star 46~ 08' 59".4 Polaris............. 206 ASTRONOMY. Apparent declination of Star 880 28t 3011.5. Apt. N. P. D. of Star = 10 31! 29".5 = 5489".5 = A Refraction (Ther. 570 - Bar. 30.013 inches)... - 55".4 Index error of Sextant.......... 4- 2' 50"' Erro~rs of excentricity &c. of Sextant...... + It 28"5 h. m. s. Apparent AR. of the Star Polaris (a Ursce JAfinoris).. 1 03 57.3 Sidereal time at mean noon at this station..... 11 00 27.1 Sidereal interval from mean noon, of Star's culmination. 14 03 30.2 Retardation of mean on Sidereal time.... - 2 18.2 Mean time of culmination of Star Polaris..... 14 01 12 Chron. No. 2440, fast of mean time at time of observation 4 29 24.8 Time by Chronometer of culmination of Star Polaris.. 6 30 36.8 The reduction of the mean time of observation to sidereal time, in the preceding example, might have been omitted by using table of JR. in arce into mean time, pages 152, &c. Thus-(lst observation) Mean time of observation........ Ih 331n 02P.5 Mean time culmination of Polaris...... 6 30 36.8 Hour angle, p, in intervals of mean time..... 4 57 34.3 Sidereal equivalents, in are.... 4h = 60~ 09' 51".39 57m = 14 17 20.45 34s = 8 31.40.3 = 4.51 p, in are........ 74 351 47".75 FoRM FOR COMPUTATION-(1St observation) 1st term. 2d term. 3d term. log cos p (- )= 9.4242480 sin p = 9.98411 " A - 3.7395327 A = 3.73953 A cosp 3.1637807 A sin p = 3.72364... = 3.16378 - 1458".1 - 1st term — 24' 18".1 (A sin p)2 = 7.44728... = 7.44728 log, = 4.38454 log i = 8.89403 A — 46~31 58.6 tang A = 0.02325 9.50509 46 07 40.5 1.85507 3d t'm =-0".32 2d term - + 1 11.63 = 71'.63 2d term -+ 1' 11.63 46 08 52.13 3d term - -0.32 Latitude -46081' 51".81 LATITUDE. 207 VIII. 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: 1. 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 Tt and El denote the corresponding quantities for the western transit. Then the two-hour angles, in sidereal time, will be, the eastern negative, t =_ T-+ E - AR -t -tT + TEl- 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 of the Star, gives the following relations: cos t cos D sin L - sin D cos L cot a cos D sin t cos tt cos D sin L - sin D cos L cos D sin VI Whence, cos ~ (t' + t) tang L tang ) Dcos (t t) If the instrument is very nearly on the prime vertical, cos i (t + t) -- cos 0~ = 1, and tang L = tang D sec. I (it - t) for the passage over the middle wire of the instrument. 208 ASTRONOMY. 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 firom the middle wire, measured on a great circle, be 15 f; f being the equatorial interval of the wire, in time. Then, to reduce the transit over a side wire, to the centre wire, f v sin (L- +D). sin (L — D) -t 5f > a The upper sign of the term:t ~5 f, 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 f = c, for f in the above, according as the error of collimation c, lies on the same or opposite sides off 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 LATITUDE. 209 a being the Azimuth of the centre wire of the telescope, supposed in collimation. This may be found from the time elapsed between the E. and W. transits of the same star, thus: cot u = tang a (1t - t) sin D. sin u sin a = cos D cosna a is taken between 0~ and 900 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 900, 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. 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. 27 210 ASTRONOMY. 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. Whein the telescope is on the prime vertical, callingP the hour angle, and z, the zenith distance of the Stat | then coss = tang D cot L sin D C3S Z = sin L 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, etc., the instrument may be reversed between the transits over each vertical; i. e., the wires on one side of the centre 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, = l diff. of sidereal time of transit over the same wire, for same position of axis. t = hour angle, illuminated axis south, tang 1 tang L- tang cos ~ (tt-]- t). cos (tj —t) LATITUDE. 211 IX. 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 altitude telescope. Compute an approximate latitude by the formula. L - ~ 180~-(/ + Al ) - +~ (z-z') where A and at are the polar distances of the south and north Stars respectively, and (z —zf) 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 its inclination is southward, or subtracting it when to the northward. This correction is found by multiplying the value of 1 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, of and et denote the readings of the object and eye-ends of the bubble, for south and north stars respectively; corrections for level = - (o' -el) — (o-e) X the value of 1 division of the level scale in arc. 2. The correction for error of meridionalposition of the central vertical wire, is found by computing the usual:"reduction to the meridian" for each star; then the difference between the reductions 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 - 4 212 ASTRONOMY. m being the reduction for stars south, and mI for stars north of the zenith. 3. The correction for Refraction is applied similarly to this last, but with a contrary sign; or, correction for refraction - (r - r') being small, no note need be taken of the state of the barometer and thermometer at the time of observation. It is sufficient to use the actual tabular quantities. Including all the corrections, the general expression for Latitude will be 180 -(A + A') (z-z') (o + e) - (o + e) 6 (M- M) (r r') 2 +-2.a+ a and b being the are values of 1 division of the micrometer and level scale respectively. 4. Should the Star be observed on one side or the other of the central wire, the reduction to the meridian becomes m- 4 sin 1l. sin 2 A. p' [6.43569741 sin 2 A. p2 p being the hour angle of the Star in seconds of time. Sine 2 A is negative when the Star is south of the equator or sub-polo. LATITUDE. 213 5. To find the value a, of I division of the micrometer, note the time by chronometer of the transit of Polaris over the moveable wire placed vertically, and set successively to, say, every hundredth division of its scale. Then let x be the angular distance from the meridian at which any reading of the screw was had; p, the hour angle of the Star at the same instant, and A its polar distance, sin - sin a sin p. 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. 6. The value b6, of 1 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 microineter, during the bubble's run over the scale, as the telescope's elevation is gradually altered, may afterwards be measured by the micrometer screw. 7. 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 Z. D. of a pair will belong as often to the N. Star as to the S. Star; for if the Z. D. of the N. 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, 2 Z - Z = o. FORM FOR RECORD AND CoMPUTATION. 4Q. Latitude, Station Observations with Zenith Telescope by Micrometers Diff. Z. D. Levels. Corrections. Date. a - - Polar A. B..cYm~i distances.' P a ~E: IBy mi- 0 ]E A c. 0B... S 1846. II I 42 36z' S. E 1 It Il Q 6640 N. 1388.31..... 32 38 10.29 {1.... } 36.2 { 49.0 /-:6.4 22.0 0 40 15.38 j j { // 6690 S. 276.5 1664. 752.01 62 21 06.28... 35.0 49.8 7.4 25.0 0.85 15.63 ~Sept.20~ 0862 N.. 255.5. 29 47 01.20.... 41.7 {42.0- 0.1 )39.0 /1.03 18.53 {{ 6966 8..... 1282.7 1027.2l 464.00 64 51 56.23.... 41.2 43.3 ~1.0 24.0 0.73 18.38 94 38 57.43 85 P210T.57 7 44.00 85 13 3.573911 28........ +0.57+.15-0.07 — 0. 57- 0.15_ — 0.107 423639.93 AZIMUTHS. 215 X. Knowing the time and the latitude of the place, to find the.zimuth of the Sun or a Star. tang I (A -+- S) - cotp ( cos (-+ - )' en sin -~ ( ~ — ) tang ~2 (A —S) _ cot p n ( ) A=: — = ( A + S). ( A- S) the upper or negative sign is used when 7 is greater than A. Where At = 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. -= the co-latitude of the place. a = the north polar distance of the sun or star. p _ the hour angle at the pole. XI. Without the use of a chronometer, by observing the altitude of the sun or stUr at the same instant with the observation of the azimuth. Let Z = the zenith distance, corrected for refraction, parallax, and semidiameter. sin k. sin ( k- A) Cos, I A 2sin Z sin, 2 k= Z + A + 216 ASTRONOMY. XII. To find the amplitude of a celestial c3jevt at its rising or setting; by amplitude is meant the complement of the azimuth, or distance from t]he 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 distance, instead of being 90~, is, on account of refraction and parallax, 900 + k. Where k = hor. refraction - hor. parallax - 331 451 - hor. parallax. For stars, the hor. par. - 0 and k - 90~ 331 45", for the sun, k = 33' 45"t - 8" 6 and k =-90 33136".4; the mean refraction and mean hor. par. are here used as these observations are not susceptible of a great degree of accuracy. XIII. To find the true meridian by the method of equal altitudes of the Sun. The instrument remaining stationary, observe the readings of the horizontal limb when the altitude of the Sun's centre, or of either limb, 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 a (t- t') where D -- D = the change in the sun's declination in the interval of the observations, (t - i/) = this interval of time, expressed in arc L = the latitude of the place. AZIMUTHS. 217 XIV. To find the azimuth of Polaris at its greatest eastern or western elongation. Cos p - tang A cot -= cot D tang L _ tang L tang A. Cos L sin A = sin A = cos D, where, the hour angle of the Star, A = the required azimuth, D= its declination, L -the lat. of the place, A = its polar distance, - - 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 X and XI, or by the formula A (in seconds) os L A- +A- sin 1t cosp tang L i The most approved method is to observe a series of azimuths of Polaris 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 t; the correction to the azimuth will be, c = (112.5) Vt sin 1,I tang A log (112.5) sin III 6.7367274. The quantities found in the tables for "reduction to the meridian (2 sin' P the meridian" k2 ] F) correspond very nearly to (112.5) t2 sin If', when t does not exceed 15'; so, by entering the table with the time from elongation, and multiplying the tabular quantities by tang A, we obtain the 28 218 ASTRONOMB Y required correction in seconds of arc. This will be found a convenient substitute for the more rigorous method. 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 id w [(w+ wt) -(e + el)] tang *'s altitude where d = the value of one division of the level scale, w-= the inclination of the level to the west, e = the inclination of the level to the east, w1 and el, the same values after reversing the level. XV. Correction for Run in Reading JMicroscopes. As it is difficult to adjust the microscopes so that five revolutions 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 m = the mean of first readings, that is, the readings obtained by turning the screw in the direction of increasing numbers from zero of the comb. m= - the mean of second, or reverse, readings. Then, (mean) Run - r = m - nm + 300, and 300. m 300 (r +- ml — 300) true (mean) reading = r r = the number of minutes and seconds to be added to the degrees and minutes of the limb. LONGITUDE. 219 XVI. Lunar distances. To determine the true distance of the moon from the sun, or a star; the appcrent distance, together with the apparent altitudes of the moon and the sun, or star, being given. Let, d = apparent distance dt -true distance H moon's app't altitude HI moon's true altitude h = sun's app't altitude h = sun's true altitude P _ moon's hor. par. at place FP moon's par. in altitude p - sun's hor. parallax p't sunls par. in altitude S = moon's hor. semidiam- S = moon's augm. semidieter ameter s - sun's semidiameter D observed distance R = refraction for moon's r = refraction for sun's alaltitude titude A observed altitude of a _ observed altitude of moon's limb sun's limb. P = 2- -. E. sin2 L; where;t = moon's equatorial horizontal parallax. E = the elipticity, log E = 7.5233789; L = the latitude of place. S = [9.43537] P St = S + augmentation H = A: St h = ais p' - p cos H pI =p cos h HI - H'+ (P'-R) t h- (r ) 220 ASTRONOMY. For a Star or a Planet. ht - h —r d- D:k: St cos h cos HC sin2 d = cos ~! (A' + H') + C ~ cos (hl' + Hf)- C The reduced distance being thus found, the longitude may be deduced from it as follows: Suppose that at 5hrs 05m 56s mean time, 29th April, 1838, at a place whose longitude is presumed to be 4hrs 45m 008 west of Greenwich, the result of observations gave the reduced distance between the sun and moon, d' = 71~ 05' 35" Mean time obs'n _ 5hrs 05n 56s Approx. long'de 4 45 00 9 50 56 approx. Greenwich m. time of observation. By Naut. Alm. at IX —= 70~ 411 3011 70~ 41' 30" (April 29th) XIIh: 72~ 071 47"1 d' - 71~ 051 35"t 1 26 17 24' 051I Increase of distance in 3hrs- 5177(1.0 8 dl - 144511 Then 5177"/: 10800":: 1445"1: x - Oh 50 145.5 Add 91hr Greenwich mean time deduced = 9hrs 50n 14s.5 Mean time at place =5 05 56.0 Longitude, deduced = 4h 44m 18s.5 LONGITUDE. 221 The reduction of this proportion is very much facilitated by the use of Proportional Logarithms, or logs. 3hrs of - given in treatises on Navigation, in conjunction with those in the Nautical Almanac. The proportion, however, requires a correction for second differences, when greater accuracy is desired, arising from the irregularity of the moon's motion. A closer approximation to the true value of the quantity x being X 3r d ttxA+Q-Bx In which B 4 the sum of the second differences, and A - the middle first difference - B; thus, Fromn the JVNautical 1lmanac, april 29, 1838. 1st difference. 2d difference. At VI -690 14' 54" IX -700 41' 3011 - 02 1936 XII 72~ 07' 47' + 1 26 17" 18 XV-= 730 33' 4611 x — 50m 14".5 = Ohrs 83736 (table page 173.) B = - 9".2; A = a - B = 5177"1 + 911.2 = 5186".2 ad = 14451"; B x- - 2".56; A + 4 B x- 5183"1.64 whence 10800s X 1445" X 583 -- 501 108.6. 5183and longitude deduced = 4 and, longitude deduced - 4hrM 44- 148.6. 222 ASTRONOMY. Reduction of the.Moon's Equatorial Horizontal Parallax to the Horizontal Parallax in any Latitude. HORIZONTAL PARALLAX. E 54' 56' 58' 60r 62/ 0, f,, If, I ft 0 0.0 0.0 0.0 0.0 0.0 8 0.2 0.2 0.2 0.2 0.2 16 0.8 0.8 0.9 0.9 0.9 20 1.3 1.3 1.4 1.4 1.5 24 1.8 1.9 1.9 2.0 2.0 28 2.4 2.5 2.6 2.6 2.7 32 3.0 3.1 3.3 3.4 3.5 36 3.7 3.9 4.0 4.1 4.3 40 4.5 4.6 4.8 5.0 5.1 44 5.2 5.4 5.6 5.8 6.0 48 6.0 6.2 6.4 6.6 6.8 52 6.7 7.0 7.2 7.4 7.6 56 7.4 7.7 8.0 8.2 8.5 60 8.1 8.4 8.7 9.0 9.3 64 8.8 9.1 9.4 9.7 10.0 68 9.3 9.6 10.0 10.3 10.6 72 9.8 10.1 10.4 10.8 11.2 76 10.2 10.6 10.9 11.3 11.7 84 10.7 11.1 11.5 11.9 12.0 90 10.8 11.2 11.6 12.0 12.4 The moon's horizontal parallax, given in the second page of each month, in the'"American Nautical Almanac," for noon and midnight, is the equatorial parallax for Greenwich mean noon and midnight; from thence it is to be deduced for the time and place of observation. The correction for latitude, on account of the spheroidal figure of the earth, can be made from the table above. Thus, supposing the hor. equat. par. to be 58'; the hor. par. in lat 52~ would be 58'- 7".2 - 57' 52".8. LOiNGITUDE. 223.lugmentation of the JMoon's Semidiameter, on account of her apparent altitude. ~I~ HORIZONTAL SEMIDIAMETER. t ~ 14'0 1 5' 0' " 15' 30"' 16' 0" 16' 30'' 17' 01' o II,, II,, 0 0.00 0.00 0.00 0.00 0.00 0.00 3 0.71 0.75 0.80 0.86 0.92 0.97 6 1.41 1.50 1.60 1.71 1.83 1.94 9 2.11 2.25 2.40 2.56 2.73 2.90 12 2.81 3.00 3.20 3.41 3.63 3.86 15 3.50 3.74 3.99 4.25 4.52 4.80 18 4.17 4.46 4.76 5.07 5.39 5.73 21 4.84 5.18 5.52 5.89 6.26 6.65 24 5.49 5.88 6.27 6.68 7.11 7.54 27 6.13 6.56 7.00 7.46 7.93 8.42 30 6.75 7.23 7.71 8.22 8.v74 9.28 33 7.35 7.88 8.40 8.96 9.52 10.12 36 7.93 8.50 9.07 9.67 10.28 10.92 39 8.49 9.10 9.72 10.36 11.02 11.66 42 9.03 9.68 10.34 11.02 11.72 12.44 45 9.55 10.23 10.93 11.65 12.39 13.15 48 10.05 10.76 11.49 12.25 13.03 13.83 51 10.52 11.26 12.02 12.81 13.63 14.46 54 10.95 11.72 12.52 13.34 14.19 15.06 57 11.35 12.15 12.98 13.83 14.72 15.62 60 11.72 12.55 13.40 14.29 15.20 16.13 63 12.06 12.91 13.79 14.70 15.64 16.60 66 12.37 13.24 14.14 15.08 16.04 17.03 69 12.64 13.53 14.46 15.41 16.39 17.40 72 12.88 13.79 14.73 15.70 16.70 17.73 75 13.08 14.01 14.96 15.95 16.96 18.01 78 13.24 14.18 15.15 16.15 17.18 18.24 81 13.37 14.32 15.30 16.31 17.35 18.42 84 13.46 14.42 15.41 16.42 17.47 18.55 87 13.52 14.48 15.47 16.49 17.54 18.62 90 13.54 14.50 15.49 16.51 17.57 18.65 224 ASTRONOMY. XVII. Longitude by Lunar Culminations. 1. Interpolation.-When the quantities in the ephemeris are given in intervals of 12hrs, and the assumed meridian is +-, or west of Greenwich, the following arrangement will be found convenient: Let a, = the moon's place, from the ephemeris, for the preceding noon or midnight, a'= the moon's place, for the following midnight or noon, a= (a + a), b the middle first difference, c -the mean of the two middle second differences, =- (C1 + C'), d - the middle third difference, e - the mean of the two fourth differences 2= (e, + el), f - the fifth differences, t the interval in seconds since the date for a,, m - the variation of the moon's place for the interval (t - 6hrs), n = the average hourly variation, n- = the true hourly variation at the instant, t, Enclosing in brackets the constant log. co-efficients, Let X =[5.3645163] (t - 6 O- 0') X = [0.42800] t (t - 12 0' 0) X"l [9.52291 XXt. Xl= — [9.6499] X' (I + 12h Om O) (t - 24"1 0' OS) Then: m = b X + c X' + dX" t + e X"t LONGITUDE. 225 n [3.25527] (b +2 m) n'= Y. B (b + y d-T f f)+(c- - e) X + d X If the corrections beyond the second differences are neglected, then m=bX + c XI n [3.25527] (b + 2 m) This will require but four quantities from the ephemeris; two preceding and two following the time t. 2. To apply this to moon culminations: Let AR = the right ascension in the Nautical Almanac of the moon's bright limb at Greenwich for the upper transit next preceding the transit observed. AR'. the observed AR of the moon's bright limb at the place whose longitude, L, is required. Assume an approximate longitude, 1, and compute from the Nautical Almanac, by means of the foregoing method of interpolation, what the increase, m, in the AR of the moon's bright limb from its last transit at Greenwich, should be for that longitude. As the correction, in this case, is to be applied to a,, instead of ~ (a1 + a1), the co-efficient X becomes = [5.3645163] 1; the other co-eficients remaining the same, merely changing t to 1. By your own observations and the Nautical Almanac, this change is (AR' - AR); then as m: (ARI - AR):: L and L = (AR' AR). 29 226 ASTRONOMY. As the moon's motion in right ascension is not uniform, this proportion is only true when the values of I and L are nearly equal; but it is supposed that the approximate longitude is known to within a minute of time. It has also been supposed that both AR and ARI are correctly determined; that the quantities in the Nautical Almanac are free from errors in the Lunar Tables, and those of nutation, &c.; and that the observed AR is corrected for error of clock and errors of position, etc., in the Transit instrument. It is to eliminate these sources of error that moon culminating stars are observed in conjunction with the moon, and that corresponding observations at points whose positions are accurately known, are substituted for the tabular values, although the elements of the Nautical Almanac give very good approximations. The complete method will be better illustrated by an example: Suppose that at a station, whose longitude is presumed to be 4h 55m 50s west of Greenwich, the following transits have been taken with a Chronometer marking sidereal time; the error of the Chronometer being immaterial, but the Transit instrument being supposed in the meridian, or very nearly so. February 18, ~' Geminorum 6h 54m 41s.75 J Geminorum 7 10 38.97 )'a 1st Limb - - - 7h 28m 06s.76' Cancri 8 03 06.11 Sum 3) 22 08 26.83 7 22 48.943 Diff. 0 15 17.817 Rate of Chronometer + 3', daily, -.0318 Corrected difference = t, 0 15 17.785 LONGITUDE. 227 And suppose the following to be the corresponding observations at Greenwich, (these, however, are from the Nautical Almanac.) February 18,' Geminorum 6h 54m 57s.41 o" Geminorum 7 10 54.36 )'s 1st Limb - - - 7h 27m 478.66' Cancri 8 03 21.44 3) 22 09 13.21 7 23 04.403 Difference = tt - = 0 04 43.257 t= 0 15 17.785 then t, - t1' observed increase in )'s AR m' - 10 34.528 In the same manner would be,-btained, for other corresponding observations, values of m'", m"', &c. Next, compute this increase from the Nautical Almanac, as follows:.flpproximate Longitude = 1 = 4h 55m 50s. I = 4h 55m 50s =177508; log = 4.2491984 constant... - 5.3645163 log X..... + 9.6137147 l - 12 hrs. = — 25450S; log — 4,.40568 log I........ 4.24919 constant..... + 0.42800 log X'..... =-9.08287 log X...... + 9.6137 log X'...... - 9.0828 constant..= + 9.5229 log X"....... -8.2194 I + 12 hrs. = 60950S; log - + 4.7849 - 24 hrs. -686508, log - -4.8366 log X'..... -9.0828 constant.. + 9.6499 log X"' -..... 8.3542 AR 1's Istlimb by Naut. Almanac. a a 17 U 6h 35m 55S.73 +- 26... 00s.54 -t++ —-- c L 7 01 56.57 - -m 09S.15 25 51.39 - 01s.06 c: Igww 18 U 7 27 47.66 c =-0 10.21 e,= - 0.81 to 00 o a) b=-+ 25 41.18 d= — 0.25 1 L 7 53 28.84 cl= —0 10.46 e,-= 0.84 -25 30.72 + 0.59 19 U 8 18 59.56 -0 09.87 ~ 25 20.85 L - 8 44 20.41 Sum of differences.. 2h 08m 24s.68 - Om 39s.69 - 0s.72 + 1.65 0 Upper left hand quantity. 6 35 55.73 26 00.54 — 0 9.15 _1.06 Gc4 o~q 0"Q Check. Sum + upper left cm ulcr Check. Sun 4 ul erleft 1 8 44 20.41 25 20.85 - 0 9.87 + 0.59 I bc, d, de, 1.. + - 1541..18 -- 10.33 - Os.25 + 06.82 ao i log b, log c, log d, loge 3.1878533 - 1.01431 - 9.3979 + 9.9138 i, e,,D log X, log X', log X"i, log X", + 9.6137147 - 9.08187 - 8.2194 + 8.3542 n ~C- o log b X, log c X', etc.. 2 8015680 + 0.09718 + 7.6173 + 8.2680 b X: c X', d X ", etc. ~ + 633.240 -t- 1.250 + Os.004 + 08.018 LONGITUDE. 229 Then, observed increase = 634.528- mn computed do. = 634.512 m observed excess = m - m o. 016. Longitude, deduced, - 411 55rm 50 -- (ml - m) -4h 55m 50),.45 If there are corresponding observations at some other well known point, say Cambridge, Mass., longitude 4h 44-n 321= 1l; compute the increase mi for this longitude, by changing the co-efficients X, X', X",1 etc., to correspond to 1t. Then (m - mi) will be the computed increase If for Cambridge to your station, and - the rate of this increase, with which proceed as above. It often happens that two observers do not use the same number of.wires, or that the same number of stars are not observed 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 European 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, I and It -- the number of wires on which each limb was observed, a and at = similar values for a star, I= -7 for the moon's limb, 230 ASTRONOMY. a a! u = -,. for one star. u= a similar value for another star,: = symbol to denote the aggregate of similar quantities, = the correction required. ~,U Then E 7,- + U and L -I +- (m — m + ) Also, calling W, the weight of each day's comparison, W — (d + a) Z' in which z is the same as- and a = u + urt +ul, etc. For the weight of the result of all the comparisons, we have X W- = ( d+ _)z Let e denote the probable error of observation, and E the probable error of the final result; then, E e (6a-+ X) z2 It 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 LONGITUDE. 231 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 byp, the horizontal parallax by w, the latitude of the place of observation by p, and the true declination of the moon by 68, 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 = 0 sin w cos q sec, in which o, is the hour angle, or equatorial interval in sidereal 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: 0 0 sin w cos ~ sec 6, and the reduction to meridian or middle wire will be 0 1 - sin w cos cp sec 6'cos 1- 0.00277 m in which m, 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. 232 ASTRONOMY. XVIII. The value of a quantity at three consecutive whole hours) T - 1, T and T + 1, being given, to find its value at an intermediate time T', and its hourly variation at that time. Attending to the algebraic signs subtract the value of the quantity at the time T - 1, from its value at the time T; and its value at the time T, from its value at the time T - 1; and the remainders will be the first differences. Subtract the first ofthese from the second, and the remainder will be the second difference. Let a - the value of the quantity at the time T; b = the half sum of the first differences; c = the second difference; and t = the interval between T and TI, expressed in the fraction of an hour, and marked negative when T' is earlier than T. Then the value of the quantity at the time T', will be a+ t b + T C And the hourly variation of the quantity at the time T', will be b + t co EXAMPLE. Given the moon's declination, on a certain day, as follows: At lOh, D - + 150 58' 50".1; at 11'1, D = 150 47' 11".0; At 12h, D - 150 35' 27".1. Required its value at 101'. D 1st differences. 2d difference. 10 -- 150 58' 50".1 11 15~ 471 11".0 - 11 391 - 4".8 - 1I 43".9 12 150 35' 27". 1 a = + 150 47' l11".0, b = 11' 4111.5, c = 4".8, t -- tb = + 4' 40".6 c = 0".4 D -A- 15~ 51t 51".2 at time T' b = - 11' 41".5 t. c- 1".9 Hourly variation at time T' - - 11' 391.6 LONGITUDE. 233 XIX. To find the Longitude of a place from an observed occultation of a fixed star by the.Moon. Let A -Moon's AR, A' - Star's AR, D Moon's declination, D' -Star's declination, A" = Moon's hourly variation in AR, D" = Moon's hourly variation in declination, X, Moon's equatorial horizontal parallax, H' - Star's hour angle for Greenwich, k _ sine moon's appt. semidiam. constant 0.2725, log — 9.43536, i Geographical north latitude of place, =' - Geocentric north latitude of place, p = Earth's radius at place. It is unnecessary to compute iqt and p separately, as p sinlt- (- -esin - =A sin sin /1 —e2 sin2 q p cos q -- - _ cos q -= B cos p,/l —e sin, q in which e =.081697 -= the Earth's eccentricity; and as the values of log A and log B may be taken from the following table, with the argument q: 0 Log A. Log B. 0 9.9971 0.0000 10 9.9971 0.0000 20 9.9973 0.0002 30 9.9975 0.0004 40 9.9977 0.0006 50 9.9979 0.0009 60 9.9982 0.0011 70 9.9984 0.0013 30 *'a —-.is..,.._,__.,...,,,___ 234 ASTRONOMY. 1. With the estimated Longitude of the place, reduce the observed mean time of immersion to Greenwich time. Let T stand for this time, and T' for the same time, taken to the nearest tenth of an hour. From the Nautical Almanac, find for the time T' by the problem on page 232, the values of A, D, All, DII, and by proportion, the value of a; and also take out the values of A', D', and the sidereal time of mean moon. 2. With the values of A, D, etc., at the time T', find the values of p, q, p' and q', from the following formulae: (A —A) cos D D-D' P23 C log B = log p + log sin D' + 4.6856 d B (A-Al); = + d, A" cos D D/l at --- c A d B A"D d' B Dtf pl- a —dl qt= bf + cl 3. To the sidereal time at mean noon, add the sidereal time corresponding to the interval that T is past noon, and from the sum subtract Al. To the remainder, apply the longitude of the place in time, by adding if it is east, but subtiracling if it is west, and converting the result into degrees, it will be H, the star's hour angle at the observed time of immersion. LONGITUDE. 235 4. Having found log p cos q, and log p sin ~P, for the place, find u, v, N, F, t and t" by the following formulae: f = p sin cq' cos DI u cos pf sin H, g= p cos (t cos H sin DI cot N = (p-)otN, cos F - (d +- - - ) sin N k k cos (N + F) t__ —u t - - tI pt, Pt' Then will't - tP + -t, be the corrected Greenwich mean time of the immersion. The difference between this and the observed time, will be the Longitude in time; west if the observed time is the earlier of the two, but east, if it is later. In a similar manner would be deduced the Longitude from the observed emersion, except, that instead of t, k cos (N - F) we find ti =. When 1the immersion and emersion have both been observed, the Longitude should be obtained from each, and the mean of the two results taken. EXAMPLE. Suppose the observed immersion of i Leonis, on Jan. 7th, 1836, at a place in Latitude 520 08' 281" N., estimated Longitude 0 h. I m. W., was 1Oh. 45 m. 53.3 sec., mean time; required the Longitude of the place. 236 ASTRONOMY. The observed time of immersion reduced to Greenwich time is, T- = 10h. 46 m. 53,3sec. Taking T' = 10.8 h. = 10 h., we easily find from the Nautical Almanac, A = 10h 20m 33s.89 D -v- 15~ 49' 31".2 A'= 12 23 26.39 D'I —t1 40 58' 38".8 A"= 122s.905 = (in arc), 1843".6; D"= - 700".5 A - A' - 2587".5; D - D' - 3052".4; ~r = 3362".0 A -A'.... log 3.41288 - D -D' =.... log 3.48464 7r = Ar. Co." 6.47340 = Ar. Co. log 6.47340 D-.....cos 9.98322 c -.9079 =9.95804 p 7 —o. o7404 9.86950 - d =.0012'=... sin 9.4124 q.9091 4.6856 B = 3.9675 A — A'..... log 3.4129 d=.0024 = 7.3804 A"=.... log 3.26567 D"..... log 2.84541Vr o.. Ar. Co. log 6.47340 r, Ar. Co. log 6.47340 D..... cos 9.98322 b" -.2084 = 9.31881a' -.5276. e. 9.72229 B = 3.9675 — B = 3.9675A".......... 3.2657 - D"...... 2.8454c' —.-.0017 - 7.2332 - d' -.0006 6.8129 p' = a' - d =.5270; q' =b' - c'- -.2101 Sidereal time at mean noon Greenwich from N. A. = 19h 04m 22s.41 Sidereal interval from noon to time T.. 10 48 39.57 5 53 01.98 Al............ 10 23 26.39 - 4 30 24.41 1 00 H= —67~ 51' 06"=- 4 31 24.41 LONGITUDE. 237 From page 233 we have log g cos' = 9.78888 log g sin 9' = 9.89538 g sin <' = 9.89538 D' cos 9.98499 f-=.7592 -9.88037 cos... 9.78888 coscq.. 9.78888 H. sin 9.96671 H.. cos 9.57635 - -.5696 = 9.75559 - D'.. sin 9.41236 v.6993 g-.0599 = 8.77759 q'.. log 9.32243- N.. sin 9.96797 p' Ar. Co. log 0.27819 d -t vq = 2.1417 log 9.15137 N - 1110 44' 10" cot 9.60062 - k Ar. Co. log 0.56463 p-u =-.1708= log9.23249 -, F -= 1180~ 53' 00" cos 9.68397d.0681 8.83311 N +F 2300~ 37'10" cos9.80241-, p —u.. log 9.23249k..... log 9.43537 p' Ar. Co. log 0.27819 p' Ar. Co. log 0.27819 t" -.3241 = 9.51068 - t -.3281 9.51597Tr - t" + t = lOh 47n 45s.6 Observed time = 10 45 53.3 Longitude of place = Im 52s.3 west. 238 ASTRONOMY. XIX. Formulo, for Probable Error and Precision. 1. Let n, n1, etc., = the results found by observation. x = their arithmetical mean; or the result deduced from these by the method of least squares, E, = (mean error), El — (mean error)l - arithmetical mean without regard to sign, m = the number of observations, r = probable error of a single observation, n, n/, etc., R = probable error of final result, x, h = measure of exactness of a single observation n, nf, H = measure of exactness of final result, x, w - probable error of an observation assumed as the standard of excellence, p, p', etc., = the weights of the determinations of several variables, = symbol representing the sum, E-,|z (x - n) E (-n) m- - m-1 r = 0.674489 E, r - 0.845347 E, 0.469360 h H h =h/m r R -- Mn PROBABLE ERRORS. 239 2. By the weight of any determination, is meant its relative approximation to the true value. It may be measured by the number of equally good observations (one of which is assumed to represent the unit of excellence) necessary to give a result equally near the true value. The weights of two determinations are to each other in the direct proportion of the squares of their relative measures of exactness, and in the inverse proportion of the squares of their probable errors: tv=- w and calling the weight of any function of the two determinations, whose weights are p and p, P- PP p -pp P +PI the probable error of the value of the function is RI - r2 —s'rr If the index or measure of precision vary as any element v, involved in any given determination, h vI or h: h: v v'.. h' -A= then will the weight become (see page 230.) P +P-1 If there be but a single variable, and this has been found by different examinations, giving the values of a at, al, a etc., with probable errors r, rf, rtt, etc., or the weights p, 240 ASTRONOMY. pt, I't, etc.; and we seek to find from them the most probable value of x, a at a"t a j -- at pt + ant p!1 + etc.,~+-r2+TU72 + etc. ap +p alp + atip" + etc. 1 et. p~~ q-+pI~~ 2 I'q + etcetc. Its weight P -p +Pt +pft + etc.; Its probable error z -t 1 + + etc. +^ ertrt Position of some of the principal Observatories, etc. LONGITUDE PLACE. Latitude From the Observatory at From the Observatory at AUTHORITIES, North. Washington. Greenwich. In time. In time. In are. 0 E s0 - E 5 s 0 | O iI 11 | ifM S | O " { E O M / Greenwich Observatory 51 28 38.2 E. 5 08 1B1.2 77 02 48.0.o...o. 6 ID. X......... O Am. Naut. Alam., Paris 1 48 50 13.2 E. 5 17 32. 1 79 23 09.9 E. 0 09 21.5 2 20 21.9 (1855 Cambridge, Mass. " 42 22 48.6 E. 0 23 41.5 5 55 23.1 W. 4 44 29.6 71 07 24.9 Cincinnati, Ohio, " 39 05 54.0 W. 0 29 46.9 7 2642.8 W. 5 37 58.0 84 29 30.8 Georgetown,D.C., " 38 54 26.1 W. 00 06.2 0 01 33.0 W.5 08 17.4 77 04 21.0 Hudson, Ohio, " 41 14 42.6 W. 0 17 32.1 4 23 00.9 W. 5 25 43.3 81 25 48.9 Philadelphia " 39 57 07.5 E. 0 07 33.6 1 53 24.6 W. 5 00 37.6 75 09 23.4 Washington " 38 53 39.3......................... W. 5 08 11.2 77 02 48.0 Washington, Capitol..... 38 53 19.9 E. 0 00 10.2 0 02 33.0 W. 5 08 01.0 77 00 15.0 CoastSurvey(1852) 242 GEOGRAPHICAL POSITIONS. Falls St. Anthony, U. S. Cottage, Lat. = 440 58' 40" Nicollet. Lon. — = 6h 12m 42 cc Fort Leavenworth, Landing.. Lat. = 390 31' 14" Emory. Lon. = 6h 18m 568 Nicollet. Council Bluffs..... Lat. = 410 25' 04"1 Graham. Lon. = 6h 22m 55s.5 c Fort Gibson, old block house. Lat. = 35~ 47' 34".8 Woodruff. Lon. = 6h 21m 00s.9 t" San Antonio, Texas.. Lat. = — 290 25' 22".0 Johnston, Lon. = 6h 33m 57s sc Paso del Norte, Plaza... Lat. = 310 44' 16" Salazar. Lon. = 7h 05'n 15s " Frontera, White's rancheria.. Lat. = 310 48' 39" Whipple. Lon. = 7h 05m 54s 68 Santa Fe.... Lat. = 35~ 41' 06" Emory. Lon. = 7h 04m 1O c Bent's Fort..... Lat. = 380 02' 22" Fremont. Lon. = 6h 54m 13s.3 C6 Fort Laramie...... Lat. = 42~ 12' 10" Fremont. Lon. = 6h 59m 10'.9 "c Fort Hall..... Lat. = 430 01' 30"1 Fremont. Lon.= 7h 29m 59.6 " San Diego, Coast Survey obs'y, Lat. = 320 41' 57".9 Coast Sur'y, Lon. = 7h 48m 53S.4 Rep't of'51. Point Conception C. S. obs'y,. Lat. = 340 26' 56".3 Coast Sur'y, Lon. = 8h Olm 42s.2 Rep't of'51. Point Pinos, Coast Survey obs'y Lat. = 360 37' 59".8 Coast Sur'y, Lon. = 8h 07m 37S.4 Rep't of'51. San Francisco, Presidio Hill. Lat. 37~0 47' 35".6 Coast Sur'y, Lon. = 8h 09mn 47'.2 Rep't of'51. Longitudes west from Greenwich.